The Reflective Educator

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Tag: math education

Six word story

In the spirit of a story Ernest Hemmingway probably never wrote, I was going to offer this as my six word story for #etmooc.

"For sale: Master’s degree, never used."

Unfortunately, it seems that this particular short story has been thought of before. It’s also worth noting that this particular piece of fiction does not accurately describe my life. So I went back to the drawing board and came up with this:

"Student died in math. Nobody noticed."

This is obviously fictional, but it better describes what has become my life’s work to avoid happening to students.

The difference between instrumental and relational understanding

Stanley Park map

 

I recently found this article written by Richard Skemp that Gary Davis (@republicofmath) highlighted on his blog . I recommend reading the whole article. Skemp describes the difference between instrumental and relational understanding, and how the word understanding is used by different people to mean different types of understanding. He also makes the observation that what we call mathematics is in fact taught in two very distinct ways. Skemp uses an analogy to try and explain the difference between relational and instrumental knowledge which I would like to explore. 

Imagine you are navigating a park, and you learn from someone else some specific paths to follow in the park. You move back and forth along the paths, and learn how to get from point A to B in the park, and you may even be able to move quickly from point A to B. Eventually, you add more points to your list of locations to which you know how to navigate. Step off any of your known paths though, and you are quickly completely lost, and you might even develop a fear of accidentally losing your way. You never really develop an overall understanding of what the park looks like, and you may even not know about other connections between the points you know. This is instrumental understanding.

Imagine that instead of navigating the park by specific paths shown to you, you get to wander all over the park. For some parts of the park you may be guided, through other parts of the park, you wander aimlessly. In time, you develop an overall picture of the park. You might discover the shortest paths between two points, and you might not, but you would understand the overall structure of the park, and how each point in the park is related to each other point. If someone showed you a short-cut in the park, you’d probably understood why it worked, and why it was faster than your meandering path. You wouldn’t worry about stepping off the path though, since even if you get lost, you’d be able to use your overall understanding to come to a place you know. This is relational understanding.

Here’s Richard Skemp’s description of the analogy.

“The kind of learning which leads to instrumental mathematics consists of the learning of an increasing number of fixed plans, by which pupils can find their way from particular starting points (the data) to required finishing points (the answers to the questions). The plan tells them what to do at each choice point, as in the concrete example. And as in the concrete example, what has to be done next is determined purely by the local situation. (When you see the post office, turn left. When you have cleared brackets, collect like terms.) There is no awareness of the overall relationship between successive stages, and the final goal. And in both cases, the learner is dependent on outside guidance for learning each new ‘way to get there’.

In contrast, learning relational mathematics consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point. (I say ‘in principle’ because of course some of these paths will be much harder to construct than others.) This kind of learning is different in several ways from instrumental learning.” ~ Richard Skemp, Mathematics Teaching, 77, 20–26, (1976)

Instrumental understanding is really useful when you have to know how to do a specific task quickly, and aren’t too concerned about how this task fits into other similar tasks. Relational understanding is useful when you want to explore ideas further, are unconcerned about your destination, and are more concerned with the process.

Unfortunately, our system tends to favour instrumental understanding too much. While it is useful to be able to get from point A to point B quickly, if one is not aware of one’s surroundings, and doesn’t get to enjoy the scenery, it hardly makes the trip worthwhile.

Story telling in mathematics

I tell a lot of stories when I teach, but not generally stories about my life or past stories of students. I use story-telling as a vehicle for explaining concepts that are difficult to understand when abstracted in symbols. 

For example, when I talk about sums of arithmetic sequences, I start with the story of the legendary mathematical genius Gauss as a kid.

Supposedly when Gauss was a kid, he was given the task of adding up the numbers from 1 to 100. His teacher expected this task to take a while, but Gauss finished it in seconds. He apparently was the only student to write down the correct response, 5050. What Gauss did was to group the additions from 1 to 100 like so:

1   + 100
2   +   99
3   +   98
     …
50 +   51

He noticed that each of these added up to 101, and that there are 50 such pairs. 50 x 101 = 5050, the answer Gauss came up with.

When I share this story with students, they tend to remember the process Gauss went through and they identify with him, not because he’s so smart, but because he bests his teacher which is a story a lot of students would like to believe. They remember the story because our minds are adapted for remembering stories.

Leading students through the more abstract proof of the sum of an arithmetic sequence formula is a lot easier when they understand where it comes from. I find story-telling in mathematics is an easy way to turn an abstract concept into something kids get and in my experience practically everything in mathematics comes from a story.

Want to explain infinite series? Tell the story of Achilles and the Tortoise. Want students to understand distance vs time graphs? Make sure to tell a story as you trace out the graph, particularly one with a lot of action you can pantomine.

Update: Here’s an example of a story being told in video form. 

Boat In The River — Question from Dan Meyer on Vimeo.

Multimedia can make telling the story easier, especially if the concept you want to share is complicated. Think of the pictures that you see in children’s books, and ask yourself, how much do they contribute to the telling of the story?

 

Story-telling is an excellent tool in mathematics education. How could you see this used in your own subject area?