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One problem with mathematical difficulty is its “invisibility”. We can’t see how other people are thinking, how they solve the problem, how they struggle. And s
by sinuhe69 6y ago
One problem with mathematical difficulty is its “invisibility”. We can’t see how other people are thinking, how they solve the problem, how they struggle. And sometimes, we cannot even explain our thinking!
That’s why making thinking visible is so important! Because only then we can learn from the others, they can learn from us and we can see where it’s right and where is wrong.
One way to making thinking visible is to apply thinking routines, like:
- what are given?
- what are the unknowns?
- what do we know?
- what do we need, how i can get there? (strategies)
- choose a strategy and explore it
- rethink the strategy and optimize it
- connect and reflex
Once we practice thinking routines, we will find it increasingly easier to explain our thinking and even be able to debug our work. If your class also applies thinking routines, it’ll get even easier for everyone to learn and understand each other.
I recommend the following books:
- Polya, How to Solve It
- Velleman, How to Prove It: A Structured Approach
- Ritchhart: Make Thinking Visible
Mathematical work is basically a series of equivalent transformations and logical reasonings, starting from the given informations to get where we want, thus once we knew the why, it should not be too difficult to explain our work because most are self-explanatory.
Hope it can help.
- quantified 6y ago+1. Across a set of rote problems that you know how to solve: take the problem and write down in any language the steps you take to solve it, then also write down why you took those steps at the points you did. When you’ve covered the set, review them all. If you tried to record a tutorial for someone else at that point to follow in your footsteps, what would you say? One of the great ways to learn is by teaching, you can simulate for yourself. But it has to come from your understanding, not just reciting the words of your textbooks and problem sets.
- mettamage 6y agoSo a twitch.tv for mathematicians might help.
- agumonkey 6y agoI'm not a mathematician and not good at it either, but something that I see a lot is that mathematics in general are very keen on sharing the conclusions in concise, if not cryptic, forms. A lot of metamathematics are lost in the process.. but I think that people that are really good in math like this way of life. But it kinda hides a part of what doing mathematics is. It should be written as books preludes.
- nathanmcrae 6y agoMy favorite math professor was great because any time he would forget the statement of a theorem or the outline of a proof he wouldn't just go to his notes to remember it, he would out-loud reason his way to the correct version. This would include the minor missteps he made on the way. Best math lecturing experience I have had by far.