We felt very nice and snug, the more so since it was so chilly out of doors; indeed out of bed-clothes too, seeing that there was no fire in the room. The more so, I say, because truly to enjoy bodily warmth, some small part of you must be cold, for there is no quality in this world that is not what it is merely by contrast. Nothing exists in itself.
– Herman Melville, Moby Dick
Variation theory: you can’t fully understand a concept, unless you understand what it is not.There are loads of brilliant applications of this in teaching. Kris Boulton first introduced the concept to me via GCSE students who were unable to identify this shape as a pentagon:
The reason? While they might have been told that a pentagon is a shape with 5 sides, what they usually see accompanying this explanation is:
and it’s this regular pentagon that sticks in their minds as a ‘pentagon’, leaving them at a loss when faced with naming the unfamiliar shape above.
So, when defining the concept of a pentagon, it’s most helpful to show numerous examples of various irregular pentagons, together with shapes which are not pentagons, so learners can develop a full concept of what ‘having 5 sides’ means and doesn’t mean. It’s another way of addressing the problem Greg Ashman highlights here:
To experts, it’s obvious that the shape above is a pentagon, due to a thorough understanding of pentagons through many experiences; yet to communicate only that “a pentagon is ‘a shape with 5 sides’ that looks like ⬠” means that many learners will link the concept only with what’s been highlighted – the regular pentagon.
In this post I’ll look at one easy way to apply variation theory in designing tasks: adding peripheral questions to tasks.
I’ve experienced this problem lots of times. For example, students not being able to identify perpendicular heights in shapes (since they can be shown as a side of the shape, a dotted line inside the shape, a dotted line outside the shape, and in any orientation!); similarly, exam questions such as ‘Bill is 10 years older than James. Write a simplified expression for their total age’ often get answered like this:
which underscores that I don’t teach the concepts of ‘simplify’ and ‘solve’ well enough.
The most immediate and natural application of variation theory is in how to explain things – illustrate definitions by showing lots of examples and non-examples – there’s a really simple way to make use of it in tasks: constantly include peripheral questions within tasks. (Another motivation: don’t just explain concepts and non-concepts but go on and test them too.) Here’s what I mean. The following task is taken from a first lesson introducing multiplying algebraic expressions. It follows a sequence of lessons on adding and subtracting terms. If your experience is anything like mine, you’ll know that in tests and beyond most learners repeatedly get these two operations horribly, horribly confused. So, it’s a natural place to focus on applying variation theory.
As you can see, the task includes both adding terms as well as multiplying them. Since the lesson is focused on multiplying terms, the questions on adding terms are thus the peripheral questions I’m referring to – questions which have been learnt successfully before, and which are related (and often confused) with the questions relating to the LO. In my explanation I applied variation theory by highlighting the differences between multiplying and adding; so now it’s being tested in the task. Some caveats: this task is NOT the first batch of questions that students should do; I would only use it following a large selection of mini-whiteboard questions, in which I’d throw in a few addition questions and expect quite a few students to get it wrong (at which point I go through the motions of thoroughly chastising them, hopefully ingraining it in their memories).
It’s as simple as that, and as a result of its simplicity, I’m increasingly applying it across all of my teaching. At AS level, throw in a sneaky arithmetic sequence or two in tasks testing summation of geometric series. In year 7, after you’ve taught another fraction operation, make sure you throw in a few questions on the previously taught operations in the main task.
How do students respond? From my experience, they start off very uncertain and unsure when they reach their first peripheral question; many hands will go up, but I simply encourage them to stay the course: ‘what’s the operation in the middle? Can we simply F + 3? What did we do last lesson?’ And soon they’re flying through, while thinking carefully, trying not to get caught out, while also expressing great satisfaction: ‘sir nice try, you didn’t catch me out on that one!’ Now, I also want to highlight: their confusion at the first peripheral question is a clear sign to me that this is essential practice. Yes, it might be confusing; yes, there is the risk of introducing peripheral questions too early, before learners have a solid-enough grasp on the main learning objective. However, we surely want students to reach a place where they can confidently distinguish similar-looking concepts and processes. Including peripheral questions is one key way of doing this. Yet in my experience, most worksheets and textbooks don’t include anything like this in their tasks.
Some practical tips from my experience so far:
- Keep the peripheral questions as simple as you can. Since the LO is multiplying expressions, now’s not the time to throw in a collecting like terms question that involves different powers and fractional coefficients. Otherwise it’s much harder to isolate the source of the conceptual confusion.
- Make sure you first include variation and peripheral questions in your explanations, too! Otherwise the resulting confusion will be chaotic. If your class are a bottom set or near it, it might be even better to leave such tasks for a plenary, or right at the end of a lesson.
- Again, don’t throw them straight into a task with peripheral questions; include a warm up task which just focuses on new aspects of the process/concept, till that starts getting established. Again, make it easy to isolate the source of conceptual confusion.
- Some possible examples where it might work well: include some simple linear equations to solve, when teaching solving quadratics – so students start to tell when they should use quadratic methods; when drawing histograms or cumulative frequency curves, add in a few ‘draw a frequency polygon’ questions; when teaching factorising into double brackets, throw in some questions where it only factorises into a single bracket; likewise with expanding; when teaching finding area of shapes, include questions using shapes they’ve learnt in previous lessons…