Does a discussion of real life examples help in teaching abstract concepts in mathematics? A recent research finding says it ain't so.

... The idea is that making math more relevant makes it easier to learn.

That idea may be wrong, if researchers at Ohio State University are correct. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives [and other such real world examples] stay in the real world and, in the classroom. ...

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. “We told students you can use the knowledge you just acquired to figure out these rules of the game,” Dr. Kaminski said.

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 other research finding is even more insidious: professor ratings on RateMyProfessor.com are highly correlated with other such evaluations!

## 7 Comments:

Abi,

See

http://en.wikipedia.org/wiki/Wason_selection_task

The first finding seems contradict this. Whether this is so or not, I am not sure. But these days much of this reporting seems to come for university publicity sources and quite a bit of based on mickey mouse studies.

Much depends on the what is being taught, and how. (Like a seasoned slashdotter, I haven't clicked on your link.) I'd teach concrete examples together with, or after, the abstraction -- not before. And in particular, if it is a concept that is widely applicable, I'd make that as clear as possible at every stage -- both while teaching the abstractions and while teaching the examples.

Unfortunately it's not just elementary-school kids who have problems with abstractions. I keep running into students with undergraduate degrees in physics or math or engineering, who can find the eigenvectors of a matrix or the "normal modes" of a system, but are lost when asked to use an arbitrary initial condition (apparently this is not a textbook problem in India): it doesn't occur to them that the point of eigenvectors/normal modes is

one can use them as a basisto represent the initial condition, and that immediately gives you the solution.Empirically, this study appears to be analogous to this 2006 paper, which showed that a very elementary process could separate people who could understand programming from those who just wouldn't be able to cut it. Note that 'motivation' didn't help either! It strikes me that there may be a (genetic?) aptitude for stringent logical thinking that can't be imparted with any amount of training?

OK I have now read the NYT article. I haven't yet read the Science article -- can't access it at home without jumping through some hoops. The train example at the start of the NYT article is a perfect example of why abstractions should come first. You can't solve that problem without first knowing how to set up an algebraic equation and solving it. But I assume that's not the actual problem that the researchers used.

Feodor - it could be that some people think abstractly better than others, but the research doesn't seem to mention such differences. It suggests that, in general, learning the abstract rules is better than learning specific examples, and even better than learning the abstractions after learning the specific examples. I'm not totally surprised by that.

Generally, when you look at "answers to problems", you have two reactions: either "that's a nice idea, I could use that", or "how is anyone supposed to think of

that?" I think focussing on specific examples leads to the latter situation, which is not helpful.Elementary example: solving a quadratic equation x^2 + bx + c = 0. The solution involves adding and subtracting the term (b^2)/4, which lets you factorise the terms involving x as (x + b/2)^2. If this is thrown at you without warning, you would say "Yes, but how is anyone supposed to think of that?" It doesn't help if you insert concrete values for b and c, and phrase it in terms of some real-world problem. But if you explain the concept of "completing the square" and how to do it in general, suddenly it becomes more obvious and more useful.

"You can't solve that problem without first knowing how to set up an algebraic equation and solving it. But I assume that's not the actual problem that the researchers used."

I am not sure of this. There are a couple of ways of thinking about this. If they start simultaneously, one can guess that when they meet the distances travelled are proprtinal to their speeds. Or to reach the mid point one needs 5 hours and the other four hours. Since one is given a head start of one hour, they should meet at the mid point.

The article does not speciify at what age which is suitable. Von Neumann is supposedto have solved a similar problem (both starting at the same time) by using infinite series.

My granddughter was taught to call me 'thatha' and my wife 'ammamma'. Soon she started using the same terms for all old people. I think that this may be considered as an example of abstraction from cocrete. It is possible that different approaches work at different ages.

There is a recent report by US Math. Advisory Panel reported in Sciencemagazine (needs subscrption)

http://www.sciencemag.org/cgi/content/summary/319/5870/1605

I am not able to access the report but from the science magazine article, it seems that there is no concensus except that some grounding in Algebra around eigth and ninth standards is supposed to be good. They also recommend at an earlier age "There's been a tendency in recent decades to regard fractions to be operationally less important than numbers because you can express everything in decimals or in spreadsheets. But it's important to have an instinctual sense of what a third of a pie is, or what 20% of something is, to understand the ratio of numbers involved and what happens as you manipulate it."

My guess is that both concrete examples and abstraction are important and at what age etc is a matter for detailed studies and cannot be decided by one or two experiments. Moreover one prescription may not suit all. For example I like abstract stuff and hate examples. Lot of mathematicians like Weyl, Von Neumann say that really good mathematics comes from concrete problems, but here we are talking about a different group of people.

Sorry; there is at least one correction to my previous comment. I think Von Neumann soved a different problem. It is like this. From A, train starts as in the previous example at 40 miles an hour and helipter starts at the other end at 80 miles an hour. When they meet, they helicopter turns back goes to B and starts again. It keeps doing it until the train reaches B. The question is what is the distance travelled by the helicopter. Mathematicians are supposed to take half a minute to one to solve this once they realize that the distance travelled by helicopter must be twice the distance travelled by the train. Some like me struggle with infinite series. Von Neumann soved it in a flash and the questioner said that he must have seen the simple solution. But von Neumann responded that he, of course, solved by summing up the infinite series.

I may be also a bit off about the last comment about'concrete problems'. It may be 'specific problems' or some such thing. I think that the articles are in "The World of Mathematics" edited by J.R. Newman but I do not have a copy of it.

The Wason selection task is about performance/understanding in a logical task, not extension to another domain, which is what this study shows. In general, I would agree with the conclusion that people who work with abstractions are able to apply them more easily to another domain. Concrete examples are good for understanding, but not for extension.

The study appears counterintuitive only because these two are usually conflated, i.e. if you understand a phenomenon, it is expected that you know the underlying principle and can apply it readily elsewhere. That is probably not true. Examples are like a GUI, it lets you access and use a system, but to actually build another system, you need to go beyond the GUI.

Extending this analogy, if you think of teaching as a form of usability engineering, then GUIs is the way to do it. Building different GUIs, until students get the abstraction, a sort of paving the way. If you think of teaching as a form of coaching, then making the student jump ever higher abstraction hoops is the way to do it. The latter is less work for the teacher, and will get results faster, but any fool with a master's can probably do it. Maybe even a well-designed program could step someone through ever fancier abstractions. Good teaching involves a judicial mix of the two.

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