I strongly agree that there is instrumental and relational teaching and learning regarding the subject of mathematics. I believe that not all teachers of mathematics who use the method of instrumental teaching are aware they are doing so and this issue must be addressed before any plans can be formulated as to how mathematics is to be taught in the future.
In an ideal world all mathematics would be taught relationally and all students of this method would thrive. However it is a well known fact that we are all individuals with different learning styles, thus does everyone have the ability to learn mathematics if taught relationally from an early enough age? A lot of students who struggle with the subject need it to be taught instrumentally so that they can pass their exams and go on to do what they want to do in life, if this is not mathematically related do we have the right to destroy their hopes and dreams by forcing relational teaching upon them. This then raises the question of if mathematics is to be taught relationally should it then be mandatory? Possibly both methods of teaching should be used but to different students in different classrooms, then the question arises of when do you decide to separate them; at what age is it appropriate to make the decision that one child cannot succesfully learn relationally, they are after all children, who are all unique and develop differently.
Relational teaching must be implemented at the start of a students education and followed through with. By the time a student reaches their teenage years, a critical time in their life, it would be harmful to their self-esteem to try to switch them from the instrumental teaching methods they are acustomed to.
There is one point in the article to which I disagree, that enlarging a students awareness of possibilities can only be a good thing. For the most part, of course, if we are not aware we cannot make informed choices. However the saying ignorance is bliss is one which I don't think should be forgotten. If opened to too many possibilities some can become over-whelmed and lose focus. We do not wish our students to feel they are drowning in a sea of knowledge and possibilities.
As mathematics teaching stands today, I do not believe the teachers have the time nor the leeway to be able to teach completely relationally, nor am I convinced that this is a good thing for everyone.
I am concerned that you are dichotomizing relational and instrumental ways of learning and making the following assumptions which I believe to be untrue:
ReplyDelete•that only the 'top' students can handle meaningful math learning
•that 'poor' math students can only manage to learn meaningless procedures, and that we should teach them these to get them through school. (What would be the point of spending years teaching and learning and subsequently forgetting meaningless procedures? Why bother teaching math in this way at all?)
•that relational learning takes longer
•that students can't change their habits but are doomed to be merely instrumental learners if they start out that way
in contrast, I would suggest that:
•everyone wants to learn things that make sense, and no one really wants to waste their time on meaningless procedures
• there is an interplay between learning HOW and learning WHY something works, so that instrumental and relational learning can be complementary
•learning understanding takes no longer than learning algorithms (and may, in fact, take less time, since it's simpler to see an overall pattern than to learn dozens of meaningless procedural steps).
What do you think at this point? I'd be interested to hear how your ideas develop on this.