Republic of Mathematics blog

Contrived versus genuine mathematical problems

Posted by: Gary Ernest Davis on: April 15, 2011

It might come as a surprise to some people, but kids are not stupid.

So what’s a little kid to make of the following question:

A black-spotted Dalmatian puppy named Orion, courtesy of Pharao Hound

“Bobby has a dog with 8 spots. Jennifer has a dog with 3 fewer spots than Bobby’s dog. How many spots does Jennifer’s dog have?”

What’s a kid to think?

Did Jennifer count the spots?

How did she know here were 3 fewer spots.?

Could we ask Jennifer how many spots her dog has?

Aren’t these sensible questions?

Much more sensible than the original “word problem”.

Kids are not fooled by these questions. They know they are not real mathematical problems. They know these problems are just invented as puzzles to test if they know what “fewer” means.

We lose kids trust if we pretend these ridiculous puzzles have anything to do with real mathematical problems.

Here’s another contrived question at a more advanced level:

“B stuffs twice as many envelopes as A in half the time. If they stuff a total of 700 (in same time) how many did B stuff?”

Well, why not ask B? Surely he or she knows how many envelopes they stuffed?

How else did they, or someone else, figure that B stuffs twice as many envelopes as A in half the time?

Is that an historical fact, always true, or is it only true in this instance?

The problem is contrived, and it doesn’t take a particularly smart kid to realize that this is a setup, being used to force them to carry out certain arithmetic operations.

It’s not a real problem, in other words: it’s a phony problem built around a mathematical task.

What’s being tested here?

The ability to convert “twice as many in half the time” to a ratio of 4:1?

The ability to use the 4:1 ratio to realize that of a certain number of envelopes, B stuffed \frac{4}{5} and A stuffed \frac{1}{5} ?

The ability to calculate 700\times \frac{4}{5} ?

And if a student gets the answer wrong, they were wrong because …?

And if the question is multiple choice (which it would be because this is a proposed SAT question) did a student get the correct answer because they went through these steps, or by eliminating some answers, or by plugging in one of the proposed answers?

In other words, if this question is testing something, success or failure in answering the question does not tell us whether the student succeeded or failed on what was supposedly being tested.

This sort of  question has nothing  – should not have anything – to do with the serious practice of mathematics.

The same structure of the problem could be used to solve  problems that really does need to be solved. For example:

“Brenda gets paid every 2 weeks, and Allyson gets paid each month. Brenda and Allison decide to jointly invest in a company’s stock. They agree that Brenda will put in twice as much money each 2 weeks as Allyson puts in each month. When they have together purchased a total of 700 shares the company will offer a dividend of per share. Allyson and Brenda want to figure how much they each will receive in dividends at that time.”

This is now a real problem for Brenda and Allyson.

Brenda and Allyson set up the process of buying shares on a 4:1 ratio – that was given before we knew anything else. When the share dividend comes due the  two women want to know how much they will get in dividends. That’s a sensible, real problem.

We run a severe risk of making mathematics appear to be a bunch of tricks to solve problems that are contrived in order to simply test some procedure or other knowledge.

Lieven Verschaffel

A leader in the use of real, sensible word problems as contrasted with contrived, and often foolish, word problems is Lieven Verschaffel at the University of Leuven, Belgium.

Here is the abstract from an article published in the Journal for Research in Mathematics Education, 1997, Vol. 28, No. 5, pp. 577-601, with Erik De Corte:

Recent research has convincingly documented elementary school children’s tendency to neglect real-world knowledge and realistic considerations during mathematical modeling of word problems in school arithmetic. The present article describes the design and the results of an exploratory teaching experiment carried out to test the hypothesis that it is feasible to develop in pupils a disposition toward (more) realistic mathematical modeling. This goal is achieved by immersing them in a classroom culture in which word problems are conceived as exercises in mathematical modeling, with a focus on the assumptions and the appropriateness of the model underlying any proposed solution. The learning and transfer effects of an experimental class of 10- and 11-year-old pupils–compared to the results in two control classes-provide support for the hypothesis that it is possible to develop in elementary school pupils a disposition toward (more) realistic mathematical modeling.

This was published over 13 years ago in the Journal for Research in Mathematics Education, an American publication that is widely recognized as the leading journal for mathematics education research.

Still, over 13 years later, and despite much research and development in realistic word problems, we are, in the United States at least, promoting contrived word problems that in my opinion signal to students we are either lying to them or treating them as stupid.

It is time to stop this nonsense, and focus on realistic word problems.

This does NOT mean such problems have to have their origins in applications outside mathematics itself.

For example, a problem involving number theory or geometry can be a realistic problem within mathematics, as are  most of those discussed by James Tanton (@jamestanton) or Alexander Bogomolny (@CutTheKnotMath).









5 Responses to "Contrived versus genuine mathematical problems"

Brilliant! This was poignant and funny… I’m forwarding your blog link to my sister (math teacher extraordinaire in Dallas). Glad to find you on Twitter’s #ff.

I’ll never forget when a pre-calculus student nailed me on this. He was complaining about some word problems I’d assigned. I said “Life comes to you in word problems.” And he replied “Yeah, but not problems like THIS.” He pointed to some problem that was clearly contrived and I had to concede he was right.

From that day on I’ve tried to give realistic problems. It’s hard to find problems that are realistic but not too complicated or tedious. I at least try to be honest when I’m artificial: we’re going to make an unrealistic assumption here because a more realistic version would take too long. But then I give them a glimpse of what the more realistic problem statement would be.

By the way, your post reminded me of this one about business risk. See the example of betting on a coin flip and the list of questions a real business person would have.

No matter how you design a math textbook, people are going to assume it’s a tome of black magic. So you might as well make it LOOK and FEEL like a tome of black magic —- at least that way, it’s cool ;)

Great post! A friend of mine actually created a video on YouTube about this exact thing. Very funny.

It’s such a good point. I generally try to create word problems that relate to the student’s career interests and use their name and their friends’ names to make it seem more relevant. Of course, this is only practical in one-on-one tutoring sessions rather than a large classroom, but it does work.

I like your post, focusing as it does on the student’s point of view.
“The problem is contrived, and it doesn’t take a particularly smart kid to realize that this is a setup, being used to force them to carry out certain arithmetic operations.” Finally, when it’s assessed, what does it tell us about the student’s thinking? What’s really being tested?

One of my colleagues is moving away from text-based ‘word problems’ and is now promoting a visual approach to presenting problems, as you can find in Dan Meyer’s “Real World Math” video.

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