Pointless points and plotless plots: The dangers of accelerating mathematics learning

Everyone likes a good movie. Good movies tell a story. They have a plot. They keep the viewer engaged and interested.

Some movies, though, are forgettable. The story line might be impossible to follow. The plot is full of holes. There are moments that are engaging, interesting, or humorous, but they seem random and disconnected. Have you ever watched a movie that would have suffered no loss if the scenes had been randomly shuffled? I thought about that after watching one particular movie. (I won’t mention the name, but it took place on another planet.) It was more a piecemeal collection of scenes than it was a work of art. Movies, after all, have to be something more than simply a collection of bits if they want to be successful, if they want to tell a story that means something and connects with their viewers.

Math classes, too, must be more than a collection of seemingly unconnected bits. Good mathematics instruction tells a story. It has a plot. It keeps the student engaged and interested. The story line shouldn’t be impossible to follow, there should be no plot holes, and the experience should feel anything but random.

Where we lose the plot

Think back to your own experiences as you learned mathematics. Was it presented as a collection of random bits or were you engaged in the unfolding of a story?

Over the years I’ve heard many times, from many people, about how mathematics doesn’t make sense. It’s a remarkable thing when you think about it, as mathematics, arguably more than any other school discipline, is entirely grounded in sense-making. That’s what it is all about. Ideas in mathematics build on other ideas in a logical fashion. Nothing just “pops up” or stands alone. It’s all logically interconnected. Something is seriously broken when students see mathematics as the subject that doesn’t make sense when it is arguably the subject that makes the most sense.

Good mathematics instruction tells a story. […] The story line shouldn’t be impossible to follow, there should be no plot holes, and the experience should feel anything but random.

Yet here we are. This disconnect, this losing of the plot, is a natural consequence of a focus on “ways of doing” mathematics while ignoring the “ways of thinking” about the mathematics. It’s what happens when we are OK with pointless points and plotless plots, with merely collecting procedures.

For example, think about how you would answer the question, “How many years was it from February 14, 1912, to February 14, 2007?” Using the standard algorithm to find 2007 minus 1912 to answer this question would not be necessary, nor even efficient, but relating the quantities through subtraction as comparison and using mental strategies would be a wonderful approach. Think about how much less powerful only knowing the algorithm (the end of the subtraction progression) would be.

Mathematical procedures are players in the plotline of mathematics, but they are not the climax of the story. We are happy they are there, as they help move the plot along more efficiently, but the story doesn’t end with them. I think, perhaps, that some of this confusion arises from where procedures are generally positioned in state standards: they tend to come at or near the end of multi-year progressions. It’s tempting to think of the end of a progression as the goal of the progression, but it is not. Knowing the procedure is not a substitute for the thinking that corresponds to the relationships of the quantities within that procedure. The progression—the whole progression—is a story to understand.

Math takes time

Learning mathematics, truly learning mathematics, and not just that which is easily measurable, takes time. It’s about understanding and connecting ideas. It’s about being able to see the plot all the way back to the beginning. It’s not a race.

Something is seriously broken when students see mathematics as the subject that doesn’t make sense when it is arguably the subject that makes the most sense.

Now, with the interruptions of learning that have taken place over the past year and a half, there are many conversations about the need to accelerate students to help them catch up. There is a clear drive to get students back on track with respect to grade-level expectations.

One can accelerate math instruction by cutting content, speeding through content, or merging content, and each carries risks to learning mathematics deeply. Cutting content risks leaving out significant parts of the story line. Will students still be able to trace the plot back to the beginning? Speeding through content makes little sense when the standards in a grade are already sufficiently challenging. Have you ever tried to watch a movie at double speed? Merging content risks continuity as well, unless it’s done with great care. Can it be done without losing sight of the plot? Possibly, but it will take effort.

Perhaps the most dangerous risk in accelerating is the potential to miss the beauty in mathematics. Not only is there a plot that we can trace back to the beginning, but there is a sort of joy that everyone should have the opportunity to experience. This is summed up beautifully by the mathematician Jordan Ellenberg in How Not to Be Wrong: The Power of Mathematical Thinking. “Mathematicians aren’t crazy, and we aren’t aliens, and we aren’t mystics. What’s true is that the sensation of mathematical understanding—of suddenly knowing what’s going on, with total certainty, all the way to the bottom—is a special thing, attainable in few if any other places in life. You feel you’ve reached into the universe’s guts and put your hand on the wire. It’s hard to describe to people who haven’t experienced it.”

Let’s not steal this chance from our students.

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