Types of Solutions and Multiple Solutions (Part 3) – What’s your perspective on multiple solutions?

In the last post, I described a lesson in which my year 8 students attempted three problems while I tried to help them see the power and elegance of algebraic strategies without favouring any method. In this post I’m going to talk very generally about different strategies in mathematics from different perspectives.

I was going to start with an in-depth discussion of my second lesson which covered some interesting problems. Instead I think I’ll pick up with the second problem on arithmagons from the first lesson. The problem is as follows:

The numbers in the squares are the sum of the numbers in the circle either side. Find the missing numbers.

A different solution to the one I presented in the last post is a standard use of three simultaneous equations:

x + y = 30 (1) , y + z = 23 (2) , x + z = 25 (3)

Making x and z the subject in equations (1) and (2) respectively and then substituting into (3) we get:

 30 – y + 23 – y = 25   and hence the unique solution of:             y = 16, x = 14, z = 9

When my students saw a second algebraic approach to the same problem we had a discussion on different methods in mathematics. I was amazed that one of my students actually believed that while there’s more than one way to approach a problem,  it is not possible to have more than one algebraic approach. She obviously saw algebra as a set of rules instead of a creative problem solving tool. After that comment I made a significant effort to model different algebraic approaches and set projects that encouraged a creative use of algebra. However I often make it clear that I do not favor one method over another. I use the analogy that mathematics is remarkably similar to writing stories.The last sentence is useful but can be quite boring, i.e. “they all lived happily ever after.” It is the story that captivates you, not the last line.

The question I have now come to ask myself is whether I should be favoring one method over another? The reason that I ask this is to do with some research I read recently comparing Chinese teachers views of mathematics to American teachers¹. It was only a small study carried out with a sample of around 20 teachers but it did make me think. I must stress before I delve into a discussion on this that I do not believe that this piece of qualitative research can be generalised to all Chinese and American maths teachers – again, it simply made me stop and think.

The 20 teachers involved in the study were asked to mark 28 students responses to different problems, giving each problem a mark from 0-4. Every student got every question right so the marks were awarded for the approach strategy and had nothing to do with the correct answer. The interesting thing was that the Chinese teachers gave much lower marks to drawing strategies. According to the Chinese teachers:

“Being able to solve a problem is good but it is just the first step of learning. Chinese teachers want students to use generalised problem solving methods that will transfer to other problem situations.”

Conversely, the American teachers’ goal was to have students solve problems no matter what strategies they use. Take the pizza ratio problem given to 6th grade students in China and America. In this problem, seven girls have to share 2 pizzas and 3 boys have to share one pizza. Do the boys or girls have more pizza each?

A conventional strategy used by many Chinese students uses an efficient equivalent fractions approach.

E.g.           Boy = 1/3 = 7/21                 Girl = 2/7 = 6/21             Girls get a lower share.

Many of the US students used much less conventional strategies.

E.g.         or             

In another study carried out to find which strategy teachers would suggest to students as ‘one of the better ways’ to solve a problem, the teachers gave varying answers². Here’s an example similar to the one recently posted by Dan Meyer on shadows. 

Problem: A building that is 24 m high casts a shadow of 18 m. At the same time, a flagpole casts a shadow of 15 m. How high is the flagpole?

Teacher A Response: In geometric problems we recommend a drawing.

Teacher B Response:

Teacher C Response: I would accept any solution that was shown to get the answer. Remember this is 7th Grade.

I think Teacher C’s response is particularly interesting. He/she seems to be implying that because they are only in 7th Grade they should be exploring different solutions. I wonder if he/she would have a different opinion if it were an older student?

After thinking about this for a while, I have called upon another aspect of learning which I believe to be absolutely axiomatic in teaching any topic. This is that all learning should be situated (i.e. it should start with where the learner is currently at). If I didn’t encourage original solutions that come from students then I may be enforcing approaches on students who aren’t ready for them or simply confusing students who prefer their own approach. I have thus decided that when multiple solutions arise in class, we will celebrate all of them. However, on the issue of transferability to other problems, I have decided to either tell students which method/skill I believe is more transferable, or ask them to think about which one they think would help with other problems. I think it is also acceptable, and necessary, that people have different views on which strategy they prefer – in this case I don’t see why I can’t say which one I personally prefer and encourage the students to do the same in a mature manner which promotes rich discussion.

What do other teachers do in this situation?

1. Jinfa, C & Cifarell, V – “Thinking Mathematically by Chinese Learners: A Cross National Comparison Perspective”

2. Cogan, L & Schmidt, W – “An Examination of Instructional Practices in Six Countries.”

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