I found this article to be simply reiterating and reinforcing the idea of relational versus instrumental learning and teaching. The discussion about creative, flexible and adaptive problem solving, rather than routine problem solving and using problem solving techniques that are suggested, i.e. "use the compensation approach to solve the following..." extends what we have previously discussed about relational and instrumental understanding. By telling students what technique to use to solve a particular problem is limiting to their mathematical creativity on one hand, but on the other it can force them to think in a different way than they may have otherwise and get them to see another way of doing a problem. I don't, however, see this as teaching flexibility, creativity and adaptivity. I see this as being helpful for some students, in particular those who struggle with recognizing what method to use, in which case I think that it is clear that these students do not have a solid conceptual understanding of the materials yet and thus need more help. I feel that if a student can recognize a method, or be able to modify a method, that will help them get to a correct solution then they should be encouraged, and given the freedom, to use it. This is, as I believe, the most important thing.
Something else that I found to be interesting that was mentioned in this article, although not very surprising, was that students with good prior knowledge had a harder time and were less successful in being adaptive than students that had less prior knowledge. This begs the question then as to how important previous knowledge really is when teaching for good relational and conceptual understanding? A method of teaching that I found myself adopting during my short practicum was an exploratory teaching/learning method where students are strongly encouraged to attempt questions with hardly any formal instruction but rather a little bit of guidance towards the direction in which they should be going.
Hi Matt,
ReplyDeleteI appreciate your comments about relational and instrumental learning. It was something that I wanted to bring up in my response, but forgot to mention. The way I see it, someone with an instrumental understanding is very dependent on the method they have learned, and are unlikely to try something different or adapt their current strategy. Someone with an relational understanding, however, will understand how problems relate to each other, and how different strategies might be changed or adapted to fit different situations.
As for the issue with prior knowledge, I think this becomes an issue because people have a tendency to get stuck in a rut. That and they will stick with what they find most comfortable. If you've been solving a problem one way and have become comfortable with this method, why change? There are certainly advantages to other strategies and techniques, and these can be explained to students. However knowing that something is "better" and actually acting on this knowledge are two different things.
Hi Matt,
ReplyDeleteI thought of relational vs instrumental learning as well! I agree with you that it's important to encourage students to use their own methods as long as they arrive at the correct solutions and are sensible methods. How did your exploratory teaching/learning method work out? Did you find that it worked for most your students? I attempted to do the same during my short practicum, but it didn't work well for some students... (I guess as teachers we'll need to be creative, flexible and adaptive in our methods too...)
-Erica
Hi Matt,
ReplyDeleteI also thought the same,that this article also emphasize on the importance relational understanding verses instumental understanding.Our duty as teachers is to build confidence in students that they will be able to find their owm methods or strategies to solve a problem. By telling them solution of the problems, limits their thinking in the sense they think that it is the only way of solving the problem.
Hi Matt,
ReplyDeleteIt's interesting that you saw this article as instrumental versus relational. I did not see it that way until I read your response, but I can definitely see it that way. I agree, which is better leading students to solving problems one way or allowing them to develop their own unique methods? I think both are beneficial for students. Those who may be stuck will need the leading, while those who are more confident might appreciate the choice.