In providing a continued focus on tasks and activities that help to illustrate key ideas embedded in the Australian Curriculum: Mathematics (AC:M), the focus in this issue is on Measurement in the Measurement and Geometry strand with a particular emphasis on highlighting the proficiency of Problem Solving. Problem Solving is described in the AC:M document as: "Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively." (ACARA, 2013).

One of my favourite problems (see Figure 1) is one that I call "Congratulations" but is also known as "Money Measurement". The problem is presented to pairs or small groups of students with little other guidance. The only editorial I add is that "Greed is good," and that they are looking for which prize will give them the greatest return.

This is an activity which exercises all four of the proficiencies of Fluency, Reasoning, Understanding and Problem Solving. For this article, we are going to focus on the proficiency of Problem Solving, but as we work through it, it becomes obvious just how intertwined the other proficiencies are within Problem Solving.

The students are given equipment: a 1 litre milk carton, ten \$2 coins, ten \$1 coins, twenty 20c coins, twenty 5c coins, a 30 cm ruler and a set of scales. (Just as a matter of interest, I have lost very little of this money working with students; working with teachers is a different matter!) Students quickly realise there is not enough of any of the equipment to solve this by using counting, that is, there are not enough \$1 coins to weigh one kilogram, for example. It is fascinating to watch how they then approach the problem of how they solve the problem.
```
Number &   Measurement    Statistics
Algebra    & Geometry    & Probability

Understanding
Fluency
Reasoning
Problem Solving

Figure 1 (Beesey, Clarke, Clarke, Stephens & Sullivan,
2003, p. 63).

CONGRATULATIONS!

You have won a prize

a 1 litre milk carton filled with
20c coins

--OR--

1 kg of \$1 coins

--OR--

a line of \$2 coins 1 metre long
(lying flat and touching)

--OR--

a square metre of 5c coins
(lying flat and touching).

Which one will you choose?

Explain how you made an estimate of each
one and show clearly any calculations.
```

George Polya (1887-1985), often called the father of problem-solving in mathematics education, articulated a four step process for solving problems (see Figure 2).

At the start of the activity, students need to select one of the parts of the problem to approach. For the sake of illustration, let us imagine they choose to investigate the 1 metre line of \$2 coins. It is fine to allow the students to approach the problem creatively but some will require some sort of scaffolding to make headway.

To make sure they understand the problem (see), it is good to have a few questions to act as prompts. Questions such as: What do you need to find out? (Often, the best way to see if the students have understood the question is to get them to tell you what they need to do, in their own language). Do you have all of the information you need to solve this problem? If not, what else do you need? What sort of answer are you expecting?

Once they have shown that they have understood the problem, they then need to decide how they will tackle it (plan). This is the part that I find the most intriguing; the time when I like to stand back, watch and listen to the strategies that the students invoke. Some questions that could be framed for those students who are finding this a bit of a challenge are: Have you seen a problem like this before? How will you find out how many coins you need to make the line? How will you find out how much the coins in the line will be worth? What mathematics do you need to find this out? What equipment will be useful?

Given the opportunity, quite a number of the students will move to the third phase of problem solving (do), first. It has been an observation that the students who tend to engage in the first two phases (see, plan), even in an informal way, are generally the students who would be considered 'better' (for me, that is that they are generally more efficient and more likely to end up with a correct answer or a pathway to a solution) problem solvers than those who do not.

The fourth phase of the problem-solving cycle is then to review what has happened and what the answer is (check). Some scaffolding questions for this part could be: How will I record my results? What units do I need to use? How should my answer be worded to answer the original question? Does my answer make sense? Is my answer close to my estimation? If it is not close, why is there a difference?

This, of course, was only one part of the overall problem, and similar steps need to be taken to solve the dilemma of how much money they will get if they choose a square metre of five-cent coins, or a kilogram of one-dollar coins, or a litre of twenty-cent pieces. At the point when the totals are calculated for all of the coins, and a decision is made about which is the best alternative, the students need to exercise an important part of being a mathematician: they need to 'publish' their ideas in some manner. This articulation is one of the best ways to determine the depth of understanding in reaching the solution and the reasoning which was applied in order to reach the conclusion. The Fluency proficiency is evident during the planning and doing stages and the Problem Solving proficiency is articulated when the students make choices, interpret, formulate, model and investigate.

I would take the five-cent coins: I wonder if I am being sensible? But that is a conversation for another time!

References

Australian Curriculum, Assessment and Reporting Authority (ACARA) (2012). Australian curriculum: mathematics. Retrieved from http://www. australiancurriculum.edu.au/Mathematics/ Content-structure.

Beesey, C., Clarke, B. A., Clarke, D. M., Stephens, W. M., & Sullivan, P. (2003). Exemplary assessment materials: Mathematics. Carlton, Vic.: Longman/Board of Studies.

Derek Hurrell

University of Notre Dame

Australia, Fremantle, WA

<derek.hurreU@nd.edu.au>
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