Tall, Dark, and Mysterious

2/28/2005

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

File under: Those Who Can't, When We Were Young, Know Thyself. Posted by Moebius Stripper at 8:28 am.

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.

10 Comments

  1. Your students sound exactly like mine. I have been accused many times of giving “IQ tess.” Once, I explained that a certain question was designed to be difficult, so that only the “A” students would get it. A student (who was an Education major) protested and said that I should expect every student to get every question right. I guess that’s what’s wrong with Education schools.

    - William — 2/28/2005 @ 8:47 am

  2. Mommy and Daddy cuddled a lot, she told me, and then the baby got inside Mommy.

    If I was told this at age 5, I would have had disturbing science-fictiony nightmares of babies possessing Mommies. Well done you for growing up to be mentally sane (I guess?)!

    And, you’re right about the tests.

    As a teacher, I do think it’s important that a student knows what is going to be tested. I don’t think that means that you have to hand feed them all the possible question types. In fact, I agree with you - I don’t think you should.

    What I mean is - If it’s important that students are able to recognise graphs and describe their associated function, the teacher ought to specifically tell them that they need to be able to do this.
    Some students - like your 5 year old self - might be fully capable of learning that, but won’t be able to make that connection for themselves: and not (always) for lack of trying.

    - Ronald — 2/28/2005 @ 9:32 am

  3. My best friend from grade school’s mom told her that “Daddy planted a seed in Mommy.”

    When we were in junior high, my friend told me that for years she had this mental image of her parents walking naked onto a stage in a big orange auditorium, where her mother bent over and her father stuck a sheaf of wheat in her butt. This is my all-time favourite misconception of where babies came from.

    As for telling students what to expect, here’s my general test-design strategy, which I’ve cribbed from another instructor at my college: around 70% of the test should be very straightforward, though not necessarily easy. Meaning, if a student does all the homework, asks for help when necessary, then they should get full marks (or close) on 70% of the test. Another 15% should be like the harder homework questions. The remaining 15% should be the “time to stretch” questions: they’re the ones that require students to, say, put together two concepts that they’ve seen individually, but not together. These are the questions that separate the B students from the A students.

    As for that graphing question, I talked at length about how the graph of y=(x-p)^2+q is a parabola with vertex at (p,q), and from that they could give a decent sketch of it. On the test I gave a picture of a parabola with vertex (3,1) and wrote “this is the graph of y=(x-p)^2+q. Find p and q.” After having spent a week talking about transformations of graphs, and giving a variety of questions pertaining to same (though not the exact one on the test), I think that my question was fair. From experience, if I were to say “you’ll have to recognize graphs and describe their associated functions,” I’d have summarily been pressured to show them examples of exactly that, which would have defeated part of the purpose of the question.

    - Moebius Stripper — 2/28/2005 @ 9:56 am

  4. Yes, many of my students believe that memorization and pattern-matching are the essence of a college education. Such foolishness the young (and some not-so-young) people have. Of course, whenever I give them a problem that differs even a little from the examples and homework exercises, I am accused of trying to trick them. No, I just want to see if they understand what’s going on! And even if I don’t give them any significantly different problems, I’ll get the complaint I got this morning: The word problem involving a kayak going upstream and downstream in a river was too different from the problem we worked in class during the review session the day before the exam, as that one involved an airplane traveling with a headwind and then a tailwind. (!)

    - TonyB — 2/28/2005 @ 1:21 pm

  5. That is an awesome story, and it applies equally to the sciences, I can assure you.

    (And Precalculus Bingo? Brilliant…)

    - dr. dave — 2/28/2005 @ 6:56 pm

  6. http://pratie.blogspot.com/2005/03/get-yer-student-evaluations-here.html
    …many of my academic friends agonize over the written evaluations students make at the end of every class. Lately there has been some generous sharing of exceptionally stupid and/or cruel student comments, sent by amused and/or bitter profs, at bitch…

    - Pratie Place — 3/6/2005 @ 3:57 am

  7. […] e a penis graph on a TI-83″. Check it out - I’m number six. And seven. Back on February 28 I did indeed mention both the penis and the TI-83+, but apparently five people d […]

  8. My name is Jay and i would like to say that i really respect what you do. I am fourteen years old and i myself wish to become a college professor in the math area. I am in ninth grade and am planning to take two math classes every year until i am out of high school. I already find math to come natural to me and therefore am in a class aimed toward juniors that are gifted in mathmatics. I think that you have a really good point and that you should keep formatting the tests in this manner. My teacher does the same thing and even though the other students hate the way she does things, i like it and think that it’s the best way to get us to actually learn things.

    Good for you and if you have any pointers for me, feel free to e-mail me.

    - Jay Sherman — 4/20/2005 @ 6:48 pm

  9. Hi, Jay - welcome and thanks for the compliments. Do you want pointers about anything in particular?

    One thing I can think of offhand is that you’ll definitely be working with students for whom math doesn’t come naturally. Take advantage of the questions your classmates ask in class - often, when I hear a question from a student, I’ll think, “hey, I didn’t even realize that someone could be confused about THAT.” As I get more experienced, I think that less and less, because I’m growing more familiar with the many layers of math that can confuse people. This is very challenging for those of us for whom math feels natural, and it’s good to be mindful of this aspect of the material. (That said, if you keep following a math-y career path, eventually you’ll be taking some math classes that you’ll struggle through - this is a guarantee.)

    - Moebius Stripper — 4/20/2005 @ 9:29 pm

  10. Thanks for the advice, i’ll make sure to use it any chance i get. I already help some of the other students in my class when they have problems and i have thought the very same thing about how there’s no way they could get confused by something like that. One of my mother’s coworkers has a son who is struggling in eighth grade math and i have offered to help tutor him; no word back though.

    I would also like to say that i am looking for pointers about almost exactly what you explained in your response. I am trying to learn a little more about how to teach someone else about mathmatics without assuming they know something that they have no idea about. In short, i want to make sure i don’t skip something they need to know.

    And thanks for the other pointers, too.

    - Jay Sherman — 4/21/2005 @ 12:12 pm

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