I like the story about Ferit's strategy. I think this is an interesting problem that extends far beyond mathematics. We all have our preferred way of doing things, and tend to stick with them longer than we should.
This happens everywhere, not just in math. For example, in rugby the need occasionally arises to kick the ball through the goalposts. We all have a slightly different style for doing this, and most of our approaches are not ideal. The goal is to line up your kick, then take a series of practiced steps to place yourself at such a distance from the ball that you can perform the same kicking motion every time. It's the same thing that American football kickers do. But many of us take such unusual steps. Instead of the simplest "2 back, 2 sideways" that the NFL kickers perform, many of us have such strange sequences. I've seen people move 4 back, 2 forward, 3 sideways...in such a convoluted pattern. This is surely not the ideal strategy, but it works for them!
And this is really the crux of the matter. Most of us don't do things in the simplest and most reliable way. But we do things the way we're used to. We do things the way we've practiced them, and after so much practice, doing it the hard way is more reliable for us than doing it the simplest way!
This raises an interesting question. Once a person has a considerable amount of practice in doing something the hard way, is it still worthwhile to teach him a new and simpler way? At what point do the gains of learning a simpler method no longer outweigh the advantages of being highly practiced?
Hi Paul!
ReplyDeleteI agree with you. We do things in certain way, not necessarily most efficient, because that’s what we feel comfortable with. What I think though, that students should learn, is being aware of the alternative route. They may not realize that there are simpler ways of doing the same thing. However, as you said, that new method may not seem so reliable to them. So once they know that there are different strategies available, then they can choose to pick a method they like. There may be a problem in the future where they realize that the other method they learnt would work better- then there has been progress made!
For the very last question above, I think saving time would be the major gain; and then there are things such as perhaps making less error by keeping things simpler, and easier to memorize or teach to others.
I couldn't have said this better myself. I really like that you were able to extend the problem that Ferit was having to something outside of mathematics. I too have the same question, at what point and at what cost should we still be trying to teach different methods to a student(s) that is seemingly "stuck" using the same method(s)? If someone is more proficient in one method over another than I think so be it. I myself have struggled with this, I could do a problem in one way, but I knew that there were faster, simpler methods to get to the same result, but when I tried to learn and use them I often greatly struggled and found it faster for myself to revert back to what I was most comfortable with. Also, what is the best strategy is a rather subjective notion.
ReplyDeleteI like your description of the rugby kicking, and I have a similar story to share as well. I can play a C major scale perfectly on the piano with the wrong fingering (1234,1234 instead of 123,12345). When I tried the right fingering, my scale sounded broken and it didn’t go well with my mind. I agree with you that most of the time, our seemingly un-smart ways of doing things become more reliable for us since we have practiced for so many times and now we can apply it freely and “efficiently”.
ReplyDeleteRegarding to your question, I think it is worthwhile to teach the new way, just to let them know that there are other ways of doing things, but we don’t want to impose it. Some people might switch while others might not depending on what their goals are. For example, I will have to change it if I want to be a piano player, but I don’t have to if I only want to be able to play a C major scale.
This issue relates to something that I talked about in my reflection. I talk briefly about a study that found that graduate students in mathematics don't always use the best strategy to solve a problem, a stick with it for a long time. I pointed out that perhaps this is because this is the strategy they are most comfortable with, and which they have found works best for them. This is of course, exactly what you have mentioned here. I think that while there isn't anything inherently wrong with sticking with familiar routines, you have to be flexible enough to use other methods too. If you only learn one method you can become overly dependent on it, and in a sense you're "putting all your eggs in one basket" because what do you do when this strategy doesn't work anymore?
ReplyDeleteHi Paul,
ReplyDeleteI agree with your comments about doing things the way we do them because we’re comfortable with them. My ways often seem to be inefficient to others, but I continue to use them because this way is the most efficient for me. This is something that we, as teachers, need to be careful of. When we are teaching, we need to ensure that we do not just demonstrate methods we are used to, but also other methods/ideas/ways of thinking that exist. The students can then decide for themselves which method(s) work best for them. Of course, this may be very idealistic in that class time is already limited and we are obligated to teach to the IRPs.
It is human nature to stick with what you are comfortable with and I do agree that if what you already know gets you there then where's the harm? The problem is that not everyone manages to get there with what they're comfortable with and the majority that stick to their comfort zone will struggle with problem solving if the nature of the problem is not explicitly what they are used to.
ReplyDeleteI believe we should all strive to better ourselves and thus to always go with what is comfortable isn't something I truly believe in as productive. It is a concern however that some may struggle with new ways of doing things but may feel forced into trying, thus not obtaining the correct results they may have obtained had they felt safe to "do it the hard way".