Numbers don't change the situation
The coordinator of first year Chemistry had a chat to me the other day about how to support students in solving word problems. The issue is that students have trouble using the words to help them decide what sorts of calculations need to be done in order to solve the problem. This issue is not new – people have been solving word problems for thousands of years, and the maths education literature is littered with papers discussing the issue. No clear concensus has been reached, of course, because there are any number of factors that affect students' ability to solve problems.
One of these many factors I only learned about earlier this year when reading the following paper: A. Af Ekenstam and K. Greger (1983), "Some aspects of children's ability to solve mathematical problems", Educational Studies in Mathematics, 14, 369--384. It's easiest to describe using the following two problems (slightly modified from those presented in the paper):
| Problem 1: A block of cheese weighs 3kg. 1kg costs $28. Find the price of this block of cheese. |
| Problem 2: A piece of cheese weighs 0.923 kg. 1kg costs $27.50. Find the price of this piece of cheese. |
The paper reported how students aged 12-13 years were asked these problems, and specifically asked what sort of calculation they would choose to do in order to solve them. What would you choose for each one?
All of the students in this study chose multiplication for Problem 1. However many of them did not choose multiplication for Problem 2, and some of them did not know at all what to do. To be clear, it wasn't that the students didn't know how to actually perform the calculation; it was that they didn't know what sort of calculation to do. Even when the teacher explicitly pointed out how similar the two problems were, many students still did not know what to do for Problem 2. Upon discussion with the students they discovered that the students were choosing what calculation to perform based on the numbers they saw, rather than on the situation described.
This was a big surprise to me. Of course, I experience students not knowing what to do and choosing the wrong thing to do all the time, but it had never occurred to me that they were making the choice based on the numbers they saw. To me the situation itself has always told me what to do, regardless of the numbers themselves -- if every kilo is worth THIS dollars, then THAT number of kilos ought to be THIS times THAT dollars, regardless of what THIS and THAT actually are. But clearly not everyone thinks this way!
The authors of the paper have a few theories for why students are confused when the numbers are different.
One theory is to do with the students' experience of word problems. For many students, the majority of problems they've seen before have involved whole numbers for at least one of the numbers involved, and so seeing decimals in both positions just doesn't fit with their experience. Moreover, they have succeeded perfectly well on other problems by focussing on the numbers. This says more about the students' schooling than the student themselves, really.
Another theory is that their experience of numbers has led them to believe certain things about multiplication and division. With whole numbers, when you multiply the answer can only get bigger, and when you divide the answer can only get smaller. Other research confirms that these ideas are very strong in children and tend to impede them having a fuller picture of what multiplication and division mean for other types of numbers. In this experiment, some students talked about how in the second problem the cheese is less than a whole kilogram and so the answer ought to be smaller than $27.50, which is in fact a perfectly correct and quite sophisticated attack on the problem. But because the answer had to get smaller, they chose to do division, because this is how you make numbers smaller.
The final theory is that many people view multiplication and division (and most other things in maths) as a procedure, partly because of the focus on procedural fluency in primary school. In this context, the procedure for multiplying decimals by hand actually is different from the procedure for multiplying whole numbers. With decimals there's all this stuff about shifting decimal places back and forth which makes the procedure much more complicated. And working with fractions is wildly different again! So it's hardly surprising that students, when faced with a problem involving decimals, will expect that the action to perform should be different.
Regardless of the reason, one thing is clear: many students are not focussing on the right thing to help them solve the problem! So one way to help those Chemistry students is to help them focus on what the words tell them about the situation, and how the situation tells them what they should be doing, rather than the numbers themselves. Because it's the situation that tells you what to do, not the numbers, and the numbers don't change the situation.