I keep hearing the following argument from different places: Why bother to teach addition/subtraction/multiplication/division? We have calculators to do that for us. Why waste kids’ time teaching them stuff machines can do?
Well, I think I can tell you why. I know I won’t convince the world, but I hope you’ll at least consider the reasons I believe teaching arithmetic is extremely important.
- Students who don’t know arithmetic will have difficulty developing their estimation skills. This is a large handicap, since students who can’t estimate can never develop a sense for a “ballpark” answer. They never learn to tell when they are on the right track.
- By not teaching arithmetic, we make arithmetic seem deep and mysterious and beyond anyone’s comprehension. These things are not difficult; they are just time consuming, but they are less time consuming for people who develop their arithmetic skills with practice. Arithmetic is only difficult for people who never do any.
- We need to know the arithmetic in order to make things concrete. Math is abstract enough without taking all the numbers away. If you don’t know how to multiply or divide, the idea of a remainder, a factor, or a quotient becomes very difficult to grasp.
- I can’t think of many things more crippling for a math student than total dependence on a calculator. Calculator dependency is already rampant in our schools even though most schools (I believe) still teach the usual algorithms for multiplication & division. What would happen if we stopped teaching the algorithms? I think our students would lose all sense of numeracy. Would you believe I have actually seen students punch in “17 x 1″ and other simple expressions on calculators? It’s true! And the one who tried to punch in “17 x 1″ actually punched in “17 - 1″ and then tried to convince me that 17 x 1 = 16 because the calculator said so. This was a college algebra student of average intelligence. She wasn’t naturally stupid, but she had learned to trust her calculator instead of her brain. What will happen when students have no choice but to trust the calculator?
Some people say that the division algorithm we teach in school is too frustrating and complicated. I can go along with that to a degree. I’m not sure if it’s necessary for kids to learn how to work with three-digit divisors. But I know this: They retain less than you teach them. So if you want them to remember how to work with two-digit divisors, then maybe we better do at least a little bit with the three-digit ones. And if they haven’t learned multiplication or division algorithms of any kind, it becomes difficult to talk about other important concepts such as prime factors, divisibility, division with decimals, and percents.
One reason people think algebra (and therefore all other math) is hard is that they get bogged down in the calculations and can’t focus on the algebra concepts. But knowing arithmetic would help solve this problem! The weakest algebra students are the ones who have to think the longest about the arithmetic. The ones who pick up the fastest are the ones who can zip through the calculations.
Even though it may be painful for a little while, the results are worth the effort. Our students need to know how to add, subtract, multiply, & divide for the same reasons they need to know how to spell even though we all have spell-checkers now.
Check out this short story by Isaac Asimov about what it would be like to live in a world where everyone was completely calculator-dependent.
8 responses so far ↓
Carnival of Mathematics IX « JD2718 // June 2, 2007 at 8:24 am
[...] (Math Notes) Addresses why we teach Arithmetic and about how we All can learn the Abcs. (elementary mathematics [...]
Susan B. // June 2, 2007 at 10:07 pm
I thought of that Asimov story as soon as I started reading this.
Very well said. So much of math is mysterious to so many people already that allowing kids to grow up without even considering basic arithmetic just makes me shudder.
lizalee // June 3, 2007 at 10:02 am
Basic arithmetics are so important to just understanding how the world works. Thank you for such a clear defense of the importance of math.
Rolfe Schmidt // June 4, 2007 at 5:59 am
I pretty much agree, but I think it is important to separate understanding arithmetic from performing algorithms. I believe that the algorithms are often presented too early and too rigidly. If a child really understands and plays around with the ring axioms for a while, estimation and algorithms will come easily. I’d go so far as to say that when a student is ready for the algorithms (especially division), they are probably ready for algebra too. There are many interesting and simpler topics in Mathematics that can be taught in the meantime.
Personally I never use the algorithms as I was taught in school. Instead I always start with the most significant digits and refine my estimates. Do the interesting part first, and stop when you know enough.
Alane Tentoni // June 4, 2007 at 6:46 am
I pretty much agree, but I think it is important to separate understanding arithmetic from performing algorithms.
I agree!! When my daughter was in 2nd grade, before she learned to multiply, she and I talked about a way to add the same number to itself many times. I gave her word problems, not 8×2 and such, but situations she could relate to, and she saw the need for multiplication before I taught her how. And we talked about division as a way to undo the multiplication, again, in the context of word problems. When the numbers got too big, I let her do it on the calculator, not as a permanent substitute for the algorithm, but as a way to let her understand the benefit of division without getting bogged down in details she wasn’t ready for. As a result, she had no trouble with the algorithm when it was presented the next year.
I believe that the algorithms are often presented too early and too rigidly. If a child really understands and plays around with the ring axioms for a while, estimation and algorithms will come easily.
Also a really good point — and you know why they are presented so rigidly: state tests and NCLB. How I hate memorized, mechanical math, but teachers resort to it because they have to cram 80+ objectives into 36 school weeks or the test scores suffer. It’s wrong.
Rolfe Schmidt // June 4, 2007 at 7:25 am
I also find that young kids respond better to word problems than to questions like “what is 8 times 2?” I don’t know if it is because the words help create models in their minds or if it is just because a conversational environment is more relaxed. But informal talk sure seems to help. It is interesting that later on word problems seem to cause so much trouble.
I won’t get started on the testing now. I could rant for ages and I’d rather talk about good things.
Why teach arithmetic » eon // June 9, 2007 at 8:06 pm
[...] a very nice post by Alane Tentoni on the topic, together with a link to a great story by [...]
lee // June 10, 2007 at 9:59 am
I had an experience a few years ago while buying several cards in a local shop. The teenage clerk added up all of my cards on her cash register and gave me the total price. I pointed out that it could not possibly be right! In total confusion, she asked how I could possibly know that! I explained that I had estimated the total. She thought about that for a while and finally agreed that maybe she should add it up again. What fascinated me about the experience was her complete astonishment at the idea that someone would actually do arithmetic in their head….
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