Wednesday, September 14, 2011

I searched eight years for this answer

When I first began teaching secondary school Mathematics, roughly eight years ago, I wondered where word problems fit into the structure of a topic: Do they come at the beginning to provide a context or at the end, where they provide an application for the acquired skills. In case you don't want to read on, I am now certain that they should come at the beginning. Still, please keep reading, I want to discuss my reasons and give one example of what I mean.

Rationale for leaving word problems till the end: Students often struggle with interpreting word problems. They add another layer of complexity. Teaching students to interpret the terminology while they master the essential skills could prove too much for some of them. This is the approach taken in most textbooks.

Rationale for teaching word problems first (and throughout): Learning needs a context and most students struggle with abstract concepts.

This year, I have become totally convinced of the second approach. Here are my reasons:
  1. Most students seem to find it easier to think concretely than abstractly. Before I start teaching linear graphs, I give my students a graph representing the cost of hiring a plumber. They can work out the hourly rate (gradient / slope of the line) and call-out fee (y-intercept). It takes a lot of training before I can give them the graph of y = 5x - 2 and ask them about the gradient and y-intercept. 
  2. When teaching Maths outside of any context, be it a game, collection of patterns or real-life situation, students form the idea that it is made up of arbitrary rules. I even know students who maintain that those rules were invented by evil adults to torment children their age!
  3. Word problems are sometimes phrased in a way that is open to more than one interpretation, yet the students expect Maths problems to have only one possible solution. I am not a big fan of this fact but that's another discussion.
  4. Word problems can become more difficult after skills practice! This is because the students know that we want them to use the skills we have just taught them. This is why you find them putting their intuition on hold and using unlikely formulae when first principles are sufficient. 
As an example of the fourth reason, consider the following problem from a current textbook:
For a period of 6 months after planting a sapling, a certain kind of sunflower exhibits linear growth. After 3 months the height was 36 cm, and after 4 months the height was 45 cm. Find the rule, connecting the height of the sunflower, H (in cm) and the time t (in months). (The answer is H = 9t + 9)

A student who understands that the gradient is the rate at which the sapling grows each month will realise that it is 9cm/per month. The textbook takes 5 lines and a formula to work this out!
The y-intercept is the initial height of the sapling. A student of mine went back one month at a time, subtracting 9cm for each month, to find that the sapling was 9cm high at the time of planting. She represented this by drawing a small table. The textbook takes another 5 lines of algebraic acrobatics to work this out.

You may be of the same opinion as me or you may be in favour of leaving word problems to the end. Either way, I would love to read your views in the comments section.


Gaik Bee


Hm, in Singapore, children are given word problems from the Grade 1 equivalent.

Rambling Teacher


Thanks for the comment, Gaik Bee.
I don't doubt what you say. It is certainly part of every primary curriculum: "Jane bought 13 lollies and 2 to each of her 3 friends. How many lollies does Jane still have?" But does Algebra instruction progress from the concrete to the abstract or are students supposed to and endless stream of simultaneous equations like:
2x + 3y = 11 and 4x - y = 1
before the x and y begin to stand for any quantities that the student can relate to?