Tactic Training
I’ve recently come up with some ‘Tactic Training’ resources. They are branded as such to encourage students to realise that mathematicians use common strategies to unlock problems. These strategies make the maths required clear. It naturally follows on from the thinking I’ve been doing about problem-solving strategies themselves being a part of our disciplinary knowledge in maths that should be explicitly taught to students.
Students sometimes have the mathematical knowledge, in terms of facts, formulae and conceptual understanding, to solve a problem, but are sometimes unable to make a first step, and it is these ‘tactics’ that help unlock the problem. It stands to reason, then, that the problems we set in order to teach the strategy must be relatively ‘easy’ in terms of the level of maths that is required. This can feel counterintuitive, with problem solving often being reserved for only the top sets or as extension work. These resources are designed to make the problem-solving strategy the purpose of the teaching episode, not just an add-on. Clearly, this can never replace topic teaching. These resources are intended as a semi-regular supplement to maths teaching, not as an alternative.
How to use a Tactic Training resource
Present the first set of problems (branded a “Problem Lab”) to students. Observe how they approach them. What do they do? Are there any they can’t do? But don’t dwell on this; we don’t want students being left directionless for too long.
Hold a class discussion about different approaches: what did students do first? What did the problems have in common? At what point did they get stuck? Which was easiest or hardest? What maths did they use?
Show students the tactic in question. I would do this in the context of worked solutions to the first problem set. The focus is not just on how to solve the problems, nor just on the first step (or the tactic). The aim is to show how the tactic reveals some element of the maths that was previously hidden and how it unlocks the problem into an application of already known maths.
For example, for the area of an L-shaped hexagon, I might say:
“If we draw a helpful line here… we can see that there are now two rectangles… we can find the area of the rectangles separately.”
Or, for a parallel-line question:
“Let’s draw a helpful line parallel to the other two… we can now see that there are pairs of alternate angles… we know that in parallel lines alternate angles are equal, so we can write the angles on the diagram.”
In an angle-in-a-polygon question where the fact it’s regular is vital:
“We can spot the loaded word ‘regular’… that means that these two lines must be equal in length… this shows us that it is an isosceles triangle, with two angles that are equal.”
After this explanation, present the second set of problems to the students, either as another independent task or one question at a time, with questioning as required. It is absolutely fine that they will now be “hunting” for the tactic; when students aren’t sure where to start, we want them hunting for a way in. We are modelling that in a more controlled way.
After this one-off lesson on a tactic, students will need opportunities to practise the tactic. We don’t want to repeat hour-long “draw a helpful line” lessons regularly. But once they’ve had a lesson like this, they will hopefully be able to apply that strategy to problems in future lessons or assessments. As teachers, we can be consistent in our language around these tactics so that students become familiar with them over time.
Creating these resources has been more difficult than I first imagined. Overcoming an ‘expertise bias’ that finds all school-level problems easy is always an issue for a maths teacher. It has been tough even to identify and name the techniques, and that’s before forming explanations and then finding suitable examples and practice questions that sufficiently showcase the tactic. I got on a roll when I was using the NCETM example of using an auxiliary line, but as soon as I was left alone without a scaffold, the “Loaded Word” Tactic Training resource proved harder to create. I think the idea is strong, but the final resource is slightly weaker, although still perfectly usable I hope.
These resources are new: very much a summer holiday project. I’ll be trying them first with my new Year 10 class in the coming academic year. As they¹ say, “No plan survives first contact with the enemy.” I’m not so very sure I have just cause to describe Year 10 as “the enemy” (yet), but the principle still holds true. I suspect I’ll adapt both the physical resources themselves and the way I use them. But I’m certainly going to give this idea a proper try this year.
Have a look at them here, I’d appreciate any feedback.
¹ I looked up who “they” were. Apparently, it was Helmuth von Moltke the Elder, which is a name worthy of such a well-known adage.
