Problem-solving is not only a prominent Maths activity, as shown in the Maths ability pyramid. It is also a discipline of its own, with its specific know-how. In other words, the specific skills of problem-solving can be learnt too. By doing so, students will not only learn to solve problems more efficiently, they will also make the best of problem-solving’s high educational value.
For Maths teachers, it means that it is possible to choose problems for students not only according to a particular Maths topic (fractions, algebra, trigonometry, etc.) but also with a view to practise one or several problem-solving skills.
In order to do this, it is necessary to identify and name these skills. This post covers 10 problem-solving skills, which you can see in action in UKMT JMC 2015 (cf. my JMC 2015 teacher’s notes).
- Economy: this means doing just what is necessary, and not more than is necessary. This is a highly educational problem-solving skill, as it not only helps students to save time, but it increases their capacity to focus on the essential.
- Example: in JMC 2015 Question 4, students do not need to complete the whole pyramid in order to find the missing number.
- Additional note: Economy is often your best friend when it comes to proof. Suppose, for example, you want to prove the Sine rule (a/sinA = b/sinB = c/sinC) for any triangle. If you choose 2 vertices, say A and B, and prove that a/sinA = b/sinB, then it works for A and C, as well as B and C. Which means you can prove the Sine rule with one rectangle only.
- Alternative strategies: this is a skill you can use to push students further, especially the most able ones who find solutions quickly. It is also a way to get students to go beyond the obvious and access conceptually superior solutions.
- Example: in JMC 2015 Question 3, finding an alternative strategy means (1) avoiding a tedious and unnecessary long division students, and (2) seeing the problem from 2 different angles: either eliminate arithmetically impossible answers or work out a quick estimate.
- Elimination: this is a particularly useful skill for UKMT Challenges and Kangaroos, and more generally for problems where students have to choose from a limited number of answers. It is the art of detective Dupin (and not Sherlock Holmes, as often miscredited): ‘once you have eliminated the impossible,…’
- Example: in JMC 2015 Question 8, there are 2 smart elimination possibilities using properties of factors and multiples. This skill can be combined with the previous one (alternative strategies), for example: find 2 elimination strategies for this question.
- Deduction: as a skill, this is about training and strengthening the ability to sort out all the available information and use it in the right order. For students, I often compare deduction as a line of sugar lumps: once you tumble the first lump, all the other ones follow. Some experts argue that this should not be called ‘deduction’, but ‘induction’ or ‘inference’. I am not a logician, and therefore not in a position to put forward any argument for or against this choice of terminology. ‘Deduction’ is a convenient choice, as popularized by our old (and contemporary) role model Sherlock Holmes and his renowned ‘science of deduction’, which is essentially picking up bits of information and going some way with it.
- Example: JMC 2015 Question 6 provides a good example of 3 elementary deductions based on geometry.
- Name / Label: this skill refers to one of the essential rights of the problem solver: the right to name things — especially things you’re looking for. Sometimes, the naming or labelling is already done, like x° in the previously mentioned question (JMC 2015 Question 6). If not, students should be well aware that they are allowed to do it from their own initiative, either because they will have to solve an equation, or simply because it helps them clarify their own thinking. Many students go ‘blank’ just because they fail to name what they’re looking for.
- Example: in JMC 2015 Question 12, it helps to name the weight of the fish (w, for example, or x, or whatever…), whether students will use fractions or algebra to solve the problem.
- Systematic list: this is a simple yet essential skill every time students tackle a question that involves numbers with specific properties within a limited range, for example: listing the first multiples of 4, or the first squares, of the first prime numbers, or cubes between 100 and 199, etc.
- Example: in JMC 2015 Question 13, students have to list all multiples of 3 between 3 and 15; in Question 11, they need to list all prime numbers up to 23; Question 19 is about cubes up to 512.
- Tree: this skill is used to sort out information in a binary logic question (for example, statements from liars and truthful guys ,as in JMC 2015 Question 17). Students easily get lost in a succession of ‘if… then…’ and ‘if not… then…’. Or if they don’t get lost, they will bother everyone else with a wordy and totally incomprehensible solution. Sketching out a Logical Tree is the answer.
- Bar modelling: this skill is one of the core tools from what is now known as Singapore Maths. Bar modelling is a great visual tool to enable students to access conceptual thinking for all kinds of problems involving arithmetics. In a purely UKMT challenge context, bar modelling would not be advised because the key to a Gold medal and further is speed. But as the purpose of these teacher’s notes is to use UKMT questions for their educational value, i.e. learning to solve problems, bar modelling is an important piece of scaffolding. For more explanation about the principle of bar modelling and how it can be used in diverse contexts, see the Singapore Maths website and more particularly this presentation (Flash required).
- Example: in JMC 2015 Question 12, it is possible to model the weight of the fish as one bar, which you subdivide into 2 sections (2 thirds and one third). Students will then more easily visualize that the first section of the bar (2 thirds of the total weight) is the 2kgs mentioned in the word problem.
- Easy way out: as a skill, this could be renamed ‘never overlook the obvious’. Sometimes, a problem comes up where the solution is made very simple just by noticing something ‘obvious’ — well, it’s obvious once you’ve seen it, obviously…
- Example: in JMC 2015 Question 18, noticing that both fractions are equivalent provides a shortcut to the solution.
- Complete the grid: this is typically used for patterns, tilings, fractional areas, etc. as in JMC 2015 Question 22.