(Don’t) Get to the answer, as quickly as possible

It took me two goes to learn to drive. And I don’t mean two goes at the test. I gave up first time around in my late teens without even having booked a theory test. Now I don’t want to put it (all) down to the instructor at the time. He told me what to do each time I got in the car, gave me a series of repetitive instructions that I had to learn, remember and then carry out. There was no other real discussion about what I was doing.

Years later, when I started up with a new instructor it all began to click, and the moment that I realized I had wasted those earlier years was when we came across the age old problem of the standing start. My understanding up to this point of how to get the car moving was drawn from previous experience – you rev the engine, put the handbrake down and release the clutch, making sure you get the timing just right. Practice, practice, practice: that’s how you learn.

After an untold number of stalls on a slightly inclined surface, she said to me, “Try it again without revving the engine, and release the clutch slowly.”

Then it hit me. The car moves forward as the clutch is slowly released, and without using the accelerator! Who knew?! This epiphany came to me, coupled with a sweeping sense of frustration. Why had my previous instructor never explored this concept with me to help me understand what I was doing?

The reason for sharing this impertinent little analogy of my early motoring exploits is that it draws parallels with the development of my own teaching practice. Back in the early days of my teaching career, I didn’t quite get it. What I mean by that is I wanted to impart my knowledge to students, and if they could recall that knowledge the next lesson, then I counted that as a success. I hadn’t occurred to me that learning could happen at a deeper level. If the students could recall the content for the exam, then that was the most important thing, surely?

In a moment of reflection, I asked myself why I had become a teacher. Was it to get students to perform as best they could in what I had essentially viewed as a glorified memory recall test? Or did I want students to understand the fundamental concepts that permeate through this beautiful subject that I was teaching them and allow them the opportunity to creatively explore the subject further themselves? You can probably take an educated guess.

In the Maths classroom, we often teach students to divide fractions by “flip and multiply”. While this may serve them well on a single question in an exam (if they can remember it), not understanding why this works becomes a barrier to progression in other topic areas. We can often be guilty of gift-wrapping a method to learn and giving it to students to use without taking it apart and exploring why it works. You can explain to a young child what the number 12 is. It looks like a 1 and a 2 next to each other and it comes after 11. But would they realize that it is the number that represents one 10 and 2 units added together? At a higher level, most students know that to find the area of a triangle, they must multiply the base by the height and the divide by 2. I would wager that a good proportion don’t understand why. Then what happens if the triangle is drawn in such a way that there is no “height”? And what does area even mean? This “get to the answer as quick as possible” approach is a short-term fix that if used in isolation can be detrimental to student progress.

There is a balance to be struck however. For one, there are many barriers to achieving a utopian goal of embedding a deep conceptual understanding in the minds of all learners, and you probably need look no further than the learners themselves. One of the biggest problems I find is that many students don’t expect to be taught a conceptual understanding of what they are learning. They want to be taught how to get to the answer as quick as possible. Does it matter if you don’t get to the answer? No, not necessarily. You may have learnt more by not getting there at all, or by approaching the task in a slower more considered manner. The end goal of the exam is often at the forefront of their thinking, and so if by the end of a lesson, or even a single activity, they don’t have a model answer or method to succeed, then they are not satisfied. They want to learn memory techniques and use flashcards. They want facts to remember and recall with a lack of thirst for understanding. The challenge we face as teachers is changing the mindset of those students who think in this way and we need to create a ethos in our classrooms whereby it doesn’t matter if you don’t get to the answer or you don’t get it right.

I think rote learning and instrumental teaching have their place, and sure there are times when a quick fix is needed. The Year 11 student who comes to me the day before their exam to ask, “How do you do percentages sir?” is not going to benefit from a deep conceptual understanding of the topic, and will probably not have the appetite for it anyhow. At some point, students will need to remember and recall aspects of what you are teaching them. But for most, there is quite a definite limit to the amount of information their over-cooked brains can hold in storage. They need our help. If they gain an understanding at a fundamental level, then these processes, facts and concepts that need to be learnt become embedded, so that they can later be deduced rather than added to the vast bank of information to be recalled. As teachers, I think we need to move away from a reliance on teaching activities that require the use of good short-term memory and incorporate more deep learning approaches.

I am a fan of the new GCSE specifications, or at least the ideas and principles that underpin them. They are designed in such a way that students need to have more than good memory recall to be successful, unlike the old specifications. It never sat well with me that a student could come away with a pass grade in Maths and go out into the working world without being any good at it. It requires students to buy into a culture of conceptual understanding which is really what we all want, isn’t it?

I’ll go back to what I said earlier: I became a teacher to impart knowledge and wisdom, not to teach how to remember.

Gareth Dudding, Maths.

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