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Entries from CollegeMath

MathInSalinaKansasPart2 23 Jun 2006 - 13:28 CatherineJohnson

re: MathInSalinaKansas

Wow.

I spoke yesterday to a mathematics professor at a university here in New York state.

When I asked him what level of mathematical knowledge entering freshmen bring to their course work, he said, "We can't assume that a student knows anything we would want him to know."

Specifically, his students can't do algebra.

They can't set up a two-variable word problem and solve it.

These are college freshmen.

Posted on May 07, 2005 @ 11:21

MathInSalinaKansasPart3 23 Jun 2006 - 13:28 CatherineJohnson

re: MathInSalinaKansas

Three sample problems from the PRAXIS 1 Content Assessment college students entering the field of education are frequently required to take:

1. Which of the following is equal to a quarter of a million? a) 40,000 b) 250,000 c) 2,500,000 d) 1/4,000,000 e) 4/1,000,000 2. Which of the following fractions is least? a) 11/10 b) 99/100 c) 25/24 d) 3/2 e) 501/500 3. Which of the sales commissions shown below is greatest? a) 1% of $1,000 b) 10% of $200 c) 12.5% of $100 d) 15% of $100 e) 25% of $40 The Educational Testing Service (ETS) describes these problems thus:

The Pre-Professional Skills Test in Mathematics measures those mathematical skills and concepts that an educated adult might need. It focuses on the key concepts of mathematics and on the ability to solve problems and to reason in a quantitative context. Many of the problems require the integration of multiple skills to achieve a solution. [snip] Computation is held to a minimum, and few technical words are used. Terms such as area, perimeter, ratio, integer, factor, and prime number are used, because it is assumed that these are commonly encountered in the mathematics all examinees have studied. Figures are drawn as accurately as possible and lie in a plane unless otherwise noted.

POGIL

POGIL, POGIL, POGIL

This does not sound good, POGIL.

I should reserve judgment.

I should, but every one of the little-red-light thingies on my Constructivist Nightmare Detector is flashing wildly, and all the sirens are going off—

So I’m not doing a very good job of reserving judgment.

POGIL is a student-centered method of instruction that is based on recent developments in cognitive learning theory and results from classroom research that suggest [sic] most students experience improved learning when they are actively engaged, working together, and given the opportunity to construct their own understanding. POGIL emphasizes that learning is an interactive process of thinking carefully, discussing ideas, refining understanding, practicing skills, reflecting on progress, and assessing performance. In a POGIL classroom or laboratory, students work on specially designed guided-inquiry materials in small self-managed groups. The instructor serves as facilitator of learning rather than as a source of information. The objective is to develop skills as well as mastery of discipline-specific content simultaneously. (Emphasis added)

OK, that does not sound good.

homeschool mom with common sense-y

I'll get to the professor’s various posts on POGIL as soon as I can.

I do want to read them.

But in the meantime, there's one homeschooling mom on the thread (son has LD) whose comments make sense to me:

Pat says clearly that your Socratic style doesn't work for [him]. Why do you then believe that it does work? You (rightly) want [him] to learn problem solving, but just because your method of teaching ... works for others doesn't mean it works for [him]. Maybe [he] needs repetition, repetition, repetition of the underlying content before [he's] ready for process. Other students may grasp the process after going through the underlying solutions three times, or six times, but maybe [he] needs thirty times.
You model the problem solving that you want [him] to do-- Where have you seen a problem like this? What rule did we use?-- but in a sense that's no better than modeling the actual rule for Problem 13. You want [him] to intuit that [he] is supposed to be asking [himself] those questions. But what if, as seems to be the case, [he] isn't intuiting that? Then [he's] not learning anything.

The bad news here is that, clearly, constructivists are giving lots of workshops to math professors.

Even worse, math professors are attending them.

inflexible knowledge, flexible knowledge, and expertise

One of the problems with constructivism (and, apparently, with POGILism) is that it tries to teach higher-order problem solving first, instead of second.

That option probably isn't on the menu.

According to Daniel Willingham, knowledge is always inflexible before it's flexible. You can't hopscotch over the inflexible stage by teaching process, or asking students to discover addition.

Problem solving and critical thinking seem to grow out of extensive practice of surface, shallow, inflexible knowledge.

I’d like to know more about how this happens.

At a minimum I’d like to know what cognitive psychologists (psych department cognitive psychologists, I mean) understand about the process at this point.

And I hope that Robert, who writes the brightMystery blog, will join us at KTM once in awhile to think about these things.

update

WelcomeRobertTalbert

CorePlusAndDecliningMathSkills 09 Jul 2005 - 02:45 CatherineJohnson

I'd read about the disastrous introduction of Core-plus in Michigan, but I don't think I've seen this study (pdf file) that Anne Dwyer has attached to Barry Garelick's BarryOnCorePlus page.

Here's the abstract:

As part of a study involving over 3000 Michigan students, it was found that students arriving at Michigan State University from four high schools which began using the Core- Plus Mathematics program placed into, and enrolled in, increasingly lower level courses as the implementation progressed. This conclusion is statistically very robust | the existence of a downward trend is statistically signi cant with p < :0005. The grades these students earned in the mathematics courses they took are also below average (p < :01). ACT scores suggested the existence but not the severity of these trends.

'placed into, and enrolled in, increasingly lower level courses as the implementation progressed'

more t/k

I'm struck by the fact that the decline in students' skills was not picked up by the ACT.

I'm assuming this may support my 'don't trust the tests' postulate.

Actually, 'don't trust the tests' may be a theorem, not a postulate.

MathProfCrochetsHyperbolicSpace 20 Jul 2005 - 10:28 CatherineJohnson

Fantastic story in the TIMES today:

Dr. Taimina, a math researcher at Cornell University, started crocheting the objects so her students could visualize something called hyperbolic space, which is an advanced geometric shape with constant negative curvature....
...balls and oranges, for example, have constant positive curvature. A flat table has zero curvature. And some things, like ruffled lettuce leaves, sea slugs and cancer cells, have negative curvatures.
This is not some abstract - or frightening - math lesson. Hyperbolic space is useful to many professionals - engineers, architects and landscapers, among others.
[snip]
Math professors have been teaching about hyperbolic space for decades, but did not think it was possible to create an exact physical model.
[snip]
(This is my favorite part)
Dr. Taimina was a good candidate to create a better model. As a precocious child in her native Latvia, she tried her elementary school teacher's patience. When her fellow second graders did not understand a math lesson, she recalled, she would jump up and yell, "I can't stand these idiots," prompting her teacher to send notes home.
By high school she had settled down, and was most impressed by a teacher who was known to keep his advanced students laughing and engaged. When she became an educator, she decided that no student, regardless of aptitude level, would feel out of place in her classroom. One way she assured that was by using everyday objects to explain theories. (She was known for peering so intently at the oranges at her local grocery to see if she could find perfectly round ones to use in her geometry class that she scared the clerks.)
[snip]
In 1997, while on a camping trip with her husband, she started crocheting a simple chain, believing that it might yield a hyperbolic model that could be handled without losing its original shape. She added stitches in a precise formula, keeping the yarn tight and the stitches small. After many flicks of her crocheting needle, out came a model.
One professor who had taught hyperbolic space for years saw one and said, "Oh, so that's how they look"....
[snip]
A year after she created the models, she and her husband gave a talk about them to mathematicians at a workshop at Cornell. "The second day, everyone had gone to Jo-Ann fabrics, and had yarn and crochet hooks," said Dr. Taimina. "And these are math professors."
[snip]
As an adult, when terrified artists started showing up in her math classes to fulfill their degree requirements, she signed up for a watercolor class, thinking, "Then I will know how they feel."
Now when students tell her they simply cannot understand math, she pulls out one of her paintings and says, "I learned that in three months." Then she might pull out one of her crochet models.

DimensionalAnalysis 25 Jul 2005 - 20:05 CarolynJohnston

DanK brought up dimensional analysis in this thread, and it's such a useful idea that I thought we should have a thread to explain what it is, and talk about it and its possible uses in math education.

Here's a very simple example, where dimensional analysis can help you get the right answer.

Suppose a man drives 60 miles in 50 minutes. How fast is he driving?

There are two answers a kid is likely to come up with: the first (and correct) one is 60/50, but a kid might very well come up with 50/60 and not notice he's made a mistake.

Here's how dimensional analysis could help this student get the right answer: he knows he wants a rate for an answer; distance per unit of time. If he thinks of the 60 as '60 miles', and the 50 as '50 minutes', then his two choices are:

(60 miles)/(50 minutes) = 60/50 miles/minute

(50 minutes)/(60 miles) = 50/60 minutes/mile.

This gives him more context to help him choose the right answer. Miles per minute are units that make sense for this answer: minutes per mile don't.

In addition, dimensional analysis is the tool to use to make unit changes. If the question requires the answer to be given in miles per hour, then 60/50 is not the right answer, because the units are miles per minute. How to do the conversion to miles per hour?

As with converting fractions to have common denominators, the trick is to multiply the answer by 'one'. In this case, the conversion factor will be (60 minutes)/(1 hour). (You see why this is really 'one'?)

Thus the answer in miles per hour is:

(60 miles)/(50 minutes) x (60 minutes)/(1 hour).

Notice that (60 minutes/1 hour) is actually 1, expressed in different units in the numerator and denominator!

Now for the trick. Move the units around a little, just as though they were numbers in fractions being multiplied, and you get

(60 miles/1 hour) x (60 minutes/50 minutes).

Now the minutes cancel in that second term, and you are left with 60/50 (otherwise known as 6/5) as a dimensionless number. (A dimensionless number is a number without any units attached. For example, all ratios are dimensionless).

So the answer is: 60 miles/hour x 6/5, or 72 miles/hour.

There's even more that you can do with dimensional analysis. As Dan points out, it's a very handy concept, but hardly any math text uses it to the fullest extent they could.

At the undergrad level, it's something engineers and scientists learn explicitly. They have to know it in order to make unit conversions. I was a graduate student when I learned it in a geochemistry (i.e., thermodynamics) class; I had already had a complete undergraduate math education. I taught that whole class of geochemists how to do differential calculus; in return, they taught me dimensional analysis, and I think I got the better end of the deal.

So: when are kids ready to learn, and to start using, dimensional analysis?

Manipulating dimensions is a lot like manipulating fractions, and largely uses the same skills. You can't add dimensioned quantities, for example, unless the dimensions are the same: for example:

x miles/hour + y meters/minute = x+y miles/hour

doesn't make any sense unless you first convert the y term to miles/hour. Identical units can cancel (as the first example showed, when I canceled minutes in the numerator and denominator). So right about the age Ben and Christopher are now -- tennish or elevenish -- is about the earliest kids could really start using it, and it's also about the time that math texts stop emphasizing units (as DanK pointed out).

Plus, if the parents don't know it, how can they teach it?

Once again, it's the internet to the rescue.

TwoMathEdBlogs 27 Jul 2005 - 19:26 CatherineJohnson

Stephanie just sent me a link to a fascinating list of prerequisites for college math, which includes a terrific Comments thread, at Tall Dark and Mysterious, a blog written by "Twentysomething curmudgeon seeking employment teaching college math in BC."

And btw, these are not prerequisites for a serious college math course:

A year ago, I would have posted that list under a heading more along the lines of “Things Students Should Know By Grade Nine”, but alas, experience as extinguished such optimism on my part.

This is long, but it's so valuable I'm quoting the entire list, which I'll probably 'archive' over on....the 'math lessons' page? Another Content Question for the folks at Information Architecture, Inc. (Definitely read the Comments section as well):

Based on my experiences, students graduating from high school should, in order to succeed in even the most basic college math classes:
1.Be able to add, subtract, multiply, and divide fractions. Moreover, they should understand that the horizontal bar in a fraction denotes division. (Seem obvious? I thought so, too, until I had a student tell me that she couldn’t give me a decimal approximation of (3/5)^8, because “my calculator doesn’t have a fraction button”.)
2.Have the times tables (single digit numbers) memorized. At minimum, they should understand what the basic operations mean. For instance, know that “times” means “groups of”, which will enable them to multiply, for instance, any number by 1 or 0 without a calculator, and without putting much thought into the matter. This would also enable those students who have not memorized their times tables to figure out what 3 times 8 was if they didn’t know it by heart.
3.Understand how to solve a linear (or reduces-to-linear) equation in a single variable. Recognize that the goal is to isolate the unknown quantity, and that doing so requires “undoing” the equation by reversing the order of operations. Know that that the equals sign means that both sides of the equation are the same, and that one can’t change the value of one side without changing the value of the other. (Aside: shortcuts such as “cross-multiplication” should be stricken from the high school algebra curriculum entirely - or at least until students understand where they come from. If I had a dollar for every student I ever tutored who was familiar with that phantom operation, and if I had to pay ten bucks for every student who actually got that cross-multiplication was just shorthand for multiplying both sides of an equation by the two denominators - I’d still be in the black.)
4.Be able to set up an equation, or set of equations, from a few sentences of text. (For instance, students should be able to translate simple geometric statements about perimeter and area into equations. ) Students should understand that (all together now!) an equation is a relationship among quantities, and that the goal in solving a word problem is to find the numerical value for one or more unknown quantities; and that the method for doing so involves analyzing how the given quantities are related. In order to measure whether students understand this, students must be presented, in a test setting, with word problems that differ more than superficially from the ones presented in class or in the textbook; requiring them only to parrot solutions to questions they have encountered exactly before, measures only their memorization skills.
5.Be able to interpret graphs, and to make transitions between algebraic and geometric presentations of data. For instance, students should know what an x- [y-]intercept means both geometrically (”the place where the graph crosses the x- [y-]axis”) and algebraically (”the value of x (y) when y [x] is set to zero in the function”).
6.Understand basic logic, such as the meaning of the “if…then” syllogism. They should know that if given a definition or rule of the form “if A, then B”, they need to check that the conditions of A are satisfied before they apply B. (Sound like a no-brainer? It should be. This is one of those things I completely took for granted when I started teaching at the college level. My illusions were shattered when I found that a simple statement such as “if A and B are disjoint sets, then the number of elements in (A union B) equals the number of elements in A plus the number of elements in B” caused confusion of epic proportions among a majority of my students. Many wouldn’t even check if A and B were disjoint before finding the cardinality of their union; others seemed to understand that they needed to see if A and B were disjoint, and they needed to find their cardinality - but they didn’t know how those things fit together. (They’d see that A and B were not disjoint, claim as much, and then apply the formula anyway.) It is a testament to the ridiculous extent to which mathematics is divorced from reality in students’ minds that three year olds can understand the implications of “If it’s raining, then you need an umbrella”, but that students graduating from high school are bewildered when the most elementary of mathematical concepts are juxtaposed in such a manner.)
7.More generally: students should know the basics of what it means to justify something mathematically. They should know that it is not enough to plug in a few values for x; you need to show that an identity, for instance, is true for all x. Conversely, they should understand that a single counterexample suffices to show that a claim is false. (Despite the affinity on the part of the high school text I am working for true/false questions, the students I am working with do not understand this.) Among the educational devices to be expunged from the classroom: textbooks that suggest that eyeballing the output of a graphing calculator is a legitimate method of showing, for instance, that a function has three zeroes or two asymptotes or what have you.

also added to the list by commenters:
I would add estimation and verification to that list. Students should know the difference between a sensible and nonsense answer.

Another blog by a college calculus professor: Learning Curves

WichitaBoyOnMath 31 Jul 2005 - 22:15 CatherineJohnson

We have an embarrassment of riches! At least 2 great comments from WichitaBoy, and Ed sat down and wrote out his constructivism-as-psychoanalysis thoughts, too.

Here's one of WichitaBoy's observations:

"Writing is organizing." Now there's a great thought I can take to the bank.
Here's one back for you: professional mathematics is organizing. You have vague thoughts, you notice a vague pattern, and you try to organize your thoughts, to nail down the pattern, to really clarify what's going on beneath the hood. When you've nailed it completely, when you understand with perfect perspicacity the essence of the pattern, then you've got a proof of a new theorem. If you've really organized it, you've got a theorem that goes in the "Book of God".

There's this, too, in response to my saying that what is brilliant about Saxon Math--the structure--is largely invisible:

Read Confucius or Socrates. The ideal teacher should be able to fade into the background like the Cheshire cat. And so with the ideal textbook.

BasicCollegeMathematics 02 Aug 2005 - 01:14 CatherineJohnson

A round-about path from Vlorbik to Tall, Dark, and Mysterious to Basic College Mathematics at mathnotes.com.

Scroll down.

CostOfCollege 23 Aug 2005 - 12:36 CatherineJohnson

After spending the last two days obsessing over college costs, I woke up this morning to a Wall Street Journal op-ed titled, Why Does College Cost So Much? by Richard Vedder, author of Going Broke By Degree: Why College Costs Too Much (probably subscription only).

This fall's probable average 8% increase at public universities, added onto double-digit hikes in the two previous years, means tuition at a typical state university is up 36% over 2002--at a time when consumer prices in general rose less than 9%.
In inflation-adjusted terms, tuition today is roughly triple what it was when parents of today's college students attended school in the 70s.
Tuition charges are rising faster than family incomes, an unsustainable trend in the long run. This holds true even when scholarships and financial aid are considered. One consequence of rising costs is that college enrollments are no longer increasing as much as before. Price-sensitive groups like low-income students and minorities are missing out.
A smaller proportion of Hispanics between 18 and 24 attend college today than in 1976. The U.S. is beginning to fall below some other industrial nations in population-adjusted college attendance.

Vedder lists 6 reasons:

1. rising demand 'exacerbated by soaring third-party payments....When someone else pays the bills, we become less sensitive to price.'

2. lack of market discipline 'How many universities advertise that they are cheaper than their peers?'

3. de-emphasizing undergraduate instruction 'Government subsidies and private gifts given to support affordable undergraduate instruction are often spent elsewhere.'

4. price discrimination "Universities have discovered what airlines realized a generation ago--and they increasingly charge the maximum the customer will bear. They have raised sticker prices, giving discounts (scholarships) to those who are sensitive to price. Increasingly, these discounts go not mainly to low-income students but to talented students prized by universities seeking to improve ratings on the athletic field or in the U.S. News & World Report rankings." [Oh, swell. I have never in my life bought a first-class ticket on an airplane, but I will be buying a first-class ticket to college.]

5. stagnant (falling?) productivity 'There are now six non-teaching professionals for every 100 students, up from three a generation ago. Unless teaching and research have soared in quantity and quality, which seems unlikely, productivity has fallen.'

6. 'rent seeking' behavior: better lives for the staff Salaries of full professors at research universities are up well over 50% in real terms since 1980. Mid-six-figure salaries are becoming commonplace for superstar faculty, coaches, and university presidents. Teaching loads have fallen...' [full disclosure: I'm not exactly heartsick over this development.]

The solutions portion of the op-ed is shorter than the problems portion. (No surprise there. Personally, I'd rank the More-problems-than-solutions principle right up there with Newton's Law of Gravity.)

He says this situation can't go on forever, because costs can't continue to rise faster than incomes forever.

Then he suggests vouchers.

Community Colleges vs. Kaplan

Christian, who works with Jimmy & Andrew & watches WWE wrestling with Christopher, told us about his friend who was teaching at a community college.

He quit to work at Kaplan, where he makes more than he did at his community college. He's earning $40,000 to $50,000 a year at Kaplan & has health benefits to boot.

Alan Greenspan on rising inequality
rising inequality, part 2
rising inequality, part 3
median income families UCSC students
another statistics question
channeling the Wall Street Journal
Financial Times on US college costs
Economist on US higher ed
The Economist on rising inequality in universities

NumeracyAtTheUniversity 14 Sep 2005 - 18:16 CarolynJohnston

Bernie pointed out an article in our local rag today on 'innumeracy' among college students at the University of Colorado (I'm posting the link I found, but be warned that the site will ask you to register. Registration is free).

Here are snippets from the article.

Douglas Duncan, a University of Colorado astrophysicist, is among a cadre of CU professors committed to using real-world analogies to fight scientific ignorance and innumeracy, the mathematical equivalent to illiteracy.
Duncan was about to ask a few hundred CU students to answer a question the other day. Moments earlier, he had reminded his audience that surface area is, for boxy objects, more or less the square of height, and that volume is the cube of it.
On the Duane Physics auditorium's big screen, introductory astronomy students faced the following quiz: If an adult elephant is twice the size of an adolescent elephant, how much bigger is the adult in terms of volume? Multiple choice answers: a) twice, b) four times, c) eight times, d) sixteen times.

Only 57% of the students got the right answer (one of the things I've snipped here, by the way, is the fact that Duncan wrote the book on Clickers in the Classroom, quite literally).

"He's making progress," Carl Wieman said of Duncan's efforts. But Wieman said 90 percent of the students should get such a question right. [side note: Carl Wieman is one of two Nobel Laureate physicists in Boulder; they jointly won the Nobel for the invention of Bose-Einstein condensate. Now Wieman is running a physics-for-kids program on weekends. Boulder isn't all bad].
Duncan uses the elephant scenario as a way to bring home the concept of the cooling of orbital bodies. The Earth has 16 times more surface area than the moon, but it has 64 times the volume.
"So the Earth's core is still hot and the planet is alive," Duncan said. "The moon is dead."
Duncan uses everyday concepts to make unfamiliar scientific ideas resonate. Talk about cubing diameters much less cubing radius and multiplying it by four-thirds times Pi and eyes glaze. Remind students that cupcakes cool faster than cakes and they nod in recognition.

My thought was that they should have learned this thing about cupcakes in junior high school science. But then I ended up really having to think about the next problem.

Steven Pollack, a CU physics professor whose research focus is improving the education of physicists, says the problem also happens in reverse. Physics students often can't conceptualize or explain the results of the equations they so breezily manipulate.
Some students can quote Newton's third law, Pollack says, but can't explain which vehicle feels more force in a head-on between a Mini Cooper and a UPS truck. (Both experience the same shock, if not the same damage).

This guy is describing yours truly now. That was me; I could do math all day, but physics was magic juju. Real Physicists do a kind of intuitive hand-wavy math that never feels rigorous enough to me, but that meets their needs perfectly. My intuition about space and time and nature and the behavior of physical objects is almost always wrong, which is why I prefer rigor.

Now, I don't know if this is right or not, because it's PHYSICS and not math, but here's my take on this problem. If one assumes that the Mini and the truck were going at the same speed, and also that the collision were to bring both vehicles to a dead stop, then the force felt by the truck would be greater because its mass is greater, and the deceleration of the two vehicles is the same (from 60 mph to 0 mph in a split second). Force is mass times acceleration.

But I wouldn't think that they'd come to a dead stop. My intuition would tell me that the truck would decelerate more gradually, i.e., continue forward for a little (albeit at a slower pace), and that the mini would actually end up going backward as a result of the crash, i.e. instantaneously decelerating from 60 mph to -20 or so mph. My thought then is that the force applied to each vehicle would be equal, but the deceleration is not. Can someone tell me if my reasoning is wrong?

The reform doesn't stop with the astronomers and physicists at CU. Even the biologists are yammering on about the evils of rote learning.

Michael Klymkowsky, a CU professor of molecular biology, runs a Web site called Bioliteracy.net. He and others are working to improve students' ability to truly understand key biological concepts.
Klymkowsky said he thinks the lack of science and math smarts among U.S. college students stems from failures in the higher education system.
He is working on a set of essay questions whose answers demonstrate a deep understanding of biological concepts, not just rote learning. An example: "Describe the role of random events in evolutionary processes."

Even CU journalists are going to have to get technically literate.

Paul Voakes, dean of CU's School of Journalism and Mass Communication, recently published a book, "Working with Numbers and Statistics: A Handbook for Journalists." At Indiana University in 1999, he developed a first-of-its-kind course in mathematics and statistics for journalism students.
Too often, Voakes said, journalism students have been "fleeing as fast as they could from math and science since middle school."
"We have to clear out those cobwebs and remind them that they really are good conceptual thinkers, not only in writing and with images but also in problem solving," Voakes said.

I wonder whether that handbook is any good?

0805852492.01._AA240_SCLZZZZZZZ_.jpg

partial product division in Everyday Math
fighting innumeracy at CO
conceptual understanding vs numbers

CalculusBookRecommendationNeeded 15 Sep 2005 - 17:27 CatherineJohnson

A lot of good stuff in the comments I want to get pulled up front, but since I have to go into the city today, there's no time at the moment.

I'll just get this posted, from Anne:

Speaking of good books, does anyone have a recommendation for a good calculus book? I still have mine from college, but I have a hard time understanding the proofs. It seems like there are steps missing.
In the book Countdown about the math Olympiad, the author mentions that one of the mathematicians who constructs the problems is Russian. This mathematician says that his calculus book was only 100 pages long, but it was excellent and had no extraneous information. I wonder if anyone has translated a Russian calculus book.

I've been wondering the same thing, and ktm needs a recommendation to post as well.

So if you've got suggestions, please let us know.

I have two, potentially.

Calculus Made Easy by Sylvanus P. Thompson (and updated/revised by Martin Gardner) This is a classic (always a good sign), and people rave about it. I don't know whether it has proofs, or whether the idea is to give people conceptual understanding without formal proofs.

Also, believe it or not, the University of Chicago School Mathematics Project, the same folks who are responsible for EVERYDAY MATH, had a longrunning project translating foreign math textbooks into English. I'm not sure I can track down what's happened to the list; it seems to have moved to the American Mathematical Society, but I can't find it there at the moment.

I know I did once track it down...so I assume it's still findable.

If someone else comes across it before I do, could you post the link?

Thanks.

translation from the Russian

Calculus of Variations by I.M. Gelfand & S.V. Fromin

Is this the one?

update

Bernie & others say the Gelfand book is an advanced text. (I didn't have time to read the blurb yesterday.)

CalculusTexbookRecommendations 11 Oct 2005 - 13:52 CatherineJohnson

I love Amazon reviewers. Half my life is based on these folks.

Amazon reviewers of calculus textbooks, I've just discovered, are a different animal. Not fully domesticated, I'd say. So it's gonna take me a while to cess them out.

Here's one fellow I'll probably add to my pantheon:

If you are a serious student of Calculus, go get Anton's Calculus. I am a Math teacher in Malaysia and a long time user of Anton's Calculus since his 3rd edition. I teach Calculus the traditonal way because in my country we are still new to the computers. Prof Anton has written books in his previous editions in a lively and refreshing manner that I could read his book again and again without getting bored. I may be old-fashioned, but as a fan of Anton, reading his latest 6th editions is such a delight, and only recently I have just learned how to make use of software like Maple, I could see Anton's Calculus paving my way into new explorations, as his new book says, Calculus: A New Horizon indeed. Buy Anton's Calculus, I am sure you will not regret.

update

It's nice to see college kids are also learning nothing:

In Calculus I, I was taught using computer programs how to solve Calculus problems but never actually learned Calculus. This put me in a tough spot when I had to start Calculus II and didn't know what I was doing. In this course we weren't allowed to use calculators and everything I learned in Calculus I became useless. Fortunately, I came across this book and I was able to teach myself Calculus in a matter of days. I also tried several other Calculus supplements and the only one I can recommend is "How To Ace Calculus" and its sequel for anyone taking Calc II & III. Whether you're dumping a fortune into an education on brushing up on some old math this book is the only supplement you need.

this guy is hilarious

I also spend a huge amount of time cruising Amazon's listmanias. Here's one called So you'd like to... Learn Calculus and Analysis, And Really Understand It! by one Billy Smorgasbord, a resident, it seems, of Oxnard and Antarctica.

There is an bothersome and fairly intimidating phenonemon which is widespread among mathematics teaching and textbooks. For want of a better term, we might call it "Mathematical Macho". Now, when in the grip of this mysterious phenomenon, it seems that people get the idea that it is necessary that a deep subject like mathematics be really difficult to learn, and that there should be an effect of "weeding out the weaker students" alongside that of actually teaching the stuff.
To be fair, I should mention that, over the years, I have observed an impressive number of attempts (whether or not these were made wholly in earnest will be left to the reader) by numberless (pun somewhat intended) and often quite well-esteemed authors and, even, a whole venerable organization (this called the Mathematical Association of America), to make the subject more palatable, and perhaps even interesting, to a wider audience than yet before.
Nope, sorry, fellas. Thus far things just haven't worked out all that well.
Yup, I've seen 'em come and go, alright. Witness the sometimes abysmally constructed explanations in "Calculus Made Simple" by Silvanius Thompson, the scarifying "rigorous" language purveyed by most MAA textbooks, the quite awful wording and quite annoying imbedding of mathematical syntax within text to be found in Boas' celebrated "A Primer of Real Functions", the spotty development in Schey's "Div, Grad, and All That", et cetera. We won't even go into that astonishing and original artfulness (arguably for the delectation of brilliant student and scholarly peer, not for the now-terrified beginning reader) made of the subject in Apostol's highly-regarded two-volume masterpiece.

Billy's list, to my untrained eye, seems pretty useful, and thus far Amazon reviewers mostly second his opinions. However, his listmania on Yup, You Really Can Increase Your Intelligence opens with a book by Robert Sternberg, a red flag for me. Years ago I read a popular book on intelligence by Robert Sternberg many years ago that I thought was pretty dumb.

Plus which, until I'm persuaded otherwise, I'm rejecting out of hand Billy's opinion of Calculus Made Easy. Any book that's been continuously in print for over a hundred years gets on my short list.

(I own the book, and the introduction alone is worth the price of purchase. Haven't tackled the calculus yet.)

So Billy's on probationary status.

Billy's guide says that 23 of 26 people found this guide helpful. Read 13,887 times

I'm going to have to start paying attention to how many people read listmanias.

f6b7124128a002be707b5010.M.jpg Billy?

URLs for listmania & 'so you'd like to guides'

Top So you'd like to guides

Top Listmania lists

So you'd like to... Learn Quantum Mechanics Via Worked Problems and Solutions!

point of comparison:

35 of 35 people found this guide helpful. Read 4,060 times.

while I'm on the subject

Newt Gingrich has 14 pages of book reviews on Amazon.

I bought a book on Saving the Giant Panda he recommended. Very cool pictures.

No calculus recommendations, as I recall.

how not to title your So you'd like to guide

So you'd like to throw your writing career out the window

9 of 11 people found this guide helpful. Read 317 times

CalculusRecommendations 22 Dec 2005 - 16:49 CatherineJohnson

OK, I've collected a handful of recommendations.

Michael Spivak

First, check out the Comments thread on calculus books.

Here's one interesting comment:

Michael Spivak's books are good, as is Tom Apostol's Calculus. Personally, I prefer Spivak. They are both Americans by the way. G.H. Hardy's A Course of Pure Mathematics, and Richard Courant's Differential and Integral Calculus are both classics which are very good, but probably not for everyone. Those are all longer than 100 pages. If you are looking for brevity then you can try out Dan Bernstein's(another American) "Calculus for mathematicians" which is only 12 pages. Find it here: More Mathematics .
None of these books are typical of what you will find in the modern science/engineering calculus courses. If you want something along those lines, then I'd recommend Salas, Hille, and Etgen's Calculus: One and Several Variables.
Fomin and Gelfand's book considers calculus of variations as opposed to calculus of real variables(i.e. "standard" calculus). It's a good book, but probably not what you are looking for.

People love Spivak.

oops. Just clicked on 'See all 60 customer reviews.' Some people love him, some hate him.

Here's Apostol.

Purcell, Varberg & Rigdon

I've asked both David Klein & Barry Garelick for recommendations.

Here is Klein:

I'm not up on calculus texts. I use a standard book (one of many) along with others at CSU Northridge called, CALCULUS WITH ANALYTIC GEOMETRY, 8th ed., by Purcell, Varberg, and Rigdon. It has its faults, but isn't bad. The theory part is good, but it needs more medium level difficulty problems and more graphing examples (without calculator assistance). [One Amazon reviewer loathes it; the other likes.]
Worth avoiding in my opinion is the so-called "Harvard Calculus" books:
Calculus Reform—For the $Millions by David Klein and Jerry Rosen (you'll have to register to open this pdf file, but registration is free)
WHAT IS WRONG WITH HARVARD CALCULUS? by Jerry Rosen and David Klein

Subsequent editions have remedied the worst of the deficiencies, but I would still avoid it.

(I should add that I think Carolyn somewhat liked reform calculus. She's in transit at the moment, but when she chimes in, I'll either edit out this comment, or add hers as needed.....)

Ivan Niven

Barry's first suggestion, which comes from Dick Askey of the University of Wisconsin, is Calculus: An Introductory Approach by Ivan Niven.

I'm sorry to say I've bought the one and only used copy available at Amazon, but there are 2 copies available at Alibris.

Niven wrote his book in 1961, before graphing calculators.

Lipman Bers

Another recommendation from Barry:

Calculus by Lipman Bers, which I ordered the minute I read this Amazon review:

I had come across this book in the university library.
Before that I had been getting excellent marks in Calculus by mechanically going thru the rules in my mind. This book changed all that and gave me a proper perspective on the discipline.
The explanations are clear and this book is eminently suitable for self study.
Recommend this book whole-heartedly at least for the first and second years of calculus.
This was about twenty-five years ago ! But it's just as relevant now.

Barry says Bers was a Russian mathematician emigre to the U.S. who was familiar with Russian textbooks.

Thomas's Calculus

Another possibility might be an early edition of Calculus by George B. Thomas, now in its 11th edition.

Barry says Thomas's Calculus was a college staple for years, and is not easy.

I'm having trouble finding out when Thomas died, so I have no idea which editions of Thomas' Calculus were revised after he was gone.

.....oh, here's a clue, in an Amazon reader review:

I've used both Stewart's Calculus and Thomas'. Interestingly, Thomas has been writing calculus books for a LONG time and i've picked up several editions in the used book stores, because from the first time i bought a Thomas calc book back in Jr. High for my own self interest, i was a fan of his style.
His style is that of the old-school American text book authors who wrote in a clear, concise manner of English, using tangible and visual examples. Those old writers still thought of much of the material as novel, and were appealing to a more agrarian society of students.. especially the young and booming field of engineers. This is lacking in today's texts. The only drawback is that some old texts are much too impersonal and use the passive voice for everything, which can make them very difficult to read at times.
Thomas' recent editions (at least - i can not recall for the 60's era editions) are not only formally clear, but easy to understand and read. Here are the ways in which Thomas' book beats Stewart's book....
[snip]
Thomas' book is in fact probably the best calculus textbook around. I've looked at many many of them, and fraknly, none of them are this complete and well developed... The funny thing is, Thomas' book was one of the best decades ago. It has only gotten more exhaustive and more mature!

This reminds me of Carolyn's post about the early books in a field, Don't teach in a monotone

Thomas has 5-star & 1-star reviews. Very mixed.

James Stewart

Lastly, Barry reports that James Stewart's texts, which teach graphing calculators, are being used a great deal. Barry says Stewart's books are 'fairly good.'

The two big ones seem to be:

Calculus : Concepts and Contexts (with CD-ROM, Make the Grade, and InfoTrac)

and

Calculus

Mixed reviews, expensive as the dickens.

off-topic: Arnold Kling

I just found all of Arnold Kling's Amazon reviews....

'the calculus page'

No idea if this is worthwhile: calculus.org: THE CALCULUS PAGE

BernieOnCalculus 14 Sep 2005 - 22:37 CatherineJohnson

First off, I've become very wary of Amazon's reader reviews ever since I realized that they remove negative comments in order to boost the ratings of the books. That's not kosher. [Catherine speaking: I posted 2 5-star reviews on Amazon that have disappeared, so I'm not sure Amazon has a systematic policy against negative reviews....]

Ok, what's the big deal about Calculus? Why are there thousands of Calculus books and none of them any good?

The reason is that the subject is simultaneously too big and too deep. And there's really no good way to split it up into manageable digestible pieces.

If you want to understand a computer, say, you can split it into pieces (power, case, motherboard, plug-in cards) which are you can then study and understand separately. But with Calculus, learning the subject is more like approaching a huge ship in the fog. At first you don't have any idea what is there. Then a few points become clear, but they are disconnected and make no sense. Then a few structures show themselves, and gradually, very gradually, the whole thing starts to come together. It takes much more energy and much more determination to carry through with such a program than with simpler subjects. So most people don't carry through with it, and it becomes a filter, a flunk-out class.

Linear algebra is a much more useful subject which is amenable to being broken into manageable chunks, and perhaps for this reason it doesn't carry the same mystique as Calculus.

Let's lay out what Calculus is in order to make this clear. It consists of two new operations called "differentiation" and "integration"--roughly analogous to subtracting and adding--both of which are based on a totally new view of the world, called "limits". Limits are a pretty deep concept, much deeper than is generally supposed or understood by most people taking Calculus. In fact, I would venture to say that most people taking Calculus never really grasp limits and, as a result, end up more confused and resentful about mathematics than when they started. Moreover, limits cannot be tackled until one has already achieved a certain mastery of both algebra and geometry, for they entail a melding of these two subjects. Both subjects must have been learned down to the "have it at my fingertips" level before limits will start to make sense.

To be perfectly honest, the problem is even worse than that, because I think it's fair to say that in some sense the human race doesn't really understand Calculus yet. This is because, although there is complete agreement on what basic Calculus is and how to use it, there is still sharp disagreement on what the logical underpinnings of it should be. It's really kind of like Quantum Mechanics in this regard, and that makes it quite unlike all the other kinds of mathematics young students have ever seen, which is all cut and dried.

So, to take the larger view once more, Calculus has three aspects which the student must master more or less simultaneously: 1) the mechanics of integration and differentiation and limits, 2) a philosophical understanding of limits, 3) the thing we discussed yesterday--an understanding of the underlying meaning of the formalism of Calculus in terms of real-world problems. Because there is so much interconnected stuff to learn, the connection between formalism and real-world meaning is even more tenuous, and must be held in even greater abeyance, than is the case with standard school mathematics. The student must suspend disbelief for a much longer period than ever before. Which means that there are inevitably many more Calculus students who get left by the wayside than occurs in elementary mathematics.

It is generally accepted among mathematicians that the hardest part of learning Calculus is 2), the philosophical part, and therefore the teaching of Calculus is usually broken into two subjects, taught to two different groups. "Mechanical Calculus" (high-school Calculus) is taught to students who are deemed too hopeless to ever really learn it deeply. Almost all standard Calculus taught to freshmen college students is of this kind. The students are only taught the basic formulas for differentiation and integration and some of the applications are shoved down their throat. Limits are hand-waved and never really explained, and most students don't realize there's a problem. They're just left with a vague feeling of uneasiness. If they're engineering students, then they are drilled on the applications for another 3 or 4 years, so that they become quite good at them, without worrying too much about what it all means. It works, why worry about it?

For students believed to be budding mathematicians, the whole subject is taught, with an emphasis on the meaning of limits and being able to deeply understand the logical underpinnings of the whole enterprise, i.e., to do proofs. Applications are only lightly touched upon. That's the audience Apostol's book is written for. That's a completely inappropriate book for almost all people.

The mechanics of Calculus, i.e., the basic formulas for integrating and differentiating, aren't really that big a deal except for one fly in the ointment. They are operations applied to functions rather than operations applied to numbers, which is all that the students have ever seen before. So even here there is a philosophical hurdle, because it's hard for people to think of functions as objects. We are used to thinking of functions as the "verbs" of mathematics, not the "nouns", so operating on them seems very strange and most young students probably never really grok it. It's yet another philosophical nut to chew on before one can really understand what one is doing with Calculus. It takes time for that fact to sink in.

The single most important obstacle precluding most students from mastery of Calculus is that they don't really have any idea what functions are when they start Calculus. And that's usually because they don't have a firm grasp of algebra. This, however, is a solvable problem. I personally would reorganize the curriculum so that a year is spent just messing with functions before Calculus is tackled.

But of course that runs headlong into the problem that people in high school and college--unlike students in elementary school--have very little desire to suspend disbelief: if they can't see an immediate payoff for what they are learning right now, they don't want to learn it. This leads to a quandary for the teachers/professors, namely, in order to motivate the students they have to tell them the applications. But in order to do the applications, the students need the full machinery of differentiation and integration. This leads inexorably to the continual cycle of Calculus "reform" which changes textbooks every couple of years, seeking to do the undoable by squeezing in years of difficult philosophical struggle and mechanical practice into far too short a time period.

There's also the problem that many of today's soccer mothers and fathers want to push their children into Calculus as quickly as possible in order to put another feather in their own cap, so they have no tolerance for an extra year "wasted" on learning functions. But that's a subject for a different thread.

WilliamKSmithCalculus 16 Sep 2005 - 12:16 CatherineJohnson

Here's another recommendation from Barry Garelick:

Calculus with Analytic Geometry by William K. Smith (also available at Amazon)

I've already ordered my copy.

Have I mentioned I'm planning to take calculus?

Well, I am. I'm planning to take calculus.

But first I have to 're-take' algebra & geometry. Then trig, which I've never studied.

You folks here at ktm are helping me so much. Even though I'm a writer, I can't locate the words to describe what you've given me. The reason I can't 'locate the words,' of course, is that I don't actually know what I'm learning from ktm. I study & absorb what people say, but then forget the source of my new knowledge once it's been assimilated into my store of old knowledge. I'm left with the hazy feeling that 'I'm learning a huge amount from the Commenters at ktm.'

So I'm going to start taking notes. God is in the details.

thank you!

integers! integers!

So Christopher's math class started integers on Monday—a topic he knows virtually nothing about—and he's having a test tomorrow. He is way not prepared, so I'm busy today writing an Integer Lesson. Probably won't be posting much (though I may have a couple of things from Barry.)

I'm taking a moment to make one more plug for Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa, though.

I could probably add & subtract integers in my sleep. (Though I did have to do some review last year when I first re-encountered the topic, which I take as a sign that my knowledge was more procedural than conceptual.)

But last night, after working with Christopher for awhile, who was semi-lost (I don't think he could pass a test at this point) the Math Fog rolled in.

This is the good thing about working with people who know less math than you do. Concepts and procedures you thought you understood turn out to be not quite so clear. I assume that's what Bernie meant when he said the other day that he'd realized there were aspects of reciprocals he hadn't thought about (if I've got that wrong, Bernie, I'll change it!) Carolyn has said something similar at times. I'll be asking her about some elementary concept that, for her, is as simple as breathing in and out, and suddenly she'll see why Ben--or anyone else--might get confused.

lost in translation

This is another one of those constructivist insights that's been lost in translation.

For me, and I think for most teachers & writers, teaching or writing about a subject always forces you to understand it far better than you did.

Radical constructivists conclude from this that children should explain all of their answers in words.

I'm pretty sure that's wrong, because math is not language. Math is math. A child who can explain his answer by showing the mathematical steps he took to find it has produced a proper mathematical explanation as far as I'm concerned. (Russian Math & the Chinese teachers in Liping Ma all offer mathematical explanations & demonstrations.)

But what really bothers me about the 'explain your answer in words' business is that it puts the onus on the child to teach himself. The teacher doesn't have to work and fight and struggle to find the right words; the child does. I know that's wrong.

While I'm on the subject, why don't I just go ahead and take umbrage at the suggestion that a child is capable of explaining math in words?

Writing is hard. Writing well is extremely hard. Finding the words to explain any mathematical concept well is a vast and ambitious undertaking in itself, not a toss-off in the middle of a homework assignment or state assessment. (I'm seriously against the extended response (pdf file)requirement that's taken over IL state rubrics. At least, for the time being I am. [update 5-14-06 sorry, link no longer works])

back to Russian Math

I shouldn't be putting words in people's mouths, so if I've misunderstood Bernie or Carolyn I'll issue a CORRECTION.

In the meantime, why don't I just return to quoting myself.

It's true for me that when I work with a child for awhile, I realize I don't understand things as well as I thought (or hoped).

After Christopher went to bed, I got out Mathematics 6 and turned to the section on adding & subtracting integers.

The first thing that struck me was the fact that this topic appears at the very end of the book. Prentice Hall Pre -Algebra^* opens with integers, and I question that. I question it not based on any profound grasp of pre-algebra as a coherent whole. I question it on grounds that Nurk & Telgmaa are geniuses, and they put adding & subtracting integers last, not first.

I'm sure they have their reasons. (I intend to figure out what their reasons were.)

Reading through Nurk & Telgmaa's discussion, I realized why I was confused. I think I realized why Christopher was confused, too. I hope so.

We were both, I believe, stumbling over this type of problem:

5 - (-7) = ?

Both Saxon Math 8/7 & Russian Math teach addition & subtraction of integers using the number line. Saxon's lessons were particularly strong, I thought.

But when I tried to untangle myself by resorting to the number line, I got stuck.

Start at zero, move five to the right, then.......then what?

What was my next move? My very next move, without renaming or re-expressing - (- 7) as + 7 ?

I was stuck.

Reading through Mathematics 6 I realized that the problem is something Wayne Wickelgren & his daughter Ingrid have raised: the same letter or sign has been made to stand for two different things.

There are two 'minus signs' in 5 - (-7). One means 'opposite,' and the other means 'subtract.'

One means 'perform an operation' and the other doesn't (I don't think. Is 'taking the opposite of a number' considered an operation? I don't know.)

In any case, for both Christopher and me, 'subtract' and 'take the opposite of' are two different things.

Mathematics 6 has a formal demonstration of the fact that:

5 - 7 = 5 + ( -7 )

This is something I think I figured out on my own many, many years ago. I've been using it ever since to de-confuse myself when dealing with long lines of integers to add & subtract. At some point, if I'm getting confused about whether I can or can't use the commutative or associative properties, I just turn the whole thing into addition.

Reading Mathematics 6 I realized that's what needed to happen with 5 - ( -7):

5 - ( - 7) = 5 + [ - ( - 7) ]

Voila!

Christopher and I both understand that 'the opposite of the opposite' is the number you started with originally; the opposite of the opposite of 7 is 7. (This wasn't an especially hard idea for Christopher, but the number line really nails it down.)

Once you convert '5 minus negative 7' to '5 + the opposite of the opposite of 7' it's in a form Christopher understands, and can do.

AND it's in a form you can perform on the number line, if you like or just want to check.

5 - ( - 7) =

5 + [ - ( - 7) ] =

5 + [ 7 ] =

5 + 7

Once you've converted a 'double negative' subtraction problem into addition, you no longer have an anomaly, The One Subtraction Problem That Cannot Be Done On A Number Line.

We'll see how it goes. This morning I had Christopher quickly rewrite 12 subtraction problems as addition problems. (I haven't explained to him why a subtraction problem can be rewritten as an addition problem, and I don't know whether I'll get to that today. I haven't closely studied Mathematics 6's presentation to see whether I can introduce it 18 hours before the test.

Fortunately, Ed had already introduced the idea that 'subtraction is addition' last night, when he used the addition-of-debt-to-debt (a concept that is not foreign to our household) to show Christopher that:

- 7 - 7 = - 14

I think he had a lesson in Saxon on subtracting a positive from a negative being the same thing as adding a negative to a negative, so he probably had some knowledge to build on before Ed gave him the add-one-debt-to-another example.

It's the minus-minus issue that's throwing him.

I hope.

one last thing

Looking at this, it strikes me I'm also going to have to create some problems that I ask Christopher to 'simplify'—'simplify' defined broadly as 'write it in the simplest possible correct way that will allow you to recognize what the computation is and do it.'

For instance:

-7 + 5

He probably needs some practice rewriting this as 5 - 7.

I'll see.

I'm also going to try to put together an incredibly simple 2 - 1 type problem that he can always solve quickly when he gets jumbled up. Something like this:

1 - ( - 1) = 2

-1 -1 = -2

-1 - ( - 1 ) = 0

He hasn't learned the Polya line about how 'For each complicated problem you can't do, there is a simple problem you also can't do.' I realize it's not clear that you can explicitly teach problem solving, but I'm going to have to try. He's got to learn the strategy of creating a super-simple version of a hard problem in order to see how to deal with the hard problem SOON.

^*new title: Prentice Hall Mathematics: Explorations & Applications

keywords: subtraction negative minus absolute value subtraction is addition integers extended response

CalculusWorksheets 21 Sep 2005 - 20:52 CatherineJohnson

Central Lakes College has calculus worksheets, too.

Here's one. (pdf file)

Unfortunately, they've posted a link to a set of calculus notes they characterize as ++great++, but they're off-limits to me.

Yahoo's list of math links

Yahoo math links

self-instructional mathematics tutorials

This site, self-instructional math materials, looks interesting:

The following mathematics tutorials development as part of the project, Increasing Students Success: Addressing Prerequisite Mathematics Assumptions in Introductory Non-mathematics Courses, funded by The Fund for the Improvement of Postsecondary Education. (project #P116B60125)
Various introductory courses at six universities have been selected for this project. One goal is to provide self-instructional mathematics tutorials for individuals who may need review of certain topics. This self instructional approach will:

let you move at your own pace.

provide you with additional review (if necessary).

let you know how well you are doing.

Currently the none interactive versions have been developed. While some do not have a lot of graphics, the review materials 3, 4, and 5 are fairly graphic intensive and may take a few minutes to load. Interactive versions are currently being developed and will be added to this site at a later date.

keywords: Yahoo math links calulus worksheets self instruction self teaching teach yourself

RealAnalysisTutorial 22 Sep 2005 - 15:57 CatherineJohnson

Barry Garelick sends this link to an online tutorial in real analysis by Bert Wachsmuth of Seton Hall University that he says is impressive. Apparently there are unfinished sections on the site, but what is there is excellent.

I hope he'll be able to steal time to finish the work.

Thanks, Barry!

I'm posting these resources on the math supplements page, which is listed on the sidebar as our favorite math supplements for kids. As soon as we can get to it, Carolyn & I will revise some of those links (an online tutorial on real analysis isn't for kids...) and add a link specifically on constructivist curricula. At the moment we have nowhere to list constructivist curricula, their problems, and resources for dealing with those curricula, such as the page Carolyn found listing all of the Connected Math projects.

BernieOnTrigAndCalculus 22 Sep 2005 - 16:59 CatherineJohnson

Boy, I can't even keep up with my own blooki; I don't know what makes me think I'm going to get through a math course or two or three.

Here is Bernie's comment on trigonometry and calculus. (I'm also going to figure out how to make sure these things don't get lost, so I would appreciate suggestions. I've got most of Barry's book recommendations logged on the Recommended Reading page & entered in the book-style index, but I don't have a separate page of advice and recommendations for.....what it takes to study math and succeed at learning it.

No, you don't need to take Trig before taking Calculus. They're completely unrelated. You can skip Trig entirely if you want to.
There's a reason why Trig is required before Calculus. Trig, among other things, gives you some down-to-earth examples of functions which are not simple algebraic formulas. Most students don't realize that that's what they've been given, but they have.
There is danger here. Those teachers who want to get to Calculus quickly or who are thinking that Calculus is the more important subject will teach Trig completely from the function-theoretic point of view. While that is an important part of Trig, it is a beautiful subject in its own right which can be taught completely without reference to functions.
Unlike Calculus, I've used Trig many times in engineering applications.

keywords: advice for studying math calculus trigonometry functions course sequence

EngineeringSchool 13 Nov 2005 - 18:46 CatherineJohnson

Via joannejacobs, Confessions of an Engineering Washout:

Interesting.

The United States contains a finite number of smart people, most of whom have options in life besides engineering. You will not produce thronging bevies of pocket-protector-wearing number-jockeys simply by handing out spiffy Space Shuttle patches at the local Science Fair. If you want more engineers in the United States, you must find a way for America's engineering programs to retain students like, well, me: people smart enough to do the math and motivated enough to at least take a bite at the engineering apple, but turned off by the overwhelming coursework, low grades, and abysmal teaching. Find a way to teach engineering to verbally oriented students who can't learn math by sense of smell. Demand from (and give to) students an actual mastery of the material, rather than relying on bogus on-the-curve pseudo-grades that hinge upon the amount of partial credit that bored T.A.s choose to dole out. Write textbooks that are more than just glorified problem set manuals. Give grades that will make engineering majors competitive in a grade-inflated environment. Don't let T.A.s teach unless they can actually teach.
None of these things will happen, of course. Engineering professors are perfectly happy weeding out undesirables with absurd boot-camp courses that conceal the inability of said professors to communicate with words. Fewer students will pursue science and engineering majors, and the United States will grow ever more reliant upon foreign brainpower to design its scientific and manufacturing endeavors. I did my part to fight this problem, and for my trouble I got four months of humiliation and a semester's worth of shabby grades that I had to explain to law schools and employers for years. Thousands of college students will have a similar experience this fall.
So engineering is suffering in this country? It deserves no better.

I have to say, I've given this some thought myself.

I love math, and I'd like to learn more of it.

But I'm not sure there are teachers out there who can teach me.

I'm a self-taught kind of person; I'm constantly diving into new subjects & figuring them out.

But I'm finding math is harder to self-teach than the other subjects I've tackled thus far (and the list includes autism & neuroscientific research).

I think I've mentioned that Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa makes the Singapore Math series look like a remedial text. A good remedial text, but remedial nonetheless.

If I could use Russian textbooks like Nurk & Telgmaa's, I could learn college math.

I don't know whether I can learn college math from our own texts. Or from our own teachers.

back to Stevenson & Stigler

I suspect that the 'teaching gap' in engineering departments--and I suspect there is a teaching gap--goes back to Stevenson and Stigler, who found that Americans universally see math achievement as being (largely) a matter of innate ability, not effort.

I've never met anyone, apart from Bernie, who sees math as first and foremost a matter of hard work. (And I may be misstating Bernie's position, too.)

People--including mathematically talented people, I'd say--see math as a matter of native ability, talent, genius. I see it exactly the same way, or I did. I had to wrench myself away from this view in order to teach Christopher & me.

When you see math talent as something a person is either born with or not--and in fact math professors are going to be people who were born with math talent & plenty of it--how is that going to affect your teaching?

It's going to tell you teaching isn't what makes the difference.

Overachievement U

I am a firm believer in overachievement.

In fact, AND THIS IS A NONPARTISAN BLOG, LET ME REMIND YOU, overachievement is a quality I vastly admire in Hillary Clinton, who is the hardest working, most overachieving public figure I know. (I saw her give a speech 6 years ago, and it was something. The distance she'd come from the Clintons' first campaign was remarkable. You could see the hours and hours of hard work, on the stage.)

When Ed and I were gearing up to request the Big Switch for Christopher, from Phase 3 to Phase 4, I was a nervous wreck. I had been flatly told, by one of the two Middle School guidance counselors, 'He's a three.'

Our school--everyone in it--thinks kids are ones or twos or threes or fours, and, truth be known, I thought the same. I felt like a delusional over-reacher asking that my child, an Obvious Three, be Crowned a Four.

When we raised the issue with his Phase 3 teacher, she blanched. She'd been singing Christopher's praises, telling us he was the best student in her class, but when we said, "We'd like to move him to Phase 4" she was shocked. She had no idea we were going to raise this possibility.

She had no Mental Construct saying the top student in a Phase 3 class maybe ought to move to Phase 4.

"I've never thought of Christopher as a 4," she said.

I should stop and add that she was (and is) a terrific teacher. I don't tell this story to complain about Christopher's math teachers last year; that's not the point. The point is that Stevenson & Stigler are right; Americans think of math talent as a strange, unique, built-in form of genius.

After that meeting, which had gone terrifically well, since the teacher had rapidly & correctly worked through the logic of moving Christopher and had then advised us to do it sooner rather than later, I was still nervewracked. I couldn't stop thinking about how hard we'd had to work to get him to the top of his Phase 3 class. I couldn't stop thinking that Christopher's math progress was the product of work, not nature.

I couldn't stop thinking he was really a 3.

Ed said, 'We want him to be an overachiever in math. That's our position.'

That was a help.

I'm pretty sure we need to start thinking of math ability as a Spectrum Talent.....some people have lots of it, other people also have lots of it, too, but not at the 'learn it by smell' level of the whiz kids. This second group, the 80 to 90 percentilers, need teachers. Good ones.

The big bulk of people in the middle have whatever level of natural math ability the big bulk of people in the middle do. Singapore's students probably tell us what level of math achievement the big bulk of people in the middle have when they've got a good curriculum & good teachers.

I guess what I'm saying is: Confessions of an Engineering Washout tells me that we have a math teaching problem at the professional level as well as the elementary, middle, & high school level.

I think we need to think of math the way we think of athletics.

Yes, a brilliant athlete is born with something the rest of us aren't.

But none of the greats get there on their own.

They all have coaches--good ones--showing them how to do what they do.

Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent

EngineeringSchoolPart2 28 Sep 2005 - 23:38 CarolynJohnston

I read the whole thread about "Confessions of an engineering school washout" here at KTM before reading the original article and all the comments at Joanne Jacobs' site.

I find that I agree with everybody, pretty much.

1. The author is a whiner.
2. Engineering schools do overload their students.
3. Professors at research universities typically do not care as much as they ought to about whether their students are learning, or pay as much attention to their teaching as they ought.
4. Professors and TAs often do not speak English as well as one would wish.

I think item 1 needs little explanation. I think Kern is having a hard time with his first experience of failure; he thought he was the cat's pajamas, and got smacked down hard. I sympathize, but by now I consider the occasional failure routine. It's a natural consequence of overreaching one's limitations -- whether innate or circumstantial -- in life. I hope I don't sound as whiny as he does when I talk about my failures.

As for item 2, it is a fact that engineering schools overload their students, and some are growing concerned about it. At the school Bernie and I taught at (Florida Atlantic University), there was consternation because the number of courses needed for an engineering major had grown to the point where it was impossible to complete them in 4 years (in the end, however, noone was willing to identify any courses that could be sacrificed to keep the program within its boundaries). Engineering school not only requires a lot of courses, the courses tend to be tough.

I did a math major as an undergrad, and engineering school was famously tougher, no doubt about it.

As for professors, they are incentivized to excel at research rather than at teaching. In my experience, this is as true at teaching-centered colleges (such as Bernie and I taught at) as it is at research universities. When professors lose tenure, they do so because their research was poor, not because their teaching was inadequate. Does anyone know of a single counterexample -- someone who did bangup research but was fired anyway for poor teaching (unless the person was so disliked that any excuse to oust them was seized upon)? I'd like to know. It is well worth researching for the good teachers at a school. All schools pay lip service to the importance of teaching; very few really hire and fire on that basis.

At a teaching-focused college, you can be fairly sure that you'll get a genuine (and perhaps even interested) professor for most of your classes. But at a research university, you'll generally find that the grad students that do the teaching have good domain knowledge. The grad students that I went through school with were often considered by the students they taught to be better teachers than their professors; they were less arrogant, more available than the professors, had adequate domain knowledge, and were very conscientious.

I feel that it is a student's job to take responsibility for his own education, and I wonder whether Kern fully did that. But, sometimes, hours of useless head-banging with a math text in the library can be circumvented by one good explanation from a teacher. I think students have the responsibility to utilize every tool available for learning, and one of those is their teacher. Students have the right to expect a good explanation when one is needed.

To that end, weak english is a problem that is very difficult to work around. It's best to avoid such teachers as much as possible, but it's not always possible. Departments generally try not to load themselves up with professors and TAs who can't speak english.

The question of whether a college education is actually worth $40,000 a year tuition is one best not asked around here.

Catherine worries about not being able to find a teacher who can teach her math. I worry more that she will find teachers who can't keep up with her quick mind. I think the Johnston family motto applies here: life is full of vain hopes and groundless fears. This one is a groundless fear.

Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent

TourDeForce 11 Jun 2006 - 12:41 CatherineJohnson

Engineering school is a rude awakening for most college freshmen. Many students are surprised to learn that their previous thirteen years of formal schooling have not adequately prepared them for the rigors of engineering school. Sadly, about 2/3rds of them, some very bright motivated students, won't make it through the program. This is what you learn by the end of freshman year:

1. You had been coddled the past thirteen years by your K-12 teachers. You were mostly spoon fed the material, at a slow pace, and then tested on how well you could regurgitate the exact same material back to the teacher in the exam. Rarely, if ever, were you required to apply the knowledge you had learned to solving new problems you hadn’t seen before. As a result, you could, and probably did get by, without mastering the concepts as well as you should have. You are finding out the hard way that most of your knowledge is still at the inflexible stage. This would be most apparent in...

2. Algebra: A course you took four years ago and didn’t learn well enough is coming back to haunt you now in calculus. Calculus seems much more difficult than it did when you took it last year in high school. This is because the pace is twice as fast and the exams require more than a regurgitation of what was taught (or rather won't be, see below). You see, mathematics is brutally cumulative. Calculus is really 10% calculus and 90% algebra (which includes a healthy does of trigonometry and geometry); and, the calculus step isn’t all that difficult usually. Most of the difficulty lies in either setting up the calculus step or finishing the problem after the calculus step. Calculus isn't all that difficult provided you've mastered algebra.

In high school, they allowed you over the algebra bridge without paying the full toll and you’re paying the price now, especially if you hobbled over on your graphing calculator. Anyway, you’ll need to know calculus and algebra cold if you expect to pass Physics I next semester. But this is going to be close to impossible because...

3. Your professors don’t teach and you can barely understand your TA’s poor English. This is more of an expectation problem; you’re still expecting to be coddled like you were in high school. Now you are expected to read the new material on your own and attempt to solve the problems before coming to class. This is a feature, not a bug.

By teaching yourself, you will be forced to understand and master the material, assuming you are doing the homework problems beforehand. Which you haven’t been doing because there just isn’t enough hours in the day to teach yourself and then do every problem assigned in every class. So you dutifully copy down the answers that the TA gives you during the class review all the while thinking “hey, that wasn’t so hard, now that someone’s showed me.” But, “understanding when explained by others” is not the same thing as the “ability to explain to others” which will become brutally apparent...

4. When you fail your first exam. The first test you’ve ever failed in thirteen years. You crammed the whole night before, but the test was too hard and too long. Goodbye unearned self-esteem; hello magic number 7. Seven is the number of things you can hold in working memory at one time. Partially learned knowledge uses more of these seven slots and takes longer to process than fully mastered knowledge. Your brain is being tested to its capacity for the first time and it's not prepared. You’ll become casual acquaintances with magic number 7 this semester and good friends next semester in Physics I because...

5. All those damn physics equations. Your brain is full. It feels like every time you learn something new it’s pushing something else out – like your name and your address. Spring semester brings with it Chemistry II (which requires you to remember everything you learned in Chem I), Calculus II (also brutally cumulative with Calc I), Computer Programming (learning new languages isn’t easy, especially when that language is C++); English Composition (your only easy class, too bad you have to do a term paper that’s twice as long as anything you’ve ever written before); and lastly Physics I, which will be...

6. The course that you’ll blame when you transfer to business school. Physics I – the rock upon which many engineering education ships have foundered. Two reasons – word problems from hell and the magic number seven. Physics is your first real test in your education career. It tests how well you are learning not only physics (under a withering course load of other difficult courses), but also how well you previously learned algebra and calculus. It is the latter two that will be your demise because you need every brain cell you can muster to learn physics today.

If you’re expending too many brain cycles recalling how to do the necessary calculus (most likely because you don’t sufficiently know the underlying algebra) sooner or later you’re going to meet the magic number seven. Meeting the magic number seven is like running out of active memory. You become overwhelmed and inefficient. Eventually, it all ends in tears (or an extra year of college after you’ve transferred to a nice soft major like human resources, communications, women studies, etc). So you lash out and look for someone to blame...

7. Like your college engineering department. Wrong. The train was slipping off the tracks well before they came into the picture, most likely sometime in elementary school. Don’t blame them because the train finally derailed at their station. Don’t be like the drunk who’s looking for his lost keys under the streetlights because that’s where the most light is. A career in engineering or in one of the hard sciences was effectively foreclosed to you by the 8th grade,. Most likely, you would have been none the wiser had you stayed in the soft fuzzy land of almost every other undergraduate field of study. Everyone would have been happier too because, well, you don’t know what you don’t know. Anyway, you can at least find solace in the words of Homer Simpson when he said to Lisa and Bart after they failed: “Kids, you tried your best and you failed miserably. The lesson is, never try.” But why blame yourself when you can blame the real culprit...

8. Your rotten K-12 education. Oh sure, they meant well; but look what happened. You see, you’re not part of the lower half of the bell curve who probably shouldn’t be pursuing a career in engineering or the hard sciences anyway. Nor, are you part of the two standard deviations and above gang that have the ability to succeed and compensate for a rotten education. No, you’re part of the curve that needed a good education to succeed and you didn’t get it.

And, it wasn’t a single chop that lopped your head off; rather it was death by a thousand tiny paper cuts. The accumulation of thirteen years of inefficiencies and unsound practices that prevented you from mastering and over-learning the material you needed to succeed in a rigorous college curriculum. Instead of teaching you content and facts and making you practice until automaticity, your well-meaning teachers were feed a bunch of scientifically and cognitively unsound educational fads -- constructivism, discovery learning, child-centered education, and social promotion to name a few. They all sounded so lovely in theory, yet in practice have consistently failed to adequately teach students as you have just found out. The hard way.

This advice may have arrived too late to help you; but it is not too late for that kid who just started kindergarten who lives down the street. This article is really for his or her parents, but they probably need to hear your story first before they begin to take it seriously. After all, you believed everything your K-12 educators told you and your parents, and look what happened.

- contributed by Kenneth DeRosa, October, 2005
(Note from Carolyn: this essay has been rewritten slightly, by its original author, with links added -- Carolyn).

That's going straight into the Math Writing Hall of Fame.

update

the magical number 7, plus or minus 2

Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent
grandmasters and the magical number 7

Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing

MSimonOnEngineeringTeaching 30 Sep 2005 - 21:28 CatherineJohnson

I'm pulling this to the front, because while I understand engineering is hard work, requires lots of tough courses, etc., I'm completely unsympathetic to the idea that engineering professors can't be expected to teach. By teach I mean actually put material across to students in such a fashion that they come away from class knowing & understanding more than they did when they came in.

(I'm also unsympathetic to the idea that an engineering degree should require more courses than can be completed in 4 years, and I'm going to carry on being unsympathetic in spite of the fact that I know nothing about engineering or engineering degrees. Sometimes I get like that. I have my reasons; maybe I'll get to them later.)

I skipped all that college stuff (it did go too fast in some subjects, too slow in others) by teaching myself engineering. I have worked for some large aerospace companies and hardware/software I have designed is protecting you in flight.
In fact some math routines I designed were flying on the F-16. May still be.
BTW I went to one of the top science and math high schools in the country. Omaha Central.
I might add that the US Navy knows how to teach. They cram about 2 to 3 years of engineering training into 6 months of theory and 6 months of practical application. Once you have your specialty down. Mine was electronics. However, I knew that so well that I was often teaching the course and helping the slower students pass.
Being a radio amateur at age 13 helped a lot.
In any case the Navy went faster but for me was easier. Why? The instructors knew their subjects backwards and forwards. If asked for an explanation they could give one. They worked hard to get inside the minds of the students to figure out what the student's problem was. They cared.
Why? Because they were graded on how well they taught the material. They lost their jobs if they didn't do well. No tenure.
++++
I got P-Chem in my first year of college.
I found it rather easy. I hit the wall in multi-variable differentials. (which I now get)
Heat transfer and fluid flow (which I got in the Navy) some find very hard. I sat in the back of the class reading motorcycle magazines and occasionally correcting the UC Berkely Physics Professor's mistakes. Now there was a hoot. The prof rarely called on me. I made him look bad. Still, he was quite good.
The #1 problem in our teaching corps is tenure.
++++
And yet. College was not for me.
So what if it takes 6 or 7 years to learn engineering. Shouldn't desire and tenacity count?
Such desire worked for me. But I had to do it outside of school.
Being outside of school did help me.
When microprocessors were new and there were not enough teachers to go around I taught myself. School can teach you how to learn with help. Learning on your own teaches you how to learn with no help. It ought to be valued more.
In fact learning with no help is exactly what you want on the frontiers.

The US Navy knows how to teach.

If the US Navy can have instructors who teach, universities can, too.

I don't care who teachers are, whether they're full professors or T.A.s. They need to be able to teach. We can talk all we want about the purpose of research universities being research, not teaching. What pays the bills--what keeps research universities running & funded by taxpayers & tuition paying students & parents--is the fact that they award degrees to college & grad school students.

That means teaching.

MathProfessorsVsComputerScienceProfessors 17 Sep 2006 - 01:14 CatherineJohnson

Very interesting comment from Lesley Stevens:

Tangential to the "math brain" discussion, my husband has made a very interesting observation.
A smidge of background here: He has always been one who has no fear of questioning or correcting his instructors, something that many of his primary school teachers didn't much care for, as you can imagine. He has a double major in mathematics and computer science and he'll graduate with his B.S. this spring. (He is 31, finishing his degree after a 10 year hiatus.)
What he has noticed is that while his CompSci and gen ed instructors often resent being corrected, his mathematics instructors do not.
His theory is that people who do math are accustomed to being wrong. They make mistakes all the time, and it's easy to do when working a complex problem on a blackboard. He thinks that you pretty much can't do math all the time and still maintain an infallibility complex, or superior attitude towards students. Especially since math is a young person's game, and most math professors are already past their "peak" in math ability, and know it.
In addition, in "soft" liberal arts areas, or conversely, extremely complex areas like programming, mistakes may not be obvious, or may be open to some debate. In math, an instructor can't wiggle around a mistake. If he has added 6 to 7 and gotten 14, that's just wrong, end of story.
What I think I'm getting at here is that making math easy for students through "no one answer", etc. is not helpful because it delays an understanding that math is hard for everybody including people like my husband, and that the best mathematicians in the world make mistakes all the time. This understanding actually makes me feel a lot better about my own anxieties about math.
Oh, and as for "math brains", my husband's major the first time around, before the 10 year break, was Philosophy.

This discussion has been a revelation to me. I'm going to keep all the URLs handy so I can print out these comments out and/or send the links to friends, teachers, & administrators as needed.

The vast majority of people simply assume, without even realizing they are assuming, that doing math comes naturally to the select few AND that those select few are the ones who ought to be doing math, and who deserved to be put in Phase 4.

I was just this afternoon talking to a mom whose son was moved from Phase 4 to Phase 3; according to figures I was given, 35% of Irvington's Phase 4 5th graders failed the Phase 4 placement test at the end of 5th grade, something most parents don't know. Most of these children switched to Phase 3, though some parents refused the move. I know of two; there may be others.

All of this gatekeeping activity is based on the explicitly stated judgment that 'he/she doesn't belong in Phase 4.'

It's an essentialist argument.

I was already off the boat for the whole 'He's a three' business, thanks to Wayne Wickelgren, and to Ed ("We want Christopher to be an overachiever.")

Now I'm seriously off the boat. And I'm armed.

Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent

MalcolmGladwellOnHarvardAdmissions 06 Oct 2005 - 16:03 CatherineJohnson

After Malcolm Gladwell's recent article on universal health insurance I thought he'd jumped the shark.

Turns out I was wrong.

GETTING IN: The social logic of Ivy League admissions

question

Am I going to spend $18.48 ordering this 672-page book?

0618574581.01._AA240_SCLZZZZZZZ_.jpg

I hope not.

UPDATE 9-23-2006: For once, common sense has prevailed around here.
I do not own a copy of this book, and I do not intend to own one in the future.

no common sense-y and no running backs, either

Then there's The Game of Life, which I would read, except it would depress me.

We have ZERO football players in this house.

That's too bad, because I, for one, completely get the point of being a football player, having read Betty Harragan's Games Never Taught You early on.

Oh, well. I'll stick with my Original Plan of encouraging Christopher to go out for track.

question number 2

Are there any track star Captains of Industry?

on playing Notre Dame

I love this observation, from Kaus:

Like many New Yorker policy articles, Gladwell's reads like a lecture to an isolated, ill-informed and somewhat gullible group of highly literate children. They are cheap dates. They won't think of the obvious objections. They won't demand that you "play Notre Dame," as my boss Charles Peters used to say, and take on the best arguments for the other side.

(Do we still have debate teams in U.S. high schools?)

I hope so.

ProblemsRemedialCollegeFreshmenCantDo 27 Nov 2005 - 16:09 CatherineJohnson

Notice that all of these questions involve fractions. Fractions & percents.

I remember being shocked when Carolyn first told me that the barrier to college kids succeeding in algebra was fractions. I had no idea. Then Bernie told me the same thing, and since then every math professor I've ever talked to has said, 'Students can't do fractions.'

Of course, this made sense, since at the point where I met Carolyn I had recently discovered I couldn't do fractions, either.

That's not quite right.

I could do fractions.

I couldn't teach them.

Waukesha County Tech is short for Waukesha County Technical College. These are college freshmen we're talking about.

source:
Journal Sentinal 10-6-2003

DetroitFreePressArticle 23 Oct 2005 - 12:49 CarolynJohnston

LoneRanger found this article on college students taking remedial math at Wayne State.

What's the price of leaving high school unprepared? Ask Chelsea Stephanoff, a Wayne State University student who is spending nearly $600 this semester for a class that won't count toward graduation.
Why? Her math skills were poor enough that even after four years of high school math, she was placed in a remedial class.
"Math is not my strong point at all. I'm horrible at it. I have a hard time focusing on it," said Stephanoff, a fourth-year student from Shelby Township who wants to be an elementary school teacher.

So Michigan is apparently looking at instituting stiffer graduation requirements for high school students.

Last year, the Cherry Commission on Higher Education and Economic Growth recommended a rigorous curriculum that includes four years of English, three of math, three of laboratory sciences, three and a half of social studies and two years of a foreign language.

Such requirements wouldn't have done Chelsea Stephanoff a lot of good though, if she took four years of math in high school and still can't do math to save her life.

Like Ken says, the train is apparently derailing earlier than that.

article from Lone Ranger on remedial ed in MI colleges
more from Lone Ranger's MI article

KumonCenterLogPage 17 Nov 2005 - 14:17 CatherineJohnson

Spent a good 3 hours at KUMON today. What a trip.

Only three white people showed up for the whole afternoon, & Christopher & I were two of them.

Then there were two black kids.

After that?

Foreign nationals. Asians & Indians. And the Asians came as couples. That right there blew me away. The only time, in Irvington, you see both parents turn show up for an extracurricular activity, it's soccer or baseball.

Not only did both parents show, they were dressed. One mom was wearing patent leather flats. I can't even remember the last time I saw a pair of patent leather flats. She looked like the kind of woman you see shopping in the Prada outlet at Woodbury Common. (If the kind of woman you see shopping in the Prada outlet at Woodbury Common doesn't instantly call an image to mind, think: Asian, young, great-looking, chic, and rich.)

There are lots of foreign nationals in these parts, it seems, but they don't mix in much, or integrate. I keep hearing from other parents things like, 'they send their kids to Japanese school on Saturdays.' Which has always sounded like an urban legend to me. Japanese school? What is Japanese School? Where is Japanese School? Can I see the building from the road?

Now I'm thinking: Japanese school.

I better look into that.

Christopher passed the test for 3rd grade, and flunked 2nd grade because he was too slow. (Speed and Accuracy, the Kumon mantra.) So he has to spend this week reviewing 2nd grade math facts, then take the achievement test again next Saturday. Assuming he passes, he starts grade 3

As for me, I was a Calculating Whiz.

When I handed in my first test, the owner said, "That was fast."

When I handed in my achievement test he said, "Done already?"

Then he put me in fourth grade.

More later.

KenDeRosaOnCalculus 30 Oct 2005 - 21:49 CatherineJohnson

from KDeRosa Who you callin’ crazy now?

Rudbeckia Hirta of Learning Curves writes about her latest crop of college math students. It’s not pretty.

First the Gen-Ed kids:

Unfortunately for my gen-ed class, many of them cannot do algebra at all. Again, these students are bright and interested. Good students; they seem to like the class. Almost all of them hard-working and with excellent attitudes. … If these students were as serious about their high school educations as they are about their college educations, their lack of algebra skills cannot be entirely their faults.
And yet, the most missed question on the test (tripping up between a quarter and a third of the class) was missed because of algebra. They can set up the equation, but they can't solve for x.

No surprises here. They know almost no algebra. If you look at the sample exam questions given, you’ll see that she’s testing very basic algebra. Of course, the calculus students know more algebra. But do they know enough?

Allow me to point out that almost all of the calculus students can do algebra; their mathematical problems are fairly minor. Some may have to work at it more than they should, and we have had some parentheses issues, but, overall, the calculus students can use the distributive law and do not do stupid things with radicals and exponents. They are wary of rational expressions and a bit shaky on trig identities, but they are as skillful and well prepared as any students that one might recall from some mythical glory days back when freshmen could do algebra.

So their algebra skills aren’t exactly carved in stone. And, how much more of their algebraic knowledge is still at the inflexible stage? The wheels on the train are getting wobbly. Will the train stay on the tracks all through Calc I? And then II? We’ll find out in Physics.

TheShoelaceProblem 15 Dec 2005 - 15:53 CatherineJohnson

Now that Doug has solved my helmet problem, ^* I'm hoping someone can solve my shoelace problem.

A couple of years ago the then-director of special ed (we're on our 3rd in 7 years) told me to forget about teaching Andrew to tie his shoes. Forget about it as in: forget about it for good. It's not going to happen, don't speak of it again.

Naturally this was my cue to decide Andrew would be learning to tie his shoes come hell or high water.

[pause]

Wow. Hell or high water. I've been saying hell-or-high-water most of my adult life, and until Hurricane Katrina it hadn't occurred to me what the first person to say come-hell-or-high-water was actually talking about.

He was talking about teaching his autistic kid to tie his shoes in the midst of torrential rains and major flooding.

Which reminds me: possibly the only good thing about ageing is that you get to find out the true meaning of sayings. Most sayings come from dogs, I find, except for the ones that come from square dancing. Wolf it down, dog your heels, dog days, dog eat dog, let sleeping dogs lie, and so on. Pretty much the whole lot. Dogs have had a big influence, being our co-evolutionists and all.

What comes from square dancing, you ask? Back to square one comes from square dancing.

Speaking of which, we were talking about:

tying shoes

Andrew is now actively interested in tying his shoes, and is making progress.

But I can't remember the easy way of tying shoes his aide showed me a couple of years ago. (She's not his aide anymore, or I'd ask her.) And I can't find it on the internet.

I may have now reconstructed it for myself (discovery knowledge! that's the ticket!) But if anyone knows how it's done, I'd appreciate hearing from you.

^*not to mention my number line problem, my fraction problem, and my distributive property problem

update

wow!

Look what KDeRosa found!

You guys are amazing.

RudbeckiaHirtaOnCalculus 02 Nov 2005 - 19:15 CatherineJohnson

...answering these questions:

In terms of teaching calculus to college freshmen, it would be easier if they had NOT taken calculus in high school. Or, rather, it would be best if the students who took calculus in high school all—or almost all—scored well enough on standardized tests so as to not retake calculus in college. It's the retaking that sucks (and the "wasted" year).

Why do you say this?

Are these kids hard to teach....or are you saying they're wasting their time re-taking calculus....??

And do you think it would be better to take neither of the AP calculus courses?

If so, is there a math course you'd prefer these kids take senior year?

The first issue is one that we read here all the time: teaching by "exposure" versus teaching to mastery. If you want the students to know calculus (not to have merely "taken" calculus), then you want them to have been in a class that expects mastery. Some high school classes do, and some high school classes don't; some college classes do, and some college classes don't. But most importantly why spend two years on a task that most students are capable of doing in one?
(I believe it took Newton two years to INVENT calculus. Of course he was working on it full time because everything was closed due to some plague.)
The other problem with re-teaching is that the students think they already know everything. This leads to a few common problems: the students think they know everything already, so they are reluctant to put effort into learning (coming to class, doing homework)—instead trying to get by on what they already know. If they took a sub-optimal high school calculus class, the teacher may have treated the foundational material (which is very abstract and difficult for students) as "unimportant"; the students often pick up on this attitude. During certain parts of the class, the point of the lesson is to understand a certain idea (the definition of the derivative and its connection to slope), and students who have already taken calculus will instead choose to (incorrectly) answer the question by using an easier calculation taught later in the course. (It's not that I'm against "short cuts"; the point of the lesson is to understand what's going on behind the scenes of the shorter calculation.)
In terms of the AP calculus classes, I really like the BC calculus. (Biased I am, as I took BC calculus myself during the 1989-90 school year at Niskayuna High School in Niskayuna, NY.) There are very few circumstances where I would recommend AB calculus. There are some obvious ones (teacher, schedule logistics, etc.) but aside from that, the only other time I would recommend AB over BC would be for a student who has struggled greatly in precalc, who is planning on studying the humanities in college, and who is planning on attending a college where part of the gen-ed requirements can be fulfilled by scoring well on the AB calculus exam. However, a BC calculus course that prepares students to take the BC exam is a fine opportunity.
In terms of 12th grade math offerings, that would be a fairly place-dependent recommendation. If the school has a corps of students who finish 11th grade ready for a real calculus course, then something like BC calc would be a canonical recommendation. Otherwise it then becomes an issue of looking at WHY aren't there students ready for 12th grade calculus (school too small for critical mass? ineffective programs? something else?) and making decisions based on that. There is a lot of interesting mathematics that can be done at the high school level, and the "right" course will depend on both the school and the students.

people hate learning things all over again

The other problem with re-teaching is that the students think they already know everything. This leads to a few common problems: the students think they know everything already, so they are reluctant to put effort into learning (coming to class, doing homework)—instead trying to get by on what they already know.

As far as I can tell, this is a huge, 'foundational' problem at all levels.

My Singapore Math kids are in open revolt at having to learn to do bar models when they can already solve a problem doing simple calculations (i.e. subtracting). Christopher, too, runs amok whenever I ask him to 'back up' a step.

My neighbor and I were talking about this one day, about why it's so aversive—and aversive is the word—to 'go back to the beginning,' or to 'do things more than one way.'

I came up with a theory, which I have now forgotten.

It will come back to me.

discovery ≠ memory

That's something I've been meaning to point out.

I've seen constructivist educators claim that discovery is a memory aide. Something you discover for yourself you remember better.

I've also seen cognitive scientists say this is rubbish, and I can tell you it's rubbish without recourse to PubMed.

Nonfiction writers are constantly forgetting their discoveries (which we call 'ideas'); this is why writers carry notebooks and pencils around; this is why some writers will actually get up out of bed in the middle of the night to write down middle-of-the-night discoveries on a piece of paper.

If you don't write your ideas down, they're gone.

As a matter of fact, new ideas are far less sturdy than old ideas, precisely because they are new. They haven't been rehearsed, by definition.

what's the code for the 'does not equal' sign?

Remember: it pays to ask a question!

never mind

I found it here

KDeRosasPageOnMathematiciansFindingCommonGroundWithConstructivists 04 Nov 2005 - 02:14 CatherineJohnson

What is important in mathematics?

Direct Instruction math

Ken has also managed to find some sample pages from Connecting Math Concepts, which is a direct instruction curriculum (which I believe was designed by Engelmann??)

Ken will tell us...

Exercise 2: Writing Fractions
Level B, Lesson 108 (Presentation Book)
Lesson 30

Wayne Bishop compares Saxon, CMC, Sadlier-Oxford, & Everyday Math

here

key words: SRA direct instruction sample lessons

ProofOfCreativity 10 Nov 2005 - 20:26 CatherineJohnson

Paul Miller left this link.

TheDivisionsOfMathematics 15 Nov 2005 - 13:03 CatherineJohnson

A ktm guest left a link to a terrific web site called The Divisions of Mathematics, and says that "You can follow the links there to find out what some of the fields in statistics are."

I'm posting the link in the 'book-style index.'

This is incredibly helpful for me. When I first started teaching Christopher I was constantly trying to figure out the various genres & subgenres of the field.

$mathmap.gif$

NSF map of math

Here's the NSF's breakdown of the field:

Algebra and Number Theory
Topology and Foundations
Geometric Analysis
Analysis
Statistics and Probability
Computational Mathematics
Applied Mathematics

hmm

I have to say, for me these categories raise as many questions as they answer, which I suppose was inevitable.

Good starting place, though.

ah-hah

The perils of scanning.

If I'd read this first, I would have understood:

Another way to divide the portions of mathematics is by level of complexity. Elementary topics include arithmetic and measurement; intermediate topics include simple algebra and plane geometry. From there we may pass to somewhat more complex topics built upon these: trigonometry, "advanced" algebra, analytic geometry, and calculus.
This website is limited to topics more advanced than these; little mention will be made of topics which are typically not considered (except in their most elementary aspects) until a student has progressed through some University studies. Our intended audience at the site is the person who has already studied some mathematics courses beyond these at the university level, although in this tour we try to be more inclusive.

JamesMilgramOnLongDivisionAndTimeLagInMath 15 Nov 2005 - 21:42 CatherineJohnson

from James Milgram's talk:

In mathematics many skills must be developed for many years before they can be used effectively or before applications become available.
First of all, I claim that taking—even asking to take [long division] out of the curriculum—shows a profound ignorance of the subject of mathematics. The point is, in mathematics, many, many skills develop over an extended period of time and are not really fully exploited until perhaps 10, 12, or even 15 years after they've been introduced. Some skills begin to develop in the first or second grade and they do not come to fruition or see their major applications until maybe the second year of college. This happens a lot in mathematics and long division is one of the key examples.

Students cannot understand why rational numbers are either terminating or (ultimately) repeating decimals without understanding long division.

Long division is essential in learning to manipulate and factor polynomials.

Polynomial manipulation and factoring are skills critical in calculus and linear algebra: partial fractions and canonical forms.

I regard the repeating decimal standard as relatively minor, but some people seem to think it is important. The next topic is critical and almost everyone thinks it's minor (Laughter). Long division is essential to learning to manipulate polynomials. Without it, you simply cannot factor polynomials.
So what, you ask? Again, this is a question that doesn't come up until the third year in college. At this point the skills that have come from long division through handling polynomials become essential to things like partial fraction decomposition which is important in calculus but finds its main applications in the study of systems of linear differential equations, particularly in using Laplace transforms, which is the critical construction in control theory. It is also essential in linear algebra for understanding eigenvalues, eigenvectors, and ultimately, all of canonical form theory -- the chief underpinning of optimization and design in engineering, economics, and other areas.
[snip]
What happens when you take long division out of the curriculum? Unfortunately, from personal and recent experience at Stanford, I can tell you exactly what happens. What I'm referring to here is the experience of my students in a differential equations class in the fall of 1998. The students in that course were the last students at Stanford taught using the Harvard calculus. And I had a very difficult time teaching them the usual content of the differential equations course because they could not handle basic polynomial manipulations. Consequently, it was impossible for us to get to the depth needed in both the subjects of Laplace transforms and eigenvalue methods required and expected by the engineering school.
But what made things worse was that the students knew full well what had happened to them and why, and in a sense they were desperate. They were off schedule in 4th and 3rd years, taking differential equations because they were having severe difficulties in their engineering courses. It was a disaster. Moreover, it was very difficult for them to fill in the gaps in their knowledge. It seems to take a considerable amount of time for the requisite skills to develop.

key words: gapology
James Milgram on long division & time
can you cram math: learning a year of math in 2 months
NYU math major
overlearning
remediating Los Angeles algebra students
Inflexible Knowledge: The First Step to Expertise by Daniel Willingham
Matt Goff & Susan S on remediating gaps
Anne Dwyer on diagnosing gaps & request for 'gap' stories
formative assessment and Richard Nixon
Terminator

IsThereAShortageInEngineeringJobs 24 Nov 2005 - 19:46 CatherineJohnson

The Wall Street Journal had a story last week about whether there is or is not a shortage of engineers to fill the available positions.

The jist of the article is: there's isn't.

Many companies say they're facing an increasingly severe shortage of engineers. It's so bad, some executives say, that Congress must act to boost funding for engineering education.
Yet unemployed engineers say there's actually a big surplus. "No one I know who has looked at the data with an open mind has been able to find any sign of a current shortage," says demographer Michael Teitelbaum of the Alfred P. Sloan Foundation.
What's really going on? Consider the case of recruiter Rich Carver. In February, he got a call from the U.S. unit of JSP Corp., a Tokyo plastic-foam maker. The company was looking for an engineer with manufacturing experience to serve as a shift supervisor at its Butler, Pa., plant, which makes automobile-bumper parts.
Within two weeks, Mr. Carver and a colleague at the Hudson Highland Group had collected more than 200 résumés. They immediately eliminated just over 100 people who didn't have the required bachelor of science degree, even though many had the kind of job experience the company wanted. A further 65 or so then fell out of the running. Some were deemed overqualified. Others lacked experience with the proper manufacturing software. JSP brought in a half-dozen candidates for an interview, and by August the company had its woman.
[snip]
The dueling perceptions of engineer shortages lie behind some big policy debates in Washington, fueling emotional clashes over immigration policy and the future of well-paying jobs in America.
Under the H-1B temporary work visa program, U.S. employers are permitted to hire foreign nationals with knowledge and skills deemed to be in short supply. The visas are valid for up to six years and are currently capped at 65,000 per year. Business groups, led by the Electronic Industries Alliance, argue that they need the foreigners because they can't find enough skilled U.S. engineers and technical workers. American engineers, particularly those who are unemployed, complain that the H-1Bs take away their jobs.
[snip]
In fact, the number of students graduating with a bachelor of science degree in computer science rose 85% from 1998 to 2004, according to figures compiled from universities by the Computing Research Association. The number of bachelor degrees in engineering rose to 72,893 in 2004 from 61,553 in 1999, according to the American Society for Engineering Education.
Unemployment among engineers was 2.5% in 2004, in line with the 2.8% rate for all professional occupations. In 2003, 4.3% of engineers were unemployed compared with 3.2% for all professionals. The figures don't include people who gave up looking for work in their profession. From 2000 to 2003 engineering employment fell 8.7%, according to an analysis of Bureau of Labor Statistics data by the Center for Labor Market Studies at Northeastern University in Boston.
[snip]
Some elite companies have an even higher applicants-to-jobs ratio. Microsoft received résumés from about 100,000 graduating students last year, screened 15,000 of them, interviewed 3,500 and hired 1,000, says a spokesman. The software maker receives about 60,000 résumés of every kind monthly, and currently has 2,000 openings for software-development jobs.
[snip]
Companies often draw up extremely narrow job descriptions, recruiters and staffing managers say, causing searches to get drawn out. One cause: the rise of online job sites, which makes it hard for company executives to personally review every candidate. To screen out the hundreds or thousands of résumés that pour in to a posting on Monster.com or Yahoo HotJobs^?, companies use software filters to look for keywords. In engineering, those keywords typically describe machinery or computer fields in which expertise is sought, such as C+++, server/stepper and CAE schematic.
[snip]
The detailed demands aren't confined to software jobs. Mr. Sylvester was asked to find a mechanical engineer to oversee a heating, ventilation and air-conditioning system at a hospital. "A pump is a pump and a duct is a duct, but they wouldn't even look at candidates who had HVAC experience in a mill instead of a hospital," he says.
[snip]
James Murphy, 60 years old, of North Hills, Calif., sees the phenomenon from the other side. He holds a master's degree in mechanical engineering and worked for major aerospace companies doing dynamic load analysis -- figuring out what forces would cause an aircraft to break. Later he worked at Continental Airlines using computer algorithms to optimize flight scheduling. Laid off in 2001 from his position doing computerized inventory for a music wholesaler, he estimates he has sent out 10 résumés a week. He has had two job interviews in the past year, both with aircraft manufacturers. Neither led to an offer.
"There is now a string of requirements for an engineering job," says Mr. Murphy. "Years ago there would be one major requirement, with x, y and z nice to have. The worst thing about this emotionally is reading about the 'shortage' of engineers."

the Asperger moment

One employer demand that flummoxes many engineers is the need for "soft" skills -- working in groups, communicating and writing. In August, Cornell University hired a speaker to instruct its engineering students in "etiquette and interpersonal skills." (Hints: Don't crumble crackers into your soup or blot your underarms with the dinner napkin.)

I've had this thought myself

Many executives who contend there's an engineer shortage today predict it will get worse over the next decade as baby boomers begin to retire. This summer a report from a business consortium called for doubling the number of science and engineering graduates by 2015 to fill a projected gap. But crystal balls about labor markets tend to be cloudy. In the mid-1980s, the National Science Foundation predicted "looming shortfalls" of some 675,000 scientists and engineers in the following two decades. They never materialized.
"Every few years there is a spurt of panic that we won't have enough engineers in five years," says Paul Kostek, a systems engineer in Seattle who recently got a job at Boeing after working as a consultant for a decade. "And I say to myself, gee, I'll still be here."

factoid

More than half of engineering doctoral degrees awarded in the U.S. go to non-U.S. citizens, according to the National Science Foundation. U.S. citizens earn the majority of bachelor's and master's degrees in engineering. An earlier version of this column incorrectly said that the bulk of all engineering degrees awarded in the U.S. go to foreign nationals.

sources:
Behind 'Shortage' of Engineers: Employers Grow More Choosy by Sharon Begley Wall Street Journal November 16, 2005
Outsourcing Fears Help Inflate Some Numbers

KtmGuestOnMillenialGeneration 24 Nov 2005 - 19:45 CatherineJohnson

One aspect of the problem is the heightened expectations of people who grew up and/or were in college during the craziness of the dot-com boom.
Robert W.Wendover and Terrence L. Gargiulo have written an excellent article -- actually a chapter excerpt from their upcoming book -- on the topic of generational differences in the workplace.
[boldface added] The Millennials are coming of age in an era of technology and rapid change. Many of them honestly wonder why machines don't do many of the mundane tasks they are asked to perform in entry-level positions. They have been heavily influenced to believe that every job should match the same level of stimulation they receive from a video game. As this generation matures into the workforce, some of these perceptions will change. But this group will also alter society's interpretation of work ethic as they go.
I found a rather negative Associated Press article about millennials in the workplace.
Then again, I also found a different quite positive article on the same subject.
As someone who reads resumes and interviews job candidates for software development positions, I have to say, the dot-com boom gave programming jobs to a lot of people who lacked either the talent or the training to do the job effectively. Now these people are out in the job market with "5 years of xyz technology" on their resumes, and it's up to the employers to figure out whether they are talented or just lucky.

from the American Management Association:

...a thumbnail description of the four generations in today's workforce and the “labels” most often attached to them.

Greatest Generation — 63 million
Baby Boomers — 77 million
Generation X — 50 million
Generation Y — 81 million

I love this part:

Baby Boomers are unique in the sense that they have given birth to one and a half generations.

This reminds me of a Roseanne joke.

I decided to skip having kids and go straight to having my own grandkids.

She was 42 and pregnant at the time.

the Millenials

Because the Baby Boomers produced two waves of children, the youngest generation in the workforce is a product of both younger Boomers and the older half of Generation X. Terms associated with them include Generation Y, Generation WHY, Net-Geners, Nexters, and Echo Boomers to name a few. Over time, the term “Millennials” has become the preferred moniker.

These people have got to be confused....

The Matures, for instance, grew up in the midst of war-time shortages and economic depression. They have always worked hard and paid their dues.
Even in better times, they have continued these ways simply because this is the ethic with which they feel most comfortable.
Baby Boomers came of age in the midst of tremendous economic expansion, learning to use all the convenience-oriented products that came on the market during their youth. Because of the size of their generation, they were also the focus of everyone's attention. Boomers have always put in long hours because of how closely they associate their occupation with their identity. Even as they edge into retirement, we predict that most of them will still live to work.
Having watched their parents, the Baby Boomers, put in these long hours, Generation Xers have developed a different perspective on work. They do not necessarily equate productive work with long hours. Instead, they look for ways to work smarter, resulting in fewer hours but greater output. This is the reason why Boomers and Matures sometimes accuse those in Generation X of “punching the clock.”
The Millennials are coming of age in an era of technology and rapid change. Many of them honestly wonder why machines don't do many of the mundane tasks they are asked to perform in entry-level positions. They have been heavily influenced to believe that every job should match the same level of stimulation they receive from a video game. As this generation matures into the workforce, some of these perceptions will change. But this group will also alter society's interpretation of work ethic as they go.

source:
Frequently Asked Questions and Answers about Generational Differences

SeekingAGoodPhysicsText 21 Feb 2006 - 00:49 CarolynJohnston

My stepson Colin (who wrote this post a couple of months ago, on the frustrations of his pre-calculus class) is back, and reports that he's been very pleased with the pre-calculus text that this group collectively suggested (Sullivan's Precalculus).

And we are now in need of another recommendation. Colin is stuck with a text for his physics class -- Serway/Faughn College Physics, published by Brooks/Cole -- which is, in its way, similar to the Prentice-Hall "Math Course X" series in its page splatter problems, and general Missing Of The Point. Not only that, it weighs at least ten pounds (no help for that, though).

We need a supplementary first college physics text -- non-calculus-based, preferably -- that is as good in its sphere as Sullivan's Precalculus is in its. Does anyone have any suggestions for books they've particularly liked?

PeterDruckerOnTheFuture 28 Nov 2005 - 00:34 CatherineJohnson

from Steve Forbes' Tribute to Peter Drucker (subscription probably required):

Mr. Drucker's ability to prophesy — almost always correctly — was uncanny. All of this is why he could come up with innovations that now seem commonplace, such as management by objective. He continued to admonish executives to carve out time to think and make careful decisions, to focus on one or two tasks, to delegate to others what you can't do well yourself. That's why, for example, Mr. Drucker remained a one-man shop, a soloist; he could easily have founded a large consulting firm and gotten immensely rich. But that would have gone against his profoundest instincts. He was at his best as a teacher — gathering information, gaining insights and then getting others to gain understanding. Schumpeter believed asking the right questions was more important than the answers. Mr. Drucker agreed — to a point, anyway.
Decades ago, Mr. Drucker foresaw the rise of "knowledge workers." After World War II, he realized the far-reaching consequences of the GI Bill of Rights, which enabled millions of veterans to go to college, thus leading him to predict long before computer chips and the Internet that "knowledge workers" would replace manual workers. Mr. Drucker also prophesied the breakdown of the traditional, thoroughly integrated, hierarchal industrial corporation. In the 1950s, he predicted the rise of Japan as a major economy, an astonishing insight when many experts thought the country would forever be a nation of small farmers and manufacturers of cheap, shoddy goods. He also saw Japan's subsequent troubles — an aging population and lack of vigorous entrepreneurship and worker flexibility.
Mr. Drucker long ago warned of the consequences of the rise of corporate and government pension funds, and the impact these vast accumulations of money — and thus power — would have on corporate governance, years before anyone had heard of Calpers. He also warned of a backlash from the extraordinary rise in CEO pay. "In the next economic downturn," he told Forbes readers nearly a decade ago, "there will be an outbreak of bitterness and contempt for these super corporate chieftains who pay themselves millions. In every major economic downturn in U.S. history, the villains have been the heroes during the preceding book."
Mr. Drucker also told us to expect enormous changes that will come in higher education, thanks to the rise of satellites and the Internet. "Thirty years from now big universities will be relics. Universities won't survive. It is as large a change as when we first got the printed book." He believed "High school graduates should work for at least five years before going on to college." It will be news to most college presidents and a lot of alumni that "higher education is in deep crisis. Colleges won't survive as residential institutions. Today's buildings are hopelessly unsuited and totally unneeded." All this from a life-long academic.

Ed thinks this is wrong, though he does think small colleges may disappear.

HowCanCollegeFreshmenFillGaps 30 Nov 2005 - 18:34 CatherineJohnson

from Anne Dwyer:

Going back to the question of what to do with students who have large gaps in their background:
We (someone, I forget who) once asked this question on this site: once you have these large gaps in your knowledge, can you ever catch up and close all the gaps?
I think this is especially relevant at the college level. There is a basic math course at our community college, but it goes incredibly slowly. The prealgebra class gives basic lip service to large number problems, then goes straight into algebra. Towards the end of the course, the curriculum goes back to decimals and percentages and conversion factors. But, by then, many of the students are completely and totally lost.
Then, they break basic algebra into two classes: elementary algebra and intermediate algebra.
Even with a tutor, there isn't enough time to determine where the weaknesses are and to go back and correct while the student is taking the class. This would require them to work on parellel tracks: making up gaps and keeping up in class. Everything is geared towards students keeping up in class not preparing the student with the basics for the class.

This was a conundrum for me.

I don't know the structure of mathematics well enough to be able to tell where I have significant gaps and where I don't; if I did, I (probably) wouldn't have gaps.

This is why I decided to go back and re-learn everything from the beginning. That way, I figured, whatever gaps I didn't know I had would get taken care of.

I didn't end up being able to do that, mainly because I had to keep up with Christopher. So I started in 5th grade, where he was.

I'm wondering whether KUMON would be a good idea for students in this position.

I started in Level D—roughly 4th grade—and moved to Level E after two weeks.

Algebra begins in Level H.

Each level has 200 worksheets, and you do 5 worksheets a day, 6 days a week. (I think you're supposed to do 5 worksheets a day, 7 days a week, but Mr. Liu only gives me enough for 6 days.)

If you figure roughly 7 weeks per level, I'll move from Level E to Level H in 21 weeks. I can do that and easily do everything else; my KUMON worksheets are the least demanding part of my day. So I think an ill-prepared college student swamped with remedial work could do KUMON sheets and keep up with his classwork.

I gather there are some adults taking KUMON; I wonder if any of them have written about it.

RudbeckiaHirtaAtJoannes 01 Dec 2005 - 00:00 CatherineJohnson

It's taken me awhile to put Rudbeckia Hirta together with her blog, Learning Curves.

Or rather, I should say, it's taken me awhile to keep them together....since I used to know that RH writes Learning Curves.

Then I forgot.

In any case, I've got it now.

Learning Curves is fantastic today. There's a terrific math horror story (I keep a collection), an intriguing homework story, a calculator lament, a freshman haiku, and an excellent proposal for ed research that RH may not have the patience for, but I hope someone some day will.

Here's the Math Horror Story:

A few years ago I was at JoAnn [Fabrics] (the one on route 35, just south of Red Bank), and there was a woman at the cutting table. She was holding a roll of home-dec fabric and a pattern. The clerk asked her how much fabric she wanted cut. The woman said she didn't know. She was making covers for her dining room chairs, the pattern said that each chair needed 5/8 of a yard of fabric, and she had eight chairs. The clerk didn't know either. They were not wondering whether you could get by with less than five yards of fabric if you arranged the pattern pieces cleverly. No, they had NO IDEA how much fabric she needed.

This is the kind of thing I have to have dust-ups with my husband about.

A few months ago, I was obsessing over Bad Fraction Knowledge In American Students, when Ed said, 'Nobody uses fractions.'

I'm sure he says these things on purpose. He says he doesn't, but I think he does.

Anyway, I pointed out, logically, that I use fractions all the time.

When I cook, for instance.

Say I want to modify a recipe. I will use fractions.

So Ed says, 'Nobody modifies recipes.'

Again, typical. One of the Themes of our marriage is the outlier-ness of me.

Yes, the implication is, you modify recipes.

But you're different. Nobody else does the crazed, obsessive, over-the-top, recipe-modifying, fraction-using stuff you do.

Which is hilarious, seeing as how I'm the least outlier-ish person I know. I am practically a walking cliche, I'm so mainstream. UPDATE 9-23-2006: Except when it comes to TV sci fi.

So Ed says, 'Nobody modifies recipes,' and I say, 'There's a whole big bestselling book on how to make your own mixes. You know, like my pancake mix. The women who wrote it give lectures and presentations all over the country. They appear on morning talk shows. They use fractions!'

And he stopped arguing about fractions and conceded the point!

It was great!

This is what gets to me.

Ed has an excuse.

He's sitting at his kitchen table on a weekend morning, his wife is obsessing over fractions, and he's long since lost all interest in whether American students can or cannot add, subtract, multiply, and divide fractions. He already knows they can't; it's not a burning mystery in his life.

He comes up with things like Nobody uses fractions just to liven things up. Plus he hasn't figured out the difference between over the top and outlier.

I am over the top.

I am not an outlier.

Case in point, modifying recipes. Modifying recipes is mainstream behavior, which occurs not infrequently on television cooking shows. At least, I think it does.

So Ed has an excuse when he comes up with things like, Nobody uses fractions.

But what's everyone else's excuse?

Why do we keep hearing that people don't use fractions, or don't use long division, or don't use quadratic equations in everyday life?

OK, yes, it seems to be the case that people don't use quadratic equations in everyday life.

But fractions?

Long division?

People don't use this stuff in everyday life?

Maybe Ed is right.

Maybe I live in a parallel universe where people are ceaselessly modifying recipes or purchasing fabric at JoAnn's or altering knitting patterns or buying paint or laying carpet or what have you, all the while using fractions & long division to do it.

Because obviously, this kind of thing doesn't happen in the real world.

PaulMillerAndRudbeckiaHirtaOnAssessment 19 May 2006 - 22:09 CatherineJohnson

I'm disheartened today. Watching Christopher fall apart is excruciating (all the more so given how much I know about fear and the brain), and.....

......and I've had it.

So when I got home this morning, after dealing with the THIRD car to be stuck in our driveway in two days (I'm starting to feel like Bill Murray in GROUNDHOG DAY), and found these comments from Paul Miller and Rudbeckia Hirta, I thought, There's hope. (I'll be a much more cheerful person tomorrow, or even.....later on this afternoon!)

from Paul:

One thing I've been putting a lot of thought into is how to teach to mastery in an environment where I'm on a strict schedule and have very limited time. I bet Black and Wiliam weren't thinking of people who have to jam what would be a whole year of algebra in high school into a semester.
Still, I have decided, there will be quizzes at least weekly next semester.

and from Rudbeckia:

This semester I gave twenty quizzes in calculus (the best 10 counted), and I'm thinking of giving quizzes every class next time I teach something from the algebra / precalc / calc sequence. Next time I'm going to make them VERY short, 3-5 minutes, and give them at the exact beginning of class. My bet is that the instructional face-time lost will trade well with increased studying.

Here's how I feel, reading these comments.

These comments, these actions, are a gift. A gift from two highly intelligent and educated people to the younger people they are trying to teach.

The way I'm feeling today, they're a gift to me, too.

where we are with English

Mrs. Roth can't teach our child. That battle we can handle, although the school will certainly refuse to move Christopher to another class. If I were a betting person I'd bet they end up moving him whether they want to or not, but we'll see.

Whether he goes or stays, he will never write another assignment for this woman.

Worksheets, fine; reading logs, check. But no written work. We're done.

What we need is for the principal to read Christopher's essay and tell him it's not a 'D.' His friends are making fun of him, telling him his parents are 'just saying' his essay is good, because we're his parents. All these boys insult each other all day long, Chris included. But on this issue his friends are drawing blood, which I'm sure they don't know. He's probably hurting them, too. The things they say to each other are appalling, and I have no idea what to do about it.

Advice?

Christopher's confidence is shot. He thinks he can't write, can't do math, can't do anything.

We saw this happen before, in 2001, after the attacks. He'd been an aggressive little soccer player, one of the best on the team. Then he lost his nerve. He just....stopped. On the field, he was diffident and slow. At school, he was bullied.

Ed was the soccer coach, so he was there; he watched it happen. He told me last night he's seeing the same thing all over again, only this time in academics, where it counts.

Maybe it's not like that; maybe he'll bounce back. We'll see.

question

So Mrs. Roth has to go, but the math teacher is another story.

She's very young; I think this is her first job. (back story for new readers stopping by: Her course last year was so brutal for the kids—unintentionally so—that the parents were in open revolt.)

She's a good egg. Last year must have been painful for her; the huge revisions they did to her course over the summer may have been distressing, too. Yes, it's important to have mentors and help, but having mentors and help in the context of parent fury is another story.

So....I need to push her for Christopher's sake, but I want to 'push' in a way that's positive, helpful, and likely to be listened to.

Here's what I think we need: If any of you have extra items to add, let me know

First item: I need to know, from the beginning of each chapter, what 'showing your work' means to Ms. Kahl.

Let me ask all of you: what is the work that would typically be shown for this question?

Compare using <, >, =
0.635 __ 0.365

To me, this is a simple comparison—but do teachers typically ask for work to be shown on this kind of question?

If so, does the student write a subtraction problem, or perhaps draw a number line?

I'll find out from Christopher's teacher, but I'm wondering about other peoples' experience.

I have no problem with the requirement that the kids show their work; I think it's probably good at this stage. But I've got to know from the get-go what 'showing your work' means for each given problem, so we can practice it from the get-go.

Second item, Christopher needs guided practice in class.

Christopher says that the norm is for Ms. Kahl to lecture and give an assignment. The kids do the procedure she's taught for the first time at home.

I'm sure his perception of the class and her perception of the class are going to be an imperfect match. she does have them do worksheets in class sometimes, or start their homework. I'm not sure whether either of those situations constitute 'true' guided practice, but they're probably in the realm.

Still, the fact is that he not infrequently comes home from school without a clue how to do the procedure she's demonstrated in class that day is significant. While she may be doing some guided practice, I need her to do more. Which means I'm crossing a line into the realm of telling a teacher how to teach.

Third, and most important, I need formative assessment to be happening in the class.

We have no teaching to mastery at all. Instead we have a classic 'accelerated' course, where the children are expected to be math brains, the teacher whizzes through the material, and only the strong survive. The weak fall behind, struggle to move their legs faster than they'll go, gulp down huge mouthfuls of air, pour sweat, and finally collapse in a heap. Only one grading period into the year so far, Christopher's nearing collapse. He earned a B on his first chapter test, a C on his 2nd, and, now, a D on his 3rd.

Yes, he could move down to the combined Phase 2/3 course.

He could move down and study place value. They've spent weeks on place value. I forget what they're doing now; I'll find out. It's not going to be anything he needs to spend an hour a day doing.

Here's my question: how do I broach these subjects?

These are large issues, not small. And this teacher is almost certainly in Paul's situation. She has to cover this material, and she has to cover it fast. What she's got to work with is nothing like a Singapore course where the curriculum has been painstakingly put together to allow the fastest possible progress for all children, math brains or no.

So she's up against it.

But we need these changes. We need the school and the individual teachers to assume responsibility for making sure the children have learned what they've been taught. All but the brainiest kids need this, and even the brainiest kids are going to need it somewhere along the line, too.

back to Rudbeckia & Paul

Actually, it suddenly occurs to me that I can cite Paul & Rudbeckia—especially, for my purposes, Rudbeckia's top-10-quizzes count approach.

That would be so much more humane for these kids, and so much more motivating.

Alright, that's a possibility.

what we told Christopher

The math situation is probably manageable.

Ed, this morning, read over Christopher's test and said that he's not having nearly the amount of homework he needs if he's to do the tests she's giving.

Math class lasts 50 minutes; the test had 24 questions, some with several parts. Christopher has two minutes at most to answer each question, and he has to show his work (and his handwriting is not only bad, but slow).

Now he's developed test anxiety, so he's not managing to read the questions. He must be freezing up, just not seeing the words.

The point is: if he's going to do 24-item tests in 45 minutes, he has to have more practice. Ms. Kahl sometimes sends home homework 'sets' with only 4 problems. Maybe the math brains can do 4 problems and ace a test (they probably can).

Christopher can't. If Christopher is going to do a 24-item test in 45 minutes he can't have done 4-problem homework sets. Wayne Wickelgren says children should do 30 problems a night. That's what Christopher needs to do. Thirty problems a night.

We were finally able to get through to him on this point last night—thanks to KUMON and to Saxon Math.

I said, "Do you ever flunk KUMON worksheets?"

Christopher said, "No."

I said, "Why don't you flunk KUMON worksheets?"

Christopher said, "Because I've practiced."

I said, "Because you've practiced a lot."

Then I said, "Did you ever flunk Saxon tests?"

"No."

Why?"

"Because I practiced."

"Because you practiced a lot."

Then both Ed and I said, You need to be able to do these problems as fast as you can write.

You need to be able to do them in your sleep.

You need to know them cold.

That's a simple message, and he understood it.

I hope it will finally start to sink in. Christopher thinks that if he can do a problem he knows it. It may take him 5 minutes to do one problem, but if he gets it right, he's done.

No one at the school has told him that isn't the way it works. He's had two months of "Study Skills" class and the only thing they seem to have told him about study and learning is 'Find a quiet place.'

I, of course, have been trying to get this message across for months, but, as Carolyn pointed out, we're hitting the end of parental influence.

Last night he heard us.

A couple of weeks ago I tracked down the Prentice Hall pre-algebra workbook that accompanies his text. We agreed that from now on he'll do ALL the problems on the work sheet, not every other problem, or, even worse, every fourth problem. (I'd put money on it Ms. Kahl has been told not to overload the kids with homework.)

Last night, that's what we did. Every single problem.

That proved to be a terrific object lesson.

He did one problem laboriously, taking far longer than he'd have on a test.

Then, because we were doing every problem, he did the next one— in half the time.

I said, "Look how much faster you got just from doing two problems instead of one."

He saw it.

cheeful thought

I'm going to get a grip now.

My neighbor, whose son struggled through this class last year, told me that the 7th grade book is mostly review. I think they start algebra in January, so I'm assuming they spend fall semester reviewing the gazillion procedures and concepts they learned in 6th grade pre-algebra, then make the move to formal algebra mid-year.

That's good.

I'm obviously back in re-teaching land; Christopher is losing another year of math instruction, just as he did in 4th grade.

But this time he's got KUMON, and KUMON speeds along. Yes, he's doing 3rd grade math now, but in two weeks he'll be doing 4th grade; 7 weeks after that he'll move to 5th. Slow but steady wins the race. Mr. Liu told us parents see major gains after one year of KUMON.

'You need to invest that time,' he said.

We're investing.

And this time I know I have to re-teach, and I'm starting now. I'll have the summer, too.

Then he'll have a fall semester of review with, I hope, the best teacher they've got.

So I think we can do this.

MattGoffOnTeachingAlgebra 11 Feb 2006 - 15:57 CarolynJohnston

Matt Goff, who teacher math at a small liberal arts college in, I believe, Alaska, left a great comment about his teaching methods and about how they've changed recently.

All the boldfaces are mine.

I have been reading this site for the last month or so and feel that I've gotten some good ideas and information from it. There are three separate (but related) perspectives that I take when thinking about this information:
1. I teach math at a small open enrollment liberal arts college. Many students are poorly prepared in math. In a typical year, half (or more) of the incoming students will place in Basic Math or Basic Algebra (the equivalent of Algebra I). Even after they arrive, they often struggle to be successful. For example, I have taught a class of 11 where 10 of the students had failed to complete the course previously. It has become clear to me that I could probably be a more effective teacher by making some changes (more on this later).
2. Many of the students who struggle in my math classes are Elementary Education majors. 5 of the 10 students mentioned previously were Elementary Education majors. College Algebra (they can usually make it through Basic Algebra eventually) seems to be a real stumbling block for many of them. However, beyond that, they are required to pass Trigonometry before taking a Math for Elementary Teachers course and final a Math Methods course. Reading these pages as well as having kids that are nearing school age (see below) has really caused me to evaluate how I view these students and their progress (or lack thereof) through their required math coursework.
3. I am planning to homeschool my kids (now ages 4.5 and 2.5). I'm not sure how I came to this decision, I think it has partially grown out of my own experience going through school (I was a 'gifted' student, but my mom had to regularly fight with the school district regarding my schooling. The recurring theme seemed to be concern that by accelerating I would either run out of stuff to do or get in over my head. I'm pretty laid back and I guess when I started thinking about the possibility of conflict with the school district, it felt like I might rather just teach them myself. From that starting point, the idea grew on me and now I feel like it's likely to be a very rewarding experience for them and me.)
Upon seeing over half of a class need to repeat college algebra, I felt like I needed to change some things. In all but one case, the students had not done the work that was expected of them, so in a sense, it was their fault. They are in college after all, and it seems like I should be able to assume they can take responsbility for their own learning. I have finally realized that I can't. For whatever reason, they are just not learning how to be effective learners before they get to me. I seem to be one of those folks who is highly suited for learning in the school environment. Many learning strategies either came to me naturally or were not necessary for me in the first place. I think this is why it took me so long to realize that a lack of such strategies and/or meta-learning might be why students were not being successful.
The typical approach I have preferred to take to teaching is as follows: Introduce a section and assign homework. The next class (we meet three times a week) I take questions on the homework assigned the previous class period. Typically, I would plan to spend up to half the class doing this. It's my feeling that explanations will be more effective if students have already engaged the material. Students then have until the next class period to finish up the assignment and turn it in. The remainder of the class period is spent introducing the next section. Homework is collected and graded on completeness as most of the answers to the assigned questions are in the back of the book. Tests are given at the end of each chapter and students are allowed to turn in test corrections to get half-points back on problems them missed. My reasoning for this was that mid-term tests can and should be a learning tool. I figured that if they could use the test (and corrections) to firm up the knowledge that had been weak on and mastered it by the final, that was good.
Although the better students did fine, there were some problems with this approach for others. Students did not do the homework the day it was assigned and consequently the question time was not very helpful for them. Students did not use the answers in the text as an effective study aide. I eventually came to the conclusion that some of them did not really know how to. I assumed that if students were serious about learning, they would self-evaluate (using answers in the text) and ask questions and/or do more questions to make sure they understood the material. I generally found out they did not know what they were doing when they took a test. In hindsight, it has become clear that this was probably too late for many of them; especially when their lack of progress was masked for a couple of chapters because they were getting by on half-remembered knowledge from previous courses (either in high school or college) that overlapped with the early chapters of my course. They really did not know what they didn't know, and even worse, they didn't seem to know how to figure out that they didn't know it (until it was made clear to them in the form of a failed test).
Largely as a result of frustration with the poor performance of my students and the things I have read on this site, I am trying a new approach in my college algebra class this semester. Rather than giving a class day for asking questions on homework and then collecting it the second class after it was assigned, I am giving a quiz on the homework the class after it was assigned. Already there have been a few things that I have caught that I was able to go back and explain.

(I'm pulling this whole section of Matt's comment out and boldfacing it because I JUST LOVE IT:)

I've required that each student meet with me once a week (an advantage of small schools, for sure) to go over homework (which I am requiring that they keep organized in a three ring binder along with a log of questions, time spent, and in-class quizzes). One of my goals in the one-on-one meetings is to help them figure out what they need to do to effectively learn the material. They just took the first chapter test today, so it will be awhile before I am really able to tell how succesful this approach is. So far I am cautiously optimistic.

What I like about Matt's approach is that it's sending a clear message to his students that they are expected to try to learn the material to some level of mastery after every class. Students, even in college (as Matt points out), take cues from the teacher's policies about what they are expected to do when. Whether or not a teacher intends it, a student assumes that if the teacher is giving one big test at the end of a section, then it's okay for them to try to cram on ALL the material at the end of the chapter.

The daily quiz is a lot of work for the teacher, but I've come to believe that it's a great, success-creating idea. And it can be set up so that it's a quickish grading job for the teacher. I think it is really, really worth it.

-- CarolynJohnston - 28 Jan 2006

MattGoffOnTeachingCollegeKids 29 Jan 2006 - 06:06 CatherineJohnson

As the KTM Resident Math Phobe who escaped most of college math due to a scholarship in Fine Arts, I can totally understand your students failure and mentality. Had I not gone the direction I did I would have been right there with them.
I can't speak for all of them, but for a good portion I will just say that it is and always will be The Gaps. I wish there had been offered classes all along called, "What's Your Gap?"
In many cases, students can tell you that they don't really know fractions, but like you said, even teenagers and college kids might not be able to tell you.
When it turned out that, in fact, I did have to take College Algebra I was truly depressed. I had long since given up on myself and had spent my entire childhood avoiding the unpleasantness in any way possible. That's where a lot of your students' bad habits and seeming unwillingness to meet with you come from, more than likely.
A friend of mine who planned on being a math teacher tutored me daily. She was calm, cool, and didn't judge. I remember when she realized that I really didn't know or understand the Distributive Property (something I didn't tell her about because I didn't realize that I didn't know it.) I didn't understand Order of Operations, and many Algebra 1 things. And that's how she put it. We'll go back and get those things and then you'll be fine.
Math phobes never understand they have to DO math to be proficient at it. They really think that if they didn't get it immediately then something must be wrong.
I also found that I never really learned a couple of multiplication facts. I just avoided many little things like that because I never realized their importance down the line.
A basic skills class going through Algebra 1 with the emphasis being, "your students will have gaps from time to time. What are yours?" might help those kind of students overcome their own phobias. Fractions are the big hangup for a lot of us. Conceptual knowledge was non-existent for me and procedural was weak. Math language always confused me to the point where my brain would just shut off at some point during the lecture. I never had a strong enough foundation to understand what was being said. Math language sounds like a foreign tongue and has no real meaning to the math phobe, who never really understands how it ties together.
But it really only took me a couple of weeks before I started to understand what was being said in class. I was shocked when my quizzes had A's on them. That one class erased a lifetime of confusion.
So, my unsolicited advice would be to not give up too quick on the college kids. There really is hope. As education majors they need to look this square in the eye or they will just add to the problem. They need to admit their most embarassing math secrets and fix them or they'll never help others do the same. If the focus is put that way, some might really step up to the plate. An examination of how they went off the rails will only make them better teachers.
And might I add that you sound exactly like the kind of teacher those kids need. Someone who really wants to figure it out. I wish we all had teachers like that.

And might I add that you sound exactly like the kind of teacher those kids need. Someone who really wants to figure it out.

I agree.

Matt's right; these are college kids; they're supposed to know basic math; and they should have figured out how to study and learn from a textbook by now.

But shoulda-woulda-coulda and five bucks will buy you a cappucino at Starbucks.

The fact is, college professors are trying to teach math to young people who a) don't know math and b) don't know how to help themselves climb out of the hole they're in.

At some level, what Matt is doing is Good Citizenship.

These are young people who hope to teach the next generation.

Anything Matt can do to bring up their math skills and comprehension is not only good teaching, it's a good deed.

how quickly can 'remediation' happen?

But it really only took me a couple of weeks before I started to understand what was being said in class. I was shocked when my quizzes had A's on them. That one class erased a lifetime of confusion.

This is another thing I've wondered about.

I suspect KUMON may be overkill. I'm sick & tired of doing fractions, fractions, fractions......I know how to do the four operations, and I'm now officially burned out doing them. I will do them, all of them, because that is the KUMON Way, and I'm taking KUMON's word for it this is a good thing.

But I suspect diminishing returns are setting in. (I'm just praying Level G, which I'll get to in 3 weeks, won't be LOTS MORE FRACTIONS.)

During my first Singapore Math class I was shocked at how rapidly the kids gained speed at math facts. They could improve from one class to the next, with no practice in between.

Unlearning material you've learned wrong is hard & takes time &mdsah; primarily, I assume, because you can't actually unlearn things. 'Extinction,' which is what behaviorism calls unlearning, actually means that you've suppressed the wrong response. You haven't forgotten it; it's still there.

That's why errorless learning has become important in rehabilitation of TBI & stroke — and why coaches and trainers don't let athletes learn anything the wrong way (or so I've heard).

So I doubt there are remediation shortcuts when you're trying to 'extinguish' wrong math learning.

But a lot of students are probably like Susan & me. They're suffering from gaps, not mistakes. I don't remember any Wrong Ideas I've had to correct so far. There may have been one or two, but if so they were so insignificant I've forgotten what they were. [update: no! that's wrong! I thought 7 x 6 was 43! it hasn't been easy unlearning that one]

My problems have been:

severely fragmented knowledge

limited conceptual understanding of procedural knowledge

Of these two, the fragmentation has by far been the more important of the two.

I was able to begin fixing both of these problems pretty quickly once I started re-teaching myself math with Saxon 6/5. The first few months were hard, but that's a fairly short period of time when you're trying to remediate years of bad math education.

Could a lot of college kids make up lost ground more quickly than we think?

And how would a college kid who wants to make up lost ground quickly go about this?

I don't know the answer to that.

key worsd: gapology
James Milgram on long division & time
can you cram math: learning a year of math in 2 months
overlearning
remediating Los Angeles algebra students
Inflexible Knowledge: The First Step to Expertise by Daniel Willingham
Matt Goff & Susan S on remediating gaps
Anne Dwyer on diagnosing gaps & request for 'gap' stories
formative assessment and Richard Nixon
Terminator

-- CatherineJohnson - 28 Jan 2006

InPraiseOfBayes 04 Feb 2006 - 16:34 CatherineJohnson

Carolyn wrote a post about THE ECONOMIST's recent article on Bayes & the human mind ($).

Here are excerpts from the article on Bayesian statistics they ran in September 20, 2000 issue, In Praise of Bayes ($):

IT IS not often that a man born 300 years ago suddenly springs back to life. But that is what has happened to the Reverend Thomas Bayes, an 18th-century Presbyterian minister and mathematician—in spirit, at least, if not in body. Over the past decade the value of a statistical method outlined by Bayes in a paper first published in 1763 has become increasingly apparent and has resulted in a blossoming of “Bayesian” methods in scientific fields ranging from archaeology to computing. Bayes’s fans have restored his tomb and posted pictures of it on the Internet, and a celebratory bash is planned for next year to mark the 300th anniversary of his birth. There is even a Bayes songbook—though, since Bayesians are an academic bunch, it is available only in the obscure file formats that are used for scientific papers.
Proponents of the Bayesian approach argue that it has many advantages over traditional, “frequentist” statistical methods. Expressing scientific results in Bayesian terms, they suggest, makes them easier to understand and makes borderline or inconclusive results less prone to misinterpretation. Bayesians claim that their methods could make clinical trials of drugs faster and fairer, and computers easier to use. There are even suggestions that Bayes’s ideas could prompt a re-evaluation of fundamental scientific concepts of evidence and causality....
The essence of the Bayesian approach is to provide a mathematical rule explaining how you should change your existing beliefs in the light of new evidence. In other words, it allows scientists to combine new data with their existing knowledge or expertise.
The canonical example is to imagine that a precocious newborn observes his first sunset, and wonders whether the sun will rise again or not. He assigns equal prior probabilities to both possible outcomes, and represents this by placing one white and one black marble into a bag. The following day, when the sun rises, the child places another white marble in the bag. The probability that a marble plucked randomly from the bag will be white (ie, the child’s degree of belief in future sunrises) has thus gone from a half to two-thirds. After sunrise the next day, the child adds another white marble, and the probability (and thus the degree of belief) goes from two-thirds to three-quarters. And so on. Gradually, the initial belief that the sun is just as likely as not to rise each morning is modified to become a near-certainty that the sun will always rise. In a Bayesian analysis, in other words, a set of observations should be seen as something that changes opinion, rather than as a means of determining ultimate truth. In the case of a drug trial, for example, it is possible to evaluate and compare the degree to which a sceptic and an enthusiast would be convinced by a particular set of results. Only if the sceptic can be convinced should a drug be licensed for use.
This is far more subtle than the traditional way of presenting results, in which an outcome is deemed statistically significant only if there is a better than 95% chance that it could not have occurred by chance. The problem, according to Robert Matthews, a mathematician at Aston University in Birmingham, is that medical researchers have failed to understand that subtlety. In a paper to be published shortly in the Journal of Statistical Planning and Inference, he sets out to demystify the Bayesian approach, and explains how to apply it after the event to existing data.
Patients in clinical trials will soon benefit. Bayesian methods offer the possibility of modifying a trial while it is being conducted, something that is impossible with traditional statistics. Andy Grieve and his colleagues at Pfizer, a drug firm, are intending to do just that.
Traditionally, dose-allocation trials—in which the aim is to establish the most effective dose of a new drug—involve giving different groups of patients different doses and evaluating the results once the trial has finished. This is fine from a statistical point of view, but unfair on those patients who turn out to have been given non-optimal doses. Rather than analysing the results at the end of a trial, Dr Grieve’s method will evaluate patients’ responses during it, and adjust the doses accordingly.
[snip]
Pfizer is intending to conduct a trial using this new method, and the plan is to re-analyse the data once it is completed in ways that will satisfy both Bayesians and non-Bayesians.
[snip]
Bayesian methods can also be used to decide between several competing hypotheses, by seeing which is most consistent with the available data.
[snip]
Bayes is still, however, the focus of much controversy.
[snip]
Perhaps the grandest claims made for Bayesian methods are those of Judea Pearl, a computer scientist at the University of California, Los Angeles. Dr Pearl has suggested that by analysing scientific data using a Bayesian approach it may be possible to distinguish between correlation (in which two phenomena, such as smoking and lung cancer, occur together) and causation (in which one actually causes the other).

This is why I would like to see more educational research focused on good teachers.

It's easy enough to pick out the good teachers in a school — not for me, probably, but for other teachers & administrators in the school.

I'd like to know what they're doing.

In the past, the only kind of research one could do on an individual teacher was.....Geertzian thick description or qualitative analysis of some kind.

I'd like to see lots more thick description & qualitative analysis; I'm not Frequentist-with-a-capital-F.

But Bayesian statistics strike me as being, potentially, incredibly useful for a empirical research on Individual Great Teachers.

spaced repetition

In a Bayesian analysis, in other words, a set of observations should be seen as something that changes opinion, rather than as a means of determining ultimate truth.

Bayesian statistics & false positives
Bayes & the human mind
Bayesian reasoning, intuition, & the cognitive unconscious
most bell curves have thick tails
ECONOMIST explanation Bayesian statistics
Bayesian certainty scale

Bayesianprobability

-- CatherineJohnson - 29 Jan 2006

EconomicsBlackboard 04 Feb 2006 - 20:57 CatherineJohnson

source:
Economics Round Table

-- CatherineJohnson - 04 Feb 2006

CollegePrep 09 Feb 2006 - 01:24 CatherineJohnson

via eduwonk, a link to an Ed Sector anaylsis, High Schools Failing to Prepare Many College-Bound Students for Science Careers.

factoids

science, technology, engineering, and mathematics = STEM

82 percent of high school kids say they plan to go to college, but only 51% are in college prep [ed.: awhile back I read some material from Roy Ohrbach, of U.C. Riverside, showing that often Hispanic parents have no idea their kids aren't in college prep — Ohrbach's been traveling around CA, IIRC, giving parents papers in Spanish explaining what the college track is & how to find out if your child is in it]

definition of college prep: 4 years of English, 3 years of math, science, and social studies, 2 years of foreign language, and 1 semester of computers — 31 percent of high school graduates complete this basic college preparatory curriculum

14 percent earn math or science credit in Advanced Placement (AP) or International Baccalaureate (IB) programs

about 60 percent of students who take AP tests in Biology, Chemistry, and AB Calculus get a score of "3" or better

12 percent of h.s. kids take calculus

40 percent take trig (this includes the 12% of all h.s. kids who go on to take calculus)

I wouldn't think these figures were so bad, if it weren't for the charts below. Forty percent of all h.s. kids making it through trigonometry sounds OK to me (not that I would know...)

But when you look at how many of these kids take no math at all in college — around 70 percent — that seems pretty bad to me.

Because of poor middle school preparation, tracking, inadequate guidance counseling, low-quality instruction, or a simple absence of available courses, too many students are permanently knocked off the pathway to a STEM career early in high school or even before. This is particularly true for low-income and minority students. No one tells them or their parents that by failing to enroll in a rigorous, math-oriented college prep curriculum, they're effectively making a life decision to forgo the opportunity to pursue a career as a scientist or engineer.

This isn't just a problem for low-income & minority students. It's a problem for just about anyone who majored in the social sciences or humanities.

I had no idea, when Christopher was tracked into Phase 3 math in 3rd grade, that he'd been tracked out of calculus in high school. None. (spaced repetition, I know) I had no idea that a) there is a 'math track' and b) it starts young. People with jobs like mine naturally assume that math works like everything else. You go to high school, you graduate, you go to college, you choose a major — and the major can be anything you decide you're interested in. All doors are open.

update: Tracy & Matt Goff weigh in below

CYCT%20February%202006%20-%20High%20School%20Preparation%20Chart%201.gif

from Tracy

On the topic of the importance of doing maths, I know two girls who were tracked out of what they wanted to do by not doing maths.
One was told by her guidance councillor that she didn't need Maths With Calculus to get into engineering, only Maths With Statistics. (You could take two maths courses in the last year of high school).
Another was told by the Head of Chemistry that she didn't need to do another maths course at uni for her chemistry degree. Then she couldn't do an advanced organic chemistry course because she didn't have enough of a calculus background and had to change the topic of her PhD.

here's Matt G

I would not say that it is not possible to get a degree in a STEM field without having had calculus in High School. One of my math major classmates as an undergraduate had not had calculus in High School and he did fine starting in Calculus in college (which many students need to do anyway, even if they have already had Calculus in High School). I knew at least one person while I was at graduate school who had started in the basic algebra class and worked her way up through the math program (she was a non-traditional/adult student).
It is, however, my impression that if you have (barely) made it through algebra in High School, the chances are pretty decent that in some way for some reason you have been turned off to math (and likely science). At that point it seems very unlikely that you would choose to major in a STEM related field. That is to say, I think the barrier to students entering STEM fields is mostly a matter of perception and/or expectation, rather than something fundamental and insurmountable. It may take a year or more extra, and you probably won't get your degree form Cal Tech or MIT, but there are plenty of schools where the motivated student can work through the math/science curriculum (and whatever prerequisites might be necessary) and enter a STEM field.

That makes sense to me (based in extremely limited knowledge of what it takes to succeed in college math, obviously.)

It's never struck me as likely that not taking AP calculus would knock a kid out of any kind of math at all in college. And based in Ed's view of AP history (not especially positive) I assume most AP students are going to have to repeat calculus in college.

My AP calculus goal for Christopher is almost entirely pragmatic.

I'm assuming that if Christopher sets AP calculus as his goal (which, at this point, he has) he'll work hard in lower level courses, and learn more.

I also assume that taking calculus twice is a good thing. (Maybe it's not, but for me it's been good to do basic math twice.)

Rudbeckia Hirta on taking calculus twice

Bad calculus is worse than no calculus. I'd much rather have students in my class with a solid algebra background + no calculus than those who took a purely algorithmic high school calculus classes. Just this week one of my students (in Calculus 1) told me, "I already know calculus. It's when you take the number up top and put it down in front and lower it."
But perhaps I say this because this week I am teaching the limit definition of the derivative.
[snip]
I would say that a bad calculus course would be one that emphasized the easy, algorithmic calculations while minimizing the historical context, the applications, the technical details that make it all work, and the importance of mathematical precision in phrasing and justifying statements.
A crude analogy would be a history class that was only about dates and places and names (bad) and one that involved analysis of the issues involved and their context (in addition to the dates and places and names) (good).
You can probably teach a BIRD how to take the derivative of a polynomial function. Knowing when to do it, why you can, and what it means requires a person (who probably has taken a good calculus course).
The problem that I face is that my students (who are at the dualistic thinking stage of the Perry Model) believe that their high school teacher's point of view ("Calculus is about computing derivatives and integrals") is the right one and that mine ("Calculus is a subject in which mathematical techniques were developed to solve problems relating to areas and tangents.") is not. If they came to me thinking, "In my high school calculus course, I learned a little bit about part of calculus," then it would be OK. But instead they tend to think, "In my high school calculus course, I learned calculus. And my college is SO MEAN AND UNFAIR by making me take this so-called calculus course that ISN'T REALLY CALCULUS because it contains all sorts of stupid and unimportant stuff like proofs and limits and word problems!"

I had never heard of the Perry model - it's terrific.

Ed is constantly trying to talk college undergraduates out of stage 2.

-- CatherineJohnson - 07 Feb 2006

NyuMathMajor 03 Oct 2006 - 01:13 CatherineJohnson

Ed talked to an undergraduate majoring in math today.

I guess the kid spontaneously told Ed that, "Calculators are the worst thing that ever happened to math students."

Ed said he almost burst out laughing, because next this student went on to say that nobody who used calculators as a kid can do fractions, and if you can't do fractions you can't do calculus.

Ed said this guy could have been channelling me.

The student also said that, in high school, his calculus teacher had told the students who were having trouble, "You're having trouble because you used calculators in grade school and you never learned to do fractions." It was obvious to her. He spent quite a lot of time describing automaticity to Ed, and how important it is.

Ed asked why he hadn't used calculators as a child, when everyone else was, and the answer was chilling: he hadn't used calculators because he 'was into' math, he liked it, and he wanted to do the calculations by hand.

What that tells me is that only the natural born Math Brains are going to make it through these days — natural born Math Brains who know they're natural born Math Brains.

Your basic kid is going to use the calculator if the teacher hands it to him.

Then he's going to regret it later on.

That's what happened to the other kids in his high school calculus class.

Ed asked him whether the kids who'd used calculators could catch up.

The kid didn't think so. At least, he hadn't seen it happen.

Math is hard, he said. It's hard, it takes a long time to learn, and he didn't think a high school student who'd lost that much time could make it up.

That's what James Milgram said, too.

no calculators in Irvington

I don't think any of the grade school kids here are using calculators.

One of main criterion for choosing a new math curriculum was (paraphrasing) 'constructivist approach.'

One of the other main criterion was emphasis on math facts & computation.

TRAILBLAZERS was the only constructivist curriculum they considered that stressed fluency in math facts.

(I assume they're teaching the traditional long division algorithm in spite of the fact that TRAILBLAZERS teaches 'forgiving division,' but I don't know. Nevertheless, nobody's passing out baskets of calculators.)

Good for them.

which reminds me

I had to buy Christopher an expensive graphing calculator (or something) last fall, for Middle School.

He never used it once, and then finally lost the thing.

Good riddance!

His teacher is letting them use calculators for the first time this year, to calculate circumference & area of circles. I'm not even sure that's such a good idea.

Since he's doing KUMON, though, I figure it's OK. He's incredibly fast & accurate on the KUMON sheets.

Of course, the two "Fraction Levels" - E & F - are killers.

-- CatherineJohnson - 14 Feb 2006

SampleExamQuestionsFromHell 23 Feb 2006 - 20:07 CatherineJohnson

Economics: Describe in four hundred words or less what you would have done to prevent the Great Depression.
Political Science: There is a red telephone on the desk beside you. Start World War III. Report at length on its socio-political effects, if any.
Mathematics: Derive the Cauchy-Euler equations using only a straightedge and compass. Discuss in detail the role these equations had on mathematical analysis in Europe during the 1800s.
Computer Science: Write a fifth-generation computer language. Using this language, write a computer program to finish the rest of this exam for you.
Extra Credit: Define the universe, and give three examples.

source: Sample Exam Questions from Hell

Here's a real one:

My exams this semester are going horrible. I just love that feeling in which you leave an exam and you have no clue of how well you did. In fact, I feel as if I just wasted 13 weeks of my life studying, because my exams questions generally have nothing to do with the topic that I am studying. Our Con Law exam for instance wanted us to analogize an insignificant comment that Justice Breyer made in an interview about form and functionalism and how that relates to Supreme Court Commerce Clause decisions of the past 25 years. This is a least what I thought it said.
Thanks everybody I feel better now.
Sorry about the typos. I am a little stressed.

hoo boy

That is an exam question from hell.

-- CatherineJohnson - 23 Feb 2006

GraphPaperForOurTeachers 09 Mar 2006 - 17:22 CatherineJohnson

I found a terrific cache of graph paper online this weekend at a site called Mathematics Help Central — perfect for homework or for taking notes in class.

Mathematics Central is a lot of fun:

Are you stumped on a math problem? Help is on the way! Mathematics is a challenging subject that mystifies many. Imagine the problem as a complicated puzzle that you must solve. All the pieces must fit in order for you to realize your success. This web site is devoted to helping you through your math worries! Take a look around! There's plenty of lecture notes, helpful links, personally developed graph paper, and a little section about why I love math. Enjoy!

She's posted her lecture notes.

From her 'About Me' section:

I have not always loved math. In fact, math does not come easily for me. I have to work hard for it! I suppose that's why I find it so challenging.
I am a college student enrolled full time at a southern university pursuing a major in mathematics. I am also a divorced mother of two beautiful little girls. In February 1998, (Friday the 13th of all days), my husband of five years and I were separated. At the time, my oldest daughter was almost four years old, and I was six months pregnant. I was hurt, devastated, and miserable. The divorce was painful. My self-confidence was nowhere to be found. I returned home to live with my parents because I was a stay-at-home mom who had devoted most of my time to loving my family, and simply didn't have a way to make it on my own.
My parents encouraged me to go to college. I was excited about the idea, yet a little intimidated also. It had been years since I graduated, and I just wasn't sure if I could do it by myself with two small children. My parents assured me that they would help me in whatever ways they could, even though both are disabled and are experiencing increasing health problems to date. With much thought, a little preparation, and a lot of guts, I enrolled for classes during the spring semester at our local community college.
My father, and many others I had talked to, encouraged me to go into an engineering or computer related field with a concentration in mathematics. I had never had any trouble keeping my checkbook balanced--that is when I had money in it! My first class in college was Intermediate College Algebra. I was excited and ready to go. I thought to myself, "This should be pretty easy. Probably mostly review from high school."
Boy, was I wrong! When my professor began reviewing pre-requisite material, I began to panic! I didn't remember anything! (And my teacher was so tough!) I looked for help anywhere I could find it, and I even had to ask my 15 year old nephew for help with fractions and equations.

USA Today mentioned her in a story, too.

the graphs

There are 9 different forms, each in color or black and white.

Here's a terrific homework form for kids learning functions:

"Three graphs per page with plenty of room to work problems, also."

There's space for equations & calculations, and an already-made chart for the 'input' and 'output' numbers. I love it.

(He obviously took a screen shot before he'd finished; the actual sheet doesn't have the funky mis-matched lines on the graphs.)

Here's another:

"Eight graphs per page. This graph paper is best when you have a lot of graphs to make. The graphs are
small without numbers."

This one might be great for hand-outs —

"Large x-y Co-ordinate Graph Paper. This graph paper is a must for any student doing extensive
graphing. This graph is perfect for graphing class notes quickly. The x and y axis and the rectangular
co-ordinates are already drawn to speed up the homework and study process! The minimum and
maximum x and y values are blank so that you can scale the graph to fit your needs."

Graph paper for college geometry:

This one is an original!!! This graph paper was designed for College Geometry where proofs are
involved. You can fit two proofs to a page. The "Given" and "To Prove" are labeled beside the
Statements/Reasons tables, and there is room to draw your given geometric figure. This was a
HUGE time-saver for me. The headings and color scheme are the same as Form 5A. The paper is
probably suitable for for other types of mathematics proofs, but it was not designed for that purpose.

Graph Paper Printer Program

He's also got a 'Graph Paper Printer Program' available for download.

I tried to download it, but didn't know how to work it. Apparently you're supposed to 'paste' it into your word processing program?

I have no idea how one would do that, sad to say.

$bluhorsa.gif$

'graph paper' for word problems?

I'm thinking about creating some kind of how-to-solve-word-problems template for Christopher.

He has essentially zero idea how to tackle even a simple word problem, and the state test is on March 14 - 15.

polar coordinate graph paper (pdf file)

-- CatherineJohnson - 06 Mar 2006

CollegeGraduateLiteracyRates 09 Mar 2006 - 16:27 CatherineJohnson

I think Ken left a link to this study awhile back, but I have no idea where it is, and I don't think I wrote it down anywhere, so I'lll start with this:

Reports on college literacy levels sobering
'Study: Most students unable to handle complex but common reading tasks'

More than 50 percent of students at four-year schools and more than 75 percent at two-year colleges lacked the skills to perform complex literacy tasks.
That means they could not interpret a table about exercise and blood pressure, understand the arguments of newspaper editorials, compare credit card offers with different interest rates and annual fees, or summarize results of a survey about parental involvement in school.

3 forms of literacy

analyzing news stories and other prose

understanding documents

having math skills needed for checkbooks or restaurant tips

worst area is math

The survey examined college and university students nearing the end of their degree programs. The students did the worst on matters involving math...
Almost 20 percent of students pursuing four-year degrees had only basic quantitative skills. For example, the students could not estimate if their car had enough gas to get to the service station. About 30 percent of two-year students had only basic math skills.

what they could do

from the CBS News account:

Most students at community colleges and four-year schools showed intermediate skills, meaning they could perform moderately challenging tasks. Examples include identifying a location on a map, calculating the cost of ordering office supplies or consulting a reference guide to figure out which foods contain a particular vitamin.

in a nutshell

more than 50 percent of students at four-year schools & more than 75 percent at two-year colleges lacked skills for complex literacy tasks

literacy defined as:
• analyzing news stories and other prose
• understanding documents
• having math skills needed for checkbooks or restaurant tips

can't interpret a table about exercise and blood pressure

can't understand arguments of newspaper editorials

can't compare credit card offers with different interest rates and annual fees

can't summarize results of a survey about parental involvement in school

worst area is math

20 percent of students in four-year programs & 30 percent in 2-year programs had only basic quantitative skills

students with basic quantitative skills can't tell whether their car has enough gas to get to the service station

college accountability?

There's been talk lately of holding colleges accountable. I haven't followed it, but it's out there.

The AIR study of college student literacy reminds me of an obit I read when Peter Drucker died. The writer said that Drucker had made many correct predictions in his lifetime, and that his main prediction lately had been that colleges as we know them would cease to exist. Unfortunately, I don't seem to have saved the article, but a new article by Walter Russell Mead is about the same issues:

Paying for college education is one of the biggest financial worries facing middle class and working families....
...perhaps [government] could offer an alternative: a federally recognized national baccalaureate (or 'national bac') degree that students could earn by demonstrating competence and knowledge.
With input from employers, the Department of Education could develop standards in fields like English, the sciences, information technology, mathematics, and so on. Students would get certificates when they passed an exam in a given subject. These certificates could be used, like the Advanced Placement tests of the College Board, to reduce the number of courses students would need to graduate from a traditional college. And colleges that accepted federal funds could be required to award credits for them.
But the certificates would be good for something else as well. With enough certificates in the right subjects, students could get a national bac without going to college. Government agencies would accept the bac as the equivalent of a conventional bachelor's degree; graduate schools and any organization receiving federal funds would also be required to accept it.
Subject exams calibrated to a national standard would give employers something they do not now have: assurance that a student has achieved a certain level of knowledge and skill.

This is a long excerpt, so I'll put the rest of it here.

Without having thought about it, I'm in favor of anything that increases the public's knowledge of what students are actually learning in school.

National Assessment of Adult Literacy
"A nationally representative and continuing assessment of English language literacy skills of American Adults"
key findings of NAAL

AIR press release
complete AIR study here

AP account

-- CatherineJohnson - 07 Mar 2006

MiniProblems 15 Jul 2006 - 16:33 CatherineJohnson

I've been complaining for months about the lack of word problems in Christopher's math class.

The kids memorize one procedure/rule/formula a day, do a few calculations, and march on. As a direct result, IMO, their knowledge really is rote as opposed to procedural. At least, Christopher's is. And I've had enough math talks with other kids in the class to know some of them are in the same boat.

Today I had a eureka moment reading a Comment left by Kathy Iggy:

The old math books I found (the same ones I used in grade school) have lots of what they call "mini problems" used to illustrate how a recently taught concept would be presented in a word problem. Megan likes these because of their brevity and she doesn't have to struggle with comprehension that much.
For example:
20 yards of ribbon. 1/4 used for dress. How much ribbon used?

That's IT!

mini problems

That's the concept, and the phrase, I've been looking for.

mini problems:word problems :: basic skills:higher order skills .

That's from Ken, and he's exactly right.

[update 4/23/2006: no! he's not right! Actually, he's write about using mini problems to teach word problems; I'm talking about mini problems to teach math - to teach the fundamental concept in a lesson. Awhile back I realized that word problems are the 'real manipulatives.' Now I know what I mean by that.

All concepts should be taught — illustrated — with mini problems. All concepts, every last one.

PRIMARY MATHEMATICS does this; SAXON MATH does it; KUMON does it. I'll post examples.

I've come to feel that the first word problems illustrating a new concept should be so simple children can do them in their heads.

For example, the very first ratio word problem a child does should be something like this:

Christopher bought one pencil for one dollar.
How many pencils can he buy for two dollars?

The question should be written this way, too: on two separate lines, so the child sees instantly that the first sentence is the set-up, and the second sentence is the question. Richard Brown's revision of Mary Dolciani's BASIC ALGEBRA, a book I like very much, does this for many of its word problems. I'll post some of those, too, as I get to it.

mini problems are applications

The problem with word problems is that, in the U.S., they're always hard.

Word problems are so hard people have apparently come to think that if a word problem isn't hard it isn't really a word problem.

I'm wondering if we ought to ditch the phrase 'word problem' (ditto for 'story problem') and adopt the word 'application.'

A better idea: we should think about the point of word problems.

Some word problems are written and assigned to give students practice.

Many word problems are written and assigned to assess whether students have developed flexible knowledge.

I'm talking about a third purpose, which is instruction. I'm talking about word problems designed to teach.

instructional word problems

A word problem is an application. A super-simple, starter word problem explains and demonstrates a mathematical concept by showing students how the concept is applied.

As a matter of fact, an instructional word problem shouldn't even be a 'problem.' It should just be a question, and the answer should be obvious.

A simple, instructional mini-problem should not test the child, should not challenge the child, and certainly should not trick the child.

It should teach.

examples to come

be sure to see Google Master's comment

how do you teach your child word problems?
mini problems (important)
arithmetic to algebra & mini-problems

-- CatherineJohnson - 07 Mar 2006

IvyWise 26 Mar 2006 - 23:54 CatherineJohnson

When it comes to college admission anxiety, I thought I'd seen it all.

Turns out I hadn't.

IvyWise is a college counseling business. For $30,000, families can hire the company's top counselor for two years to help students mold their high school career with an eye toward college, and to help students and parents navigate the application process.

yowza!

source:
College Admissions: Is Gate Open or Closed?
WSJ ($)

-- CatherineJohnson - 26 Mar 2006

CollegeAdmissionsSqueeze 10 Apr 2006 - 20:21 CatherineJohnson

Top Colleges Reject Record Numbers Schools Say Surging Applications Produce Unusually Competitive Year; Stanford Admits 11%! ($)

It's just getting worse and worse!

It'll be way worse in 2012!

11% worse!

It's not just the sheer number of applicants that makes schools competitive. The colleges indicate that they are also seeing large numbers of highly qualified students. The University of Pennsylvania turned away 394 of the 1,045 valedictorians that applied. Also, about 70% of applicants who got near-perfect scores in the math and critical-reading sections of the SAT were turned away, says Mr. Stetson. At Brown, 94% of admitted students this year were in the top 10% of their class.

It's all bad.

Which can only mean one thing.

Time for a glass of life-extending red wine. I want to live to see my son rejected by the top college of his choice.

-- CatherineJohnson - 08 Apr 2006

CollegeStudentWhoDoesntKnowMean 12 Apr 2006 - 19:19 CatherineJohnson

via Joanne Jacobs, a post at Right Wing Prof called I would like an answer.

After he spends 3 hours teaching a college freshman what an average is, Right Wing Prof wants to know why he's doing the job of K-12.

I know the answer to that.

In global terms, the answer is simple: K-12 is about inputs, not outputs. It's about inputs, not outputs, as a matter of law.

Students are legally entitled to receive a public education. They are not legally entitled to learn.

Legally, a public education is a set of inputs: teachers, administrators, textbooks, school buildings. Nowhere does the law state that a student is entitled to learn the content covered in school. You've probably noticed that no one ever sues a school district because his child graduated high school not knowing what an average is. We sue over everything else under the sun. We sue doctors for malpractice, we sue companies for selling tobacco, we sue McDonald's because the coffee was too hot — why has no parent ever sued a school for not teaching his child how to read or write or solve a mathematical problem?

The answer is that students have no legal entitlement to learn. Learning is an output; learning is the intended result of the inputs.

Students have a right to inputs. Students do not have a right to outputs.

UPDATE 11-8-2006: right answer, wrong question. Many parents have sued public schools in many states; the courts have universally ruled in favor of schools. If the child fails to learn, "it could be something about the kid." Therefore, courts have ruled that the school cannot be held accountable for learning. (I don't know the legal ins and outs of this. Could the school be held accountable for a broad class of students failing to learn? Could one file a class action lawsuit on behalf of non-classified kids who've failed to learn? I don't believe anyone's done it, which implies to me that someone probably tried and failed. But I don't know.)

IEPs for all

There is one exception to this rule.

Children with special needs have a federal entitlement actually to learn the material covered in school.

Each year the school must sit down with the parents and hash out a formal, legally binding document called an Individualized Education Plan, or IEP, stating exactly what knowledge and skills the child will be expected to learn in the coming year. Parents can convene a new IEP meeting any time they wish; schools must show that the student has actually learned what is listed on the IEP.

That's the law. Good luck enforcing it, of course. The reality of special ed is that it's an adversarial system, parents are called 'advocates,' and you spend huge quantities of your life fighting the school.^* In Los Angeles we went to practically every meeting with our lawyer.

Nevertheless, the law exists. Your child is legally entitled to learn.

Christopher has no entitlement to learn. None, zip. Logically, then, if he doesn't learn, it's not obviously the school's fault. Legally speaking, the school is doing what it's supposed to be doing.

So whose fault is it? It's Christopher's fault. He's not organized, or he's not paying attention, or he doesn't realize that Middle School Is Hard while Fifth Grade Was Easy, or whatever. The school and the parents both seek an explanation within the child.

I'm sure this is why we've seen the explosion in numbers of special needs students. The instant a student goes from being 'typical' to having 'special needs,' he gains a legal entitlement to learn.

I'm not saying that parents 'play' the system, though I hope some do. A bad system should be played. I think an economist would analyze the huge increase in special ed population as a case of people & systems responding to bad incentives.

Bad incentives operate below the level of consciousness, for the most part. It's not that parents and teachers consciously think to themselves: If we get him classified, the school will have to teach him. Things just go that way.

I include teachers in this category, because many teachers are frustrated by school policies requiring them to march through content students haven't mastered. Christopher's brilliant teacher in 5th grade, Ms. Duque, told me that she'd been asked to teach the accelerated Phase 4 math class but had turned it down in favor of Phase 2, which had a number of students on IEPs. She always preferred to teach students with IEPs, she said, because the IEP gave her the legal right to teach to mastery.

There are plenty of administrators who feel this way, too. I mentioned that in Los Angeles we always went to meetings with our attorney in tow.

Well, that wasn't a problem! The administrators loved her. They all went way back. You could practically see them breathing sighs of relief once Valerie showed up. They were under pressure to withhold services; once Valerie was there they knew she was going to fight them all the way and win; so they had far more ability to do what they wanted to do, which was to do their level best to see to it that kids with disabilities learned everything they could.

Parents want their children to learn, teachers want their students to learn, administrators and school boards want their students to learn. But the system is set up to cover content in a spiralling sequence, not to teach to mastery.

That produces large numbers of kids who fall behind. When they've fallen two years behind - the formal definition of an LD - they can be classified as learning disabled, which triggers a legal entitlement to be taught to mastery.

When parents and teachers both want children taught to mastery, but the system blocks teaching to mastery, the incentive to move large numbers of kids into the sole category that will allow them to be taught to mastery is immense.

This process doesn't have to be conscious, and I don't think it is conscious 99% of the time. That's the tragedy. Everyone believes the categories. I was stunned to learn from my neighbor, a clinical psychologist, that from her perspective 'learning disability' isn't a diagnosis. "Learning disability" is a legal category used by schools to assign services. (This gets complicated. The law defines a learning disability as an actual brain-based disability. However, school districts often define a learning disability as 'normal intelligence, two years behind grade level.')

People think learning disabilites are real the way diabetes or cancer are real. In many school districts, once a child falls two years behind in a spiralling curriculm, he is 'referred' for testing, which invariably finds a problem in the child, not the teaching. At that point he 'qualifies'^** for the label of 'learning disabled' and the label becomes real.

He is a learning disabled child.

It has crossed my mind that one answer to our problems with Irvington Middle School is to figure out a way to get Christopher classified.

That was one of the first questions asked by the principal at our Team Meeting. 'He doesn't have a learning disability, does he?' He sounded almost hopeful.

Life would be easier for everyone if Christopher had a classification. Come to think of it, if I hadn't done any reteaching at home, he just might qualify for a special needs classification in math by now. Dyscalculia anyone?

I'm not going to do it. I have no interest in working the system so as to have a third child 'classified.' I'm not even sure I could do it, though, knowing me, I probably could.

Nope. This is a battle for my one typical child as a typical child.

He should have the same rights his autistic brothers do.

^* Last year, in Irvington, the school board openly announced an illegal policy at a board meeting. There was a shortfall in the budget, they said, so they had come up with a plan to save money by bringing all the special needs kids 'back to district.' When a school can't meet an IEP student's needs, the district has to pay to send him elsewhere. That's expensive. Balancing a budget on the backs of special ed kids by taking them out of programs that meet their needs and bringing them back to programs that don't is illegal. Everyone hired lawyers, the district hired a superb new interim director of special ed, and that initiative came to a screeching halt. He's done a superb job creating a 'transitional' program for Jimmy. We're hoping desperately we can keep this administrator for two more years.

^** 'Qualifies' is the term used. A child must 'qualify' for 'services.' A typical child does not qualify for services.

key words: blame the student school psychologist
Pamela Darr Wright summary of Galen Alessi study
Evolving Functions for the School Psychologist
Whose Fault Is It?
educational rights of special need children versus typical children
Engelmann on Galen Alessi study
Pamela Darr Wright posted to ktm
"public school has never been about outputs..."

-- CatherineJohnson - 11 Apr 2006

OverviewOfCalculus 29 Apr 2006 - 13:58 CatherineJohnson

I was just looking up the meaning of 'delta' at Math Forum, and I found Dr. Ian's overview of calculus.

I haven't read it, but I have a distant memory Dr. Ian is one of the Math Forum people whose explanations I've liked before.

Also, I need an overview of calculus. Or I will be a couple of years from now.

-- CatherineJohnson - 27 Apr 2006

GreatestHits 11 May 2006 - 17:14 CatherineJohnson

It's The Terminator at edspresso.

-- CatherineJohnson - 11 May 2006

UnderstandingMath 18 Jun 2006 - 14:33 CatherineJohnson

from Peter Alford's website (scroll down)—

You understand a piece of mathematics if you can do all of the following:

Explain mathematical concepts and facts in terms of simpler concepts and facts.

Easily make logical connections between different facts and concepts.

Recognize the connection when you encounter something new (inside or outside of mathematics) that's close to the mathematics you understand.

Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)

Wonderful.
In terms of math and the math wars, I especially admire the first principle: you understand mathematics when you can explain mathematical concepts and facts in terms of simpler concepts and facts.
The NSF-funded curricula seem to have been trying for this idea. But they bungled it.
A person who understands something can explain it in different terms. But those terms don't have to be words - and, in the case of math, probably shouldn't be words, or at least not solely words. Alford's formulation is more sophisticated. Being able to explain something means being able to explain it in simpler concepts and facts.
I'm think I'm going to post these principles over Christopher's desk. They're universal. I've been using them for years, without having tried to sort them out or write them down. Alford has made them explicit for me. When I'm writing a book or an article, I know I'm succeeding when I can do these four things.
Seeing past the clutter — that's the big Kahuna.
Temple calls it "finding the basic principle."

Saxon Math
Saxon Math probably does a superb job of using these principles to teach math.
This year I learned from Saxon Math, for the first time in my life, that when we find areas we are always multiplying "two perpendicular dimensions." (Saxon 8/7 Lesson 82 Area of a Circle)
Of course, I sort-of knew that.....but I'd never made the connection between finding the area of a square and finding the area of a circle. (Have I mentioned my education in mathematics left a lot to be desired?)
That one observation, in Saxon 8/7, permanently changed my perception of area & volume, permanently increased my comprehension of area and volume, and permanently improved my ability either to remember area and volume formulas or to derive them when I don't remember them.
Saxon used seven sentences, illustrated by geometric figures, to make that observation. This passage embodies all four of Alford's principles:

We can find the areas of some polygons by multiplying two perpendicular dimensions.

We find the area of a rectangle by multiplying the length by the width.

A = lw [illustration of square]

We find the area of a parallelogram by multiplying the base by the height.

A = bh [illustration of parallogram]

We find the area of a triangle by multiplying the base by the height (which gives us the area of a parallelogram) and then dividing by 2.

A = bh/2 or A = 1/2bh [illustration of triangle]

To find the area of a circle, we again beegin by multiplying two perpendicular dimensions. We multiply the radius by the radius. This gives us the area of a square built on the radius. [illustration of circle]

-- CatherineJohnson - 18 Jun 2006

KarenOnTeachingCollege 05 Jul 2006 - 22:06 CatherineJohnson

I've just found Karen's response to ktm guest, who writes that "Never before have I seen a group of parents so dedicated to blame-shifting and teacher scapegoating."
I think ktmguest's comment is interesting and almost certainly true of me — although I'm not completely sure what he/she means by "blame-shifting." I assume ktmguest means that "blame" is in order; the problem isn't that I'm blaming people, but that I'm blaming the wrong people, namely teachers.
I assume this means that I should be blaming my child, or me, or both.
I've thought about this.
At some point this year I decided to "blame-shift" on purpose. That's what makes me a radical, as opposed to a reformer who eschews blogging in favor of "trying to bring about meaningful change," as our guest recommends.

how to succeed in middle school without really trying
part 2
I've mentioned that Christopher is in fantastic shape.
Other children have had a tougher time of it this year. I've been talking to parents, and the stories they tell me are distressing. I haven't asked anyone's permission to write about their children, and I think that some of the things they've gone through at our middle school are so painful that even with details disguised, it would be wrong for me to try to create a disguised version.
All I can say is that some parents feel their children are different now, after 6th grade, from what they were last summer. They aren't smiling the way they used to; their sweet faces are closed. Summer will put them right, I hope. (side anecdote: Ed came home from picking Jimmy & Andrew up at the Y last winter and told me that Jim, the teacher who runs the program — wonderful guy — had said the reason our students do so badly compared to students in other countries is that we have long summer vacations. I almost snapped his head off. If the year-round calendar "movement" picks up steam, I will march in the streets.)
Christopher's face is still sweet. He's still open, trusting, cheerful — and responsible! (How any teacher could miss the connection between responsibility and trust in the world is beyond me.) He likes his school (!), he likes his teachers, and he likes his friends. This summer he's having a blast at camp & he's even reasonably OK about his reading, vocabulary, and math program here at home.
In the spring, when the school planned a 1950s School Spirit day (I'm repeating a story I think I already left in the Comments), Christopher put together his own costume. He was so excited! Then, when he got to school, he discovered that only four children had dressed up for the day. Four. If you didn't wear a costume, you were supposed to wear the school colors, and nobody was wearing the school colors, either.
Think about it. Ed and I have produced one of only 4 children in the entire 6th grade who has school spirit.
This weekend my neighbor hired Christopher for the first time to look after her dogs for two days while they drive their son to camp.
Christopher has remembered the exact time he was supposed to go to her house, without reminding. Apparently he's fixing to become a punctual adult, a quality he didn't pick up from either of us I'm sorry to say.
It's almost as if this year never happened. Christopher is his same self.
His same self, only older and more mature. This feels like a miracle.

how to be on your child's side
Ed and I have both had the sense that our war with the school, which on the face of it sounds like a dreadful idea, turned out to be some kind of Brilliantly Counterintuitive Parenting Strategy. (sorry)
I couldn't understand it.
Then Ed said the reason war-with-the-school worked was simply that it meant we directed our anger at the school, not at our child.
Which is exactly what ktm guest objects to. In this, he/she is typical of the tone set by our own middle school. Our middle school triangulates parents against their children. We are told constantly that our children need to "take responsibility for their learning"; then, when our children get bad grades, we are encouraged to see this as a failure of character, not teaching.
This works. Parents here are tremendously responsible, hard-working people. Most of them were also good students for whom learning and good grades came easily. Suddenly they have children bringing home Cs, Ds, and Fs, and they're shocked. They know their children are brighter than a "D" or an "F" (they're right) so they conclude that the child would have earned an A or a B if only he'd studied.
Then of course we all signed our children's Contract to Improve My Grades: "I am responsible for the grades I receive. I can improve my grades by changing my study behavior." Ed and I are the only parents in the entire 6th grade, to my knowledge, who refused to allow our child to hand the contract back in.

when the baby is crying, the parents are fighting
Years ago, when Jimmy was a baby and we didn't know he was autistic, our family motto was "When the baby is crying, the parents are fighting."
Jimmy cried constantly; he was a very, very difficult baby. We didn't know how to help him, we didn't know what was wrong, we didn't know why he cried so much when other people's babies didn't.
We had as happy a marriage as anyone we knew, but inevitably, at some point, we would snap at each other. When your child suffers, your marriage suffers.
Our middle school stresses children and families. The K-5 schools never, ever did this. Never. Nor does the high school. Our middle schools is the problem child of the district.
More than once children in Christopher's class cried at school when they got their Cs and Ds and Fs returned to them in class. "My mom is going to kill me." "My mom is going to ground me."
Christopher would tell them, "My mom blames the school."
He would!
Imagine how beloved we are!
That kept him safe.

His job was clear. He was supposed to do his homework, behave himself in class and on the playground, and learn.
Those were his responsibilities.
If he did all those things and still got clobbered, we blamed the school. We intend to keep right on blaming the school if things don't change next year, under the new principal.

two moms I know
I know two other moms who took this path.
Both began the year believing that their child had to be responsible, and both adopted the school's definition of the word.
Both found their relationships with their children under stress. Anger, arguments, tears.
One was looking at the possibility that her son would have to attend summer school or even repeat 6th grade. He was failing, and the household was in an uproar.
When we talked in January, she was at her wits' end with her child.
I told her she needed to be at her wits' end with the school, not her son. She didn't believe me, so I pushed.
Finally I said, "Is there any family in town who wouldn't welcome your son into their home."
No.
I said, "J. is a good person. He is responsible. He has good character. He is doing the best he can. It's the school's job to make sure he learns the material they're teaching. They are the adults; they are the employees of the school district; they must teach him."
I didn't talk to her for a few months after that. When I did she told me that that one conversation changed her life! "We don't argue about school any more," she said. "J. comes home and he wants to do his homework. He gets right down to it. He knows he can do it."
This is what a pep talk and a $90-an-hour tutor will do for a kid!
Joking aside, she and her husband did what they had to do. The school was going to fail their child, literally fail him in his case. When they hired the tutor — and $90/hour is money they can ill afford to spend — and stopped all anger about his spacy ways, he soared. His face is still sweet like Christopher's, too.

For my other friend the shift was more gradual. She's a very strong parent. She sets firm rules & lots of them, she enforces her rules, and she expects her kids to do as they're told at home and at school. I sometimes tell Christopher that if he doesn't shape up he's going to go live with my friend for a while. She's that kind of mom.
She was pretty hostile to my blame-the-school philosophy at first.
I wore her down.
That's a joke, though there's some truth to it, I think. I'm perfectly happy to use the words "I blame the school." What I mean, though, is that I hold the school accountable — and after I've said this a few dozen times parents realize that they agree.
None of us is paying the school to teach responsibility.
We are paying the school to teach reading, writing, and math.
Over time, I think, my friend simply stopped believing the school narrative.

all your children are belong to us
Middle schools slam the gates shut. Childhood is over; parents stay out.
That's the message. I've heard this from parents everywhere.
A mom who pulled her child out of the school reminded me that last year, at the 5th grade graduation ceremony, the middle school principal told parents, "Your children are mine now."
This fall, at back to school night, he told us, "This is the year your child will stop talking to you. So come to us. Your children talk to us, and we'll know more about your child than you do."
That's pretty close to a direct quote.
If your middle school principal or teachers make sounds like this, it's time to set limits.
You don't need to be in open conflict with the school. But you do need to make clear to your child that you are still the parent. You are still the parent, you are still in charge, and you, not the school, will decide what he needs to do to be considered a responsible human being.
The school's job is to teach content.
And that's it.

Karen on college teaching
Karen's statement is beautiful.
Most Americans idealize teachers, and this is why:

I am both a parent and a college professor. My teaching philosophy is that the teacher sets the tone. I am also always mindful that as a teacher, I am modeling behavior.
Do I want them to take responsibility? Yes, I do, and I model that at every opportunity. For example, I broke my ankle last semester and was not allowed to put weight bearing pressure on it for six weeks. Just getting through the day became a challenge. However, I missed only one class and that was to have the cast put on; that appointment was dictated by the orthopedic surgeon. I also took great pains to connect the dots for my freshmen students to make sure they understood that while it was a challenge for me to be there, I was still there. I turned my misfortune into a teaching moment.
I am also mindful that while I am the teacher, I am also a student. My goal is to always be learning--in every way possible. That means I have to see the world through my students' eyes and it also means that I have to take responsibility for my own actions as well. Translated into action for me, that means that I am actively engaged in the process of learning.
For example, I can rant and rave and tell students that if they don't proofread their papers, there will be consequences. However, what I have learned from getting in the trenches with students is that sometimes it's a lack of knowing how to proofread effectively (it's a skill that can be taught), and sometimes it truly is carelessness. However, sometimes the students just don't know the rules of grammar, which is an entirely different problem. If you don't know how to use a comma properly in the first place, then proofreading isn't going to help all that much.
I also understand full well the importance of paying attention to detail. Without that skill, the students will have a hard time passing their introductory accounting class. So, in the freshmen class that I teach, my goal is to purposely and mindfully structure my assignments in such a way that I am helping the students grow that skill. Put simply, if I want my students to develop a skill or habit, then I need to teach it, and then provide opportunities for them to practice it--to reinforce the skill.
I also have the philosophy that if what I'm doing isn't achieving the objective I wish to achieve, I need to examine and understand why that is. Did I explain (teach) the concept in a way that the students understood it? Were my expectations clearly stated, or did I unintentionally surprise them? Is it them, or is it me or is there a design flaw with the system? In short, I suppose I approach such matters as possible problems to be solved. That is, I use critical thinking and problem solving skills.
Don't misunderstand me--I am both confident and competent. It's just that I am always striving for perfection--to do the best job that I can at teaching and at reaching the maximum number of students possible. I want all of my students to succeed and I want to help them do so, if they are motivated to do so. And I want them to understand that they are accountable for their actions and that there are consequences for their actions.
I don't know what grade or subject you teach, or whether your students are motivated or not, but I am curious about your method for handing out homework papers. Why is it that the students don't seem to able to pick up the papers on the way out the door? If they are typical kids, the minute that class is over, they may be focused on talking to their friends. Or, perhaps they are trying to get to their next class on time. Or, maybe they just don't care. That's a different and more difficult issue and one that would require a bit more reflection and analysis. But, assuming that they do care and are motivated to succeed, why not hand the papers out during class?
I also want my students to understand that they are accountable for their actions and that there are consequences for their actions, both positive and negative. However, I am also mindful of what I call the human motivation factor. I always want a student to believe that they can succeed if they are willing to put in the time and effort that is needed to do so. That is not the same as a harsh and punitive approach to grading.
For example, the infamous deduction of 20 points for failing to label the graph. In the first place, that seems pretty harsh for 6th graders. Did the teacher just assume that this procedure had been taught to automaticity in the earlier grades? Or, did she teach it herself? Did she provide a rubric with the consequences spelled out? Don't misunderstand me--I think that it's appropriate that this is automatic. My question is--did she teach it, or know full well that someone else had? Also, what was her objective with deducting 20 points--was it "teach a lesson?" If so, what lesson was she trying to teach? And perhaps what I'm also getting it (and what the parents are getting at) is: What is her teaching philosophy? Why is she doing what she is doing? What is she trying to accomplish, and are the methods she is using the best way to achieve this?
I would guess that the KTM readers and the IMS teachers want the same thing. We want our kids to have solid, fundamental skills, we want them to love learning, and to be respectful of others. We want them to pay attention to detail, to be careful readers, and to learn to take responsibility for their actions. In short, we want our children to have all the tools they need to be able to survive and thrive in the world as productive citizens. However, what we may not agree on is the most effective method to get there. And that, I think, is the source of frustration for many parents.

I'm going to send this to all my friends.
And I'm going to re-read it often.

-- CatherineJohnson - 03 Jul 2006

SingaporeCalculus 10 Jul 2006 - 16:25 CatherineJohnson

There isn't any.

That makes sense to me.
I've been dipping into the literature on what skills people actually need to earn a "middle class wage."
Calculus isn't one of them.
Entry-level algebra is.
IIRC, high school in Singapore ends earlier than high school here. When students graduate they've done a huge amount of work in algebra & geometry; they also seem to take a year of trigonometry.
No calculus.
No proofs, either, I don't think. (cursory impression)

search words: jobskills
Anthony Carnevale
How Computers Are Creating the Next Job Market

-- CatherineJohnson - 06 Jul 2006

CommunityCollegeStudents 05 Sep 2006 - 04:54 CatherineJohnson

At first, Michael Walton, starting at community college here, was sure that there was some mistake. Having done so well in high school in West Virginia that he graduated a year and a half early, how could he need remedial math?
At 2-Year Colleges, Students Eager but Unready
By DIANA JEAN SCHEMO

keep reading, it gets better

The sheer numbers of enrollees like Mr. Walton who have to take make-up math is overwhelming, with 8,000 last year among the nearly 30,000 degree-seeking students systemwide....
More than one in four remedial students work on elementary and middle school arithmetic. Math is where students often lose confidence and give up.
“It brings up a lot of emotional stuff for them,’’ Dr. McKusik said.
She told of 20 students who had just burst into tears on receiving their math entrance exam scores and walked out on college. Mr. Walton remembers a fellow student who failed to hand in a math assignment for the fourth time in the last week of class and learned that he would fail. The student lunged toward the professor and said, “I’ll kill you.”
“You can say whatever you want, but this really isn’t helping your grade,” the professor replied, Mr. Walton said.

this is incredibly cool —

But Mr. Walton made it through that remedial math class four years ago, ultimately praising the dean for standing firm. In June, he crossed a stage to receive an associate’s degree in computer science. Next year, he plans to earn another degree in, of all things, math.
He said he would like to earn a full bachelor’s, but hesitates.
“I’m scared to death of going to college,’’ he said. “I’ll be up to my eyeballs in debt.’’
This summer he sent his résumé even to employers demanding bachelor’s degrees and several years’ experience, hoping that his enthusiasm would compensate where credentials fell short. He sought positions that included tuition breaks for employees.
His strategy paid off with two offers, one in data entry at the community college here, a job he held on work study before graduating, and another as a technician repairing copying machines. Mr. Walton went for the second.
It offers benefits, tuition reimbursement and a salary of $22,850 a year, with extra money toward buying a new car every few years.
“I feel a little bit more — I don’t want to say confident — but maybe worthy,’’ Mr. Walton said. “Now, I feel like I’m all that, and a bag of chips.’’

Well, that's the way it's going to be around here. Step by step.

-- CatherineJohnson - 02 Sep 2006

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5 - (-7) = ?

5 - 7 = 5 + ( -7 )

5 - ( - 7) = 5 + [ - ( - 7) ]

5 - ( - 7) =

5 + [ - ( - 7) ] =

5 + [ 7 ] =

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-1 -1 = -2

-1 - ( - 1 ) = 0

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