Despite the article being informative, I think that the author missed a crucial point. That point is that instrumental mathematics can be thought of as a subset of relational mathematics. This means that if someone has a relational understanding of something, they will also have at the very minimum, the knowledge of someone who has an instrumental understanding.
There is a clear statement that relational understanding gives the student the ability to adapt to slight differences in scenarios, but I believe that sometimes, that isn't necessary. For example, let's take a look at the quadratic formula. If a person can use the equation, then they will be able to find the roots of every quadratic that exists. In addition, if that same person was taught the scenarios for b2-4ac>0, b2-4ac<0 and b2-4ac=0, then there is nothing more that the quadratic equation can offer. And in this case, instrumental understanding would span into relational understanding and ultimately, there would be no difference.
Last but not least, I think that the article could have finished with definitive solutions to allow teachers to teach mathematics relationally rather than instrumentally. Of course, the first step is always awareness; knowing is half the battle. But what about the other half? Where can teachers, who want to improve on teaching relational understanding, start? Especially since teachers already have the prior knowledge to have relational understanding when they are given information that would allow students to only have an instrumental understanding. In other words, a teacher could be teaching students instrumentally, and the scary thought is that teacher may think that the material that is being presented will provide a relational understanding to the students.
Great responses, Ed! There's lots to talk about here.
ReplyDeleteI agree with you when you say that relational knowing is truly the state of knowing. I would add that there are ALWAYS ways to deepen one's relational knowing of a topic, and this is especially clear in mathematics. Any topic or problem that one understands can always be approached in a different way or contextualized differently, and that will lead to further understanding, without limit! That can be disheartening for the perfectionists among us (because one can never know anything fully and perfectly), but it is also exciting, because there are always new insights possible.
I like your proviso that instrumental knowing is a subset of relational knowing. I also like your example of the quadratic formula as an instrumental way of knowing that encapsulates much of the relational knowing that is possible. I would only add that having a sense of how and why that formula was derived, why the sign of the discriminant b^2-4ac matters to the roots, and when one would do best to use (and not use) the quadratic formula would enhance understanding.
And yes, it's too bad that Skemp didn't continue on to discuss ways that teachers could actually foster relational understanding in their students! Once we are aware that we could aim to do this, the question of 'how' looms large! I guess we can address our own work to trying to figure this out, through discussion with colleagues and learners and through teaching experiments in our classrooms.
Well done!