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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 a

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5 - (-7) = ?

5 - 7 = 5 + ( -7 )

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

5 - ( - 7) =

5 + [ - ( - 7) ] =

5 + [ 7 ] =

5 + 7

- 7 - 7 = - 14

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-7 + 5

1 - ( - 1) = 2

-1 -1 = -2

-1 - ( - 1 ) = 0

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