Friday, 27 March 2015

Teaching Problem Solving Heuristics in Mathematics

What are heuristics?

Heuristics are techniques students can use to tackle a problem when the solution to the problem is not obvious. Some examples of heuristics are listed below and grouped into four categories according to how they are used: To give a representation e.g. draw a diagram, make a list, use equations To make a calculated guess e.g. guess and check, look for patterns, make suppositions To go through the process e.g. act it out, work backwards, before-after To change the problem e.g. restate the problem, simplify the problem, solve part of the problem Here is an example of one that involves guess and check: The perimeter of a rectangle is 42 cm. Its area is 108 cm2. Find its length and breadth. Since the perimeter of the rectangle is 42 cm, the sum of its length and breadth will be 21 cm. Make a list of the possible lengths and breadths.

Singapore mathematics syllabuses

The Singapore mathematics syllabuses, developed by Curriculum Planning and Developing Division (CPDD), Ministry of Education Singapore (MOE), have identified thirteen heuristics that are applicable to mathematical problem solving.
1. Act it out
2. Use a diagram/model
3. Use guess-and-check
4. Make a systematic list
5. Look for patterns
6. Work backwards
7. Use before-after concept
8. Make suppositions
9. Restate the problem in another way
10. Simplify the problem
11. Solve part of the problem
12. Think of a related problem
13. Use equations (Heuristics 12 and 13 are not in the primary syllabus.)

We can also treat these four ideas as four general heuristics, “use different representations”, “simplify your problem”, “approach your problem from different directions”, and “bring in solutions”.

Figure 1: Model for problem solving in mathematics (Tiong, Hedberg, & Lioe, 2005)

By definition, all heuristics have the following two characteristics: 1. Heuristics do not guarantee a solution. All heuristics do is pointing us towards possible ways in which we might be able to find our solution. 2. Heuristics do not come with specific procedures. When we use heuristics, we are required to make some judgments of our own regarding what we should do. Heuristics help us to deal with difficult problems or problems that we are not familiar with. Other than that, heuristics usually enables us to find solutions with less time and effort as compared to when we use algorithms to find solutions. Not all heuristics are the same in term of specificity. There are ones that give very general and ambiguous instruction, while the others have more specific procedures that we as problem solvers would like to follow. For example, heuristic “represent you problem differently” only gives us a very general direction to what we should do, where as heuristic “draw a diagram” tells how we can represent our problem, visually. Heuristic “draw a histogram” on the other hand gives a much more specific instruction than the previous two. Representation Simplification Pathway Bring in solution P R O B L E M S O L U T I O N To follow this heuristic “draw a histogram”, we will need to know first the procedures in which we can draw a histogram, whereas to follow the heuristic “draw a diagram” we can invent our own diagrams and the rules or procedures in which we can manipulate them; here we have the freedom to choose and be creative. Of course, we can still end using the histogram, since it might be the best representation for our problems. Here we can construct hierarchy of heuristics according to their specificity. We propose to put the four “general heuristics” from the model above on top of the hierarchy. Using the example in the previous paragraph, we have Figure 2. We should note that Figure 2 is not an exhaustive hierarchy for “representations”, we can still add in more representation into the hierarchy, such as “manipulative” into the second level, “pie chart” into the third level, and so on. Here we are just trying to get a rough idea of how a hierarchy of heuristics might look like. Since the number of heuristics are only limited by our creativity and imagination, it is not possible to construct a hierarchy that contain all possible heuristics.

Figure 2: An example of hierarchy of heuristics

Here we need to clarify first that diagram, symbol or histogram, matrix themselves are not heuristics, but “use diagram”, “use equation” and “draw table” are heuristics. Representations themselves are not heuristics, however suggestions to use representations are. Of course Figure 2 is only part of the whole hierarchy of heuristics for representations, and we can have similar hierarchy for “simplification”, “pathway”, and “bring in solution”. Heuristics on the top of the hierarchy are just very basic and general ideas that can be applied to most problems. These ideas can be further broken downward to make them more specific, making them easier to follow and apply to problems. However by doing so, we have restricted the applicability of the heuristics to only a specific few types of problems.

Heuristics Maths with specific procedures are usually less applicable than those with fewer procedures. Besides that, specific heuristics require less of problem solvers’ interpretation and intuition or creativity. As we can see in Figure 2, the “heuristics” at the bottom of the General Specific Representation Diagram Symbol Histogram Bar chart Matrix Equation Table hierarchy are pointing us to topics that we learn in mathematics lessons that come with a lot of procedures and rules governing how to create and manipulate them. After all, we can see mathematics as a big collection of tools or “representations” in which we use to solve problems.