We know students really enjoy their Maths learning at ISB. We also know that parents value the experience of visiting our classrooms virtually through Seesaw. We have written a lot about the way we teach Maths in Primary, our Mastery approach, and so in this post I would like to take a bit of a deeper dive into some of the more complex theory behind what we do.
It is important that children experience success in order to be motivated.
We want all students to be motivated in their Maths learning. We know that measures of motivation and achievement go hand in hand. However, there is some evidence that in Primary Maths, the direction goes from achievement to motivation and not necessarily the other way around (Garon-Carrier et al., 2016 as cited in Ashman, 2022). This means that in order for children to enjoy higher levels of motivation in Maths, they must first experience high levels of achievement. To simply be motivated, will not mean higher levels of success. To experience success will lead to greater motivation.
In 1995, David C. Geary, an American psychologist, proposed that we label knowledge we have evolved to learn as 'biologically primary' and knowledge we have not evolved to learn as 'biologically secondary' (Ashman, 2022).
Biologically primary knowledge is knowledge we have evolved to acquire, such as how to speak and listen. Learning to speak requires little formal instruction. Children pick this up through emersion, interactions and play.
Biologically secondary knowledge is knowledge we have not evolved to acquire, such as Maths or how to read and write. For this kind of knowledge, we need a better understanding of how the brain works.
Our brains work in a way that requires the use of both long term memory and working memory. Whilst long term memory can be considered to be infinite, working memory is very limited and can only process about four elements at a time (Cowen, 2001 as cited in Ashman, 2022).
Through explicit teaching in school, we give students the tools they need to think, to solve the problems such as those within their Maths questions. Students need to have the correct tools in their long term memory.
When attempting to solve complex problems, it is the knowledge within the long term memory that directs the attention of the working memory. This is why we have such a focus on mastering specific knowledge, so that it can be called upon from within the long term memory and used by the working memory to solve more complex problems.
For knowledge to find its way into the long term memory, it must pass through the working memory first, it cannot be ‘downloaded’. This is where NumBots and TT Rock Stars also come in, the working memory is used to solve simple Maths calculations and the information passes through to the long term memory where it can be used again and again.
Times tables are a great example of this in action.
For students who have mastered their Times Tables, when presented with 7x7, they simply 'see' 49. The number 49 is instantly accessible to the working memory as this is knowledge that has been previously acquired. Students presented with 8x6 simply ‘see’ 48 and do not have to devote any working memory (which is limited) to this.
The same is true for more simple number bonds. Children who are presented with 3 and 7 instantly ‘see’ 10 and do not have do devote any working memory to this process. The same can then be applied to 35 and 65 (making 100) and so on.
To have this knowledge within your long term memory, turns you from a novice to an expert when attempting such problems. An expert will have to devote less working memory than a novice. We create experts.
The more elements are available within the long term memory, the more complex the problem can be. After a student masters more and more mathematical concepts and they are within the long term memory, these can be called upon to solve more complex problems.
If a student has to devote any of their limited working memory to understanding that 3 and 7 equal 10 and that 7 multiplied by 7 is equal to 49, there is not enough left for the more complex elements of the problem itself.
Problem solving skills are biologically primary knowledge (Ashman, 2022). However, the biologically secondary knowledge we need to call upon to solve these problems need to be taught and accessible from within the long term memory. Therefore, it is this mathematical knowledge which we focus on first. Without mastering these foundations, they cannot be used for more complex problem solving.
In our Maths learning, we focus on cutting down on unnecessary information and we are explicit about what specific element of Maths we are teaching. We reduce the demand on working memory by focussing on a single element, such as number bonds, the part-whole model or multiplication. As students begin to master these concepts, they can be combined with other knowledge and used for more complex problem solving and reasoning.
Some may advocate for ‘productive failure’, learning from the process of attempting the problem and requiring some support to get to the answer. However, research shows that it is far more effective to provide the student with everything they need as a worked model to complete the task. ‘While scaffolding, like all guidance for novices, is better than no scaffolding, the ultimate scaffold, providing learners with all the information needed including a complete problem solution - either prior to the task or just-in-time during the task - is better still’. (Sweller, Kirshner and Clark, 2007 as cited in Ashman, 2022).
This means, it is perfectly good to show the explanation to a student either before or during the solving of the problem. This is more effective than providing scaffolding (limited help) during the process.
Ashman, G. 2022 'A little guide for teachers: Cognitive Load Theory'. Crown UK
Cowen, N. (2001) 'The magical number 4 in short-term memory: A reconsideration of mental storage capacity'. Behavioural and Brain Sciences, 24(1), 87-114.
Garon-Carrier, G., Boivin, M., Guay, F., Kovas, Y., Dionne, G., Lemelin, J.P., Seguin, J.R, Vitaro, F. and Tremblay, R.E. (2016) 'Intrinsic motivation and achievement in mathematics in elementary school: A longitudinal investigation of their association'. Child Development, 87(1), 165-175