# Divisions

Are graphing calculators introduced too early? Most of the would-be math teachers in ed class with “John Dewey” thought so till a contrarian spoke up.

. . . he really couldnâ€™t see what cognitive value of teaching students the procedure for multiplying 36 x 7 when calculators were available. I was unable to keep my mouth shut. â€œDonâ€™t you think that students need an understanding of basic procedures and that place value is an important concept?â€ â€œWhy?â€ he remarked and went on to the uselessness of learning long division at which I drew the line and said â€œHow can you say that? Donâ€™t you think the distributive property is worth talking about?â€

â€œWho cares?â€ he pointed out.

In Portland, the student representative to the school board objects to the new constructivist math curriculum based on CPM textbooks. The curriculum director responded:

“In the past you just had a calculator, a book,” (Marcia) Arganbright said. “This has strings and blocks and hooks and rubber bands, it’s more like a lab.”

Ah, the good old days when students just had a calculator and a book.

Meanwhile, educators in Maryland and Washington, D.C. are trying to focus the math curriculum on the main concepts students should learn rather than introducing dozens of math topics without teaching any of them to mastery.

In the fourth grade, for example, Focal Points trims the list to three essential skills: multiplication and division; decimals; and two-dimensional shapes.

Virginia lists 41 “learning expectations” for fourth-grade math students in its statewide Standards of Learning. Maryland lists 67 in its Voluntary State Curriculum. The District has 45 standards.

Math educators are looking closely at Singapore, Japan and other countries where students can multiply 36 x 7 without a calculator.

1. Is my generation one of the last to understand how to do math without a calculator. while I enjoy the frequent use of a calculator because it saves time, I can understand the way the thing works and how much of a time saver it is.

One thing is for sure, my daughters will know how to do math, including graphing, without a calculator. What happens if your batteries die or there is not enough light to power your solar powered calculator?

2. Kris says:

When my son took the placement test at the local community college he wasn’t allowed to use a calculator. He had been using a calculator for math since elementary school and totally bombed the the test. Fortunately they did let him retake it after a few weeks of reviewing things like long division. A little consistency between schools and districts would be good.

3. Fred says:

Isaac Asimov wrote a short story back in the 50’s about a future war that was won by the rediscovery of how to do arithmetic on paper. Coming soon to a world near you?

4. Fred says:

Asimov: “A Feeling of Power” (thank you Google).

5. Our daughter, when about five, insisted upon going to Sears for buy an item she wanted, because the TV said that “Sears has everything”. Sears did not have the item she wanted. She has taken TV advertising with more than a grain of salt ever since! I think that insisting that TV ads be held to the truth is a Very Bad Idea! Every child should quickly learn that advertisers lie. A wonderful real world lesson.

6. wayne martin says:

> Math educators are looking closely at Singapore,
> Japan and other countries where students can
> multiply 36 x 7 without a calculator.

Although doing this “right” requires some 7th or 8th grade algebra to explain, (7 x 36) can be decomposed into simpler numbers which kids should know from their multiplication tables:

(7 x (30 + 6)) => (7 x (3 x 10)) + (7 x 6)

(7 x 3 x 10) + (7 x 6)

7 x 3 = 21 (x 10) = 210
7 x 6 = 42

210 + 42 = 252

7. Catch Thirty Thr33 says:

I have no problem with calculator use…what I DO have a problem with is that it is not treated as something of a privilege that is earned after, and ONLY after, you understand forwards and backwards what you are short-cutting using a calculator. The over-reliance on calculators and other computing devices is becoming a problem.

8. Charles R. Williams says:

The biggest barrier my pre-calc students face is their inability to manipulate symbols according to the rules of algebra which of course are normally learned in arithmetic procedures.

I am also noticing more and more that students will write meaningless gibberish down on the paper while problem-solving. This is some kind of stenographic notation that corresponds to their own thought processes and what they are doing on their calculators. Of course, nobody else can make any sense of it and I wonder if they can figure it out themselves half an hour after they try a problem. Because they can read and write in the language of mathematics their textbooks make no sense to them.

Often students are hampered in making productive use of their fancy calculators by ignorance of the rules for order of operations. I think this comes from early and routine use of calculators to solve single step problems and the bad habits they pick up doing this.

9. Alex Bensky says:

This is becoming a widespread and serious problem. I offer anecdotal evidence:

A few years ago I was substitute teaching in a very affluent and generally very good suburban Detroit system. One day I had a seventh grade pre-algebra class. Let us say that math was not my long suit when I was that age. I think subconsciously I learned enough to do baseball averages and figured that was enough.

Anyway, the class was having trouble with a set of problems that required them to find between which two integers the square root of a number would lie; they weren’t required to find the square root itself and for much of the class the problems were difficult.

“Look,” I said, “if you think about it, it’s easy.” I looked at one of the problems, say 46.7, and wrote that on the board.

“Now,” I continued. “Five times five is twenty-five and that’s lower.” I wrote “25” to the left of the number. “And you know that six times six is thirty-six and that’s also lower.” Again I wrote “36” to the left. “And since seven times seven is forty-nine, and that’s higher, your answer is ‘between six and seven.'” I wrote “49” to the right of the number.

And I was stunned when it turned out about half of the class could not do basic multiplication tables. I didn’t say anything else but I wanted to say, “Gang, you should always make sure you carry spare batteries for your calculators or you won’t even be able to figure out which size of laundry detergent is the best buy.”

10. sharon says:

My 10-year-old just did 36 x 7 in her head in about 4 seconds. We haven’t allowed calculators for elementary school math, and have been using an old 1920’s math book that requires very fast crunching of large numbers (a 5th-grader is expected to “Divide 21462 by 732” and “Multiply 5736 by 3916” accurately in less than a minute).

Recently she was horrified to see an 18-year-old acquaintance–a graduate of our local high school–rely on a calculator to cube 3.

Are graphing calculators being introduced too early???? Well duh!! If it’s used as a crutch, it’s too early. Why are calculators so important? So Algegra teachers don’t have to teach arithmetic!

I’m not totally a luddite when it comes to calculators, but if I taught higher level math at the HS level, a calculator would not either necessary OR mandatory. Needing a calculator for Algebra I (or before) makes me sick.

12. Devilbunny says:

By contrast, there were some days in my high school experience (graduated ’93) when we were required to use the sine/cosine/tangent and logarithm tables *in the back of the book* in order to do simple trig. Not sure I learned anything from that, except that I’m really glad I didn’t have to do that kind of math in the days before cheap scientific calculators.