Dan K on number sense

I don’t have a good definition of number sense and have not researched any references to attempt to find one. Where I would be tempted to apply the term is in the context of error checking or “sanity checks.” In other words, there are errors that students make that you would think would not slip by them if they had more “number sense.”

For example, if you multiply a number by an improper fraction, your product should have a magnitude greater than the original number. So, if I make an error in multiplying 419 by 5/2, I might get a number like 168. Now I make errors all the time, and I don’t catch them all. This one, however, I would catch because I have enough “number sense” to know that I should get an answer greater than 419. You could argue that this is knowledge of some rule about proper and improper fractions rather than some “sense,” but then I would counter by saying that it was my number sense that led me to reach for the rule to apply.

Similarly, a story problem might tell me that a company with revenue of 1.84 billion dollars invested 4% of revenue in research and ask how much was invested in research. I might make an error and multiply the revenue number by 4 instead of .04. Seeing that my result was greater than the total revenue number would trigger my number sense to tell me something didn’t look right. In geometry, when computing the length of a hypotenuse, one should get a number larger than either leg of the triangle. In trigonometry, number sense would recognize that something was wrong if you applied the tangent function when you should have used the inverse tangent.

I guess the antithesis of number sense, to me, is when a student computes the third side of a triangle with sides of length 20 and 30—by the Pythagorean Theorem or the Law of Sines or however—and his calculator tells him that the third side is something nutty like 0.561 long. If he had any number sense, he’d know to not trust that answer and re-check his work. I guess a more specific case might be computing a hypotenuse with the Pythagorean Theorem, but failing to take the square root as the last step. A right triangle with sides 3, 4, and 25 (instead of the correct 5) makes no sense.

That’s my two cents’ (sense?) worth.

I see this kind of number sense as a side effect of good instruction: a natural result of good teaching but not the goal. It's an emergent property. That being the case, I imagine that the kind of Everyday Math estimation homework Concerned Parent's son brings home is most likely a FWOT.

Because number sense develops with proficiency, it's useful as an informal assessment. For instance, if I ask C. what 10% off a sale item is and he has no clue, I know we're in trouble.


These days C's number sense for percent is pretty good, but his number sense for the root or roots of quadratic equations is (probably) nonexistent.

That doesn't tell me we need to rustle up some worksheets on number sense.

It tells me we need more practice solving quadratic equations.