Somewhat.




comments...

1. This is fourth grade, and they're still looking at rectangular arrays. They should have gotten that in second and third grade when they first learned about multiplication, and then LEFT THE RECTANGULAR ARRAY STUFF BEHIND. At this point they should be learning the distributive property and starting to learn the (THE!) multidigit algorithm.

Wickelgren says that it pays to move into abstraction as soon as possible. This is the exact opposite of the philosophy of most constructivist curricula, and it is worth asking why he suggests it. There are two reasons: first, to move kids along in their educations at a reasonable pace; secondly, because things like manipulatives and rectangular arrays are learning aids that are best left behind as early as possible, lest they become time-consuming habits (like counting on your fingers).

2. Kids are being exhorted to break their multiplication problems up into arbitrary pieces, rather than being explicitly taught to break problems like 7x23 into 7x20 + 7x3.

3. Kids are being told they have to learn more than one way to solve a problem. Kids hate that.

And, last but not least:

4. Parents are being told explicitly not to provide answers or methods to help their kids with their homework. They are supposed to let the little rotters stew in their juices instead. It's part of the learning process, so hands off.

To this last exhortation, I say: phooey. He's mine and if I decide I want to teach him astrology, I will.

The keep-your-hands-off-him-he's-ours subtext is almost the worst thing about fuzzy math.

comments...

Hello, my name is Colin Johnston and in this post I will describe the horrors of high school algebra.

On the first day of my junior year, I stepped into my math class. I will leave names out so as not to offend anyone. I had heard mixed reviews about the math teacher that I would have this year, but people generally said he was a pretty good teahcer. As the bell rang, the class sat down and waited patiently for him to enter the room. He slowly stepped into the classroom. He looked like a smart enough guy. He then passed out the textbooks and walked to the front of the classroom. He began talking about the curriculum we would cover this year, his grading scale, etc. He then said something, however, that didn't go down as easily as the other things he had mentioned. He said "You kids have been told how to do everything in your math careers. That is, up until now. This year, you are going to learn how to teach yourselves."

What? Then why don't we just take this class at home, over the internet? What is the point of having a teacher that doesn't teach? I made these same arguments to my friend after class, but he just shruged it off. "It will probably get better as the year goes on." He said. I guess I would give it a shot.

But as this year has gone on, things have gotten worse if anything. Now, the norm for the class is come in, sit down, spend an hour correcting last nights assignment (it takes so long because everyone has so many questions), get the new assignment, and puzzle over it for 10 minutes until you finally give up due to complete lack of understanding. Such is life in the new new math, Constructivism.

-- Colin

comments...

Well, here is just about all I know about security holes:

1. Microsoft Windows has them

2. They are a Bad Thing.

OK, that's a bit of an exaggeration... I know a little more about them than that, including that they occasionally bring down whole sections of the internet for a while. But more to the point, the email told me exactly what the security problem was, and I definitely didn't want it. They told me to patch a couple of the main files with newer versions, which (as of this evening) I have done.

If anyone notices that something that worked before tonight has stopped working, it'll be a huge favor if you'll drop me a line about it. It's a bit nervewracking, having to hot-patch a site that gets as much traffic as this one does. Especially when you're just a system administrator manque like myself.

comments...

For those of you who just came in -- Colin is my stepson. He's 17, a junior in high school in an International Baccalaureate program, no intellectual slouch. He's got a math book called "Precalculus: Graphical, Numerical, Algebraic" by Demana and Waits, who are known for writing very 'technology-centric' books that make heavy use of tools like graphing calculators (with the crack investigative skills this gang has become known for, it took us no more than ten minutes to determine that Demana and Waits have financial ties with the Texas Instruments company, the dominant manufacturer of graphing calculators. In fact, Vlorbik has liveblogged one such technology-in-the-classroom lovefest featuring Demana and Waits; his article, with updated links, is here).

Yesterday, Colin blogged for us about his semi-constructivist high school math class, in which the teacher announced on the first day that the kids would all "be teaching themselves this year." According to Colin, his math classes have disintegrated into desperate question-and-answer sessions in which the kids try to get a clue about the homework that they failed to do the night before.

Now, this afternoon I was looking over a worksheet Colin had done about the domain and range (i.e., the set of all valid 'inputs' and 'outputs') of some common functions. He had correctly written down that the domain of the function

f(x) = ln(x-3)+2

is the interval (3, infinity), but he had written that the range of the function was [4, infinity), which was distinctly odd, since there is nothing special about the number '4' in connection with f(x)=ln(x-3)+2 ("ln" stands for the natural logarithm function).

"How'd you get this 4 here?" I asked him. Then I noticed something even funnier on the line above that one; Colin had written that the range of the natural logarithm function was [2, infinity).

"Wait, the range of the natural logarithm function is all of the real numbers," I said. "How'd this 2 come into it?"

"Look at this graphing calculator," he said.

Sure enough, the d*mned graphing calculator was cutting the range of ln(x) off at 2. It's not the calculator's fault; it's unavoidable that it will cut the range off somewhere, because calculator displays are limited by the resolution of the screens. It's not a problem if you're an engineer with a good understanding of the basic properties of the natural log; but in this case, it's being used as a TEACHING tool, and it's teaching Colin something completely incorrect.

This is a kid who is having no trouble with the usual intellectual challenges that domain, range and natural logarithms offer. That's a big deal, because those are notions that ALL of the students who ended up in my college algebra classes struggled with.

He could easily learn this stuff right, but he is learning it wrong. Why? Because he's got the GRAPHING CALCULATOR in his face all the time -- the authors of this textbook push it hard (and yes, it's a Texas Instruments calculator). The problem is made worse by the fact that his teacher is not paying any attention at all to what the kids are actually learning.

OK, before this, I didn't have a set opinion of graphing calculators and other forms of technology-in-the-classroom. There's been argument for years that weak students are overly dependent on their calculators, but I wasn't sure that was a reason to ban them.

And then I get this wake-up call today; calculators are actively misleading even the strong students. Even if you have a good teacher working with the kids on these problems, catching their misconceptions early, the graphing calculator is going to be throwing a completely unnecessary extra layer of confusion into the mix.

I am now officially against Technology-in-the-Classroom, and by extension the curricula that push them. A graphing calculator might be a good tool once you know what you're doing, but it's a terrible tool for teaching.

comments...

Every math problem a kid does - any bit of work that anybody does -- takes something out of him. As a kid gets older, the problems get harder and involve more steps. It's automation of the earlier, simpler steps that keep the complexity of a task from getting too big to handle.

Think about it in the context of reading. When you, as an adult, with your long-term expertise in reading, contemplate the work involved in learning something new, such as Roman history, do you consider the cost of the mechanical act of reading the words on the page? Of course you don't. In fact, given a page of text to read, you almost certainly can't help but read it. The act of reading costs you nothing.

Any component of a problem that a kid has mastered to automaticity will be discounted; in other words, completely automatic (rote!) tasks are not a drain on a kid's intellectual energy budget. The kid can focus on what he's intended to learn in that lesson, and can go further before he has to quit.

an example

In order to get specific, let's analyze what needs to be done in order to do an algebra word problem of a type that most of us don't remember fondly; mixture problems.

You have two lemonade mixtures. Mixture A is 5 parts water to 1 part lemonade powder, and mixture B is 2 parts water to 1 part lemonade powder. How much of mixture A and mixture B should you mix tin order to get a quart of a mixture that is 3 parts water to 1 part lemonade powder?

Nightmares are made of this stuff, but let's look at the steps you must take to do this problem. There are different ways to do this problem, obviously, but I would guess that they boil down to the same set of steps, more or less.

Step 1. First, you must figure out what fraction of each mixture is powder vs. water. This involves converting ratios -- such as 1 part lemonade powder to 5 parts water -- into equivalent fractional parts: i.e., 1/6 of this mixture consists of lemonade powder. Mixture A is 1/6 lemonade powder, mixture B is 1/3 powder, and the mixture to be created is 1/4 powder.

Step 2. You have to identify what you want to find; in this case, the unknown is the number of quarts of mixture A (once you know how much mixture A you need, the remainder needed to make a full quart is mixture B). You have to give this quantity a symbol, say x, with an associated unit, say quarts. This step seems trivial, but it's far from it (see the endnote).

Step 3. You must derive an algebraic relationship between the amount of mixtures A and B that you can solve for the unknown. Most reasoning methods will lead you to a conclusion similar to this one: if you have x quarts of mixture A, then you have 1-x quarts of mixture B, and the resulting mixture will have a proportion of lemonade powder that is expressed as:

1/6 x + 1/3 (1-x).

The correct value of x will have to satisfy:

1/6 x + 1/3 (1-x) = 1/4.

Step 4. Manipulate the above equation until you have isolated the variable x and obtained x =1/2.

This involves first multiplying out the terms in the above equation, then isolating x on one side of the equation, then solving. Isolating x correctly will give you the equation

1/3 - 1/4 = x (1/3 - 1/6).

You must then perform the fraction computations, and finally solve for x.

Step 5. Interpret this solution correctly to yield: 1/2 quart of each type of mixture is needed.

So, solving such a problem involves at least 5 separately identifiable steps. The "deepest" one -- the one involving the most insight and the least plug-and-crunching -- is step 3, in which the student derives the relationships among the given elements of the problem, and figures out what must be done to finish out the problem. That's the part of the problem that one would hope would take most of a student's effort and energy.

However, I've taught a lot of kids (in 'college algebra' classes) how to do this sort of problem, and step 3 is not the step that really flattens the kids. It's mainly step 4 that does that; the manipulation of the symbols in the equation, and the addition and subtraction of the simple fractions involved. Not far behind step 4 in difficulty is step 1, conversion of the ratios of the mixture's components to the fractional part that's lemonade powder. They haven't learned this stuff to the point of automaticity.

What does a student see when he looks at the steps involved in doing this problem? If he knows he can perform step 1 -- the conversion from ratios to fractional parts -- then that task shrinks to a point. The student knows he can do it without any effort, and discounts it from his 'energy budget' -- the effort he knows he'll have to expend to solve the problem. If he knows he has mastery of the algebraic symbol manipulations and fraction calculations involved in step 4, then that step also becomes one that has no cost for him. He realizes that most of the cost of doing the problem will come in step 3.

Step 3 really cannot be completely automated, as every problem is unique, so that step will always impact a kid's cognitive energy budget. The other steps should have no cost for a student; they are completely automatable. They should be the easy stuff.

Now imagine that you are a kid facing ten such problems for homework. You know that the equivalent of 'step 3' is going to be a challenge for every single problem. For the kid who has practiced the other skills to automaticity, that's the only challenge he'll have to face; and that's as it ought to be, since it's presumably that higher-level 'step 3' functioning he's trying to learn in this lesson.

But if you are a kid for whom each of these steps demands high-level mental energy, you are going to run out of steam sooner, get much less out of the current lesson, and remember (in all probability) not what you learned on this occasion, but how hard and painful each and every 5-step problem was. With memories of mathematics like that, no wonder you run from math at the first opportunity.

This is why drilling these procedural skills to the point of automaticity is so critical.

endnote

Step 2, identifying the unknown, is also a real hurdle for many students. I discovered when I was teaching algebra that it takes intellectual gumption to select an unknown and give it a variable name, and that sheer timidity made this necessary step difficult for practically all of my algebra students. Practice in identifying and naming unknowns seemed to help here -- simply having "permission" from the teacher to take this step seemed to help as well. If I were teaching now, I'd probably spend some time just drilling this one step for a number of types of equations.

comments...

Now imagine complicating the problem by giving one powder in kgs, the other powder in ounces, one liquid in quarts and the other in ccs, while asking for the solution in completely different units like what you have to do in those messy "real world" problems our educators are so fond of. Lots more steps, calculations, and places for error, while filling up valuable slots of working memory. Less room for unclear conceptual knowledge, especially in the basic skills area.

This brings up an excellent point. The buzz in math education these days is full of discussions about real-world problems, and how important it is to have the kids doing real-world problems in order to give them the sense that what they are doing is meaningful (never mind the fact that we teach them all sorts of stuff, like reading, when they are far too young to understand why it will be meaningful down the line. Math education, by golly, has to be meaningful).

Here's my feeling about that: these educators don't know what they've bitten off. They aren't tough enough for real-world problems.

KDeRosa is right; in the real world, quantities aren't given in the same units and you have to change them all before you can do anything. In the real world, half the work goes into figuring out what the problem is up front, before you can even start trying to solve it. Problems are so messy in the real world that you'd be nuts to try bringing them into the classroom and inflicting them on a bunch of kids who were just trying to pick up a few skills and a little knowledge to carry through life with them.

Here's an example of a "real world problem" from Prentice-Hall Math Course 1:

Bill scored 100%, 100%, 90%, 70%, and 60% on five quizzes. Which makes his grade look the highest: the mean, the median, or the mode? Which measure should his teacher use to convince Bill to study harder for the exam?

This is real world? Sure. Teacher's gonna say, "Well, Billy, your mean score on these quizzes was only an 84", and Billy's gonna say, "Fear not, fair teacher, for the mode was 100% and the mode, in truth, is a more accurate representative statistic for the true center of the distribution of my grades."

Buncha wimps. Here's a real world problem, from my own world of work:

A maximum likelihood estimate of elevation is to be obtained for a point, using a combination of posts from five digital elevation models with root mean squared errors s1, s2, .. , s5 meters respectively. What root mean squared error should the software automatically assign to posts on the resulting digital elevation model?

And this, by golly, is a straightforward one that anyone in my company could reasonably be expected to answer with a bit of thought. My point is that even the most straightforward REAL real-world problems are not well-suited for the classroom.

OK, ranting over. Finally, a request: please share your real real-world problems with us. They don't have to be artificially hard; if they're real (with the messiness that reality entails), we'd like to see them.

comments...

The interesting thing, as KDeRosa has pointed out, is that it's mainly verbal scores that shot up after recentering. Not math scores.

That was a big disappointment for me back when I first tracked these down. I was psyched to have my 620 Math shoot up into the 700s.

No such luck. A 620 then is a 620 now.

Instead, my Verbal score went from 720 to 790.

I find this intriguing.

update

College Board on recentering

"In April 1995, the College Board recentered the score scales for all tests in the SAT Program to reflect the contemporary test-taking population. Recentering reestablished the average score for a study group of 1990 seniors at about 500 — the midpoint of the 200-to-800 scale — allowing students, schools, and colleges to more easily interpret their scores in relation to those of a similar group of college-bound seniors."

source:
College Board Equivalence Tables
College Board conversion table, SAT 1





"For 1972-1986 a formula was applied to the original mean and standard deviation to convert the mean to the recentered scale. For 1987-1995 individual student scores were converted to the recentered scale and then the mean was recomputed. From 1996-1999, nearly all students received scores on the recentered scale. Any score on the original scale was converted to the recentered scale prior to computing the mean. From 2000-2003, all scores are reported on the recentered scale."
source:

2003 College Bound Seniors: A Profile of SAT Program Test Takers, page 3 (pdf file)

what does an SAT math score mean?

This chart is interesting to me in light of my own history with math.

I've mentioned before that I had always assumed I was "reasonably good at math."

I didn't think I was a math whiz; I didn't think I had any special talent. I thought I was 'pretty good' at math.

That's why it came as a shock to find I couldn't begin to teach a 4th grade math curriculum to Christopher - and that, in fact, I understood practically nothing about the subject.

That may be overstating it, which I don't want to do. Still, when I started trying to teach math to Christopher, I was constantly confronted by the discovery that I didn't-understand-fractions or didn't-understand-long-division or I didn't-understand-this or I didn't-understand-that.

I'm still having these discoveries over a year later.

Of course one can say that elementary math is deep; a person could continue making such discoveries forever.

That may be so, but it's not what I'm talking about. I'm talking about having done well in math as a child, and having minimal conceptual understanding as an adult.

..........

This 3rd chart shows you where I stacked up, percentile-wise at the end of high school. Bear in mind that SAT tests are taken only by kids going to college, and bear in mind that back when I was taking the SATs no one had ever heard of an SAT prep course. I walked into the SATs cold, not having looked at a math book in a year, sat down, and took the test.

Where did I end up among the college-bound population?

On math: top 10% of all girls, the top 13% of all boys. 11th percentile overall

At the age of 17, I concluded that this meant I was, yes, reasonably good at math.

Now I find out I don't know what a fraction is.



source:
2003 College Bound Seniors: A Profile of SAT Program Test Takers, page 13 (pdf file)

so what does an SAT score mean? (part 2)

The short answer is, I don't know.

What I think, based in my experience, is that it's entirely possible that SAT scores do tell us something about 'how good' a person is at math.

I think it's probably true that I'm "reasonably good" at math, or at least reasonably good at learning math. I've been able to teach Christopher and me, and I've been able to figure out how to do this under intense time pressure.

So...I don't think SAT scores 'lie.'

The SAT test was always supposed to be a test of aptitude, not knowledge. People have challenged that claim forever, but in my own experience, as an 'n of 1,' that's pretty much what the SAT tested. It tested my ability to learn math, not my ability to do math.

I think.

more SAT trends to come

KDeRosa has sent me some charts on long-term trends in SAT scores that I'll get posted as soon as I find them on my desktop!

Number 2 Pencil thread on SAT recentering

Grudge Match: SAT vs ACT


SAT tests: recentered scores
SAT scores & calculator use





comments...

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SatScoresCalculatorUse 03 Oct 2005 - 21:23 CatherineJohnson



So now that Carolyn and I have gone on record as being Against Calculators, here's the scoop on calculator usage as it correlates with SAT scores (still working on this, bear with me):


                                      SAT I            Percent      Verbal            Math
                                               Number    Percent  Male  Female     Mean        SD         Mean        SD


source:
2003 College Bound Seniors: A Profile of SAT Program Test Takers, page 11 (pdf file)

update

I just re-read this chart and the Message popped: this is How do you get to Carnegie Hall.

Look at the wording:

• Use (calculator) Almost Every Day

• Use Once or Twice Weekly or Less

They're not measuring calculator use. They're measuring practice.

I now use my calculator each and every day, because I'm studying math each and every day.

Before I embarked on my all-math-all-the-time regimen, I used my calculator once or twice weekly or less.

and, from TIMSS:

"Calculators. TIMSS found an almost linear relationship between the amount of calculator usage in school and high achievement at grade 8 and 12. The higher the grade level, the stronger the link. The more the calculator was used (gleaned from both students and teacher questionnaires) the higher the scores. This may mean that the tasks were of such complexity and required such a level of mathematical thinking that computation was a minor/simpler part of the problems.

More research should be done on the various uses to which calculators were put. Were they scientific, graphing or other types? The correlation did not hold, however, at fourth grade. Only two countries reported calculator use at grade four.

Computers. Computer use was a different story. It appeared that too much use could lower achievement. The probably key is knowing how and when to use computers. The Japanese geometry teacher in the TIMSS public use videotape illustrates a powerful use of the computer to quickly and concurrently show the construction of many variations of triangles between parallel lines in one image."

also:

"The problems on TIMSS (and on NAEP) are virtually all multi-step. This has implications for what should be going on in classrooms."
source:
The American Federation of Teachers Looks At TIMMS (PowerPoint presentation)
to find: Google "AFT" and "TIMSS" and "ppt"



SAT tests: recentered scores
SAT scores & calculator use





comments...

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DvdMoviesFromCoolHunting 04 Oct 2005 - 00:47 CatherineJohnson



On the off chance you don't already have enough stuff -

Wallflowers in Quicktime
Wallflowers in Windows Media Movie

Modular Moves in Quicktime
Modular Moves in Windows Media Movie

this one's the best

Op Art in Quicktime
Op Art in Windows Media Movie

source:
Detour DVD

how to spend it, part 2 (further instructions)

• subscribe to the Financial Times ($99/yr)

• receive Glossy Weekend Supplement, How to Spend It
  

step 3:

• After perusing FT opinion page (Philip Stephens: The Tories have to love the voters; John Kay: European monopoly laws are already fair), leaf through Glossy Weekend Supplement.

• Ask yourself, 'What possible relevance does this have to my life?'

comments...

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RussianMathProblem961 04 Oct 2005 - 01:39 CatherineJohnson

961.

A point with a coordinate of -3 moves along the number line in the following manner: First, it goes 5 units in the positive direction; Then it goes 7 units in the negative direction followed by 10 units in the positive direction and 8 units in the negative direction; Then it goes 3 more units in the negative direction and, lastly, 13 units in the positive direction.

Question: What is the final location of the point on the number line?

source:
Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa, page 255

comments...

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SATScoresDecline 04 Oct 2005 - 01:43 CatherineJohnson






Ticket to Nowhere
by Paul E. Peterson





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NAEPDataFromKenDeRosa 04 Oct 2005 - 01:53 CatherineJohnson

KDeRosa sends 2 charts, & observations:

1. The NAEP data only goes back to the early 70s and misses most of the substantial decline in scores that took place in the 60s and early 70s. Of course, the SAT data does not include the entire student population, only the college bound one and it is normalized. Nonetheless, it shows a very serious degradation of skills.

2. The high-water mark in both verbal and math scores was in 1963. 42 years later and we still haven't recovered.

3. There was some debate on why the decline in the 60s took place. One side says changing demographics are to blame. The other side says most of the change in demographics took place by 1963 with almost no decline in scores. Moreover, the decline in scores after 1963 included substantial declines among the smartest students at the top of the curve which is not explainable by changing demographics. I haven't been able to locate a good analysis of this period online. Maybe a more knowledgeable reader can help.

4. The SAT people have monkeyed around with the test in 1974, 1995, 2005 and sometime in the 80s I believe. Not sure what effect all those changes had, but I do know that SAT scores post 1995 no longer correlate reliably with IQ scores.

5. Did I mention that the teachers' unions came to prominence in the 60s. Coincidence?

6. Math scores for the NAEP and SAT exams have been rising in the past few years. Never mind that both tests have been dumbed down (no more quantitative section in the SAT). The SAT people say it's a bona fide gain. The ACT people say it's a phony gain. I guess the college math professors would know best.

It would be nice if a knowledgeable psychometrician could provide some insight.









"When the SAT was renormed in April 1995, mean scores were set at or near the midpoint of 500 of the 200-800 score scale, a process called recentering. All scores in this table reflect that process. Means after 1996 are recentered, and those for 1996 are based on recentered scores plus scores converted from the original to the new scale. Means for 1987-1995 were recomputed after individual scores were converted from the original to the new scale; means for 1972-1986 were converted to the new scale after a formula was applied to the original mean and standard deviation; and means before 1972 are based on estimates."


Laurence Steinberg's 1996 Beyond the Classroom: Why School Reform Has Failed, which chronicles the results of a massive study he headed, has one of the best discussions I've seen of this data. (He sees slacker parents as a major if not the major source of the problem.)

the 60s

This reminds me that I have to ask my mother about my grandmother's experience teaching in Springfield, IL. There was a point at which, quite suddenly, my grandmother perceived teacher quality to have plunged. I think this happened in the 60s, but I'll have to make sure.




comments...

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IntroducingTheFAQ 04 Oct 2005 - 03:59 CarolynJohnston

Please check out our new FAQ and leave your comments. Are there questions we forgot? Answers we forgot? Stupid answers? Let us know.

This is FAQ Beta -- in a couple of days I'll put it up on the sidebar (and probably remove some of the sidebar links so it'll be less busy).

Update

Speaking of stupid: here's the link: FrequentlyAskedQuestions.

comments...

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TeacherProtestsEverydayMath 04 Oct 2005 - 13:05 CatherineJohnson

comment left at SOCMM:

I am a third grade teacher and have been trying to tell my administrators that Everyday Math is not an effective math curriculum. I have taught it for three years and the students coming to me have no mastery of basic concepts. It also does not meet our district or state standards, but the administrators will not abandon the curriculum. I struggle with teaching it and then when I started researching the program, I now feel it is my duty to speak out to parents and be an advocate for my students.

Is there a reason why so many states are adopting this mediocre curriculm? Please respond!

comments...

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SampleNaepQuestions 05 Oct 2005 - 00:22 CatherineJohnson



from Our Nation's Report Card, the 8th grade test:

sample NAEP questions search tool

...and see Tom Loveless on the new, improved fraction-free NAEP here




comments...

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TeacherFamiliarityWithStandards 05 Oct 2005 - 00:45 CatherineJohnson





'Teachers who reported “no such document” are not included.'

source:
Mathematics Teachers’ Familiarity With Standards and Their Instructional Practices: 1995 and 1999, EDUCATION STATISTICS QUARTERLY, Vol. 5, Issue 1, Topic: Elementary and Secondary Education

on second thought

Judging by some of the half-baked stuff I've come across today.....a person who's fairly familiar with NCTM standards could be worse than a person who's very familiar.

I still like those 'no such document' folks.




comments...

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MalcolmGladwellOnHarvardAdmissions 05 Oct 2005 - 12:57 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?





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.




comments...

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RussianMathProblem976 05 Oct 2005 - 22:01 CatherineJohnson


976.
A rectangular park is 400 m longer than it is wide. The ratio of the length of the park to its width is 5:3. How long will it take someone walking at a rate of 2 km/h to go around the park?

question

How would a child work this problem without using algebra?

Here's the way it's done using a bar model, but I'm not seeing how you would do this without simple arithmetic.

(I hate the way this looks. I need Quark.)

Anyway, using the bar model you see that you have a bunch of equal units, and that two of these units equal 400.

Therefore, each individual unit equals 200.

So the width has to equal 3 x 200, or 600 and the length has to equal 5 x 200 or 10,000.

ratio problem

Actually, now that I think about it, at this point in RUSSIAN MATH, kids have learned ratios & proportions, so they could just solve it that way.

width = w
length = l = w + 400

5/3 = l/w

5/3 = w + 400/w



What I can't remember is whether the book has taught kids to use two variables...

comments...

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ExtendedProblem1 05 Oct 2005 - 22:43 CatherineJohnson

Find all the numbers that satisfy all of the following conditions:

1. Positive whole numbers less than 100,
2. Four more than each number is a multiple of 6
3. The sum of the digits of each number is a multiple of 4.

and what is the best way to do this problem?

We used Doug's number lines (WHICH ARE GOING TO BE GETTING A WORKOUT THIS YEAR, IT'S OBVIOUS). We labeled one number line with multiples of 4, and the other with multiples of 6. We didn't need the multiples of 6, but it made things easier to have all the multiples of 6 sitting there, where we could see them.

How does a person who knows what she's doing do this problem?




extended response problem from IL state test
extended response problem 1
extended response problem 2
extended response problem 6
extended response problems 7, 8, 9
direct instruction & the rigor conundrum
Dan's daughter reacts to extended response problem
defensive teaching of Singapore bar models
open-ended problems in math ed
problems that teach - "Action Math"
email to the principal

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ConceptualSaxon 06 Oct 2005 - 02:37 CarolynJohnston

Anyone out there still think Saxon Math isn't challenging or deep?

Here's a question on from Saxon 8/7 that Ben and I discussed for twenty minutes tonight.

Regurgitate the answer to this one, if you will. This problem addresses quantization error, which is at the heart of a lot of engineering problems.

Jeffrey's ruler is marked in eighths of an inch. Assuming a measurement is done correctly, what is the maximum measurement error possible using Jeffrey's ruler?

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LearnersAreFragile 07 Oct 2005 - 02:56 CarolynJohnston

Our discussion of this math problem, and its value as a problem weighed against the effort of doing it, got me wondering whether we shouldn't be stepping more carefully than we do when we give a learner a math problem. People are tender, and discourage easily. Teachers have to fight to keep learners absorbed and engaged and moving forward; we need to anticipate demons, and head them off.

misconceptions

Learners pick up misconceptions so easily. There have always been many ways to acquire misconceptions, but technology offers some new ones.

Bernie was telling me last night that when he was a student in trigonometry in high school, he had a good teacher. When it came time to learn about sine and cosine, she drew a unit circle on a coordinate plane, put a point on it, and drew the radius from the origin out to the point. She then labeled the angle.

x is sine of this angle, and y is cosine of the angle, she said, and labeled them that way. No! Wait!, she said; it's the other way around! She fixed the labels so that they were correct.

"She only had the wrong thing up for a minute," he said, "but ever since then I've had it stuck in my head that way and I have to think about it to be sure I've got it right."

too much work, too little payoff

He told me another story, of a more recent vintage, about a brilliant coworker of ours. This guy is a brilliant hardware engineer from Caltech, who worked at the Jet Propulsion Lab and was involved in the development of the capture system for the Magellan mission's synthetic aperture radar. This guy is always learning things -- he is one of the most intellectually lively people you could ever hope to meet.

This coworker had a copy of Bernie's book, *Numbers and Symmetry", and was working through some of the problems. One day he came up to Bernie and said, "Is this problem hard?"

As it happened, it was a problem that Bernie and Fred threw into the mix of problems, as they were writing the book, without really test-driving it first (I was there, and I can attest that while they took huge care with the text, the problems were something the editors encouraged them to add at the end of the process). "This problem was harder than I expected it to be," Bernie told me last night. "It was a challenge problem, but I just tossed it in with the others. It should have had some kind of warning on it, two stars or something."

"I think it is hard," Bernie told our friend. "It's been a while since I thought about it."

"I've been working on it for two days," he said, "and I haven't gotten it yet."

After that, according to Bernie, our friend put the book down and has not picked it up again. Bernie feels badly about it. "If I've driven him off," he said, "then there's no question about it: I've messed up. I've failed in my mission with that book. But I was younger when I wrote it, and much more insouciant."

"Yeah," I said, "I was a more insouciant teacher then too."

Insouciant says it, and I think a lot of teachers -- and textbook writers -- are insouciant. Not that teachers can't ever make mistakes; we don't have to be perfect, but we have to understand how delicate even the brightest learners are, and we have to step very carefully. Liping Ma says that teachers in China spend a lot of time discussing the most common student misconceptions with each other, and building up a large knowledge base of multiple approaches to teaching misconception-prone ideas. I really think that's where teacher's efforts ought to be.

I think one intention of constructivists is to try to take that element of fragility out of the learning process, but I don't think it can be done. Inflexible knowledge is a natural stage in the process of learning, and inflexible means brittle. We need to be paying enough attention to nip misconceptions before they flower, and we need to be watching for signs of discouragement.

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JayMathewsMiddleSchoolsMoreRigorous 07 Oct 2005 - 15:28 CatherineJohnson

I hope he's right about this: Traditional Social Focus Yielding to Academics: Instead of a Year to Adjust to Puberty, 13-Year-Olds Now Given Algebra and Other Demanding Coursework

Much of the seventh-grade achievement pressure is focused on mathematics, and Kenmore math teacher Emily Henry is preparing many students for what used to be a high school course: Algebra I.

Word said he expects more than 55 percent of this year's seventh-graders to have completed first-year algebra when they finish eighth grade, compared with 25 percent nationally. At Kenmore, 16 seventh-graders are taking algebra. The push to accelerate math instruction seems to have had a national effect. The National Assessment of Educational Progress test, a common measure of academic performance, shows that 13-year-olds had an average math score of 281 in 2004, up from 270 in 1990. English scores, on the other hand, are almost unchanged, from 257 in 1990 to 259 in 2004.

I'll remain skeptical about the increase in math scores until such time as Tom Loveless tells me the NAEP tests are assessing math skills above the 1st & second grades (pdf file).

via: joannejacobs

Irvington Middle School

I've mentioned before that, last year, our middle school's stated goal was to cut the number of students placed in Phase 4 math, the only course in which students take and master algebra in the 8th grade.

They didn't say how many students they planned to cut, and soon rumors were flying that 25% of the kids would be 'demoted' to Phase 3. Ed sent an email to the middle school math chair asking her about the figure; her reply was noncommittal, as I recall.

Subsequently I was present at a meeting in which parents directly asked the principal about his plans to cut students from Phase 4. His response—almost verbatim—was, 'I don't know where these rumors come from.'

So how many kids did they cut?

35% ^*

(It's always worse than you think.)

Here are my figures on the cuts to Phase 4, based on conversations with school personnel:

school year: 2004-2005
grade 5 class size: 155 students
phase 4 placement: 60 students
number of students moved from phase 4 to phase 3 at end of school year: 21 ^*
percent of children cut: 35% ^*

So here you have a highly affluent suburban school district, a district that spends roughly $18,000 a year per pupil, devoting time, energy, and a portion of that $18,000 to decreasing the number of students who master algebra in 8th grade.

what happened?

But here's the interesting development, and this is something parents have no idea also took place.

It's not just that 21 kids moved down.^*

Another seven kids moved up.

That's 7 kids not including Christopher, who moved to phase 4 in February. Add him to the total, and you've got 8 phase 3 kids swapping places with 21 phase 4 kids. If you had to choose just one factoid to illustrate the folly of assessing math talent in the third grade, that would be it.

To my knowledge, Irvington has never had 8 kids move from phase 3 to phase 4 in one school year. Never.

I happen to know this because, when I first raised the subject of Christopher changing tracks, I had teachers & guidance counselors saying things like, 'I can only think of one student who's moved up this year.'

Or: 'A student can always move up! It's never too late. We had one phase 3 student who just blossomed this year, all of a sudden.'

Two different people made these statements. One thought he was telling me 'No chance'; the other thought she was telling me, 'There's always a chance!'

But they were saying the same thing.

Question: How many phase 3 math students move to phase 4 in a year?

Answer: One.

down to 30%

So here's how things shape up this year, roughly speaking (there are some new kids in the district; I don't know their placements):

155 6th graders, approximately
est. 47 students in Phase 4
apprx. 30% of '05-06 IMS 6th graders on track to master algebra in 8th grade

UPDATE 9-18-2006: in school year 05/06 there were
3 sections of Phase 4 math, grade 6,
apprx 17 - 18 students per class

Meanwhile the KIPP Academy in the Bronx is reporting as many as 80% of its student body mastering algebra in the 8th grade, and passing the Regents A exam. Per pupil spending: $9,900.

I assume our new Superintendent in charge of curriculum will be taking a look at this.

salt in the wound

Last year, 80 percent of our eighth graders passed the high school level exit exam in math here in New York, the Regents, the math A (ph). Eighty percent of our eighth graders passed the high school level exam, exit exam and less than 40 percent of our kids who are coming in in fifth grade on level.

-- David Levin, Knowledge is Power Program (KIPP), Co-Founder; interview with Brian Lamb, C-span

back to NAEP

Here's Loveless:

The failures are even more alarming at the eighth grade. Almost four out of ten items (39.6%) address arithmetic skills taught at the first and second grade – six years below the grade level of eighth graders taking the test. More than three-fourths of the items are at least four years below grade level – taught in the fourth grade or lower. Yet, the percentage of eighth graders answering items correctly is an unimpressive 41.4%.

[snip]

Algebra items lack rigor at both the fourth and eighth grades. On the eighth grade assessment, the arithmetic demands of algebra items are pitched at only the mid-second grade level.

[snip]

“Really knowing algebra means being able to solve equations that contain more sophisticated forms of numbers than whole numbers. Calling these items algebra is conveying a false sense of rigor, making very simple math seem more sophisticated than it actually is,” noted Loveless.

“If students do not possess the tools to solve problems involving fractions, decimals, and percents – if students do not grasp forms of numbers other than whole numbers – then the only problems they will ever be able to solve will be mathematically trivial,” the report warns.

source:
New Study Finds That Math Items on the Nation’s Benchmark Exam Are Too Easy, Don’t Adequately Assess Skills-Eighth Graders Asked to Solve Problems Using First Grade Arithmetic

keywords: Irvington math


^* My figures are a headcount of how many students did not pass the placement test. To my knowledge, the administration approached all of these parents and expressed an intention to move the child to Phase 3. Some parents refused the move, and those parents, again to my knowledge, were accommodated; their children remained in Phase 4. I know of two such cases; there may be more.




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StevensonAndStiglerOnGroupLearning 07 Oct 2005 - 20:03 CatherineJohnson


Carolyn and I have been reading emails from a friend who's just discovered that the math teacher at her son's private school is promising to make 'prolific' use of collaborative learning in his classroom this year.

Like Carolyn, I view the word prolific, used in this context, with suspicion. (Hmm. I wonder what Google will give me for 'red flag'?)

Just in case you were wondering.

You can also access lyrics & music to The Red Flag by James O'Connell.

And that's about it.

back on topic

I'm inclined towards the position that group learning, when not used prolifically, is fine and dandy.

I think so for a couple of reasons:

• first of all, a California Department of Education study found that group learning was supported by quality research

• second: group learning, I think, can set up healthy competition, both within a group and among groups

• and third: people are natural born observational learners, and a group is, or could be, the place where a child can observe other children doing math. (No time to expound on the history of observational learning in the field of behavioral research on animals. But, if you're interested, see Alex the parrot. Alex didn't learn a thing until he had a rival learner getting the right answer & snapping up all the goodies.)

Oh, OK, I will say something about the history of observational learning in the field of animal behavior studies.

Even better, I will make this a Discovery Task!

Think and Discuss

For many years, behaviorists believed animals learned through classical conditioning, which many people call 'trial and error' learning.

What problems might an animal experience if he has to learn everything he needs to know through trial and error?

Do you think errorless learning might have some advantages for an animal living in the wild?

Bonus question

If you were a baby antelope, which way would you prefer to learn about lions?

Through trial and error, or through watching other baby antelopes turn into lunch?

Explain.

group learning in Japan

Here are Stigler & Stevenson on group learning in Asia versus America:

Perhaps the most profound difference in the way Asian and American children spend their time at school is in the degree to which they are alone versus being part of a group. American children have far fewer opportunities for group participation than do Chinese and Japanese children.

[snip]

Chicago children spend a great deal of time working on their own. The time spent at their desks filling in workbooks or handout sheets, reading, and doing other solitary activities occupied nearly 50 percent of their class time, but never more than 31 percent of the class time int he Asian cities. Conversely, Sendai, Beijing, and Taipei children spend most of their time in classrooms that were organized so that all of the children were working as a unit with the teacher as leader. Participation in lessons that involve the whole class, even in classes with many students, enhances students' feelings of group membership and reduces their sense of isolation. [ed.: notice that, in Asian countries, the entire class can be experienced as a group]

Several of the American mothers we interviewed expressed approval of the fact that their child's teacher allowed the children to work at their own pace. This practice may have benefits, but working at one's own pace means working alone, and the slower one's pace, the more time spent alone. Many times we observed a class where all but a few of the children had finished their assignment. The remaining few children struggled alone.

[snip]

When small groups are formed in Chinese and Japanesse classrooms, children are selected so that all levels of achievement as well as other characteristics will be represented in each group. In Japan, these groups are known as han.

[snip]

One Japanese teacher explained to us her method of grouping children: "I mix the groups so that each child has something to contribute. Each group should have a top student, but it would not be good to put all of the top students together. A group needs other talents as well: someone with artistic talents, someone who is good at sports, and so on."

[snip]

Witin each han, different children exercised leadership based on their particular skills.

[snip]

We understood why American children are more likely to seek other children for after-school play, why they spend so much time in their classrooms talking inappropriately to other children, and why they might not find school an especially pleasant place to be. Indeed, American children are less likely than Chinese or Japanese children to say they like school. For example, between 75 percent and 86 percent of the children in Taipei, compared to between 52 percent and 65 percent of the American children...indicated that they liked school.

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AlexTheParrot 07 Oct 2005 - 21:21 CatherineJohnson

We were training Alex to sound out phonemes, not because we want him to read as humans do, but we want to see if he understands that his labels are made up of sounds that can be combined in different ways to make up new words; that is, to demonstrate evidence for segmentation. He babbles at dusk, producing strings like "green, cheen, bean, keen", so we have some evidence for this behavior, but we need more solid data.

Thus we are trying to get him to sound out refrigerator letters, the same way one would train children on phonics. We were doing demos at the Media Lab for our corporate sponsors; we had a very small amount of time scheduled and the visitors wanted to see Alex work. So we put a number of differently colored letters on the tray that we use, put the tray in front of Alex, and asked, "Alex, what sound is blue?" He answers, "Ssss." It was an "s", so we say "Good birdie" and he replies, "Want a nut."

Well, I don't want him sitting there using our limited amount of time to eat a nut, so I tell him to wait, and I ask, "What sound is green?" Alex answers, "Ssshh." He's right, it's "sh," and we go through the routine again: "Good parrot." "Want a nut." "Alex, wait. What sound is orange?" "ch." "Good bird!" "Want a nut." We're going on and on and Alex is clearly getting more and more frustrated. He finally gets very slitty-eyed and he looks at me and states, "Want a nut. Nnn, uh, tuh."

source:
THAT DAMN BIRD






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HomerOnTheBus 07 Oct 2005 - 21:54 CatherineJohnson






I've made a pact with myself to start taking pictures of Andrew's still lifes & letter arrays.

This tableau, by the way, is no accident. Andrew carefully arranges his figures exactly as he wants them. He gets mad if the dogs bump into them, and he puts everything back exactly as it was.



keywords: Homer Simpson on the bus



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TrespassersInWonderland 07 Oct 2005 - 22:53 CatherineJohnson

from Barry Garelick:

In the 2004 National Assessment of Education Progress (NAEP) exam only 20 percent of fourth graders correctly calculated the answer to 314 x 12. Eighth graders’ performance was also disturbing: a question asked for the length of a line segment above a ruler, with one end at the 2 cm mark and the other at the 7 cm mark. Only 58 percent of eight graders got it right; and it was multiple choice. On the international front, anyone following how U.S. fourth and eighth graders fare in international tests in math (called TIMMS) have by now noticed that U.S. has come in about 14th or 15th, and that Asian countries top the list (Singapore is number one).

To put the issue of math education in context, one has to understand the prevailing attitude toward math education in this country. Two years ago, at a packed conference on math education. Jim Milgram, a math professor from Stanford, gave a talk in which he presented the following story problem which, he noted, seventh grade students in Russia are expected to solve: “Two people left their villages at sunrise and walked, each to the other’s village at constant speed. They met at noon and the first arrived in the others’ village at 4:00 PM while the second arrive at 9:00 PM. What time was sunrise?”

At this, a man sitting behind me articulated the following reaction typical of those who believe that U.S. students do not perform well in math because they are not taught how to apply it to the problems that actually occur in real life: “Who cares?”

Sentence first—verdict afterward

In May, 2005, the National Council of Mathematics Teachers (NCTM) in a statement which appeared in the Washington Post echoed the exact same “Who cares?” sentiments as the disgruntled man at the conference:

“For generations, mathematics was taught as an isolated topic with its own categories of word problems. It didn’t work. Adults groan when they hear ‘If a train leaves Boston at 2 o’clock traveling at 80 mph, and at the same time a train leaves New York...’ Whatever problems and contexts are used, they need to engage students and be relevant to today’s demanding and rapidly changing world.”

NCTM is a large organization based in Reston, Virginia which exerts considerable influence over how math is taught in this country. In 1989, NCTM published a set of curriculum and evaluation standards for math, and revised them in 2000. Some states have relied on these standards in framing their own. Such standards de-emphasize learning basic skills, are light on content and heavy on context-based learning otherwise known as “real life math”. Cathy Seeley, current president of NCTM is critical of math texts and programs that tell students "here's the rule, now do the problem" and says there is too much “teacher instruction” in the U.S. NCTM’s topsy-turvy approach to teaching math is more like “Here’s the problem, you figure out the rules needed to solve it”—an approach alarmingly similar to the Queen’s declaration at Alice’s trial in Alice in Wonderland: “Sentence first–verdict afterward.”

Some real life problems

Here’s an example of a real life problem which can be found on NCTM’s very own web site in the section called “Illuminations”:

"Suppose you have saved $63. You find a used video game system that you would like to buy. The seller is asking $180. You earn $10 a week doing odd jobs. How long will it take you to earn enough money to buy the game?”

While this type of problem has been around for years, NCTM’s suggestions for how to “explore” the problem in class is what’s different. They explain that adults typically subtract 63 from 180 and divide by 10. While this would be a preferred approach for students to have mastered by the 5th or 6th grade–the grade level for this activity–NCTM describes with particular pride a student entering 63 into the calculator (no apology offered for calculators being used here), then adding his first week’s allowance, then the second, third, and so forth until the display showed that he had at least $180 (12 weeks).

NCTM explains: "Allowing students the freedom to use strategies that are intuitively obvious to them helps them to feel more comfortable in the problem-solving process. At some stage it also helps them appreciate the efficiency of standard algorithms." NCTM does not discuss when this stage will occur. One would hope that it occurs quickly so that the calculator-aided counting-on-fingers method can be supplanted with the more efficient method that students in Japan and Singapore have mastered by the third grade.

A Word from NCTM

Recently, when asked why U.S. students suffer from an inability to perform complex reasoning and mathematical assignments compared to students overseas, Cathy Seeley responded: “We’re not doing as much problem-solving of that type as we need to be.” In another instance, she said “We can definitely learn lessons from Singapore, Japan and China. But we have to look beyond their textbooks to determine what these lessons are.”

Even a faithful NCTM adherent would not fail to notice that in Singapore’s textbooks, problems require multi-step solutions that are considerably more complex than what we expect US students to solve at that grade level. From a sixth grade Singapore textbook: 3/5 of Mary's flowers were roses and the rest were orchids. After giving away ½ of the roses and 1/4 of the orchids, she had 54 flowers left. How many flowers did she have at first?

Looking beyond the textbook as Cathy suggests allows NCTM to throw the baby out with the bath water, and to reject problems that are good by saying “It’s not the text, it’s the teaching.” In fact, in Japan, Singapore and Russia, they do teach math differently. They teach it correctly. They teach content. They teach skills and facts as a foundation upon which understanding will be built. They teach like they used to in the U.S.

Alice’s retort

In Alice in Wonderland, Alice tells the royal family “Who cares for you? You’re nothing but a pack of cards!” In real life, however, boards of education, school districts and state departments of education are bowing to a pack of cards that has made math education almost content free. Over and over, as parents, teachers, and world-class mathematicians protest how math is being taught, and tell school boards and administrators the type of content students should be mastering, they are viewed as trespassers in Wonderland. Story problems are met with groans, proclaimed not to be real life, and dismissed with a mighty “Who cares?”

“Who cares is not the point,” Jim Milgram says. “Let me give you an example of a problem that people had better care about since it will affect their very lives. Design a robot arm to select and lift items off an assembly line and place them on a second line correctly positioned for a second robot to work on them. There is no chance in hell that someone can do this if they can't do the Russian problem about the two villagers.”

Until “real life math” is recognized for the pack of cards it is, the influence of NCTM and their followers will continue, as will the unmistakable and irreversible harm to our children, many of whom do not know how to multiply two-digit numbers without a calculator, nor how to use a ruler.

-- BarryGarelick - 06 Oct 2005


Trespassers in Wonderland can be found here, and is indexed here and here.

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NAEPPanBalance 07 Oct 2005 - 23:46 CatherineJohnson







Does anyone know how to find out the percentage pass rate for particular NAEP items?

never mind

This is exactly the kind of real-world problem that Turns Kids On, and helps them Make Connections To Their World.




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NaepNumberLine 07 Oct 2005 - 23:53 CatherineJohnson






Yes, it is a NAEP number line!


sigh

Christopher just got this problem severely wrong.

I'm gonna be using a LOT of number lines this year.


(From the 8th grade test)







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NaepFindAPattern 08 Oct 2005 - 00:09 CatherineJohnson










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NaepFractionProblem 08 Oct 2005 - 00:22 CatherineJohnson










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NAEPScaleProblem 08 Oct 2005 - 00:30 CatherineJohnson












I just want to know how many kids used their calculators.

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NaepWordProblemMultiplyAndDivide 08 Oct 2005 - 00:39 CatherineJohnson












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NaepDrawSquare 08 Oct 2005 - 00:51 CatherineJohnson

how old is an 8th grader?

13, right?

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AharoniArticlePart1 08 Oct 2005 - 04:06 CarolynJohnston


This article by Ron Aharoni, which appeared in the Fall issue of American Educator, is brilliant. Catherine and I have both read it, and agree that there is enough in this article to chew over in multiple postings. So this one, I guess, will be the post that launches our discussion of it, and we'll tease out its excellent content gradually in more posts.

Aharoni, who is a math professor in Israel, got involved with a friend in a project to promote math education in elementary schools. His friends warned him off it, saying that elementary education is a whole different ballgame from professional mathematics. He went anyway.

He came in with some preconceived, rather idealistic notions of how elementary school math lessons should work, but wised up fast.

The banner I was carrying at that time was that of "experience". The children should experience abstract concepts concretely, I thought, after which the abstractions should occur by themselves. I took the kids out to the playground. We measured lengths of shadows and compared them to the lengths of the objects themselves, then used this information to calculate the height of trees according to their shadows. (This idea is borrowed from Thales, who was born in the 7th century B.C.) Then we measured the length and width of the classroom in various ways to find how many floor tiles fit into one square meter, and what the ratio was between the length of the classroom in meters and its length in tiles.

I learned the price of conceit the hard way: most of my lessons were a mess.

Aharoni discovered exactly what I did, when I started getting involved in my son's education; having a Ph.D. in math didn't mean I knew a darn thing about how to teach it to elementary school kids: I didn't. Not elementary school mathematics. It was different from any sort of teaching I'd done before. I was totally nonplussed, and Aharoni expresses it for both of us very well:

But what surprised me most was that I learned mathematics. Actually, a lot of it. This would not be the case had I gone to teach in a high school. The mathematical concepts there are known to a professional mathematician. In elementary school, it's the teaching of the most basic principles that counts; the nature of numbers, the meaning of the arithmetical operations, the principles of the decimal system. About these, it is rare for a mathematician to stop and think.

He addresses, in his article, the fact that the most common operations have multiple meanings. We adults get so used to moving between these meanings that we conflate them all in our minds, and when asked about the difference, can't even recall that there is a difference. Here's an example from Aharoni's article:

I was experienced enough to know that such confusion almost always originates from having skipped a stage. In this case the missing stage was the understanding that subtraction has more than one meaning. There is the meaning of diminution, where objects are removed: I had 5 balloons, 2 of them burst, how many do I have left? But there is also the meaning of comparison of quantities, where nothing disappears: There are 5 children in a group, 2 of them are boys. How many are girls? Or perhaps: How many more green apples than red apples are there? In these cases, too, the exercise is one of subtraction, but the meaning is different.

The various meanings of subtraction are an example of a fine point that has to be taught explicitly. Skipping this stage will result in later difficulties with word problems.

Another example of this conflation happened when Catherine asked me, and a couple of other math types, what the meaning of 'partitive' vs. 'quotitive' division was. She'd come across the concept in Liping Ma's book, which claimed that Chinese teachers could generate word problems involving fractions that were of either type, whereas most American teachers couldn't.

Well, mathematically there's no difference; division is division. Partitive vs. quotitive is just a pedagogical distinction that a teacher needs to know, in order to be sure that she can generate word problems that cover the whole set of possible problems, so that the kids will understand that the division operation is the same for all.

I didn't see the distinction at first; neither did any of the other mathematicians Catherine asked. We didn't need to in order to do our own work; but teaching elementary math is a whole different ballgame.

And here's an insight that I just love:

When I started teaching in elementary school, I was convinced that precise formulations and the explicit naming of principles was a matter for grownups. Children should learn things on an intuitive level, I thought. One of the greatest surprises that awaited me was to realize how wrong I was about that. Children need precise formulations. Such formulations consolidate their knowledge of the present layer and make it a safer basis on which higher layers can be built. Moreover, children love "adult" formulations and notations, and are proud of being able to use them. First-grade children who learn the notation "1/2" are happy to discover the notation for "1/3" by themselves.

It makes complete sense to me: grade schoolers realize that knowledge and skills are power, and grownups have the knowledge and skills. They don't want to learn dumbed-down, intuitive formulations of problems; they want to do what YOU do. It reminds me of how my Dad used to leave papers around with derivatives and integrals on them; I thought they looked like the coolest thing, and I wanted to grow up and do what he did.

More to come from both Catherine and me.


Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)

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RoshHashanah 08 Oct 2005 - 11:29 CatherineJohnson





After all these years, I'm still vague on the exact dates of Jewish holidays. (I'm still vague