3: Word Problems
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All the evidence suggests that the shapes of reality are mathematical.
George Steiner (1929– )
The previous chapter focused on aspects of the arithmetic of pure numbers - mostly without any surrounding context. However, our mathematical experience does not begin with pure numbers. At school level, mathematical concepts, and the reasoning we bring to understanding and using them, have their roots in language. And in real life, every application of mathematics starts out with a situation which is described in words, and which has to be reformulated mathematically before we can begin to calculate, and to draw meaningful mathematical conclusions. Word problems play an important, if limited, role in helping students to appreciate, and to handle the subtleties involved in
the art of using the mathematics we know
to solve problems given in words.
This art of using mathematics involves two distinct - but interacting - processes, which we refer to here as “simplifying” and “recognising structure”.
- To identify the mathematical heart of a problem arising in the real world, one may first have to simplify - that is, to side-line details that seem unimportant or irrelevant, and then simplify as much as possible without changing the underlying problem (e.g. by replacing some awkward feature by a different quantity which is easier to measure, or by an approximation which is easier to work with).
This “simplifying” stage is well-illustrated by the tongue-in-cheek title of the classic textbook Consider a spherical cow ... by John Harte (1985):
Milk production at a dairy farm was low, so [... ] a multidisciplinary team of professors was assembled. [... After] two weeks of intensive on-site investigation [... ] the farmer received the write up, and opened it to read [... ] “Consider a spherical cow . . . ”.
The point to emphasise here is that the judgements needed when “simplifying” are subtle, depend on an understanding of the particular situation being modelled, and may lead to a model which at first sight seems to be counterintuitive, but which may not be as silly as it seems - and which therefore needs to be explained sensitively to non-mathematicians.
In contrast word problems by-pass the “simplifying” stage, and focus instead on “recognising structure”: they present the solver with a problem which is already essentially mathematical, but where the inner structure is contextualised, and is described in words. All the solver has to do is to interpret the verbal description in a way that extracts the structure just beneath the surface, and to translate it into a familiar mathematical form. That is, word problems are designed to develop facility with the process of “recognising structure”, while avoiding the complication of expecting students to make modelling judgements of the kind required by the subtler “simplifying” process.
Because word problems focus on the second process of “recognising structure”, they tend to incorporate the relevant mathematical structure isomorphically. The underlying structure still needs to be identified and interpreted, but the interpretations are likely to be standard, with no need for imaginative assumptions and simplifications before the structure can be discerned. For example, if a problem in primary school refers to an unknown number of “sweets” to be “shared” between six children, then the collection of “sweets” is isomorphic to a pure number (the number of sweets); and the act of “sharing” is a thinly veiled reference to numerical division.
The story in a word problem may be a purely mathematical problem in disguise. But the art of identifying the correspondence between
the data given in the story line, and
the mathematical entities to which they correspond and
and between
the actions in the story line, and
the corresponding mathematical operations on those mathematical entities
is non-trivial, and has to be learned the hard way. The first problem below illustrates the remarkable variety of instances of even the simplest subtraction, or difference.
As in Chapters 1 and 2 the “essence of mathematics” is to be found in the problems themselves. Some discussion of this “essence” is presented in the text between the problems; but most of the relevant observations are either to be found in the solutions (or in the Notes which follow many of the solutions), or are left for readers to extract for themselves.