Just completed our annual Teachers’ Convention. It was not particularly eventful for me, but as always, I walked away with something to think about.
On Friday, I caught two sessions with playwright-mathematician John Mighton. Mighton is an interesting character. After an undergraduate degree which, in his words, left him with the impression that he wasn’t much good at anything, he put his energy into learning to write. He developed a modestly successful reputation as a playwright in Toronto, but still found by his early 30s that this was a tough way to earn a living. He made ends meet by tutoring kids and discovered that he was able to learn (re-learn) mathematics if he managed to break the learning into small, concrete steps. And he found that his pupils learned the same way. This led Mighton to graduate school and a career as a mathematician.
Somehow, Mighton maintained his interest in student learning, and put together the basic structure of Jump Math, which has grown to a not-for-profit organization championing Mighton’s vision and cooperating with university researchers to provide a research base to support, refine and change the program.
I’ve linked to the organization above, so I won’t repeat their basic information. Rather, I’ll recall and reflect on Mighton’s sessions.
Mighton hits on some themes that I’ve reflected on in the past—mainly the virtues of practice and mastery—but he takes it further. Jump Math is predicated on the principles that
· Scaffolding is essential to learning.
· The teacher is responsible for making the scaffolding increments small enough so that every student can make every move.
· Every child must climb every rung of the scaffold before the class can move forward.
· Success is its own reward.
· Every child (with enough cognitive capacity for language) can make significant progress in elementary mathematics.
· Competence precedes understanding.
On the one hand, this is a pretty unsurprising list of principles (do remember that this is my reading of the talk, not necessarily the voice of John or of the organization). On the other, it’s about as un-trendy as you can be in contemporary education.
Before I go further, a note about the Jump Math materials. You can buy stuff from them, but the teachers’ guides (Grades 1-8) are available for free on the website. Mighton emphasized that the student materials are quite uninteresting, as they are nothing more than overly-large practice sets. They’ll save you time and effort, but they are the least important part of the program. The teacher’s guides articulate the recommended scaffolding for the classroom lesson. If you only have one thing from Jump Math, Mighton says, make sure it’s the teacher guide. This seems eminently sensible to me.
So how might a teacher prepare a lesson utilizing the Jump ideas?
First you need to know where on the continuum of background knowledge and skills each child in the classroom lies. This is relatively straightforward. Mighton used the miniature whiteboards that are common in classrooms today. (Actually, it was a sheet of white paper inside a plastic folder; participants wrote in dry-erase marker on the folder.) The Do Now activity is simple and relevant to the activity. If today’s lesson, for example is the addition of simple rational expressions, then I need to know if each student is comfortable adding fractions. So I might put a simple fraction addition on the board, and ask each student to perform the addition on the white boards and hold it up. We’re talking simple 10-second stuff here. When each student holds their board up (no exceptions; no one is allowed to opt out) then you can see if you’re ready to move forward. If anyone requires attention, do it now. Again, you move together in a group.
Side note: it’s easy to see Mighton’s background in theatre. When an audience is unified, there is an energy in the room, that far exceeds the energy of individuals working separately.
The key part of the lesson is “what’s next?” When the simple addition is successfully performed, add a SMALL bit of extra complexity to the problem. The idea is that even fairly simple mathematics requires a surprisingly large number of small pieces for it to be sensible. If I am going from 2/3 + 4/5 to (x+1)/(2x-3)+5/(x-6), I should do it in baby steps. Make sure that every idea of common denominator, gathering of like terms, reducing fractions, etc. is in place. If I give all at once, I can be sure that some will get it (eventually) and some will not. The belief behind Jump is that everyone can and should get each step before moving on.
I’ll not belabour the lesson further, as I think the point of the lesson is reasonably clear. In a sense, I think every teacher is onside with much of the above, but there are still moments of uncertainty. Am I going to slow? Am I going to bore my brighter students? What if someone doesn’t catch on? Do I have time for all of this?
Of course, the answers to these questions are found in practice, not theory.
The other family of concerns is with the “richness” of the problems. There is a powerful movement in mathematics education that asserts that openness, richness and exploration are the keys to mathematical learning. I’ve looked at this in the past, with my reflections on a video by Dan Meyer. Mighton is even more skeptical than I am about the virtue of open problems for most students most of the time.
Ultimately, the sessions have provoked me to look harder at my scaffolding and to be more precise in my progression. I have always taught in a similar way, but I have not been as scrupulous as I might have been about always taking small steps and always ensuring 100% understanding before moving on. In my defense, I teach high school academic mathematics, and students are capable of storing anomalies for now, and resolving them later. Or have I been assuming too much? How big do these steps need to be? I will report back.
Finally, Mighton offers a view of inclusion that is curiously out of sync with most established views. In most of the literature on inclusion (see my earlier entries on UDL, for example) we plan for multiple entry points for students with multiple means of participating in activities at their own level. Jump suggests that this is excessive. Jump Math claims that we can harness the group dynamic (Weber’s “collective effervescence”) and differentiate by working together at all times. If this is so, then much can change in education—well, mathematics education at least. It is likely that this structured approach is well suited to mathematics because of its rigid internal structure; learning to read is likely a different sort of cognitive experience. But that’s a talk for another day.