I happened to be waiting at the sports centre and was watching a group of kids being shown how to use the climbing wall. There were about 10 kids of late primary/early secondary age. The wall was one of those that had the little different coloured hand and foot holds everywhere. The kids had helmets and harnesses on and took it in turns with the the instructor, who clipped the rope on the harness and talked the kid up the wall. One of two of them went up like Spiderman, some went up steadily and a couple struggled all the way.
But here’s the thing. They all reached the top. The ones who looked like born athletes and the ones who looked like they struggled to get out of bed in the morning. Some would now be able to get up quicker, reaching for hand holds high above them, others would still struggle, however hard they tried. Some required the instructor to continually guide and encourage them, other needed the instructor to tell them to slow down and concentrate on their technique. But they still all got to the top. Every time. And I’d say that they all got to the top quicker by the end than they had at the start.
I was thinking about this when planning some lessons for two different classes and working out how to differentiate them both within and between the groups. And, having thought about it, I decided not to. Both groups needed to learn how to find the angle in a triangle using trigonometry. They were all clearly at different stages in their maths journey, but they all needed to do this. Some would find it easy, some would find it hard. Some would peak at that point; others would go on later to use the techniques with 3D shapes.
So I took the same approach as the instructor in charge of the wall. I provided all the support necessary through worked examples for the least capable of the students to get to the top. And they did. The ablest flew through the exemplars and onto further examples, where they often combined steps. Some of them had to be encouraged throughout – “Look at the triangle in the example, what were the different sides called, how do they match up to the ones in this triangle” and some had to be slowed down to ensure they hit all the points a marker might expect to see. But that was the plan. Everyone got to the top.
The differentiation was not by task, or by outcome (unless you count quantity over quality) but by how much of the support we were able to take away from each student. For some it was none, for others it was all, they could look at the triangle and write down the function they needed to put into their calculator to get the answer. Those at that end of the scale would clearly be able to do similar problems presented to them in future. Those at the other end would (maybe only initially) require the support of the worked examples to get them going.
All students succeed. They feel the good feels that come from having got the answer right. Their attention is not on “I got it wrong” but on “What bits of this process do I have to learn more to get quicker”.
I’m thinking this is now my planning routine. What do the students need to be able to do? What worked examples will take them there? How detailed does it need to be (I’m working on no more that 6/7 steps at the moment)? How generic can the exemplar be (does one example cover the use of Sin, Cos and Tan or do I need one for each)?
The benefits of this is that the planning and the “instruction” are the same thing. The students benefit by having the exemplar. I benefit by being able to use the exemplar across a wide range classes. And trig doesn’t change much so it’ll still be useful in 10 years time. The other benefit is that the examples I want the students to try fall straight out of the exemplar. I’m also thinking that keeping a similar structure across topics will also support learning.
The downside is that the upfront effort feels at the moment a bit more than I would like, but I suspect that’s because I’m over-thinking each example. Once I hit the right level (and find the right tools) it will be much quicker. Also, I guess some might feel that this is a very transactional approach to mathematics. That’s not the intention. I’m still there to provide the deeper understanding of the mathematics and how it joins up. My thought is that a student will be more likely to listen to the proof of why trig works once they know they make it work themselves. Once they know it’s not just something meant for other students.
This methodology is clearly not optimum for some aspects of mathematics, or at least some aspects mandate a looser approach with much more explanatory input by the teachers. Some areas are clearly less amenable to this more algorithmic approach.
When I get more confident with the content I’ll share some on here.