There's a class of simple math problem that most fourth graders will get right and most tenth graders will get wrong. You can come up with any number of examples but I've found this one makes the most lasting impact:

You have four kids and thirteen puppies. How many puppies does each kid get?

The fourth grader will correctly answer "three with one left over." More often than not the tenth grader will answer "three point two five." At this point, I would remind the student that no one should ever get a fourth of a puppy and therefore this is one of those times when it makes more sense to talk about division with remainders.

Asimov (and, I'm sure, many others) observed that mathematical progress often came down to finding ways to allow people to solve more problems with less thought. Mathematicians like Leibniz and Gauss come up with elegant notation that allow us to do much of our work mechanically. Ideally this frees us up to think about more important questions, but sometimes it lets students get good marks and high test scores in math without thinking at all.

Eventually the habit of answering math questions without thinking about them will lead to problems (it's difficult to mindlessly shamble through real analysis), but for K through 12 it can actually be an advantage not to spend to much time asking questions and dwelling on implications. This is particularly true for standardized tests which tend to be time-constrained and focus on problems that can be solved mechanically.

Of course, I'm not suggesting that movement reformers are in favor of the dismemberment of cute little puppies, kittens or any other loveable pets, but, as in other cases, I wonder if they've really thought this through.

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