Organised excellently by Philosopher-Mark, in the style of Moral Maze, George and I debated the following motion, in front of an audience of 30 Yr12 students and a handful of teachers:
The best way to learn problem-solving is to problem-solve
For the motion, my argument. Some of these points are not really that relevant, but might be useful:
- I think that a more general statement is true: “The best way to learn x is to do x”. It is true of learning an instrument, learning to swim, learning to write, to draw, to play sport… Of course there is a place for learning from experts and standing on the shoulders of giants, but the majority of your time should be spent doing. Why should problem-solving somehow break this rule?
- I would rather inspire students to love maths than to train them to be excellent at it. If they are motivated, then they can learn skills from any number of sources (the internet and friends are both amazing things).
- One counter-argument might be that “First, they have to master the basics”. In other words, it is a waste of time to dedicate valuable lesson-time to problem-solving if the students can’t even add numbers correctly. I have three responses. Firstly, what constitutes “the basics” is relative and therefore meaningless. Is multivariable differentiation a basic skill? Maybe at university. Is knowing about place value a basic skill? Maybe at primary school. At any stage there will always be new skills to learn. If you waited for a student to master the basics, you would be waiting for a long time. Secondly, order of delivery implies order of importance. If you start with building skills, and tack a problem-solving lesson on at the end of term, you are sending a clear message to students. “If I had to choose, I would rather you can correctly use the cosine rule than can think independently and flexibly in novel situations”. Thirdly, it would be ridiculous and cruel to expect a student of music to only play scales as a child, waiting till they were 18 before allowing them to play actual music (stolen from Lockhart). Similarly, surely, for Maths.
- Another question might be “What is the role of the teacher then?” If the best way to learn is to do, then does the teacher become irrelevant? No. The teacher carefully chooses problems at the correct level and of the correct flavour to challenge and inspire particular students. The teacher observes first, and models second to correct key misconceptions. The teacher spots mistakes and decides whether to intervene or not – some mistakes are worth stopping instantly and some should be left to flourish.
- My three grounding texts are “A Mathematician’s Lament“, “If this is the headache…” and “Mathematical Etudes“.
Against the motion, George’s argument (correct me if I got it wrong!):
- I know the paths, streams and hills of my childhood intimately. If you wanted to go for a run there, wouldn’t you want to ask me for a favourite route – one that is enjoyable and doesn’t end in muddy fields and barbed wire? By analogy, I know physics intimately. There is a place for getting lost, to ensure you remain aware of the vastness and complexity of science. Most of the time, however, you just “want to get home” without getting lost. My role as a teacher is to guide you, to teach you how to read the map of Physics. I will use my expertise to help you avoid the muddy field of negative signs and to navigate over the barbed wire of resolving forces.
- If you are allowed to get lost too often, you lose joy. As an example, a student wrote “There is a proof, but we’re not smart enough 😦 ” at the end of summary of a problem-solving session. Getting home feels good, and is therefore motivating.
- George and I both view ourselves as experts. I got the impression that George was able to model, clearly and helpfully, the solution to any problem that his students were working on. On the other hand, I have been giving my students problems that I cannot solve (mostly because they are so open as to have one particular solution). Is it empowering to give students a problem their teacher can’t answer, or does it inevitably lead to everyone being confused?
- When the students leave the realm of George’s map, how will they find their way in the wide world?
- George was far more successful in capturing the imagination of the audience with his “muddy fields” image, and by restricting himself to one clear position. Consequently, there was a large swing from the beginning to the end of the debate, from “agree” to “disagree”. Well done!
- If I was to restrict my argument to a story, as George did so effectively, it would be from when I was 16, learning to become a sailing instructor. The sky was grey, the sea was choppy, I was cold and tired. I was trying to tow 5 boats behind a powerboat, but first had to tie them on. My fingers were fumbling with a bowline, trying desperately to get some slack on the rope. It took me 10 frustrating minutes to complete the knot. Only at that stage did my assessor, who had been watching me throughout, step in and tell me “You were trying to use the wrong knot. Next time use a round-turn and two half-hitches”. Apologies for the arcane sailing references, but I have never forgotten when to use each knot – it was absorbed into my gut through being allowed to make mistakes and fail.
- It is excellent that we both agree on the importance of problem-solving, and are passionate enough to organise a successful debate.
- A number of students asked excellent and eloquent questions. Training students in the art of teaching is important, since it gives them the tools to give constructive feedback to their teachers. If either George or I ever stray too far away from the happy middle-ground, hopefully our students can bring us back.
- Some students said “Maybe it is good to get lost in Maths, but good to be guided in Physics.” Is this due to a genuine difference in disciplines, or because they have been brainwashed by their teachers?
- George mentioned that he starts to have a horrible time when lost out in the wilderness. I specifically seek out the feeling of being lost when out running – I love the unexpected discoveries. Is this just a case of different personalities giving birth to different pedagogies?
- There are two ways you can love a subject. You can love the rush of getting lots of things right and being good at it. Or, you can love the feeling of challenge and exploration. Only one of these is sustainable – eventually everyone stops being good at their subject, once they get to a high enough level.