…or why the current narrative of mathematics education isn’t working.
But first, an exercise:
“In a valley shaded with rhododendrons, close to the snow line, where a stream milky with meltwater splashed and where doves and linnets flew among the immense pines, lay a cave, half-hidden by the crag above and the stiff heavy leaves that clustered below.”
Be sure to study this story, because we’ll be assessing your understanding of it in next week’s quiz.
“The beet is the most intense of vegetables. The radish, admittedly, is more feverish, but the fire of the radish is a cold fire, the fire of discontent not of passion. Tomatoes are lusty enough, yet there runs through tomatoes an undercurrent of frivolity. Beets are deadly serious.”
No, you won’t be able to use your book on the test. We expect you to know the story by now. We’ve been working on this one for the past two weeks!
“First the colors.
Then the humans.
That’s usually how I see things.
Or at least, how I try.”
Be sure to think about how this story relates to your life. We all use stories every day!
“Only three people were left under the red and white awning of the grease joint: Grady, me, and the fry cook. Grady and I sat at a battered wooden table, each facing a burger on a dented tin plate. The cook was behind the counter, scraping his griddle edge with the edge of a spatula. He had turned off the fryer some time ago, but the odor of grease lingered.”
Rewrite this story in another context. Be sure to change the characters’ names and the setting to make it new and different.
“Ships at a distance have every man’s wish on board. For some they come in with the tide. For others they sail forever on the horizon, never out of sight, never landing until the Watcher turns his eyes away in resignation, his dreams mocked to death by Time. That is the life of men.”
Now that you’ve seen several different stories, please tell me how they are connected to one another. What are the common relationships between them? How do they show the same idea in different ways?
These are the opening paragraphs from five books plucked somewhat randomly off of my shelves at home. As I said earlier, I only keep books that I really love, so there is something in all of these stories that is compelling to me. If you have read and recognize one or more of these books, perhaps you have a similar feeling towards the story (or stories). However, if you don’t know these books there isn’t enough down on the page to allow you to connect with them in any sort of real way before I interjected my assessment comments.
I know we’ve all had moments like these in our classes. When our students are jamming, the mathematical story is flowing, but due to factors beyond our control we have to assess too soon. Your right foot slams down on the pedal, the brakes screech, and before you know it the pleasant journey your class has been on is swerving out of control. I’ve done it more times than I’d like to admit, and it never ends well.
There is a pace to a good story that works. That is just plain right. There is a natural rise and fall of tension and calm. Even in a real page-turner, there are these moments of calm–just never at the end of a chapter. In his most recent post (well, one of the three) Avery talks about benchmark moments along the way, in which students feel success as they work towards a larger goal. These are the kind of moments that occur in a narrative. The overall tension remains unfulfilled, but there are little obstacles that are resolved as the story moves forward.
Unfortunately, the 50-minute class period is not conducive to having students develop a full and compelling narrative arc. This is something that may take more than one day, the same way a good book rarely gets finished in an hour. In the event that the story isn’t going to be over at the end of the period, the ideal scenario would be that we leave kids at the end of the day with a story that is interrupted at a good “chapter break”, a place where they are dying to know what happens next. We should break things off with a clearly defined question and the promise that tomorrow, they will be able to resolve the mystery.
Instead, what is more likely to occur in a typical classroom is that, once it’s been established that the story will be unfinished, the teacher steps in–like an omniscient narrator–and summarizes the ending for everyone. Instead of leaving the room with a question, students leave with an explanation that didn’t come from their own work. That they didn’t have to grapple with. This reinforces the idea that some people are good at math, and they understand clever “tricks” that make everything easy, and therefore that if they aren’t one of those people, they aren’t good at math.
But we don’t have to be slaves to a schedule. Just because the clock and the bell tells us that class is over doesn’t mean we have to wrap everything up into a tidy package. We can leave things until tomorrow. I know, I know. It’s really hard to leave things hanging. To let students leave the room with misconceptions. However this is not the same thing as leaving without a sense of closure. Remember, it is possible to resolve little things along the way. Have students explain what they’ve figured out so far, what their current thinking is. I like to call these “summary statements.” And have the class develop a question, or a series of questions, for what they want to explore next. Then you have a game-plan for your next class period. And the plan is to continue the story where it left off, not to interrupt the narrative, summarize it and then replace it with a new narrative. Stories are complex. They take time to develop. Learning is also complicated, and it also takes time. I think the biggest challenge for many teachers is finding a way to give students the gift of having enough time to process new ideas. To develop understanding at a pace that feels comfortable and manageable. Not rushed, not too slow. But just right.