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A lot of problems here. (1) "This study shows that all kids can learn math when taught effectively". Well, as the definition of "teaching effectively" is that

by ajarmst 11y ago

A lot of problems here. (1) "This study shows that all kids can learn math when taught effectively". Well, as the definition of "teaching effectively" is that students learn from it, this observation really isn't all that startling. We could do a similar study to prove that all children arrive at school when driven there. What would be startling would be if the same technique worked equally well with all students (and this one won't). (2) Ah, it wouldn't be a paper about educational techniquese without some Gladwelling (which I define as using a vastly over-simplified and facile interpretation of scientific findings in order to advance a puerile generalization). Yes, research shows that connections in the brain are modified when learning. It would be shocking if learning didn't have an effect on the brain. By FMRIs and other imaging is far,far too blunt to detail what is happening in any useful detail. Taking a scientific finding that "people seem to use these particular areas of the brain when learning" as justification for an entire philosophy of pedagogy is absurd. And, for the umpteenth time --- multiplying two numbers is no more "Mathematics" than learning how to drive a screw is "Mechanical Engineering". And encouraging children to explore driving a screw with a hammer, a pair of pliers, their teeth, a teammate and then having them do a class presentation on their findings does not help them in their later careers as mechanics. Far better to show them the most effective technique, let them practice it, and move on. "Exploration" learning is only effective when applied to skills that already have mental infrastructure provided by evolution. You learn your first language by exploration. You learn to walk by exploration. Riding bikes and mathematics were never evolutionarily selected for, so we have to learn them by, well, learning them.

Jtsummers 11y ago
> Far better to show them the most effective technique, let them practice it, and move on. This can work for the base levels of what most folks think as mathematics, namely arithmetic and algebra. But it starts falling apart as students get to more advanced algebra and mathematical topics. Memorizing a particular algorithm does not lend to understanding mathematics. In particular, the parallels between algebra (and some arithmetic) and geometry are non-obvious and not easily conveyed via a series of exercises and then jumping to the next topic. See the visualization examples of `18x5` in the linked article. Letting students see the correlation between the numbers and the objects (squares and rectangles) greatly helps their understanding of the subject. Discussing it reinforces this experience and gets them to see it beyond the exercises they've been doing, abstracting the concept beyond a handful of cases and to the general case. > Riding bikes and mathematics were never evolutionarily selected for, so we have to learn them by, well, learning them. There are many ways to learn and to teach. Teacher-tell, student-regurgitate is one method. And it works for some (many) subjects, at least in getting facts and figures into students heads. Teacher-tell, student-explain gets the students past thinking about facts and thinking about whys and hows of the world. (You can expand tell beyond spoken word to shown, demonstrated, etc.) Teachers can take it further by starting on a mathematical topic, and directing the student conversation and discussion towards understanding (I think this is the example I'm looking for, graph theory for eight-ear olds: http://jdh.hamkins.org/math-for-eight-year-olds/ http://jdh.hamkins.org/math-for-eight-year-olds/).
- jwdunne 11y ago
  To supplement this comment, How to Solve It by Polya is a nice little handbook on the education of mathematics. I'm reading through this at the moment and I've learned quite a bit. It's especially useful to see how I have unknowingly applied some heuristic to a problem I've solved in the past. As a side note, I'm finding both Calculus Made Easy and Concrete Mathematics incredibly useful. Thanks for the suggestions.
geomark 11y ago
"Far better to show them the most effective technique, let them practice it, and move on." Wasn't there a recent post on HN about something being expanded in Japan called "productive failing"? It's the opposite approach - let them struggle and then show them a/the way to solve the problem. The claim is that method is more effective.

by ajarmst 11y ago

Jtsummers 11y ago

> Far better to show them the most effective technique, let them practice it, and move on. This can work for the base levels of what most folks think as mathematics, namely arithmetic and algebra. But it starts falling apart as students get to more advanced algebra and mathematical topics. Memorizing a particular algorithm does not lend to understanding mathematics. In particular, the parallels between algebra (and some arithmetic) and geometry are non-obvious and not easily conveyed via a series of exercises and then jumping to the next topic. See the visualization examples of `18x5` in the linked article. Letting students see the correlation between the numbers and the objects (squares and rectangles) greatly helps their understanding of the subject. Discussing it reinforces this experience and gets them to see it beyond the exercises they've been doing, abstracting the concept beyond a handful of cases and to the general case. > Riding bikes and mathematics were never evolutionarily selected for, so we have to learn them by, well, learning them. There are many ways to learn and to teach. Teacher-tell, student-regurgitate is one method. And it works for some (many) subjects, at least in getting facts and figures into students heads. Teacher-tell, student-explain gets the students past thinking about facts and thinking about whys and hows of the world. (You can expand tell beyond spoken word to shown, demonstrated, etc.) Teachers can take it further by starting on a mathematical topic, and directing the student conversation and discussion towards understanding (I think this is the example I'm looking for, graph theory for eight-ear olds: http://jdh.hamkins.org/math-for-eight-year-olds/ http://jdh.hamkins.org/math-for-eight-year-olds/).

jwdunne 11y ago
To supplement this comment, How to Solve It by Polya is a nice little handbook on the education of mathematics. I'm reading through this at the moment and I've learned quite a bit. It's especially useful to see how I have unknowingly applied some heuristic to a problem I've solved in the past. As a side note, I'm finding both Calculus Made Easy and Concrete Mathematics incredibly useful. Thanks for the suggestions.

geomark 11y ago

"Far better to show them the most effective technique, let them practice it, and move on." Wasn't there a recent post on HN about something being expanded in Japan called "productive failing"? It's the opposite approach - let them struggle and then show them a/the way to solve the problem. The claim is that method is more effective.