Do we need to keep asking ourselves “why?”?
Hang on a minute, I thought we decided that conceptual understanding was the way forward. That Khan Academy was a no-no because it relies on purely instrumental understanding. But now this Table Tennis Superstar is saying that in fact, practice makes perfect and therefore conceptual understanding is really neither here nor there. So could Khan Academy have the potential to be a useful tool to allow students to practice maths to their heart’s content (much like the 24 hour table tennis hall)!? Oh wait, don’t panic, Willingham is here to get us back on track. He tells us that understanding is actually important (phew!). That if we want our students to really be able to remember knowledge then we need to encourage them to think deeply about this. This is therefore highlighting the importance of understanding not only in terms of being able to use the knowledge we learning more effectively but also to be able to retain this knowledge for longer.
I have just tried out the memrise website for the unit circle. I have memorised some ratios, yay! But what do they mean? If there’s one thing I’ve learnt throughout this course it is to ask the question “WHY?”. Memorising a bunch of ratios allows no conceptual understanding simply instrumental. I’m not sure this is the type of understanding I would like my future students to have, so in terms of which I would implement in my future teaching career, I would certainly air on the side of in-depth understanding. In my own experience also, I used to memorise maths formulas etc. just before my GCSE maths or physics exams. This worked well, I remembered them in the exam and knew which numbers to put where. However when I come back to cover the topics a few years later, I have no recollection of them, my cues were not of a high quality. Perhaps if I had gone through the proofs of the formulas or looked more into what they actually meant, I would have a longer richer memory of them. Having said this, I suppose that repetitive memorising does have its place. Past generations drilled their times tables continuously throughout their educational career. As a result of this my mum’s mental arithmetic is sooo quick compared to my sister’s or mine. Therefore I can’t help wondering if it does have its place/use within mathematics education.
I understand what Mr Syed and Carol Dweck are getting at with their fixed vs growth mindsets and the unquestionable evidence from studies they have provided us with. I agree that as teachers, we are going to have a tougher time teaching a student with a fixed mindset than a growth mindset. However, we aren’t teaching students simply to pass exams? Are we? We want to show them the beauty of maths, as the likes of Boaler, Meyer and Lockhart encourage us to. Is repetition beautiful? Or just boring? Is this where we’re losing students? With the never ending pages of textbook questions which have no context or real-life application. Perhaps instead, we have a duty to ensure that the Maths we teach interests students and enables them to think deeply in a way that Willingham suggests.