Devlin's Lament: the symbol barrier

(Cross-posted at Out In Left Field)

In an article in the most recent issue of American Scientist entitled "The Music of Math Games," Keith Devlin (head of the Human-Sciences and Technologies Advanced Research Institute at Stanford University and NPR's "math guy") says that learning math should be like learning to play the piano. In doing so, he recalls (but does not credit) Paul Lockhart's Lament ("A piano student's lament: how music lessons cheat us out of our second most fascinating and imaginative art form"), which I blogged about here.

Though Devlin is no literary virtuoso, not all of what he writes here is mushy metaphor. He begins with a discussion of educational software, and here his points are clear and consistent with my own experience. Most "math games" and "math education" software programs I've seen don't make mathematics an organic part of the games or activities. Instead, math problems--mostly arithmetic problems of the "mere calculation" variety--are shoe-horned into non-mathematical situations. Here they serve simply as tasks you must complete before moving through the current non-mathematical activity or on to the next non-mathematical activity.

As Devlin writes:

To build an engaging game that also supports good mathematics learning requires... understanding, at a deep level, what mathematics is, how and why people learn and do mathematics, how to get and keep them engaged in their learning, and how to represent the mathematics on the platform on which the game will be played.

The same is true of language learning. Most linguistic software taps only superficial aspects of language, and, as I know from personal experience, it takes great effort to build a program that does more than that.

Where I begin to part ways with Mr. Devlin is in his discussion of traditional math and what he thinks is an excessive emphasis on symbols:

Many people have come to believe mathematics is the memorization of, and mastery at using, various formulas and symbolic procedures to solve encapsulated and essentially artificial problems. Such people typically have that impression of math because they have never been shown anything else...

...

By and large, the public identifies doing math with writing symbols, often obscure symbols. Why do they make that automatic identification? A large part of the explanation is that much of the time they spent in the school mathematics classroom was devoted to the development of correct symbolic manipulation skills, and symbol-filled books are the standard way to store and distribute mathematical knowledge. So we have gotten used to the fact that mathematics is presented to us by way of symbolic expressions.

This approach to math, Devlin suggests, is at odds with the resolutions of a "blue-ribbon panel of experts" serving on the National Research Council’s Mathematics Learning Study Committee ("Adding it Up: Helping Children Learn Mathematics," National Academies Press, 2001). In Devlin's words: these resolutions hold that math proficiency consists of:

the aggregate of mathematical knowledge, skills, developed abilities, habits of mind and attitudes that are essential ingredients for life in the 21st century. They break this aggregate down to what they describe as “five tightly interwoven” threads. The first is conceptual understanding, the comprehension of mathematical concepts, operations and relations. The second is procedural fluency, defined as skill in carrying out arithmetical procedures accurately, efficiently, flexibly and appropriately. Third is strategic competence, or the ability to formulate, represent and solve mathematical problems arising in real-world situations. Fourth is adaptive reasoning—the capacity for logical thought, reflection, explanation and justification. Finally there’s productive disposition, a habitual inclination to see mathematics as sensible, useful and worthwhile, combined with a confidence in one’s own ability to master the material.

Ah, "21st century skills," "habits of mind," "conceptual understanding," "real-world situations," "explanation," "disposition"...--all this makes me wonder about the ratio of mathematicians to math eduation "experts" on this blue-ribbon panel. (It should be noted that Devlin himself is not, strictly speaking, a mathematician; he holds a Ph.D. in logic from the University of Bristol, and, while affiliated with Stanford, is not a member of the Stanford math department.)

Standing in the way of these lofty goals is what Devlin calls the "symbol barrier":

For the entire history of organized mathematics instruction, where we had no alternative to using static, symbolic expressions on flat surfaces to store and distribute mathematical knowledge, that barrier has prevented millions of people from becoming proficient in a cognitive skill set of evident major importance in today’s world, on a par with the ability to read and write.

To the rescue comes... Devlin's math education software program:

With video games, we can circumvent the barrier. Because video games are dynamic, interactive and controlled by the user yet designed by the developer, they are the perfect medium for representing everyday mathematics, allowing direct access to the mathematics (bypassing the symbols) in the same direct way that a piano provides direct access to the music.

Devlin's notion that a well-designed math video game can help students meet the National Academy's goals for math education rests on two assumptions. One is that students can achieve a sufficient level of mastery in mathematics without symbols. The other is that playing such video games is to math what playing the piano is to music.

To address the first claim, Devlin elaborates the analogy to music:

Just how essential are those symbols? After all, until the invention of various kinds of recording devices, symbolic musical notation was the only way to store and distribute music, yet no one ever confuses music with a musical score.
...
Just as music is created and enjoyed within the mind, so too is mathematics created and carried out (and by many of us enjoyed) in the mind. At its heart, mathematics is a mental activity—a way of thinking—one that over several millennia of human history has proved to be highly beneficial to life and society.

But there's an important difference between math and music--and a reason why no one confuses music with a musical score. Music has a privileged place in subjective experience. Along with sensations like color, taste, and smell, it produces in us a characteristic, irreduceable, qualitative impression--an instance of what philosophers call "qualia." Just as there's no way to capture the subjective impression of "redness" with a graph of its electromagnetic frequency, or of "chocolate" with a 3-D model of its molecular structure, so, too, with the subjective feeling of a tonic-dominant-submediant-mediant-subdominant-tonic-subdominant-dominant chord progression. Embedded in what makes music what it is to us is the qualia of its chords and melodies.

Like most other, more abstract concepts ("heliocentric," "temporary"), mathematic concepts don't generally evoke this qualia sensation. What makes math beautiful are things like eloquence, patterns, and power. Unlike a Bach fugue translated homomorphically into, say, a collage of shapes, mathematical concepts can be be translated into different representational systems without losing their essence and beauty.

Devlin argues that while we might write down symbols in the course of doing real-life math, it is primarily a "thinking process," and that "at its heart, mathematics is a mental activity—a way of thinking." I agree. Indeed, math is much more appropriately compared with thoughts than with music. But this makes math symbols the mathematical equivalent of linguistic symbols. While thoughts, like math, can be expressed in a number of different symbol systems, you need some sort of symbol system in order to represent your own thoughts and to understand the thoughts of others.

This is especially true of abstract thoughts--and of abstract math. As Devlin himself admits, "the advanced mathematics used by scientists and engineers is intrinsically symbolic. "What isn't intrinsically symbolic, Devlin claims, is "everyday mathematics":

The kind of math important to ordinary people in their lives... is not, and it can be done in your head. Roughly speaking, everyday mathematics comprises counting, arithmetic, proportional reasoning, numerical estimation, elementary geometry and trigonometry, elementary algebra, basic probability and statistics, logical thinking, algorithm use, problem formation (modeling), problem solving, and sound calculator use. (Yes, even elementary algebra belongs in that list. The symbols are not essential.)

OK, but what does this mean for education? Are we going to decide before the end of middle school which students are going to become scientists, engineers, and mathematicians, and only help those students scale the "symbol barrier"? For a barrier it certainly is, as Devlin himself notes: "people can become highly skilled at doing mental math and yet be hopeless at its symbolic representations."

But Devlin is too busy appreciating the (well-studied) math skills of Brazilian street vendors, who do complex arithmetic calculations in their heads with 98% accuracy, and supposedly without the help of symbols (even mental ones?), to realize the educational implications of the fact that "when faced with what are (from a mathematical perspective) the very same problems, but presented in the traditional symbols, their performance drops to a mere 35 to 40 percent accuracy." No, not everyone is going to become an engineer. But not all non-engineers are going to become Brazilian street vendors.

It's ironic how deeply Devlin appreciates the difficulty that "ordinary people" have with the symbol barrier without appreciating what this says about their educational needs:

It simply is not the case that ordinary people cannot do everyday math. Rather, they cannot do symbolic everyday math. In fact, for most people, it’s not accurate to say that the problems they are presented in paper-and-pencil format are “the same as” the ones they solve fluently in a real life setting. When you read the transcripts of the ways they solve the problems in the two settings, you realize that they are doing completely different things. Only someone who has mastery of symbolic mathematics can recognize the problems encountered in the two contexts as being “the same.”

Instead of seeing this as a reason for exposing children to mathematical symbols early and often, Devlin sees this as reason to create computer games that somehow teach math non-symbolically.

He calls this "adaptive technology," a term that should raise red flags. In a recent blog post, I wrote about how assistive technology often becomes yet another excuse not to teach basic skills. Kids with dyslexia struggle mightily with the symbol system of written language; should they instead learn everything through text-to-speech and speech-to-text devices, and never learn how to read and write?

Devlin makes a few other strained comparisons to the piano:

The piano metaphor can be pursued further. There’s a widespread belief that you first have to master the basic skills to progress in mathematics. That’s total nonsense. It’s like saying you have to master musical notation and the performance of musical scales before you can start to try to play an instrument—a surefire way to put someone off music if ever there was one.

No it's not; it's like saying you have to master simple scales and exercises before you move on to Rachmaninoff.

The one difference between music and math is that whereas a single piano can be used to play almost any tune, a video game designed to play, say, addition of fractions, probably won’t be able to play multiplication of fractions. This means that the task facing the game designer is not to design one instrument but an entire orchestra.

Can one create a video game that functions "as an instrument on which a person can 'play' mathematics?"

Can this be done? Yes. I know this fact to be true because I spent almost five years working with talented and experienced game developers on a stealth project at a large video game company, trying to build such an orchestra.

What does Devlin's software do? The last two paragraphs of this article function as an extended but not very informative infomercial. Here's the most informative excerpt:

Available in early March, Wuzzit Trouble is a game where players must free the Wuzzits from the traps they’ve inadvertently wandered into inside a castle. Players must use puzzle-solving skills to gather keys that open the gearlike combination locks on the cages, while avoiding hazards.

Puzzle solving? As I argue in my last post on math games, existing games already offer some version of this, and it isn't math. This, indeed, is one of the other problems with so-called math education software.

Devlin suggests his software is different:

Unlike the majority of other casual games, it is built on top of sound mathematical principles, which means that anyone who plays it will be learning and practicing good mathematical thinking—much like a person playing a musical instrument for pleasure will at the same time learn about music.

Wuzzit Trouble might look and play like a simple arithmetic game, and indeed that is the point. But looks can be deceiving. The puzzles carry star ratings, and I have yet to achieve the maximum number of stars on some of the puzzles! (I never mastered Rachmaninov on the piano either.) The game is not designed to teach. The intention is to provide an “instrument” that, in addition to being fun to play, not only provides implicit learning but may also be used as a basis for formal learning in a scholastic setting.

If you say so. But I wonder how much it will cost schools (and society) to find out whether this latest incarnation of "math education" software helps prepare students to become mathematicians, scientists, engineers--or Brazilian street vendors.

can elite students do arithmetic?

Looks like the answer is no. (pdf file)

After a conversation with a "well respected mathematician who was heavily involved with K-12 mathematics education," W. Stephen Wilson re-analyzed the results of the arithmetic test he gave his Calculus III students at Johns Hopkins in 2007. The unnamed mathematician had told Wilson that fewer than 1% of college students would be unable to work a multiplication problem by hand, so Wilson took a look:

He was a little off on his estimate.

In the fall of 2007 I gave a 10 question arithmetic test to my 229 Calculus III (multi-variable calculus) students on the first day of class. Among other things, this means they already had credit for a full year of Calculus. The vast majority of these students were freshmen and ... and the average math SAT score was about 740.

[snip]

Seven of [the problems involving multiplication] were each missed by 8% or more of my students and 69 students, or 30%, missed more than 1 problem.

These are high achieving, highly motivated students (remember the 740 average SAT math score). These are disturbing numbers for them, but I suspect the numbers are much much worse among college freshmen with an average math SAT of 582, and, from Table 145 of Digest of Education Statistics 2009 we see that the average SAT score for the intended college major of engineering is 582 in 2008-2009.

Anyway, there is no real purpose to this paper except as a resource for me. It does suggest, very strongly, to me, that we have lost the pro-arithmetic war. This is a revelation to me and it calls into question what I will do next year with my big service course. I now feel compelled to assume that [my students] are chronically accident prone or they really are arithmetically handicapped. It isn’t clear that there is a difference. The question remains, how can I teach serious college level mathematics to students who are ill-prepared?

For passers-by, here's a quick run-down of Wilson's original observations:

As another experiment, Wilson gave a short test of basic math skills at the start of his Calculus III class in 2007. The results predicted how students later fared on the final exam. Those who could use pencil and paper to do basic multiplication and long division at the beginning of the semester scored better on the final Calc III material. His most startling finding was that 33 out of 236 advanced students didn’t even know how to begin a long division problem.
Back to Basics for the "Division Clueless"
DECEMBER 6, 2010 | BY LISA WATTS

counting-all vs counting-on

When first learning to solve simple arithmetic problems (e.g., 5 + 3), children typically rely on their knowledge of counting and the associated procedures (Siegler & Shrager, 1984). These procedures are sometimes executed with the aid of fingers (finger counting) and sometimes without them (verbal counting). The two most commonly used counting procedures are termed min (or counting-on) and sum (or counting-all; Fuson, 1982; Groen & Parkman, 1972). The min procedure involves stating the larger (max) addend and then counting a number of times equal to the value of the smaller (min) addend, such as counting 5, 6, 7, 8 to solve 5 + 3. The sum procedure involves counting both addends starting from 1. Occasionally, children state the value of the smaller addend and then count the larger addend (the max procedure). The development of procedural competencies reflects a gradual shift from frequent use of the sum and max procedures to frequent use of min counting.
Numerical and Arithmetical Cognition: A Longitudinal Study of Process and Concept Deficits in Children with Learning Disability
David C. Geary, Carmen O. Hamson, and Mary K. Hoard
Journal of Experimental Child Psychology 77, 236–263 (2000)

Why does academic language have to be so opaque?

Thank God for Google.

Here's what I think they mean.

When children first begin to add numbers, they rely on sum or counting all, which a Michigan State website defines thusly: "if given the problem 5 + 2, the student may count 5 on one hand and 2 on the other hand, and then count all fingers to get 7."

Then, as children progress, they switch to min, or counting on. With counting on, the child realizes he doesn't have to count the first number; he can start with the first number and start counting from there (hence counting on). So, if the child is adding 5 + 2, he starts with the number 5 and counts two more: "5, 6, 7."

Brian Butterworth describes it this way:

Counting on from first. Some children come to realise that it is not necessary to count the first addend. [When adding 3 + 5] they can start with three, and then count on another five to get the solution. Using finger counting, the child will no longer count out the first set, but start with the word ‘Three’, and then use a hand to count on the second addend: ‘Four, five, six, seven, eight’.
The development of arithmetical abilities
Brian Butterworth
Journal of Child Psychology and Psychiatry 46:1 (2005), pp 3–18

According to Butterworth, counting on from larger is stage 3:

Counting on from larger. It is more efficient, and less prone to error, when the smaller of the two addends is counted. The child now selects the larger number to start with: ‘Five’, and then carries on ‘Six, seven, eight’.

As to the timing, Butterworth writes: 

There is a marked shift to Stage 3 in the first six months of school (around 5–6 years in the US, where this study was conducted (Carpenter & Moser, 1982). Stage 3 shows a grasp of the fact that taking the addends in either order will give the same result. This may follow from an understanding of the effects of joining two sets, that is, taking the union of two disjoint sets.

This answers my question about Dr. Wilson's observation that children with dyscalculia use inefficient strategies for simple addition.

Parker on the equal sign

On the subject of American students not knowing what the equal sign signifies, Parker writes:

I told my spouse the other day that I was convinced that many of my chemistry students did not understand what the equals sign meant and therefore they couldn't handle equations. It was just a hunch, but maybe I wasn't crazy. When they rearrange an equation they tend to just move letters around.

I don't know how we could screw up teaching this stuff.

I'm curious about this -- what happens in a chemistry class when students don't understand the equal sign?

Chisenbop tutorial

just posted to the precision teaching list: Chisenbop tutorial

corn on the cob

Have been meaning to post this --

A few weeks back David Letterman showed a print ad that reads:

Corn cobs 4 for $1.00
Works out to 25¢ a piece

pop quiz, part 3

discount

The Scribner Arithmetic

I was just talking to a woman here who is working on a transparency project (I'm in!). She got her start in all this when her district adopted TERC and rejected all forms of acceleration for gifted kids; now she's asking local townships for budget docs and the like.

Never occurred to me that the math wars can also have consequences for local government.

heh

She mentioned that when she looked back at her old textbooks, as well as her father's books, she was impressed. Her dad used Scribner Arithmetic in the 1950s.

Does anyone know anything about it?



The Scribner Arithmetic: Book 5

I wish I could remember the math books my school bought when I was in 2nd grade, I think. I remember thinking they were beautiful. Not just beautiful but elegant, though I don't think I knew to apply that word to a book at the time.

what is an average, anyway?

A friend of mine asked me to walk her through mean, median, and mode --- and it came to me, thinking about it, that I don't exactly know what an average is beyond the obvious.

Another issue: I'm so sick of my own child (& everyone else's) being lost in group means that I've come to feel some real antipathy towards the very concept of a group average. Meanwhile the concept of a personal average makes sense and seems obviously useful ----

Here are my questions.

What is useful about averages? 

What do averages tell us?

And why did the calculation of averages come to be so important culturally?

Playland

Yesterday C. and I went to Playland for their annual developmental disabilities day, also known as our annual Playland fudge day. Just inside the entrance, there's a fudge stand that sells the best fudge I've ever consumed, and C. and I binge on the stuff once a year.^*

The price is $3 per piece, or buy 4 pieces & get one free.

I told the young person behind the counter that I wanted 15 pieces, and she began methodically filling up the boxes. Methodically .... and .... slowly. At some point in this process, possibly thinking I could speed things along, I said, "So that's $36, right?"

And she said "No." She didn't seem to be exactly sure how much 15 pieces would be, but she didn't think it was going to be $36.

I said, "You get 1 free with each 4, right?"

"Yes."

"So I'm paying for 12 pieces and getting 3 pieces free."

She looked confused. I went over it again, but she still didn't think $36 sounded right.

Finally I said, "I want to buy 3 boxes of fudge, with 4 pieces in each box and 1 extra free piece," and that did the trick. Twelve dollars a box.

I don't think she knew how to work backwards from "15 pieces of fudge" to 12 pieces of fudge at $3/each + 3 extra free pieces.

Meanwhile, here are the "mathematical practices" required by the new Common Core standards (pdf file) for Kindergarten:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.

^* Have I mentioned the fact that I've de-veganed myself? Well, I have.

Math on a farm

Jane wrote:

I suspect that the data that say the vast majority of people don't use math is inaccurate. I suspect that every small business owner is constantly using basic algebra. We have a family farm and constantly, daily use algebra. My contractor calculated angles, areas, amount of paint, flooring, labor and materials necessary to remodel our house. Our banker uses a calculator, but easily and obviously has the mathematical fluency to have a real time conversation about prices, quantities, exchange rates and whether certain data sources are reliable. The sales people we deal with calibrate machinery to determine how much material we need to use on a field. Of course, I check their calibration calculations as well. Yes, we all use tools, but without the understanding of how the math fits together, the tools would be black boxes and we wouldn't know when a number didn't make sense. And we are constantly figuring out, how much seed to put on a field, if a chemical needs to be applied at a certain flow rate and dilution, how fast does the tractor move. If we can x crop and it costs z to grow and we might sell it for a range of y1 - y2, Which crop should we plant.

My dad was a farmer. He always had a slide rule handy on his desk. I was fascinated by it.

I don't know why he had it, and I can't guess because I've never learned to use a slide rule.

I'd like to.

Ron Aharoni on calculations & calculators

Calculation isn't just figuring out the result of an exercise: it is figuring out the decimal representation of the result. Therefore, the ability to calculate is in fact tantamount to a profound understanding of the decimal system. This is one of the reasons why calculation is so important, and why it should not be replaced by a calculator.

Arithmetical operations can be calculated in many ways. The methods currently taught in school are the result of generations of thought, and much wisdom has been invested in them. Most are based on writing the exercises vertically, so that the ones digits are one above the other, the tends digits are one above the other and so forth.

Calculations are based on the knowledge of the addition and multiplication tables -- the sums and products of numbers smaller than 10. These must be memorized. The addition table should be well established in the first grade, and the multiplication table in the second or third grade. In addition, the children should be familiar with the rules that govern the operations, such as the distributive law and the rules of change.

The operation of division is the most difficult to calculate. On the other hand, the algorithm of division, called "long division," includes fundamental principles and therefore it should not be passed over.

Arithmetic for Parents: A Book for Grownups about Children's Mathematics
by Ron Aharoni
p. 95

arithmetic versus algebra

Consider the set: {1, 3, 5}
:: If elements of this set can be selectively added together to yield some number q, what is its maximum?
:: Are there any odd values between 1 and q_max that q cannot hold?
:: Are there any even values between 1 and q_max that q cannot hold?

Consider the set: {1 , 3 , 5 , 7 , 9 , 11}
:: What is the maximum of q?
:: What odd values between 1 and q_max can q not hold?
:: What even values between 1 and q_max can q not hold?

Consider the set: {1 , 3 , 5 , 7 , 9 , 11, 13}
:: What is the maximum of q?
:: What odd values between 1 and q_max can q not hold?
:: What even values between 1 and q_max can q not hold?

Consider a finite set of odd numbers, {1, 3, 5, 7 ... n}
:: What is the maximum of q, in terms of n?
:: Find the odd values between 1 and q_max that q cannot hold, in terms of n.
:: Find the even values between 1 and q_max that q cannot hold, in terms of n.

Consider the set: {-6, -4, -2, 1, 3, 5, 7}
:: What is the maximum and minimum of q?
:: What odd values between q_min and q_max can q not hold?
:: What even values between q_min and q_max can q not hold?

Consider a finite set: {-2n, -2(n-1), ... -4, -2, 1, 3, 5 ... 2(n-1)+1, 2n+1 }
:: What is the maximum and minimum of q?
:: What odd values between q_min and q_max can q not hold, in terms of n?
:: What even values between q_min and q_max can q not hold, in terms of n?

is arithmetic math?

I was kibbutzing with a compatriot in the high school parking lot last night. He's a math person who is an administrator with a major college. Very knowledgeable.

He told me he wants kids to be taught math, not arithmetic; "arithmetic isn't math."

I've heard that before but still don't know what it means.

Speaking of arithmetic, Hung Hsi Wu's article "What's Sophisticated about Elementary Mathematics" (pdf file) is out!

And remember Ron Aharoni: What I Learned in Elementary School.

Wu: Arithmetic to Algebra

here

I was just taking a look at the link, and it reminded me of the time C. asked me, "Does algebra have numbers or only letters?"

I remember posting that question & laughing about it; then Tracy W explained to me that that was a correct question/observation....

maths dunces

Jenny left a link to this story:

When the Bamberger family opened a haberdashery 65 years ago, they insisted their staff use mental arithmetic to price up customers' purchases.

Despite the arrival of calculators, that attitude has remained unchanged over the intervening years.

But now the family finds itself facing an unexpected maths problem - most youngsters it would like to employ are incapable of working out sums in their heads.

Colin Bamberger, 82, whose parents founded the Remnant Shop in 1944, said that less than one in ten applicants are now able to solve basic maths problems without turning to a calculator or till.

In the past, around eight in ten made the grade.

Mr Bamberger, who stills runs one of the family's two stores, yesterday blamed the decline on falling education standards and over-reliance on the pocket calculator.

He said: 'Most of the youngsters who come to us for jobs are unemployable because they are not numerate.

'It is a sorry situation and a poor reflection on the academic qualities of young people these days. I think it shows modern teaching methods are sadly lacking.

'It is all very well using calculators but if you have not got some idea what the answer is, how do you know if you have pushed the right button? It's so easy to make a mistake.

'It was much easier finding staff a few years ago when everyone coped with working out simple maths in their heads.

Around eight out of ten people who came to us for work were capable of doing it in the 1950s and 1960s - but now it is less than one in ten.

'You ask them how much they would charge for nine metres of material at £9.90 a metre and they fiddle about for ages.'

He said that mental arithmetic was essential in his shops because, if customers queried the final bill, staff could scribble their calculations on a piece of paper to show them how they arrived at the sum.

[snip]

Robert said that even if applicants were 'massive at marketing, super at sales or even Alan Sugar's next apprentice - if they can't add up quickly in their head we won't have them'.

'My grandfather could add up a column of 50 figures in old pounds, shillings and pennies - including ha'pennies and farthings - in a matter of seconds,' he added. 'He used to insist that any staff we took on could do the same and we have carried on that practice.'

Maths dunces who don't make the cut: Haberdashers have to reject nine out of ten applicants because they can't add up

arithmetic

I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours I found it quite enthralling.

--Agatha Christie (1890 - 1976)


Arithmetic for Parents by Ron Aharoni
What I Learned in Elementary School by Ron Aharoni (in American Educator)

from ktm-1 (you may have to hit refresh a number of times):
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

pop quiz

Are You Qualified to Teach Elementary School Math? (pdf file)

Just what I wanted to hear!

from Lynn G:

FWIW, we started Primary Math 6A today. Kicking and screaming ensued.

Then we opened yesterday's mail and got a very pleasant surprise. My daughter's CMT scores (CT's state testing) arrived and were very good. She got a perfect 400 in math. Math is not her favorite, or strongest subject, but it's one we've focused on at home at length. I see the score as reflecting the coherence of the Singapore Math system.

After much oooohhing and ahhhing, I was able to get her to start 6A.

Right now, she doesn't seem to have any real weak areas. She's doing pretty good in decimals and fractions, so we're just going to charge ahead with the whole program.

That's where I am -- and I can see exactly how this curriculum gets a student to the point of perfect scores on state math exams.

The Primary Mathematics series is the coherent curriculum (pdf file) par excellence. One thing leads to another, and you have problem solving applications from the get-go.

C. has probably done 50 word problems in his entire math-learning career, if that. He has no idea how anything relates to anything else. He's been getting by on brute memory.

Here's how bad it is.

We've been doing the percent lessons in Singapore Math 5A.

That quickly became so "natural" a learning situation that, when I told C. yesterday morning I wanted him to do an entire page of problems, he looked at the page and said, "OK, that looks easy."

And it was easy. He finished in 5 or 10 minutes & got everything right.

The fact is, going back to 3A will probably be fun, especially since the Premack principle is working so brilliantly that yesterday morning (Saturday), at 10 am, C. came into my office, sat down, and said, "What do I have to do today?"

So I think we've got the think-like-a-behavior-analyst aspect of things down, which will free C. to enjoy being able to comprehend, learn, and do 3rd grade math.

That is already happening, in fact.

Yesterday, after he whipped off his one-page problem set, he said, "That Singapore method is really good."

He was referring to finding the percent equivalent of a fraction by multiplying numerator and denominator by the same number.

This is the state of his math education.

Multiplying 6/20 by 5/5 to get 30% is "that Singapore method."

this will give you some idea

When I told Ed I was going to start C. back in 1st semester 3rd grade Singapore Math, he said, "I think that's a good idea."

This brings to mind all those occasions when I wished doctors or special ed teachers would downplay my concerns. Having one's concerns downplayed seems to be a core experience of so many parents of autistic kids, and yet it has never - not once! - happened to me.

No, when I show up with my autistic kids, it's Clear the exits, here they come!

Back when I first started my teach-our-son-math project, Ed was mildly dismissive. He didn't see the big deal; C. was a smart kid; he'd learn math the same way Ed had learned math; etc.

That was irritating.

Ed has become progressively less dismissive as time has gone by, which is also irritating! By last year I was hearing regularly that C. "has no conceptual understanding at all," or "has no idea how to do X," or "doesn't know anything about Y."

I would usually say something like, "He does have some conceptual understanding," or "he does have some idea how to do X," or "he knows something about Y" -- sounding for all the world like a person failing to reflect adequately on her practice.

So, as I say, enough's enough.

I can't keep slicing & dicing it. My son does not know arithmetic, and that's that. (Does not know arithmetic and yet is one point away from "Meets standards with distinction" on the state test. Which is pretty much all you need to know about the NY state math exam.)


speaking of which

I ran across my copy of the New York State Learning Standards for Mathematics 2005 the other day:

Every teacher of mathematics, whether at the elementary, middle, or high school level, has an individual goal to provide students with the knowledge and understanding of the mathematics necessary to function in a world very dependent upon the application of mathematics. Instructionally, this goal translates into three components:

• conceptual understanding
• procedural fluency
• problem solving
  

I happen to agree with this list.

Not one of them is true of the situation around here.