How to better teach maths in physics using cognitive load theory
The 2016 changes to GCSE’s resulted in both more maths content and more challenging maths content dropping into the GCSE sciences for all pupils. This has caused serious challenges for teachers in making this harder content accessible. We now must teach what used to be A level physics content to our GCSE groups including teaching foundation groups to be able to rearrange the equation ‘final velocity2 – initial velocity2 = 2 x acceleration x distance’! Having delved into Cognitive Load Theory and worked example research I’ve found a number of useful teaching strategies that have helped to both stretch my most able and support weaker students in being able to access this more challenging maths content.
The first strategy I am going to talk about is ‘example-problem pairs’. In the past when I taught maths in science, I would start by going through some worked examples together with students and I would then set them loose on a worksheet of conventional problems. This is a good way of teaching, however worked example research has identified more effective ways of sequencing these examples.
Example-problem pairs consists of giving students a single worked example followed by a to-be-solved problem. Then another slightly different worked example is given followed by another to-be-solved problem and so on. Each time another worked example comes up it can add an extra layer of complexity. See below a section of a worksheet with example-problem pairs incorporated into it:
Example-problem pair 1 have the same problem structure. Example-problem pair 2 then require students to convert 2km into 2000m before calculating speed. Example-problem pair 3 mix things up by requiring students calculate distance. I might then have example problem pair 4 require students perform a minute to seconds unit conversion or switch the problem type to calculating time and so on.
A number of studies(*1) have compared example-problem pairs against conventional problems. These studies found that students who solved example-problem pairs took less time on the target problems than subjects who studied a block of worked example followed by a block of conventional problems. Crucially, students studying example-problem pairs also submitted more accurate solutions than students receiving blocked worked example then conventional problems. Example-problem pairs have been shown to be a more effective means of developing problem solving in studies in maths, physics, statistics and computer programming (*2). One of the more surprising findings is that even if students solve the more conventional problems overall the example-problem pairs groups performed better on post-learning tests(*3). This may be because even where a student solves an extra problem their working memory capacity is overallocated which interferes with their ability to store what they learnt in their long-term memory.
From this research and my own practice, I have found benefits including:
- Students can be reminded of what a good solution looks like. With example-problem 2 students can be reminded that they need to first calculate the unit conversion, (2km x1000 = 2000) then use that 2000m in their speed calculation in the second line.
- More able students can be more independent. If they get ahead of others in a worksheet, they don’t have to wait for the teacher to tell them how to solve a more complex problem. They can instead study a worked example and learn to solve it correctly themselves.
- Less able students have more support in worksheets. If you taught conventionally using blocks of worked examples followed by blocks of conventional problems, weaker students are not going to be able to follow what you are doing as the worked examples become more complex. By breaking up the worked examples with problems, students can consolidate their knowledge through each example-problem pair before studying the next more complex worked examples. Through this they can grasp what makes each problem type different.
Example-problem pairs are just the start of what can be gained from cognitive load theory and worked-example research. Over the coming weeks I will be adding further articles on faded-examples, goal-less problems, how to use ideas from cognitive load theory to help with rearranging equations, how to support high ability pupils and more.
By Simon Palmer
Physics teacher and Deputy Team Manager of Science at Carr Manor Community School Leeds
*1 Sweller (1998); Sweller & Cooper (1985); Sweller et al. (1998)
*2 Atkinson et al. (2003)
*3 Renkl et al. (2002)
Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2, 59-89.
Sweller, J., Van Merrienboer, J.J.G. & Paas, F.G.W.C.., 1998. Cognitive architecture and instructional design. Educational Psychology Review, 10(3), pp.251–296.
Sweller, W.M. and J., 1998. Learning to Solve Compare Word Problems : The Effect of Example Format and Generating Self-Explanations. Cognition and Instruction, 16(2), pp.173–199.
Atkinson, R.K., Renkl, A. & Merrill, M.M., 2003. Transitioning From Studying Examples to Solving Problems: Effects of Self-Explanation Prompts and Fading Worked-Out Steps. Journal of Educational Psychology, 95(4), pp.774–783.
Renkl, A. et al., 2002. From Example Study to Problem Solving: Smooth Transitions Help Learning. The Jounal of Experimental Education, 70(4), pp.293–315.
The Anthony Waterhouse Fellowship
Simon Palmer was awarded an IOP Anthony Waterhouse scholarship to work on his project "Improving the teaching of mathematics in physics, using cognitive science research". This grant, of upto £3,500, enables practising teachers in UK secondary schools or colleges, to develop an idea they have about physics teaching.