A New York Times editorial on December 6 called for improved math instruction, calling the current system “broken.” Although I agree we could be doing a better job of teaching math, the suggestions in the editorial showed a striking naiveté about what it will take to improve.
The editors of the Times offered four suggestions:
The editorial ignores the fact that 1 and 4 will be meaningless without 2 and 3. And it grossly underestimates the difficulty of implementing 2 and 3.
The editors’ idea regarding how to implement “early exposure to numbers” is to ensure better access to preschool. But that won’t do it because “exposure” won’t do it. Math is not learned like a language. Children can learn vocabulary and more complex syntax by mere exposure. They can’t learn math that way.
Still, the editorial is right to argue that very early learning is important. I’d argue it’s the key to math reform.
That National Math Panel concluded that three things need to be in place for students to be proficient in math: (1) you need to have memorized a small number of math facts to automaticity; (2) you need to know standard algorithms that apply to standard problems and; (3) you need a conceptual understanding of number, and of what algorithms do.
American kids are okay (not great) on math facts and okay (not great) on algorithms. On conceptual understanding, they are terrible. A student can get by for a time by memorizing algorithms and when they are to be applied, but eventually, not understanding what you’re doing catches up with you, meaning it affects your success in math.
But lacking conceptual understanding makes people hate math long before that. What could be more boring than executing algorithms you don’t understand? And if you don't understand what you're doing, why wouldn't you conclude "I'm not really good at math?"
This conceptual understanding ought to start in preschool with ideas like cardinality and equality. “Very early exposure to numbers” is not going to do it. That doesn’t mean taking what we had been doing in first grade and asking kids in pre-K to do it. That means putting activities into pre-K (e.g., games and puzzles that emphasize the use of space) that will provide a foundation for conceptual understanding so that first-graders will be in a better position to understand what they are doing. (Though first grade math will also have to change for that happen.)
The other recommendation that the editors of the Times get wrong is “Better teacher preparation.” They focus on high school, noting that many teachers of physical sciences did not major in these subjects. That may be a problem, but it ignores a much more serious teacher problem.
Most American teachers—like most American adults, including me--don’t have a deep conceptual understanding of math. They are a product of the system we are trying to change. You cannot teach what you don’t know.
So what’s my recommendation for American mathematics?
We need to pay much closer attention to preschool and to early elementary grades. That will entail developing methods of helping children understand the conceptual side of math—methods we now lack. It will also entail professional development to train teachers in the conceptual side of math.
The size of this undertaking is massive. But it directly addresses the issue encapsulated in the editorial’s title: “Who Says Math Has to Be Boring?” The editorial focuses on the idea that it’s boring to do things without knowing why you’re doing them. So the proffered solution is real-world application. But I think a worse problem is not understanding how math works, being asked to execute algorithms with no understanding of what is really happening. That’s a heavier lift but will ultimately be more rewarding.
The editors of the Times offered four suggestions:
- A more flexible curriculum (especially better integration of engineering)
- Very early exposure to numbers
- Better teacher preparation
- Experience in the real world (perhaps via collaborations with industry)
The editorial ignores the fact that 1 and 4 will be meaningless without 2 and 3. And it grossly underestimates the difficulty of implementing 2 and 3.
The editors’ idea regarding how to implement “early exposure to numbers” is to ensure better access to preschool. But that won’t do it because “exposure” won’t do it. Math is not learned like a language. Children can learn vocabulary and more complex syntax by mere exposure. They can’t learn math that way.
Still, the editorial is right to argue that very early learning is important. I’d argue it’s the key to math reform.
That National Math Panel concluded that three things need to be in place for students to be proficient in math: (1) you need to have memorized a small number of math facts to automaticity; (2) you need to know standard algorithms that apply to standard problems and; (3) you need a conceptual understanding of number, and of what algorithms do.
American kids are okay (not great) on math facts and okay (not great) on algorithms. On conceptual understanding, they are terrible. A student can get by for a time by memorizing algorithms and when they are to be applied, but eventually, not understanding what you’re doing catches up with you, meaning it affects your success in math.
But lacking conceptual understanding makes people hate math long before that. What could be more boring than executing algorithms you don’t understand? And if you don't understand what you're doing, why wouldn't you conclude "I'm not really good at math?"
This conceptual understanding ought to start in preschool with ideas like cardinality and equality. “Very early exposure to numbers” is not going to do it. That doesn’t mean taking what we had been doing in first grade and asking kids in pre-K to do it. That means putting activities into pre-K (e.g., games and puzzles that emphasize the use of space) that will provide a foundation for conceptual understanding so that first-graders will be in a better position to understand what they are doing. (Though first grade math will also have to change for that happen.)
The other recommendation that the editors of the Times get wrong is “Better teacher preparation.” They focus on high school, noting that many teachers of physical sciences did not major in these subjects. That may be a problem, but it ignores a much more serious teacher problem.
Most American teachers—like most American adults, including me--don’t have a deep conceptual understanding of math. They are a product of the system we are trying to change. You cannot teach what you don’t know.
So what’s my recommendation for American mathematics?
We need to pay much closer attention to preschool and to early elementary grades. That will entail developing methods of helping children understand the conceptual side of math—methods we now lack. It will also entail professional development to train teachers in the conceptual side of math.
The size of this undertaking is massive. But it directly addresses the issue encapsulated in the editorial’s title: “Who Says Math Has to Be Boring?” The editorial focuses on the idea that it’s boring to do things without knowing why you’re doing them. So the proffered solution is real-world application. But I think a worse problem is not understanding how math works, being asked to execute algorithms with no understanding of what is really happening. That’s a heavier lift but will ultimately be more rewarding.
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