National Math Panel Formed

Interesting stuff. I'm but a humble MSc in pure math, but I agree with just about everything you say.

My recent horrible experience was getting $5 more in change than was warranted because the register was wrong and the cashier couldn't grapple with the ugly reality and recognize that "the machine was wrong."

If anything, I'd be a bit more puritanical about calculators. They're great for saving time on operations which you already understand ... but you should still be able to recognize when something like a simple multiplication result just doesn't feel right. (After all, you still have to push the right keys.) Used as a crutch, they're the devil's plaything.

But I wouldn't want you to think I felt strongly about it.

And I'll end with an illustrious name in "new math" - Tom Lehrer. ("Don't worry. Base 8 is just like base 10 ... if you're missing two fingers.")

I read a lot of these ScienceBlogs and I sometimes ask naive questions as a non-scientist. But in this case I feel that I must respond from the position of (semi)expert. I too have an MS in Math and before that a BA in Math Ed. I taught Math for 15 years in everything from 5th grade through Calculus. I supervised the elementary school curriciulum and teachers for a while. And then I spent 25 years in Computer Programming/Analysis/Management. My experience revealed that very few people - including Math teachers, college graduate programmers, people in business, have a real understanding of Arithmetic. As I see it the problem is -still - too early an introduction to using algorithms to get the right answer. The important concept that "New Math" introduced, which became perverted, was that concrete experience and understanding of how numbers work must precede introduction of any algorithm. Students are not taught the most fundamental concept - the difference between a number and a numeral. They need to understand how the numbers work before they are taught the shortcuts that the symbols allow. They are taught to memorize (indoctrination - just like in church) rather than to understand WHY 7 X 9 = 9 X 7, WHY the long division algorithem works, etc. The same thing is true at every level. Once Arithmetic has been learned and understood it is the concrete level that must be used to introduce Algebra, and so on. The problem has never been with that theory, it is with the fact that the people teaching arithmetic never understood those concepts themselves.

This got me curious because i'm taking an auxiliary math course at university and i'm scheduled to take many other math courses during the next few years and so far, we practiced a lot of thing but the logic behind the rules failed to sink in (ok, i may not explain it well but Karl is), is there a (or a few) good book(s) explaining the logic and rules you can recommend ?

Thanks in advance

I'm a computer science major (junior in college) so forgive my analogy here but I think of it like a computer program. There are two ways to go, either you can have a lookup table (rote memorization) or you can have a function (theory/algorithm). They each provide advantages and disadvantages and based on the circumstances, either one could prove more useful. When I was in middle school and high school I definitely had the multiplication table stored in my brain as a lookup table. I just had to think of two numbers and the answer was there. It was very VERY fast but it used lots of memory. Now, as I've moved on to more advance subjects and space in my brain has become more and more limited, I'm forced to store the multiplication table as an algorithm. When I need to know 8x7, my brain doesn't always simply throw out the answer, sometimes I have to run the numbers in my head. Ive traded a small processing delay for increased capacity. I think at some point, depending on the frequency of use, thats a given with most people and math. In the average persons life you just dont do that much math that requires split second response time.

I am probably betraying my bias here, but is there a good argument for not having more mathematicians on the panel? Of the sixteen members listed here, only three are mathematicians (as far as I can tell from their instutitional affiliations). Of those three, one is from Harvard and another from UC-Berkley.

Frankly, I think they would have been better served replacing one (or two) of the 'education' or psychology specialists with mathematics professors from mid-sized state universities. Those folks are more likely to see the so-called fruits of the elementary/high school labors in teaching mathematics and are in a better position to point out just how little the average student actually does know. Even students who take calculus as their first math course in college are often woefully underprepared in algebra.

With respect to Karl's point about algorithm's being introduced too early, I'm not sure that I agree. Rather, I think the problem is not reinforcing those algorithms with two things:

1) tying the results of an algorithm back to the original problem or context so that students can see that the algorithm is not the answer. I think students can develop a concrete understanding of arithmetic (and algebra) at the same time that they are learning to use algorithms; the two are complementary, not contradictory.

2) practicing those algorithms over and over and over and over. The denigration of practice as "drill and kill" has allowed students to get by with very weak skills. I have seen more and more students writing the equivalent of "sqrt(a+b) = sqrt(a) + sqrt(b)" and I think such mistakes are due in large part to a lack of practice. Students should repeat doing the right thing so many times that it becomes an automatic response.

When employing most algorithms in arithmetic and algebra, thinking should be done before the algorithm (what should be done?) and after the algorithm (what do I do with the result?), not during the algorithm.

I agree that rote learning is probably necessary for the absolute basics, but I think a lot of the problem people have graduating from arithmetic to higher maths is that conceptual understanding becomes much more important but it often isn't taught that way. I was fortunate enough to have an excellent maths education (I was raised in Britain if you're wondering about the spelling), and I was always perplexed as to why people from other schools had such problems with algebra - it seems it's not just a British phenomenon if Richard Cohen's infamous column is anything to go by. Algebra has always seemed perfectly simple to me, while other relatively simple mathematical concepts were much harder for me to grasp. When I talked to people who admitted to such difficulty, it turned out they were taught algebra without any mention of variables or why it works. It was just a set of seemingly arbitrary rules for manipulating equations. That can't be the right way to teach it.

Replies:
Alain wrote: "is there a (or a few) good book(s) explaining the logic and rules you can recommend ?"
See if you can find a 3rd, 4th, 5th, 6th grade book that talks about commutative associative, transitive. Commutative: 3+4 = 4+3. Addition and multiplication are commutative, subtraction and division are not.
Associative: 3+(4+5) = (3+4)+5. Again, add & mult are, subt & div are not.
Transitive: 3X(4+5) = 3X4 + 3X5.
Knowing these principles, and the place value concept of the decimal numeration system makes mental arithmetic and therefore all following math much more understandable.

DFX wrote: "In the average persons life you just don?t do that much math that requires split second response time."
It's not a question of split-second. It is a question of having a basic understanding. Two examples: I once talked to a woman about her job function. She said that at a certain point in the process she moved the decimal over two places. I said - Oh, you're dividing by 100. She said, no, I just move the decimal point. No understanding.
Another time, ordering custom sized Venetian blinds, I asked how precise the width measurement needs to be. The reply: to the nearest fraction. No understanding.

Kipli wrote "sqrt(a+b) = sqrt(a) + sqrt(b)" and I think such mistakes are due in large part to a lack of practice"
I disagree. I think that such mistakes are due entirely to a lack of understanding of the concept. I think that this epitomizes the problem - teaching algorithms too soon.

Ginger yellow wrote: "they were taught algebra without any mention of variables or why it works. It was just a set of seemingly arbitrary rules for manipulating equations. That can't be the right way to teach it."
I agree 100%.

I think this a clear opportunity for the Bush administration to show how their particular brand of respect for technology and science can affect this situation.

First, get Michael Crichton into the White House to explain about the liberal numbers conspiracy.

Second, we need to change away from our current ARABIC number system. When you use ARABIC numbers, the terrorists win. We're an empire, we should be using Roman numerals.

Just imagine how the elimination of negative numbers and 0 could improve our current budget deficit situation.

Long ago, in the fifth grade, we learned "invert and multiply" for dividing fractions. I think the pineapple upside-down cake that was served had something to do with our success.
In our modern world, the cashier at McGrease King takes your order by pressing buttons bearing pictures of the food item, hits the big green button, and the machine spits out your change. But taking this example as a model of how most people don't need arthmetical abillities is very misleading--I see frequent demonstrations of people geeting frustrated because they don't know how to figure how much fertilizer to apply, or how to measure 100mm with a ruler graduated in 16ths of an inch, &c.
One cannot be blamed for being skeptical about Bush's support of math & science. The president that called for teaching intelligent design creationism in schools may be just as likely to think there is some Bible-based alternative to the Pythagorean theorem. Jim's right--have you ever seen a person with -10 fingers?

Thanks you very much Karl for mentioning these words, I found a useful ressource (the math forum http://mathforum.org/ at Drexel University) by googling on them, they also did a number of books http://mathforum.org/pubs/dr.mathbooks.html which look very interresting.

Alain

I was trained in the traditionalist mode, like most here, although I was certainly hit with "New Math" along the way. My sons' elementary school uses a more constructivist approach (the "Investigations" system.) The emphasis is on the kids developing solutions rather than learning algorithms. Calculators aren't used. I wouldn't say that there is too much abstraction in this method, so perhaps it isn't the same as other constructivist systems that you've seen.

My friend teaches in the middle school that is fed by this elementary and she has noticed both good and bad things about the kids taught through "Investigations" when compared to kids taught more traditionally. On the good side, the constructivist kids are much better at problem solving -- they can approach a new problem with confidence, while the other kids have trouble applying their algorithms in new situations. On the other hand, the non-traditional kids aren't as facile with calculations as their peers.

That last issue is being addressed by some "bridging" work before the kids leave elementary school. They are being taught the traditional algorithms at that point.

While I'm not a complete convert, I think that algorithms should be balanced by much more of this kind of constructivist teaching -- there is no dichotomy here. Just like language arts, where you need a combination of whole language and phonics. Not every student learns well in the same mode and teaching, especially in the elementary grades, needs to accomodate that.