Appropriate math questions

I had an interview for a middle school math position (why they called me, I dunno, I’m sure I was just filler), and had to bring a sample lesson plan.

I always create worksheets when asked to bring lesson plans, since it gives a better idea of how I teach. I read the standards and came up with this cartoon as a starting point:

I use buying decisions all the time in math classes, because the phrase “Pretend that it’s your money” has a near-magical effect on kids who would otherwise ignore word problems.

So I’d kick off a discussion by asking the kids to identify the difference between the two questions. Most kids of any ability will realize that the girl’s question has one answer, while the boy’s question has several.

From there, I drill down to the boy’s question. (click on the image for a slightly better view). Don’t get too quibbly on the definitions—math purists are insanely annoying.

Now, here’s the thing: this is not a lesson on functions. This is a lesson on variables and generating tables. It’s seventh grade math, not pre-algebra or algebra.

But in order to get kids to think about variables and tables, I find it’s helpful to start with a familiar situation and lead it back to math. In order to explain the difference between the girl’s question and the boy’s question, I have to introduce functions, which leads us to variables.

In my opinion, teachers don’t spend enough time leading kids through the process of identifying the definition of the unknowns (because trust me, if you say “identify the unknowns”, half the class says “Three Cheeseburgers!” and suddenly you’re in a Bill Murray skit.) and generating a table of solutions to the problem. Teachers see these as obvious, as mere activities that we use to solve problems. But low to mid ability kids see nothing obvious about defining variables and generating tables. I’ve been slowly realizing this over time, and it crystallized into my data modelling unit in my Algebra II class. This activity, which I drummed up for an interview (but will undoubtedly use at some point), was my attempt to move it back to a pre-algebra state.

Many teachers who only work with high ability kids would see this handout as ludicrously easy and in fact, I’d hand this worksheet to my top seventh graders and tell them to go it alone, while I talked the rest of the class through. Top kids should figure this out fairly quickly, and on day 2, when the rest of my kids are still practicing simple situations, they’d be on to more advanced scenarios (giving two points and figuring out the rule, giving them a table, and graphing).

But many teachers who work with lower ability kids work hard to find meaningful questions and yet miss the mark by making the question too complex. I was talking to one of my curriculum instructors, and he asked me about a lesson he was planning to give to middle school students in a low income district. He wanted to give kids fractions like 7/20, 8/25, 9/30 or 8/30, 9/35, 4/15 and have them just ballpark whether the fraction was greater or less than the obvious value nearby. So is 7/20 greater than or less than 1/3? and so on.

I told him that he was dramatically overestimating their ability, and recommended he start by doing the same thing with halves. So a list of numbers: 2/5, 9/17, 5/11, and so on. Greater than or less than one half? The kids who can do that quickly, move onto numbers around 1/4. Only after a few iterations should he give the strongest kids the exercise he was thinking of, which most of the kids wouldn’t be able to do, full stop. But he could build the weaker kids ability and fluency simply by getting them to think about greater or less than one half. I advised starting with the much simpler exercise. If I was wrong, he could have the more challenging activity ready to follow up, no harm done.

He took my advice and reported back. Most of the kids never got beyond the over/under 1/2, finding it challenging and meaningful. Most of the teachers he was working with had been sure the activity would be too easy.

I do think it’s important to set open questions for all kids, not just the top students, to let them tussle with before you settle into working problems for practice. However, the questions must be tailored to student ability, and math teachers dramatically overestimate the ability of their kids to work with these questions in a meaningful manner.

It gets old to be told that acknowledgement of low cognitive ability is “setting expectations low”. One of the comments on my “myth” post was “One question that I have is the extent to which students are playing Whack-a-Mole in order to get easy classes for a week or two.” My original response was rather rude, so I changed it to a simple “No”. It boggles my mind that anyone would think kids are faking it.

At this point, some teachers are remembering the time they taught their algebra intervention kids about logarithms, or complex numbers, or trigonometry cycles. I’m not saying you can’t teach it. They just won’t remember it. It will be as if they’ve never been taught.