In the April 25 issue of the New York Times, there was an article entitled, “Study Suggests Math Teachers Scrap Balls and Slices”, about research that was reported in Science Magazine on the same date, entitled “The Advantage of Abstract Examples in Learning Math”.

To give you an idea of how journalists reacted to this story, there were 148 results, (15 pages) in a google, search with the exact 8 words from the NYT article title, including articles in the District Administrator and the Christian Science Monitor. Interestingly, there were only 71 entries, (7 pages), with the exact 8 words of the title from Science Magazine. The Times did not cite the title of the Science article.

I received three emails, from friends, about the NYT article. The Times article left me with more questions than answers so I decided to read the Science article. I found out that I had to pay $10 to view the article, or subscribe at an annual rate of $99 to $144. Thank goodness for the public library.

I was not able to discern whether the Times reporter had read the Science article. He did interview one of the researchers/authors, Jennifer A. Kaminski. The language of the Science article was rather technical. For example, one form or another, of the word “instantiate” was used 46 times in the article that was less than two pages long. I had not heard the word before and had to look it up. There was also some math. I will attempt to explain the two articles, from my math-teacher perspective.

The New York Times article starts out:

"One train leaves Station A at 6 p.m. traveling at 40 miles per hour… Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn".

The author of the Times article and Dr. Kaminiski are using the scariest example of a word problem from our high school math class, thus tapping into the deep-seated math phobia of the average American. The train example is irrelevant to the research reported in the Science article, unless high school teachers around the U.S. are taking students out to the train tracks armed with stopwatches, or setting up the Lionel electric train in the classroom. And besides, the way most of us learn about the train example is in the dreaded Word Problem section of the algebra textbook, after we have learned all of the techniques for solving linear equations. In other words... the train problem shouldn't come before we have the algebraic tools to solve it.

From the Science article:

“Instantiating an abstract concept in a concrete, contextualized manner appears to constrain that knowledge and to hinder the ability to recognize the same concept elsewhere; this, in turn, obstructs knowledge transfer. At the same time, learning a generic instantiation allows for transfer, which suggests that such an instantiation could result in a portable knowledge representation.”

Well this seems obvious: Children learn to count by first counting objects, (manipulatives in teacher lingo) often their fingers (“concrete, contextualization”). They get a little older and learn that symbolic math is more practical when dealing with large numbers. They learn abstract techniques such as the multiplication and division algorithms, (“generic instantiations”), for dealing with larger quantities and more complex problems. Trying to do division with multi-digit numbers, without knowing a division algorithm, addition facts and times tables, by counting on one’s fingers, will hinder anyone.

From the Times article:

“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.”

Translation: Cut a cookie in half, cut a sandwich in half, divide a group of 4 jelly beans into two equal groups. Do you now understand that 1/2 + 1/2 = 1? Probably not.

“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.”

Translation: (Ignoring the fact that trains are irrelevant) We understand that a the pizza has twelve pieces and there are 4 of us, each one gets 3 pieces, if the sharing is to be equitable. We still may not know that 1/4 x 12 = 3, or that 1/4 = 3/12. But was the pizza good?

"The researchers said they had experimental evidence showing a similar effect with 11-year-old children. The findings run counter to what Dr. Kaminski said was a “pervasive assumption” among math educators that concrete examples help more children better understand math".

Reaction: If this is a pervasive assumption in American schools, and teachers are wasting hours having students count M&Ms, pizza slices, and jelly beans, then there is definitely a problem. Do we want our children to understand fractions or be connoisseurs of pizza? Manipulatives, vis a vis math education, should be used as a bridge to learning abstract mathematical language and processes, and be directly related to understanding the concepts taught.

On the other hand, I give you a possible counter-example as food for thought: The abacus is a manipulative sometimes used in schools. It is also used by millions of people to make calculations. At what point does a manipulative hinder learning? Can we say the use of the abacus has hindered people in their ability to learn math? The abacus is a manipulative as well as a calculating device. Then what affect does the use of a calculating device have on learning math? It looks like we may need another study.

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