**Makes Me Want To Kill Myself With Occam's Razor**

Adventures in new math. (Hey, kiddies, just grab a calculator!)

*Thanks, Norman!*

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**AMY ALKON,** APPLIED BEHAVIORAL SCIENCE AUTHOR, SPEAKER, NATIONALLY SYNDICATED COLUMNIST

## Amy Alkon

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**Makes Me Want To Kill Myself With Occam's Razor**

Adventures in new math. (Hey, kiddies, just grab a calculator!)

*Thanks, Norman!*

I didn't watch the whole video, but I watched a bunch of it. The funny thing is that the algorithms she doesn't like are far more similar to the way I (headed to the Caltech Math PhD program) do multiplication and division than the 'standard algorithm.' I'm willing to believe that the standard algorithm is a better way to teach kids, and more reliable as an algorithm. But I don't think I've ever seen one of my fellow math majors use the standard algorithm to work a problem; I stopped using it when I was in grade school.

Assuming that the new algorithms really don't teach kids as well, my assumption is that it's a result of people who are good at math trying to teach the most efficient way to do it, rather than the least error-prone or easiest to teach. I barely remember how to do long division any more because I always do it the way she doesn't like (they're equivalent, of course). But I suppose the fact that that works best for me doesn't mean it's also the best for the average fourth-grader.

Jadagul at June 14, 2008 1:39 AM

The alternate approaches made me tired. I remember the pre-new-math joy of learning how to multiply and divide large numbers. And the simplicity of it. You get kids to be into math by making it enjoyable, not totally tedious and complicated from the start. There's plenty of time for that!

As for the merits of the alternate approach, if you're doing it in your head, sure, break it down -- as I did for my 7-year-old neighbor in trying to help him learn how to easily multiply 12 x 8 (10 x 8 and then add the results of 2 x 8) in his head.

Amy Alkon at June 14, 2008 2:12 AM

If it's in my head, I do various tricks to simplify and approximate. But on paper the standard algorithm is easy. That's a direct benefit of the way we write numbers - try doing these sums in Roman numerals. Actually, I was also taught a multiplication method by breaking down one of the numbers. We called the method "practice" but never used it much. It is very useful when you are multiplying imperial measurements and pre-decimal money (eg how much are eight and a half bolts of cloth at 176 yards, at £7 17/6 per 10 feet?)

I think it's a mistake to try to teach people how to understand before teaching them how to do. (The teaching staff in my department - computer science at university - are divided about 50-50 on the issue.)

Once people can do basic arithmetic, it's a lot easier to explain the theory because you can show examples and have something concrete to talk about and pin the ideas on. Some people never get past the basic arithmetic, but then at least they're be able to count, whereas if you do theory first they end up with nothing but a feeling of stupidity and a dislike of the subject.

Norman at June 14, 2008 3:15 AM

Norman: exactly. For someone with enough mathematical sophistication to hold a degree in math or computer science, the easiest and fastest way to compute stuff is to pull out lots of tricks. (There was one guy at my high school who never got tired of the fact that I could multiply any two-digit number by 99 instantly in my head). But no one with that degree of mathematical sophistication is studying fourth-grade math. I think the core problem is people who are very, very good and experienced at math trying to teach people who are average and inexperienced to think like people who are very good. There are some things to be said for that approach, but it also has major limits.

Jadagul at June 14, 2008 3:28 AM

What's amazing about American schools is that I somehow graduated high school without ever taking a grammar class. I have an innate knowledge of grammar and syntax; probably from reading voraciously as a child; but, I couldn't tell you what the subjunctive is if my life depended on it, and I probably couldn't "diagram a sentence," either, unless that just means saying which one is the verb and which one is the adverb.

Amy Alkon at June 14, 2008 7:32 AM

Grammar's trickier because there is no complete formal agreed grammar for English (or any natural language, afaik). But it is useful to be able to discuss one's language, and that means knowing what nouns, verbs, etc are, even if we can't agree on all the finer points.

An example of why grammar is hard: "more than one car was there." Since there was more than one, should the verb not be plural, ie "more than one car were there?" I don't think the subject is "one car" because that would parse as (more than) (one car was there) which isn't even a sentence. The sense is clearly (more than one car) (was there), similar to (many cars) (were there) or (exactly one car) (was there). This kind of thing is hard to formalise.

Norman at June 14, 2008 8:05 AM

I think the telling evidence for newer(?) methods of teaching arithmetic (or math) and reading was explained to me by a co-worker's wife. She has spent the last thiry years as a remedial teacher (and sometimes tutor). In most cases, the pupils she and her fellows see have to start learning to read the old-fashioned way (phonetics). Similar traditional methods are required for numbers. She is fond of saying "Too many concepts, not enough drills."

If remediation relies so strongly on these older methods and reading and maths skills seem to have dropped as they were abandoned, why do U.S. teachers or the people designing the course of instruction not see the problem? Do all the countries whose students have superior math and reading skills use these more "modern" techniques?

sirhcton at June 14, 2008 8:33 AM

the use of calculators is doing to math what texting is doing to the english language. Kids cannot do either.

I am a little older than most posting here I think, my high school years predated the invention of the calculator. We did geometry and trig and calculus manually. Oh, we were allowed to use a slide rule in class, but not on tests (usually). Frankly I can do most arithmetic problems in my head - so I never bothered trying to learn the use of the slide rule - which was almost like torture.

I recently assisted in tutoring a 9th grade math class, and was amazed at the students' reliance on calculators - and the fact that they had no clue waht the answer should be. If there was input error it didn't matter - whatever the machine said was the answer they wrote down. I saw one girl answer a problem with a number containing a decimal, even though the problem was multiplying whole numbers. She could not grasp that if you are multiplying whole numbers, the answer - by definition - has to be a whole number!

When my kids were in school, and had issues with math, I taught them the old-fashioned way, they were then able to translate it to give the teacher the answer they were looking for in the way the teacher wanted it presented, because they understood the basics.

steveda at June 14, 2008 10:00 AM

I wouldn't even describe some of the methods in the video as "algorithms" since they seem to rely on luck. They also leave the page in such a mess that I don't see how anyone can check the workings.

Peasant (or binary) multiplication is quite neat. See Wikipedia for an explanation. I wonder why they don't teach that method.

Wikipedia also says the grid method, where you draw a grid and sum along diagonals, was introduced to Europe by Fibonacci's "Liber Abaci" in 1202. That was the book that brought in positional decimal notation to replace Roman numerals. It was a great step forward at the time, but adopting it in 21st century looks like a great step backwards.

Norman at June 14, 2008 10:02 AM

Like Jadagul, I'm a science professional who uses the "new" algorithms in the video to do math in my head. It's incredibly useful when dealing with decimals, because I can get the answer to whatever accuracy I need so much faster than writing out the whole shebang.

I don't know which way is better to teach, because I hated learning math by rote. I'm lucky I was introduced to the real logic behind math before I decided I was an artist, or worse. Those who will use math to balance their checkbooks may benefit from the standard method, but those who will live with math, and really see beyond the surface, will eventually use the "new" methods.

Josh at June 14, 2008 10:29 AM

I spend my life taking things apart in my head, and I think it's essential to explain what it means to multiply a number; ie, that's 10 ten times to explain how 100 divides out to a kid. Still, this is a complicated mess for somebody just learning to multiply.

Amy Alkon at June 14, 2008 10:58 AM

My aunt Linda was a sweet person, a bit formal, and taught 4th grade. She had a question for me about teaching math: Can you subtract a negative number? I'm a math/physics college grad.

Her teaching guide used a mailman analogy to teach addition and subtraction. You start with $10. The mailman delivers a check for $5, so you have 10 + 5 = $15.

If the mailman delivers a bill for $8, then 15 - 8 = $7. Also, you can think of this as adding a negative number, so you can also write it as 15 + (-8) = $7.

The problem is that the mailman never takes away a bill, so how can you subtract a negative number? Other teachers at her school didn't have an answer.

After some discussion, she thought it was arbitrary that subtracting a negative is like adding the positive, and it didn't fit the mailman analogy, so she would just not teach it. An inquisitive 10 year old was told that you just can't subtract negative numbers.

I will take the leap to generalize. School is teaching mathematical subjects the same way as other subjects. They are presented as a collection of facts to memorize, with no understanding of the patterns or meanings that the facts are supposed to illustrate. The failure of the textbooks shown in the video (aside from the travelogue approach for filling in 50 pages) is that they don't teach fundamentals (what is it physically like to multiply and divide), and they don't teach compact algorithms derived from the fundamentals.

They teach a middle view that obscures both the fundamentals and the algorithms, leaving math as a large collection of facts that are mostly alike and hard to apply. A mystery best left to the people who magically "get it" from above.

Sorry to say, without a deeper understanding, having a calculator is no help. You punch in a few numbers, make mistakes, get wild answers, and sign for a mortgage with terms that can't make any sense to you.

Andrew Garland at June 14, 2008 4:08 PM

Who needs multiplication? Just shift and add in base-2.

Paul Hrissikopoulos at June 14, 2008 5:48 PM

My kids have been taught with "Investigations." Some of the criticisms are valid -- the students tend to be slower in solving problems. I really disagree with her about the "confident" issue, though.

There are some advantages to the program. What the woman in the video didn't explain is that the students develop their own methods for solving problems, not just the one awkward way that she described. This helps students get a better number sense. It also helps them be more confident when faced with a never-before-seen type of problem. One of the drawbacks of the traditional algorithms (or these new ones that are proposed) is that if you aren't good at generalizing, it's often harder to find the right algorithm to solve the problem. I'm sure you all remember some terror at "word problems!" Kids who have been through Investigations are much more confident in these.

That said, I believe that traditional (or new) algorithms should be taught, alongside an Investigations-type approach.

Andrew above has some good points about the failures of traditional teaching methods.

ArtK at June 15, 2008 8:16 AM

I always did well in math. except calculus. i did alright in that, but never really understood it. anyway.

my friend teaches 5th grade "new math". i recently tried to help her correct some papers. i had absolutely no idea what the problems were talking about, couldn't follow them if my life depended on it. (and i liked "word problems")

and how long is confidence going to last - and what good does it do - if they can't understand real life math - like their mortgages and their checkbooks?

what i think is particularly stupid is we look at how poorly our kids do in school in comparison to india/japan/whatever and instead of looking at what they do different and copying them, we try to make our education system simpler. our kids are not innately stupider than the rest of the world, we just teach them to be.

kt at June 15, 2008 11:10 PM

What's education for? First, how to be a consumer in this society; second, how to be a producer in this society; third, how to change this society. These don't correspond exactly with primary, secondary, tertiary education, but there is a close match.

Norman at June 16, 2008 8:38 AM

yeah, maybe, norman, but at this rate we will be consuming welfare, producing babies on welfare, and changing society - but not in a positive way.

kt at June 18, 2008 1:38 AM

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