Everything I Ever Needed To Know About Setting Math Tests, I Learned From My Five-Year-Old Self

Every semester, I get complaints from students who take issue with my testing style, which requires them to avail themselves of cognitive functions more complex than those of memorization and pattern-matching. The standard complaint is a variation of “some of the questions on the test are different from the ones you gave on the homework, and that’s not fair.” I have a variety of explanations and analogies I use for this purpose; these days, I tend to explain that I want to see my students apply what they are learning in my class, and I point out that there are very few jobs in which one’s boss will assign only tasks that are identical to ones that he or she has outlined step-by-step before . None of those jobs, I tell them, require university educations. In fact, I point out to the students to whom this is relevant – many of my students are in my class so that they may have career options beyond the dull, low-paying jobs at which they worked for a decade or so.

But I’ve never used the most apt analogy available. My most compelling reason for testing beyond the exact material I presented in class has its roots in my early childhood.

When I was four and a half, five years old, my mother was pregnant with my brother. Five is an awkward age for this sort of thing – it’s old enough to see through the stork explanation, but too young to really understand the nuts and bolts of conception. My mother’s first-pass attempt to negotiate that murky territory between Satisfy Child’s Curiosity and They Don’t Need To Know Everything Quite Yet was, I presume, a pretty standard simplification of affairs. Bypassing the nitty-gritty of it all, Mom’s explanation segued right from foreplay into conception: Mommy and Daddy cuddled a lot, she told me, and then the baby got inside Mommy.

This made sense to me at first, and I, wanting a baby of my own, decided to apply my new knowledge. I don’t think I’ve ever greeted my father as affectionately upon his arrival home from work as I did that week during my mom’s third trimester, and both of my parents were quite touched by this display of love until they discovered my ulterior motive. The limitations of the cuddling story thus revealed, my mother sat me down beside her, and explained the whole damned thing to me, or close enough. I don’t remember exactly how she presented the story, but I do recall that it involved a penis and a vagina and a sperm and an egg and a uterus, and all sorts of other cool stuff I had never heard of before. I was fascinated. (I was also, I should mention, touched by how much my parents must have wanted babies if they were willing to go through all that to have us. Not just the pregnancy and labour part, but the rest of it, too.)

This was a real mouthful, especially for a child not quite five, so during the course of the story, my mother would periodically ask me questions to ensure that I was following:

So tell me, honey, where does the baby grow?

The u-ter-us.

Good, sweetie, that’s right!

– and then she’d move on to a story about the umbilical cord or somesuch.

My mother was wonderfully impressed with how well I was taking it all in, and she regaled her friends and relatives with stories of how her precocious not-quite-five-year-old daughter actually understood how babies were made.

This all exploded a few days later, when I approached my mother earnestly with a question. I’d been thinking about this a lot, I said, and I understood all the stuff about how the baby gets made, and how it gets fed, and where it comes out, but there’s something I just don’t get.

What is it, sweetie?

Mommy, I said, when it’s time for the baby to be born, how does it jump over the sperm and the egg?

A few years ago, when my mother told me this story, she reflected at length on how she’d really, truly believed that I understood the human reproductive process, and then – that question. I could repeat parts of the story back to her, but I had missed the point utterly.

I know now, however, exactly how my mother felt. I don’t have a five-year-old and a negative-point-seven-year-old, but I have four math classrooms full of first-year students. Many – nay, most – of them put long hours into their homework, and a majority are reasonably proficient at producing answers to homework problems, as long as their friends, their tutors, or their instructors have worked out identical examples before. Many of those same students have no true understanding of simple algebra. They can repeat things back to me, but they have no concept of the hows, the whys of it all. I’d be a negligent teacher if I allowed that group to excel without addressing the huge gaps in their knowledge.

And this is what accounts for my (firm! non-negotiable!) philosophy of giving tests containing questions that differ ever so slightly from the examples from class and from the homework. In testing my students thusly, I’m not trying to trick them; I’m merely trying to gauge how well they know the material, rather than simply how well they can memorize the limited set of problems I do in class. In one class, for instance, I discussed how one can produce a graph by seeing how the function differs from a simpler, more familiar function. I did several examples involving parabolas, absolute value, and square root functions, and I assigned some homework on the topic. On the test, instead of giving them a function and telling them to give me a graph, I gave them a graph and told them to find the function. This allowed me to distinguish between the students who merely sleepwalked through the assignment, and the students who actually thought about it.

This approach has made me unpopular with a vocal contingent of my students, many of whom see no tension between their bald admissions that they can’t add fractions, and their routine claims that they deserve A’s and B’s in my class. I will not bend on this issue, and on occasion I toy with the possibility of telling them the story from my childhood that gave rise to my dogmatism.

Yes, I’d say to them, you did the homework, and you can do questions that are identical to the ones I did in class. Questions like that test if you know that the baby grows in the uterus.

But getting you to repeat the homework questions that I showed you isn’t enough to convince me that you realize that the baby doesn’t jump over the sperm and the egg, and that’s at least as important.

At some point, my mother brought home Where Did I Come From, which I still think provides a marvellous, age-appropriate, and quite thorough explanation of above. I highly recommend it.

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