Ferret's story was very touching. To have that confidence in your own abilities is what we all want for our students, but to know as the teacher that that confidence is leading to missteps is a difficult thing to deal with. Do we step in and point out the error or do we lead the student to see the error for themselves? I feel it is much better to lead the student to see the error for themselves. The other question that comes to mind is if Ferret is using this strategy incorrectly how many other students are also using it incorrectly?
Another aspect of the article that I want to touch on is the idea of correcting a students method based on the efficiency of it. If a student has found a strategy that works for them and they feel comfortable using, in my experience they are less likely to switch to a different method. For example, my grade 10's became comfortable using algebraic methods to find the x and y intercepts of a graph and when they were introduced to how the calculator could aide them they were very resistant to using it, even though it is a tool that speeds up the process. We have to question the purpose behind introducing a method that we feel is more efficient. Is the purpose of mathematics to produce efficient problem solvers? Or is the purpose to produce students who are comfortable solving problems not matter what method they may use? I would prefer the second.
I agree with you that we want to help our students build up their confidence but at the same time we don’t want to see their overused strategies leading to missteps. It might be actually helpful for them when they found out themselves that their “super strategy” didn’t work for a particular problem, rather than we telling them first.
ReplyDeleteWhen solving math problems, I think we all tend to choose the strategies that we are most comfortable with rather than the ones that are most efficient. We have to remind ourselves that the students learn in different ways and the strategy appears to be efficient to us might not seem efficient to them. So, same as you, I prefer the second one too.
Nadine,
ReplyDeleteYou have raised an interesting point here, which is namely what makes a method "better"? In many cases efficiency is highly desirable. Being able to solve problems quickly can make the overall process more enjoyable, it can reduce the chance of errors (fewer steps in which to make mistakes), and of course it can leave time for students to do other things like do MORE homework problems (or have fun in other ways, which ever they prefer :P). However efficiency is not the only goal, one could argue that there are many more important goals in a math class.
That being said, I think we should still encourage students to use different methods. Not simply because new methods might be more efficient, but because new methods might be applicable in different places. To use your example, using algebra to find the x-intercepts of lines or quadratics is great, but what about cubics or other higher degree polynomials? Unless you're comfortable with logarithms finding x-intercepts of exponential functions is also very difficult.
Additionally, if students rely too heavily on one method, I fear that they might loose sight of what they are really trying to do. An x-intercept can loose it's meaning if you focus only on the algebra. Not of all students, but I know that there are some who will just do the calculation blindly and without thinking. Mixing up the methods might be a way of preventing this from happening.