I am not a teacher yet but I am a parent, and when I read the article of Guida De Abreu & Tony Cline I thought I could contribute to this blog as a mother. And since I am from Pakistan I can totally relate to most of the examples and statements of Pakistani kids and teachers quoted in this article.
My seven year old son is attending a private schools here in Toronto. This school is following Cambridge curriculum which is quite rigorous as compared to IB and Ontario's curriculum. I have been teaching my son at home and following the Pakistani curriculum as well. Asian curriculums are generally even more rigorous. However, I chose to teach my son partially because I know that teaching him at home will improve his overall understanding of mathematics. He is able to handle the difference in problem solving techniques. He chooses the method that he is comfortable with and generally the one that is more efficient too. I can see the benefits of this approach because he is much more at ease with what is being taught at school and is being appreciated by his teachers as well.
My experience suggests that we open up our children’s mind by introducing them to the possibility of solving the same problem in more than one ways and that a particular method for problem solving should be chosen because it is more efficient, not because of its association with a certain class. Alternatively, kids can be given the opportunity to choose among the methods that they find more interesting and easy to use.
Hence what we need to think actively about is how to counter the valorization tendencies. One way to achieve that could be by introducing practices that promote the idea that it is very normal to have diversity in problem solving methods. We can encourage kids to share their home learnt methods. In other subjects, parents are usually invited in a class to read a book that interests them or their kids. I don’t see any reason why the same activity can not be introduced for mathematics. If we are not judgmental about other methods they could be an asset in the process of learning.
Hi Enza, I hear your argument and I believe you mean we should valorize the backgrounds and home learning methods of our students and thus bring it to par with traditional schooling methods (e.g, standard algorithms). I also believe that the varying ways in which students learn and understand mathematics should be allowed to flourish in the classroom. If this is practiced we would find more students creating (from exploration) methods understandable to them as an individual instead of memorizing methods forced upon them by home or school. This in turn, like your son, would also become the most efficient way for the individual because it would logical and have meaning to them.
ReplyDeleteHi Ezna and JWallace,
ReplyDeleteI agree that we need to place more value on different ways of solving questions. Math class should not be about learning the way to solve a question. Instead, we should be teaching out students to understand the math, and part of this involves understanding how different approaches represent the same process and lead to the same answer. There are many ways to solve a math problem, even within a particular culture. Just an hour ago I was teaching students how to solve a proportion, and I realized that there were 3 different ways I could think about the question. The method that I chose to use was not the right method, it was just the one I felt most comfortable with at the moment.
As Ezna mentioned, it is worthwhile to teach many ways of solving a problem so that students can use the one that is more efficient for the situation. For example, when we teach students how to solve systems of equations, we teach both substitution and elimination, even though either can be used. Students are capable of recognizing that the two methods serve the same purpose, and by teaching them both ways, we are also teaching them to think about the problem and plan the most efficient way to solve it, instead of just working through the memorized steps.
Basically, what I am trying to say here is that having multiple ways to solve a question is not a burden of the teacher, it is a teachable moment. If the goal is to really understand math, not just do math, then questioning and comparing different methods is a very important and useful exercise for our students.
Rohini
A few thoughts on the Valorization article that are kind of related to what Enza said:
ReplyDeleteI think it was ingrained in my head from a pretty young age that there was only one way of doing math and this was simply due to lack of exposure to alternative skils/strategies/methods. I think this could be partly due to practicality issues for the teacher but could also be due to a simple lack of understanding. Many teachers, in my experience, do not feel comfortable teaching mathematics, especially at the elementary level where they are asked to teach all subjects. It is difficult to teach a subject if you do not feel comfortable with the material. When someone uses a different math strategy it’s often just a variation of something that is taught slightly differently in school but still maintains the math ‘rules.’ I recognize this and for me it’s easy to say that they should continue to use this method because I understand why it works and why it’s acceptable. For someone that does not have this understanding it would be difficult for them to say this, because there is always the uncertainty that they may be wrong. As a result there might be the tendency to discourage something that is different, but possibly just as acceptable.
Whenever I come across something that is different I always say “if it works, go for it, but I’d like for you to understand why that works.” A situation that comes to mind involves solving equations, and some students have learned a balancing method to solve more complex equations where as others have learned the ‘rules’ to solve a complex equation. When I develop the concept I emphasize how an equation is like a balance and we perform operations on the equation that maintain its balance while isolating the variable (this can easily be done using algebra tiles and paper and pencil methods). The ‘rules’ are a variation of this method but I do not believe they demonstrate what an equation means and as a result there is a limit to how someone understands solving an equation. I learned how to solve equations through this method, but I never really understood what I was doing. I think that if you understand something it’s quite easy to see how it’s applied to a multitude of differing situations.
Many of the algorithms I learned in school I learned through memorization. It was only as an adult that I began to understand what I was doing and was introduced to many different ways of doing the same math. I think that some of the common algorithms, especially with respect to place value, are sometimes introduced at too early of an age.
With regards to the valorization article, I think that if you focus treating all individuals with respect and dignity then whatever skill, knowledge, and attitude they bring with them to school should be recognized and respected (so long as it’s not damaging to themselves or others).