There's wonderful value in learning math as a group: students can help one another, and every day be reminded that everyone can understand math.
But there's wonderful value in learning math as an individual: each student can spend time struggling with whatever puzzles bamboozle him or her.
Each of these has its advantages and disadvantages — we need a synthesis.
The synthesis that many American schools choose often doesn't convince students that everyone can learn all of K-12 math, and also doesn't give every student a collection of math puzzles that bamboozle them.
Our basic plan:
We spend about an hour a day on corporate math lessons — in K-8, using the JUMP Math curriculum. (The JUMP approach excels at rapidly breaking down complex big ideas into understandable tiny ideas, and then helping students arrange these tiny ideas together.)
We also, though, have students individually play with off-curriculum math puzzles (starting with the puzzles of James Tanton), with the goal of finding puzzles that still stump them after (say) 10 minutes of focused struggle. We have them collect those problems in their personal Deep Practice Book, and help them make small-breakthrough after small-breakthrough. (We don't just tell them the answer.)
Because we understand that all memories deteriorate after time, we have students regularly re-solve (and re-approach) all the problems in their Deep Practice Books — perhaps once a week. As time goes on, and they re-solve a puzzle three, four, five, or more times, something wonderful can happen: problems that had once been unsolvable become easy, and even obvious. Puzzles that had been vexing and hateful become delightful and friendly.
And every once in a while, the student will realize that there's a much more beautiful way of unravelling the puzzle. This is a momentous discovery: it's as if the clouds roll back and a trumpet reveille sounds from the heavens! How often did moments like that happen in your K-12 math experience?
As students engrave these puzzles into their brains, they'll grow to love math more, and understand it much more deeply than is now possible in the conveyor-belt math approach that most schools use.
We think it's possible for all students to perfectly understand the K-12 math curriculum.
We think it's possible for all students to grow to love (at least in small part) the process of doing and understanding math.
We think math can become the place in the curriculum where students most develop their growth mindset — that they see math is something difficult that they can do. We can shatter the myth that only some people can understand math.
If you walk into our classrooms, you might see:
Walking into our classrooms, you might stumble upon one of our whole-group math lessons: expect to see the teacher posing a score of tiny questions to a focused group of students. During independent work time, you might see a student frowning intently as she tries yet another way to solve an especially diabolical puzzle. When she fails (again!), she goes through a problem-solving methodology, to better tease out any clues that could help her crack the riddle.
Some specific questions:
- Are James Tanton's puzzles good to start in grade school?
- What other good options are available for mathematical puzzles?
- I mention here developing a single, standard problem-solving methodology. That would be great — if we could train the kids in just one, then performing it would become a habit, one that could extend what our kids are able to do throughout the rest of their lives. But: what problem-solving methodology should we try out? (I do have the beginnings of this, and will be working on it over the summer. Presumably, too, the kids in the classroom can slowly evolve an even better one!)
- Lee, are the kids in your class too diverse in ages/math abilities to do any kind of group lessons with? Or are there enough little kids to jump into JUMP at the lowest levels? An alternative (that still uses JUMP) is just to have kids work through the workbooks themselves, with the teacher giving assists when needed. Corbett Charter School did something like this (though not with JUMP), and an amazing math teacher that both you and I know suggested that this "kids working by themselves" method might suffice to help kids learn the whole K-12 math curriculum (though she didn't think it was ideal).
- Math circles haven't come into this description at all. They're magical. Where should they fit?