*why*aspect. Alternatively, Instrumental understanding is knowing a rule and when to use it, but not why. He calls it "rules without reason."

In the article he goes on to share a few different analogies. One was quite helpful in which he drew a comparison to music. He said instrumental understanding was like knowing the notes and the staffs on paper, but not having those figures attached to sounds. Relational understanding was like learning the notes in conjunction with the sounds. Consequently when asked to write a melody the students taught relationally, would have a far easier time writing a melody. I fully agreed with this analogy; of course reinforcing the meaning of notes with auditory information would increase the fluency and skill of students writing music. Music class would in intensely boring if nobody heard any music.

The analogy that I didn't agree with was when he compared understanding to two different ways of navigating a new city. Instrumental understanding was compared to just getting directions to and from essential places, which can lead you to be incredibly lost when you make a wrong turn. Relational understanding was like exploring the city on your own and creating a mental map so you can recognize multiple ways of getting from A to B. While I appreciate his analogy on some level, I still think it's problematic. His description of exploring the city to find your own routes relies on doing so without help from an expert. It's purely self discovery. However, I believe he wants teachers to be teaching the relationships and alternate routes to get the answer. I'm not sure that I agree with the explicit teaching of these relationships though. I think it's sort of intriguing when I don't learn every detail about a rule and then I find it out on my own when I continue learning about the subject. It s almost like an incentive to learn more so you can get the satisfaction of learning why a rule works by yourself. However, I am extremely intrinsically motivated, so I understand why not all students would have the motivation to look for connections by themselves. But I also think it's a good life lesson to not spoon feed every connection to students. They need to be able to recognize connections on their own.

As teachers perhaps we could set up questions that will inevitably lead to discovering connections, but we should leave it for them to fully explore. For example when discussing slope, many kids mix up the slope formula and divide the difference of x coordinates by the difference of y coordinates (like x/y), instead of the correct y/x. To aid the relational understanding of this topic, I could ask my students to compare slope formula to the well known trick "rise over run." Hopefully they would make the connection that the difference in y coordinates is the numerator because it corresponds to the rise. Similarly the x corresponds to the denominator because it is the run between the two x values. So this quick examination of the slope formula and what it corresponds to on a graph is a simple way to build a connection that will help students remember the slope formula. But I think it's important to not answer the question for the students, but rather pose the question and leave them to find the relationship. I think the self discovery will create independence and hopefully build some intrinsic motivation to find the relationships in upcoming lessons.

Another thing that bothers me about Skemp's outright preference for relational understanding is it's inefficiency. Even if there are a million ways to do something, I think there is something to be said about knowing how to do it with ease. I think math can be explained in so many ways, but since it values efficiency, we should teach with a preference to the efficient way to answer questions. If nobody valued efficiency, no formulas would be created. Formulas are mountains of work that are consolidated so that we can use them quickly and accurately. I think that idea is discounted in the relational understanding model, but it shouldn't be. Our students don't need to reinvent the wheel, just generally learn the concept of it so it can help them later on.

Overall I'm not completely convinced that explicitly teaching relational understanding is the best method. I think instrumental instructions has its merits as well. Relational understanding could be beneficial when students are completely struggling with a concept, or as an extra independent enrichment, but I don't think teaching instrumental tricks to help kids remember certain concepts is terrible. I think there's a time and place for both levels of understanding.

Nice self-debate that works through important examples and interpretation. 5Cs +

ReplyDeleteAs you move on, It might be good to consider what does this balance look for. The difficulty in mixing, is that one seems familiar to students and the other less so, so they focus on the familiar. Starting from scratch it might be different. There have been some studies that show the difficulty of getting at relational understanding in areas where students have instrumental understanding. On the other hand - I see you guys do that all the time!

I really liked your case for instrumental understanding. I was completely on the side of teaching mostly relational understanding until I read your post. And now I think we should teach both. At least give students a chance to develop a relational understanding the first time a new topic is covered, and then show them the rule so they can gain some instrumental understanding. Since everyone learns differently maybe some students need to learn the instrumental first because they can grasp the relational understanding. As you put it, guide them with questions so they can use their instrumental understanding to discover the relational understanding on their own time.

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