Variation and Example Spaces

I‘ve been thinking a lot about Craig Barton’s wonderful book How I Wish I’d Taught Maths and have been scanning three of his new websites, Variation Theory, Same Surface, Different Deep Problems, and Maths Venns, as well as some research and other books on variation, and a lot of online commentary, in anticipation of starting to implement these ideas in some way.

Writing Algebraic Expressions

As I was reading the last page of Mr Barton’s Book, I was working on instruction around writing algebraic expressions, so this is the topic kind of hovering next to me wherever I go, waiting for when I have time to dig in. This topic is a little more fraught than the purely procedural examples that have been circulating, so it’s worth exploring how variation can be applied to something a little looser.

What does writing algebraic expressions involve (for a beginner)? Well, if I force myself to ignore what other people think writing algebraic expressions involves (essentially ignoring standards and any written material on the topic), then I would say that writing algebraic expressions means to write something like s + 2 or 2 + s when presented with a question like “How old in years will Sam be in exactly two years?”1

This, then, I would call the first example in my example space. Or, rather, an example of an example in the example space—because, if this example is any good, then I will use it as an instructional example to start and leave it out of variation work, which is about PRACTICE, not instruction.2 So, something like this, with the brilliantly simple Silent Teacher method, mentioned in Barton’s book (and a few other places), though without the natural pauses and instructions for students to copy down the correct worked example used during a normal classroom implementation of this.

Try This One

Write an algebraic expression to model the situation.

How old in years was Sam exactly 10 years ago?

I would include a follow-up to this process, here involving a discussion around (a) the idea that the resulting algebraic expression represents an answer to the question of how old Sam will be—it’s just that one part of that expression is not known, (b) asking students to check that the answer makes sense, here by substituting different values for s and comparing the result to the situation, (c) the idea that any letter can be chosen for the variable, and (d) perhaps drawing a visual model of the result (an annotated number line). Some of these could be packaged into the instruction and question above, of course—or perhaps I’ll decide to split this up even more, considering how much “in addition to” I’ve now done about this—but I think that, in general, leaving room for a stepping back step at the end of this is a good idea, to catch the kind of overflow that is difficult to squeeze into expositions like this.

And Now Enters Variation

The paired problem here has opened up a dimension of variation—using addition or subtraction in the expression, so we can play with that during Intelligent Practice (really love that phrase). Technically, the instruction was open to all four operations, but I think it makes sense to focus exclusively on addition and subtraction, leaving multiplication and division expressions for another round.

Here’s what I cooked up.

  1. How much money in dollars did Sam have if he got exactly 10 dollars?
  1. How much money in dollars did Sam have if he got exactly 10 cents?
  1. How much money in dollars did Sam have if he got exactly 2 dollars?
  1. How much money in dollars did Sam have if he lost exactly 10 cents?
  1. How much money in dollars did Sam have if he got exactly 1 dollar?
  1. How much money in dollars did Sam have if he lost exactly 1 dollar?
  1. How much money in dollars did Sam have if he got exactly 50 cents?
  1. How much money in dollars did Sam have if he lost exactly 2 dollars?
  1. How much money in dollars did Sam have if he got exactly 25 cents?
  1. How much money in dollars did Sam have if he didn’t lose or gain any money?

After this, it might be good to have students cut out the strips and place them on a number line.

It’s interesting how much my experience and training rebels against this process. What I want to get to, right away, are the difficult and ambiguous situations. In particular, I started with, and then rejected, a variation sequence involving height: How tall in inches will Sam be if he grows 2 inches? The subtraction variation is bound to confuse: How tall in inches was Sam if he grew 2 inches? That’s tricky.

But knowing about and looking out for those tricky and ambiguous and interesting situations can serve you well creating instructional routines like this. It shows you where you’re going—and your example space can be richer and broader. And if you’re serious about implementing minimally different variation like this, it shows you how far away your knowledge really is from a beginner’s. You just have to learn to have more sympathy for learners who are encountering mathematics for the first time that you’ve seen a gazillion times.

  1. It’s important to me—at the moment, at least—that the examples in this example space should also involve identifying the correct unknown, rather than simply recording the unknown, as would happen with a question like, “Sam is s years old. How old will he be in 2 years?” or with an exercise of the form “2 more than a number.” In both of these cases, the unknown is entirely exposed.
  2. This is an important aspect of variation that I worry will be lost on U.S. teachers. Intelligent practice can’t happen, beneficially, until some acquisition has happened. In 20 years, I haven’t seen a robust public discussion about acquisition. The rhetoric around instruction in the States treats it as just one long assessment, though almost no one realizes that’s what it has become.

Mr Barton’s Book

It wasn’t too long ago—not even three years—that I finished reading David Didau’s terrific book (this one), so I still remember the excitement that I felt reading it, and watching all of the silly certainties of common wisdom in education being dismantled in front of my eyes, making way—I could only hope—for pedagogical practices informed by a real science of learning.

I felt a similar excitement reading Craig Barton’s book How I Wish I’d Taught Maths, because in this book, at long last, are many of those practices in one place, constructed, as readers will see, next to the debris of familiar canards and shallow reasoning that once guided parts of Barton’s teaching.

It is not a book full of proclamations about “best” practice. But you will find in this book a beautiful translation of the science of learning to the classroom. And far from the drudgery that one may imagine this to be, the joy of effective explicit instruction, for both teacher and students, comes through in every chapter of the author’s writing. It is serious, thorough, humble, and humane. And accessible: perhaps the greatest pleasure in reading it is knowing that you could turn around and start to implement many of these practices in short order—or, perhaps, that you already do these things, but don’t know why you should stick with them or how you could improve on them.

I have a lot of underlines and margin notes, but I think these three snippets together, from the chapter on problem solving and independence, are my favorites. The section starts, as they all do, with what the author used to think:

I used to love the sight of my students struggling through problems. Scratching heads, heavy sighs, and even the snap of a pencil thrown down in frustration were the soundtrack to learning. . . .

And then we are introduced to one of these problems, Question 23 from this paper (PDF), along with a deep concern for how novices will handle it. Contrast Barton’s new diagnosis below with common wisdom—that students ask why they are doing math because it is boring, tedious, procedural, or not relevant to their lives.

The task of choosing cards and calculating their totals may prove so cognitively demanding that novices do not have any spare cognitive capacity to recognise patterns. They do not realise that it is not the actual totals that matter, but whether those totals are odd or even. They just carry on regardless. Moreover, students are so consumed with the minutiae of the problem that no cognitive capacity remains to consider the global picture—why are they doing this? The result is that the novices may end up with an assortment of lists and totals, but not actually do anything with it—the fact that this is a probability question was pushed out of working memory long ago when the first set of cards was being processed.

As you might imagine, since the diagnosis is different from that received from common wisdom, the prescribed treatment is different too:

Before I set students off to work independently, I ensure they have enough domain-specific knowledge to solve problems on their own.

Although the snippets above are certainly grist for my mill, How I Wish I’d Taught Maths is not an ideological tome. It is eminently practical, taking the best ideas from all corners of the educational universe, squeezing them through the filter of cognitive science, and setting them in the right proportion to create a firm foundation that any educator—and especially any math educator—can use and build on. I highly highly recommend it to anyone who wants to strive for better in teaching and learning.