I spent the whole day reading other people’s blogs psyching myself up to do this, so now I only have a few minutes to do it, so this one’ll be short.

In April 2008 the NY Times published a little article announcing a study by researchers at Ohio State University claiming to find that “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.” The study was published in no less a forum than Science Magazine. Knowing that something must be terribly wrong, I found myself inexorably drawn to read the Science article for myself.

“The Advantage of Abstract Examples in Learning Math”
by Jennifer Kaminski, Vladimir Sloutsky, and Andrew Heckler
Science 25, April 2008, Vol. 320. no. 5875, pp. 454-455

The article summarizes the author’s findings that when they taught a certain mathematical concept through concrete examples, students did not do a good job applying the concept in a novel situation, but when they taught it through an abstract representation, the students successfully transferred the concept to a new situation. They concluded that “If a goal of teaching mathematics is to produce knowledge that students can apply to multiple situations, then presenting mathematical concepts through generic instantiations, such as traditional symbolic notation, may be more effective than a series of ‘good examples.'” (p.455)

The bad news is that the actual details of the study design are not found in the article. The good news is that they are found in a supplemental thingydoo on the Science website. The bad news the article costs $15 to download. The good news is that the supplemental materials don’t cost anything. Even better, since I paid for and printed out everything and it’s next to me right now, you can benefit from what I found out without even leaving this site.

If you’re a math teacher you already know there must be something rank in here somewhere. The authors are actually claiming to have scientifically shown that it’s better to introduce a math idea in a way totally disconnected from reality than to base the idea on anything the students already understand about the universe. Well, a look thru the online support material makes the rankness available to all:

First of all, when the authors say they gave students abstract training in one condition and concrete training in a different condition, what they mean is that in both conditions, the students stared at words on a computer screen. In the abstract condition, the words went like this:

“On an archaeological expedition, tablets were found with inscriptions of statements in a symbolic language. The statements involve these three symbols: {circle} {diamond} {flag} and follow specific rules.

“Rules for combining symbols.

“Rule 1. The order of the two symbols on the left does not change the result.

For example {diamond}, {flag} -> {diamond}

is the same thing as {flag}, {diamond} -> {diamond}


In one of the three concrete conditions (the other two were similar), the words went like this:

“A pizzeria takes orders for one, two or three slices represented on individual cards as {1/3 of a pizza}, {2/3 of a pizza}, and {3/3 of a pizza}.  Multiple orders are placed at a time; and the cook systematically burns a portion of each group order.  Antonio needs help to determine how much pizza is burned.  There is never more than 1 whole pizza burned.  So the burned amount will always be {1/3}, {2/3} or {3/3}.

“Rules for finding how much pizza is burned stated by Antonio:

“Rule 1.  What I order first or second doesn’t matter.  The same amount gets burned.  For example, if I order this {3/3} first and then this {1/3}, then this much {1/3} is given to us burned.  The same thing happens if I order this {1/3} first and then this {3/3}.  We get this much {1/3} burned.”

In other words, the instruction in the two conditions was substantively identical, except that in the “concrete” situation there was some additional distracting information.

But the whole point of using concrete examples in teaching math is to connect the mathematical structure to a comprehensible reality in the students’ world, and this was not done at all, in either condition.  So the question that the study authors, and the NYT article, addressed themselves to (“Is it better to use concrete or abstract teaching methods?”) wasn’t answered.  The question that got answered was “Do people learn better when you don’t throw in distracting info?”  (Oh, you mean they do?  You’re kidding, right?)

I’ve already made myself late but I want to tell you more detail just to have it on record, so I’ll post again soon.