Day 3 of 30: Number Sense? Do You Have It? Can You Get It?

One of my biggest problems in the olden days, i.e., ten years ago or so, which were still "new" teaching days for me, was a lack of Number Sense among my students. This was before the advent of mandatory class sets of calculators, in those dark, Byzantine days when I used to have them -- Horrors! -- work it out! However, with the introduction of decade-old technology (by the standard of those days) ... not much changed. Sure, some problems were alleviated, but others rose to take their place. And the new ones were more confounding because they were less likely to notice something went wrong.

Before I continue, let's come up with a working definition.

What is Number Sense?

Basically, it's the ability to estimate an answer, to "ballpark" a figure, to look at a solution and know if passes the Smell Test, whether or not the answer is on the money. It should be a set of alarm bells ready to go off in your head when the cashier hits the wrong button or the waiter "accidentally" carries an extra "1" to the hundreds column. (New Year's Eve, back in the 90s -- my wife caught it when the "per-person" share of the bill was over $10 more than anything that anyone at the table had ordered! Even with drinks, tax and included gratuity, something was amiss!)

I had a simple check back in those old days. Tell them not to call out because I was going to call on someone. Then I wrote a problem as long division: 4 into 82, and waited about 30 seconds. (They didn't have to do it as long division, but many took it as a suggestion.) Instead of calling on someone, I raised a hand and asked, "Who got 2.5?" Five or six hands would go up, along with a couple of should-I-shouldn't-I hesitations.

I then showed them where the mistake was. There'd be "oh"s and "gotchas" and laughs at how people could be so silly. And then they'd make the mistake again. The thing is that had I asked them for 4 into 80 or 4 into 84, they would've gotten it right. They would know 4 into 80 isn't 2, and there's no 0 to forget in 84. Maybe it was the question, maybe it was how it was framed.

The problem for me is that I know that me students have some number sense. The problem is getting them to use it the right way. For example, when I introduce word problems, I tend to start with simpler problems with "easier" numbers. This makes it easy for them to see that they are getting the right answer. Unfortunately, their Number Sense is telling them that the answer if 14, 15, and 16 and that's all they need to know. They can't explain why it's right (other than guessing and checking), and they don't see the point of doing it my way. Then I try a harder problem, and they're stuck because they didn't learn the correct way to approach a problem.

Algebra isn't about being able to solve the problems you did in Junior High. (Or earlier, if you're teaching Algebra in Junior High. And if you are: God Bless You! Me? Never again.)

Scientific calculators have eliminated some of the problems, but there are new ones. For instance, there's little point in quizzing them on the Order of Operations if the calculator will do it for them. I can test their calculator knowledge by making the key entry a little trickier (e.g., fractions with subtraction in the denominator), but they won't know until they get their papers back that their answers are incorrect. And then there's the biggie, the one that kills them with evaluating any quadratic: squaring a negative integer.

We all know that (-3)² is +9. If I were to say this out loud, I would say "Negative three squared is positive nine." But the calculator has to know that you what "negative three, squared" and not "negative, three squared". This is made more confusing by the fact that there are separate keys for subtraction and the sign and the fact that the sign is considered part of the number (it's what makes it negative), but the convention is (and the calculator follows it) that if there is a "minus" and an exponent, the exponent wins. The problem gets compounded by the fact that even if the student is aware that a negative squared should be positive, if they evaluate -3² + 5(-3) - 14, they won't know what to expect for an answer, so no alarm bells go off.

Score one against the calculators.

How can we get student to develop this without endless drills? I don't know. Practice helps. Endless drills? After a decade of this, I'm starting to wonder if there isn't some merit in them, other than the fact that good students are so nice, they "help" the unmotivated students. (I'll be nice, too, and say "unmotivated".) When you see numbers in action more and more, they start to make sense. But even older kids aren't immune.

I have junior and seniors in a Financial Algebra class. On a recent test, a couple of students gave me incorrect answers like "$6,170" for the total of a bar graph. All of the numbers in the graph ended in "00". The answer couldn't end in "70". My first guess was confirmed when I realized that they were off by $630 -- they entered $700 as $70 and never noticed the mistake. Imagine doing that on your taxes! (We filled out 1040EZ forms a month or so ago.)

Likewise, these students still have problems with complicated fractions in graphing calculators. I have to add extra parentheses to formulas because they forget to use them otherwise. The students working with the four-function calculators don't have this problem. They break things down step-by-step, but they fall behind because they are doing so much work. However, a few of the students don't have to worry: they have the latest operating system on the calculator that enters fractions and exponents so that they look like you expect them to, without requiring the extra parentheses.

It looks good. I just wonder what new problems it will create.