... it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations ... Dr. Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler ... performed a randomized, controlled experiment. ... Though the experiment tested college students, the researchers suggested that their findings might also be true for math education in elementary through high school ...

In the experiment, the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.

Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. ... The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

“They tend to remember the superficial, the two trains passing in the night,” Dr. Kaminski said. “It’s really a problem of our attention getting pulled to superficial information.”

## Friday, April 25, 2008

### Maybe manipulatives aren't the answer?

This NYT article suggests that

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## 4 comments:

Thank you!

It's always nice to have research to back up my gut insticnts.

What are the odds that this will change the way math is taught?

No idea. Either way, I've found some of your hands-on and kinaesthetic lessons fascinating, and I still want to try them.

It's no trivial question whether these results generalize to younger learners, I think. Maybe a formal and abstract approach is more effective for those who have any chance of accessing that at all - but is it self-evident that young children can do that? Is the choice really between abstract understanding and concrete understanding, or rather between the latter and none at all? Would be worthwhile to read the actual study, I guess, but that will have to wait for now.

Is the choice really between abstract understanding and concrete understandingI think the goal is abstract understanding - and then being able to apply that abstraction to the concrete again.

concrete examples may provide scaffolding, but they are just that - crutches until the actual knowledge is built.

my (completely unscientific) guess is that poorer learners benefit from having some concrete examples, but will take correspondingly longer to break away from those examples to be able to understand the abstraction.

Interesting. I want to read the original article in Science, and so am missing having the access to everything that I did in college.

Were the manipulatives used to demonstrate the rules or to discover the rules? How abstract did the manipulative group go before they were tested? What about the amount of time each group had? For now I'm assuming they were equal, but everything I'm told says that discovery lessons take more time. Maybe that is because it's more often used for poorer learners, but going from concrete, to abstract, to concrete is more steps than going from abstract to concrete.

I'm also curious who remembers more a year later when I try to build on lessons that they've had before. "A brief refresher" sometimes takes forever...

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