kitchen table math, the sequel: Ron Aharoni on calculations & calculators

Tuesday, March 9, 2010

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


farmwifetwo said...

The basics have to be memorized. How can you read a book if you don't know what the words mean? By not teaching the "rote" part of math you aren't giving language to what's going on.

My eldest is doing well in math, because he doesn't sit in class going... "I don't know what 7*6 is". Instead he's thinking "ok, for fractions I do this, that and then...." because he didn't get stuck the moment the teacher said "add, sub, mult or divide".... He already knows how to do this.

You can't build a tree... without the roots. It'll die.

Allison said...

I think it's better to say

math facts must be known to automaticity.

I didn't learn math by memorizing it; I just knew how arithmetic worked, saw it in my head. I didn't learn to read by memorizing lists of words, either. I knew what the words meant by having heard them used. the method isn't as important as the result--automaticity.

Whether you learn those facts by actively memorizing them, or if you deduce them or infer them, whether you get them from games or manipulatives or copywork is not as important as that they are known to automaticity.

Allison said...
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Anonymous said...

Like Allison, I had one son who just learned things on the spot. I used very little memorization with him. If I did, he got it almost immediately.

My other son needed tons more, but he also would get to the point where he could recall the facts quick enough to finally work other problems and not be bogged down.

I think kids are mostly in between my two, but they still need that automaticity.

Hey Allison,

I'm looking over the Ross program summer application with the attached problems. This program isn't just for geniuses, is it? I certainly can't tell, but I'm trying to look into it. I wish one of you guys lived nearby.


Allison said...

re: Ross: I seem to have a really skewed idea of what a genius is, so perhaps I can't answer that question.

I know of Ross people from the math/comp sci theory/physics community, whether ugrad, grad or research facilities, so that's conditioned on the people who liked/did well at Ross, most likely, rather than the set of people who went in.

The people I know who went to Ross (or PROMYS or Hampshire) were bright, certainly, and in each case, the people I know said it taught them a great deal. Several said, effectively, that math camp was the place where they learned that there were people MUCH brighter than them, and that they would have to actually work HARD in life if they wanted to succeed. A few credit it with teaching them how to think. A couple said it was the place where they learned they didn't have what it takes--didn't want to work that hard (was that really the truth of what they could have taken away from it? I don't know.) So it's really important to think about how the experience of being outclassed/outperformed would be taken by your kid. Would they love having nerds around them? would they love working on problems together with other bright kids? Would they feel fear or intimidation to where it was a negative experience?

Allison said...
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ChrisA said...

Good luck with that calculation thing.

CassyT said...

Love Aharoni's book.

I'm doing a book study next year with a couple of schools. At one school, the 1st & 2nd grade teachers have asked to study the Aharoni instead of Elementary Mathematics for Teachers (Parker & Baldridge). They felt it would better suit their needs as lower elementary teachers. I expect some interesting conversations happening with both the books.

One of the teachers came to me in December or so and said, "Did you know there are two different types of subtraction? There's part - whole and take -away!" She was so excited, after 4 years of teaching Singapore Math to discover that in the materials. (I didn't mention the third meaning of subtraction, comparison, but I think she'll figure that out in the 1B materials.)

CassyT said...

"math facts must be known to automaticity." Nice & concise.

I did a seminar yesterday for teachers and showed them a card game to use with students for practicing multiplication & division facts. One teacher was stumped with 9 x 7, and had to be fed the answer by the audience.

Catherine Johnson said...

hey Cassy - quick question

Of the approximately 2000 schools using Singapore Math, what is the breakdown? (Do we know?)

private/public/charter ----??

And do we have a sense of how many homeschoolers are using Singapore Math?

Crimson Wife said...

Catherine- purely anecdotal, but I would say that about 20% of homeschoolers I know are using Singapore, about 20% Saxon, and the remaining 60% use a wide variety of different programs. Right Start, Math U See, Miquon, Horizons, Christian Light, Abeka, ALEKS, Stanford EPGY, and Teaching Textbooks are all fairly popular.

Anonymous said...

Miquon sounds like another constructivist program but I haven't seen any reviews by mathematicians (probably because unlike TERC and Everyday, it is not forced on the school system)


Crimson Wife said...

Most of the Miquon users I know use it as a supplementary program. One common combo is Miquon and MEP.

Catherine Johnson said...

Crimson Wife -- Thanks!

I was wondering about that.

Catherine Johnson said...

Good luck with that calculation thing.