Everyone is taught at school that 83 × 84 = 87; until this week, I had always considered it essential to explain to students why this is true before they attempt to incorporate it into their mental schemas. Over the past week I have had separate conversations with two people whom I respect who argue for the exact opposite: students should commit this rule (i.e. the generalisation ax × ay = ax+y) to memory before I explain it to them.
I have a visceral aversion to the idea of committing something to memory before one knows why it is true. However, through my reading over the last year, in particular What if everything you knew about education was wrong? by David Didau, I am aware that it is always helpful to interrogate what feels right in education. I have been doing this recently, and many times my strong feelings have been wrong, what has seemed to be self-evident has been exposed as erroneous when put under the microscope.
There are good reasons for me to think that in the learning process, memorising something before knowing why it’s true can be unhelpful. A significant number of maths undergraduates, myself included, will have learned a proof by heart for an exam, typically this is because one can’t follow the logic of the proof, and the imminent exam necessitates memorisation. Whenever I have done this, I have forgotten the proof almost immediately after the exam, and it would seem that no learning has taken place though this memorisation. Like most of the population, when at school I learned trigonometry by memorising the mnemonic SOHCAHTOA. Although this has stuck with me and enabled me to pass the GCSE exam, I felt it hindered my further understanding of the topic at A-level, and it wasn’t until I revisited the topic on my PGCE that I came to know why the mnemonic made sense. It struck me that memorising it had been unhelpful in learning trigonometry.
In Didau’s book, he quotes Kirschner et al:
If nothing has been changed in the long-term memory, nothing has been learned
Sometimes, memorising does not lead to a change in the long-term memory and nothing is learned; sometimes memorising leads to a change in the long-term memory but what is learned is unhelpful. However, just because memorising is sometimes unhelpful in learning something it doesn’t mean that it is unhelpful. Let me re-imagine the SOHCAHTOA example in a way that could lead to students both knowing it and knowing why it makes mathematical sense, with the goal of ensuring that they could pass the GCSE exam and use this knowledge to improve their mathematical mental schemas allowing them to move further along the path towards mathematical expertise. Here I am going to bring in Didau’s own definition of learning from his above mentioned book:
The ability to retain skills and knowledge over the long term and to be able to transfer them to new contexts
Accepting this definition, I think that learning Trigonometry at GCSE with SOHCAHTOA would require a two-step process: A. the students know why it makes sense ; and B. the students commit it to memory. The first step is necessary in terms of learning the topic, and the second step is necessary for passing a high stakes exam. It is the order of these steps that I am considering.
Refocusing on teaching 83 × 84 = 87 and the general ax × ay = ax+y, the key question for me is this: if students have committed these to memory before I explain it would this be a better way for them to learn than if they committed it to memory after I explained it?
Here are my observations about my experience of teaching this over the last 9 years:
- There has always been a sizeable proportion of students who will not grasp my explanation as to why it is true.
- In the time in the run up to the GCSE it is important to me to ensure that all students have memorised the rule regardless of whether they know why it is true or not.
- Some students will come to know why it is true and have memorised it very quickly. It has seemed to me that knowing why it is true has helped those students to add it to their mental schemas very quickly.
Upon reflecting on this, and writing this blog, it does seem to me that requiring students to memorise it should come first. My reason for this is that it seems that my explanations do not work for all students, and can perhaps cause confusion, this strikes me as being particularly important for those students who think more like novices than experts. I do think that if students are going to end up thinking more like experts than novices, and be able to build on this knowledge then I need to explain it as well. The question becomes how and when do I best explain it? I will leave that for another day.
The paper Didau quotes from is:
Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching
Paul A. Kirschner, John Sweller, Richard E. Clark (2006)
Ideas for this blog post have come from
- this blog post by Dani Quinn
- a conversation with my colleague James Colver
- books by Didau, Daniel T. Willingham and E.D. Hirsch Jr