Research School Network: Why sometimes the best goal is no goal … Using cognitive load theory to support maths teaching

Out of all the maths I teach, there’s one type of problem that a) I enjoy completing the most and b) would always surprise me that students found difficult: angle problems. For example, take the question below:

I love this type of question because there are a few different ways we can approach it but ultimately, all roads lead to Rome/‘x’. It’s a great way of showing maths not as discrete ideas, but this interconnected web of fundamental principles. The problem though, was that my students didn’t always share this outlook…

The challenge

I knew my students could answer this question. I’ve seen them answer questions that involve the component aspects of this question before, and wonderfully so. But for some reason, they would get to a question like this and just draw a blank, ​“forget finding x sir, I don’t know where to start”. It appeared that, despite possessing the requisite knowledge, students couldn’t even attempt, let alone answer the question. But why did this happen? One suggestion comes from cognitive load theory (CLT). One of the findings from research about problem-solving is that, when presented with a problem, novices tend to use a strategy called ​“means-ends analysis” [1] which involves thinking about the end-point (i.e. solution) and trying to work backwards to figure out where to start. This places a ​“load” on our thinking, because now we’re having to process [2]:

1. The end-point
2. The starting-point
3. How to get from the starting-point to the end
4. The information from the question (e.g. the diagram, values, text etc.)

And finally, they have to execute! This cognitive load is significant and often just too much, resulting in students feeling overwhelmed and displaying opt-out behaviours (e.g., giving up, lack of attempting, visible signs of frustration).

Why sometimes the best goal is no goal…

So, for the question above, if our students can’t immediately see the path from the starting-point to the end, they are likely to be unsuccessful. But what if there’s no end-point?

What Sweller and colleagues found was that, ​“A nonspecific goal eliminates the possibility of using a means-ends strategy to solve the problems.”[3]. By replacing the specific goal (e.g. ​“find x”) with a nonspecific goal (e.g. ​“find as many of the missing angles as you can”), we remove the option of students being able to use a means-ends strategy. So, all we have to process now is:

1. The starting-point
2. The information from the question

This removes the requirement of a) having to figure out the end-point, b) having to find paths to the end-point, and c) having to check whether the working we’ve done has got us any closer to the end-point. In doing so, it lowers the cognitive load and means we can focus more on the concepts we do know and how we might use them, rather than the answer we don’t know and how we might find it.

So next time, try letting students ​“fill in the gaps” rather than directing them somewhere specific.*

For more on cognitive load theory in maths, check out this blog from Greenshaw Research School.

*BIG caveat: students must already be familiar with knowledge needed for the problem.

References

[1] Sweller, J., 1988. Cognitive load during problem solving: Effects on learning. Cognitive science, 12(2), pp.257 – 285. 

[2] Robbins, A. (2019). The goal-free approach: My talk from CogSciSci 2019. [online] Reflections on Education. Available at: https://adam-robbins.com/2019/06/01/the-goal-free-approach-my-talk-from-cogscisci-2019/

[3] Sweller, J., 1988. Cognitive load during problem solving: Effects on learning. Cognitive science, 12(2), pp.257 – 285. p263