Problem solving and reasoning are important proficiencies in the Australian Curriculum for mathematics. Let’s confirm our understanding of what problem solving and reasoning entail, unpack how the two skills are related and explore ways to help students master them.
Problem solving is the ability to connect and apply mathematics to a specific situation or context. It involves interpreting information, identifying relevant mathematical terms, concepts or strategies and formulating a plan to find a solution efficiently and effectively.
Reasoning is the thinking required when solving a problem. It is the connective tissue between mathematics and the real world and explains why solutions are reasonable or effective. Students use reasoning when they recognise and identify important mathematical information, and when they check their solution is reasonable.
If problem solving seeks to find an answer, reasoning justifies why the answer makes sense. Although a problem will always contain reasoning in some capacity, problems that require students to articulate their reasoning are more challenging.
Consider a problem that involves the cost of cupcakes, where three cupcakes cost $9 and five cupcakes cost $15. A student might be quick to work out the cost for seven cupcakes, but explaining why their answer is correct or evaluating whether their answer is reasonable in the real world requires higher level thinking.
Here are some effective ways you can help students hone these valuable skills:
While the number of mathematical problems is endless, many problems can be solved by drawing on just a handful of strategies. It’s important to include the explicit teaching of problem-solving strategies as part of your maths lessons, such as the 10 iMaths Problem-Solving Strategies. By frequently revisiting the strategies and highlighting the name and process of each one as you use it, students become familiar with the essential principles of a given strategy and start to identify when they can apply them. Tip: Refer students to one of the iMaths Problem-Solving Strategy videos at iMaths Online whenever the opportunity arises.
You can also reinforce problem-solving strategies by encouraging students to share and explain their problem-solving application. A peer-explanation approach can provide aha moments for students, as they hear other students discuss different ways of tackling the same problem.
Since mathematics can sometimes feel abstract and disconnected from the real world, it is important for students to see where and when mathematics is found and used in real-world situations. Give students opportunities to connect mathematics to the real world by incorporating context-based learning. In iMaths investigations, this real-world context is a focal point and in itself provides ample opportunities to apply problem solving and reasoning in an engaging, relatable and meaningful way.
Although it can be subtle, a simple but effective way to elevate a problem-solving task into one that requires reasoning is to use a cognitive verb in the question. Cognitive verbs such as explain, compare, justify, prove, determine and evaluate help students articulate their reasoning by guiding them through the process of thinking.
Since cognitive verbs promote reasoning, it is important to point out and unpack these verbs when your students encounter them. For example, the word compare in a question requires students to consider the strengths and limitations of options, whereas the word explain would prompt students to give a detailed account of the given information to arrive at the solution. Ensure students understand the subtleties of what is being asked of them before they provide their reasoned answer.
When a student provides an answer or idea take the opportunity simply ask why or how. To ensure the focus of the question is on reasoning, not just correctness, consolidate the answer first and then ask for further information. Use follow-up questions to tease out students’ thinking and help them articulate their thoughts. For example, ‘You’re correct, but how did you know?’.
Tip: use the ready-made Communicating and reflecting questions for each iMaths investigation in the Investigation Teaching Plan at iMaths Online.
Switch things up for your next investigation and have students film the process. Not only will this add an extra level of engagement, incorporating a video component is a fantastic way to draw out students’ communicating and reasoning skills.
Are there students in your class who believe they are simply ‘not good at maths’? Investigations are an ideal platform to foster a growth mindset. Use these tips to encourage students to reframe their thinking and focus on growth and learning.