If you ask parents what heuristics are, the most common answers you would get will be:
- “Heuristics are difficult Maths questions”
- “Heuristics are those Maths word problems with the * ”
No doubt, some Maths word problems can be slightly challenging, but that’s not what heuristics is all about. As a result of this misconception, many parents of school-going children are often terrorised by today’s Maths syllabus.
Over time, the goals and aims for learning Mathematics have evolved, and inevitably so have the teaching and learning methods.
For those who are intimidated just by the word heuristics, Michelle Choo, Associate Lecturer at Marshall Cavendish Institute and established author of several Mathematics books, demystifies heuristics for parents
What are heuristics?
The word heuristics originates from the Latin word heuristicus, which is equivalent to Greek heur (ískein) + Latin –isticus –istic and means to find out or to discover.
To put it simply, heuristics are a set of problem-solving rules that help us discover the best and most practical ways to solve problems.
Mathematics is more than the study of number and patterns. The learning outcome is to develop one’s logical thinking and ability to solve complex problems.
As such, heuristics play a very important role in mathematical problem solving, which is fundamental to mathematics learning.
Heuristics play a very important role in mathematical problem solving
Singapore Mathematics, a problem-solving curriculum
With problem solving as the core of our Mathematics framework, the use of heuristics is inevitable. Of course, heuristics are not used independently, but supported by the five inter-related components:
- Attitudes
- Metacognition
- Skills
- Processes
- Concepts
Figure 1: The Singapore Mathematics curriculum framework. Image courtesy: The Ministry of Education. It has been taken from the MOE Primary Math Syllabus 2013
The Singapore Mathematics syllabuses, developed by Curriculum Planning and Development Division (CPDD), Ministry of Education Singapore (MOE), have identified 13 heuristics that are applicable to mathematical problem solving. These are:
- Act it out
- Draw a diagram / model
- Use guess-and-check
- Make a systematic list
- Look for patterns
- Work backwards
- Use before-after concept
- Make suppositions
- Restate the problem in another way
- Simplify the problem
- Solve part of the problem
- Think of a related problem*
- Use equations*
(*Covered in the secondary syllabus.)
These 13 heuristics can be grouped into four categories based on their applicability. They are:
- To give a representation. Draw a diagram, make a list, use equations.
- To make a calculated guess. Guess and check, look for patterns, make suppositions.
- To go through the process. Act it out, work backwards, before-after.
- To change the problem. Restate the problem, simplify the problem, solve part of the problem.
How to teach and learn heuristics?
Frequent use of heuristics can greatly improve a child’s problem solving performance. It is extensively recognised and accepted that heuristics should be and can be taught. The crucial point here is that heuristics are about ideas, not routine steps.
However, not all heuristics can be taught explicitly in class.
So what would be an effective way to teach heuristics and what are the conditions that are conducive to learning heuristics?
Play
Children as young as 18 months can pick up heuristic skills through play. This form of play, also known as ‘Heuristic Play’, offers children a world of unlimited discovery.
During play, as children interact with objects and their environment, they generate ideas and learnings. This helps them develop connections and come to new and better understandings and ways of doing things. (Principles Into Practice Card 4.3- Creativity and Critical Thinking)
As complex as it sounds, heuristic play is actually very simple and inexpensive. Here are a couple of ways you can encourage it:
- Introduce a “treasure basket” with natural and household objects in it. The more objects in the basket, the more choices children have to deal with – Should I pick this up? What can I do with this object? Can this object help me with…? In this instance, children learn how to choose, which is essential in Maths problem solving in later years.
Tangram can aid the development of your children’s problem solving skills
- Have a go with the tangram. The tangram is an ancient Chinese puzzle that can aid in the development of children’s problem solving skills. In a simple game of tangram, players are expected to form a specific shape based on an outline or silhouette given. The use of tangrams provides children with open-ended explorations. There is usually more than one way to construct a shape, and it is through this game that children exercise divergent thinking. Tangram also helps children develop Mathematical concepts and skills such as fractions, geometry and spatial visualisation.
Observe and study how others solve problems
By observing how others solve problems, one can analyse if the methods used are indeed effective and practical enough to be executed.
Studying the different strategies used by others to solve the same problem enables the problem solver to be more flexible and more competent.
The problem solver will learn to appreciate the fact that there is more than one way to solve a problem and this will help develop his reasoning about the whys and the hows of certain strategies are used to solve problems.
This method is also far more motivational and beneficial as compared to just being shown the steps in solving the problems as it develops curiosity, confidence and open-mindedness.
Telling someone how the problem was solved helps one develop their own heuristic skills
Tell someone how the problem was solved
Sharing the strategies used to solve a problem can lead to discussion and critique of selected methods. The process of verbalising makes one’s thinking visible, which helps in evaluation of the selected problem-solving method. By teaching others how to solve problems, one becomes more confident and more motivated too.
Now that we have understood what heuristics are and how to develop heuristic skills in our children, let us understand how heuristics can be applied to solve problem sums.
If you want to know how to apply heuristics to solve challenging problems, head on over to the next page.
How to apply heuristics to solve challenging problems?
The following are examples of some the heuristics mentioned earlier and how they can be applied to solve some simple non-routine problems.
The table also demonstrates how a mathematical word problem can be solved by using different heuristics strategies and how a problem may require more than one heuristics strategy to solve.
Heuristics can enhance the chances of solving a problem.
However, it does not guarantee that a solution can be found.
Also, there is often more than one way of solving a problem. More complex problems tend to require multiple heuristics strategies to solve them. The use of suitable heuristics usually results in more efficient ways of solving a problem.
Hopefully you are more familiar with heuristics now and will be able to work with your children to help them learn this way of problem solving.