This blog is the third in a series of three to support you in how you might use one task to explore the Thinking and Working Mathematically (TWM) characteristics from the Cambridge Primary and Lower Secondary Mathematics curriculum framework.
In the first blog we explored what the TWM characteristics are and looked at examples to help us understand them in mathematical contexts. In the second, we highlighted how you can include these TWM characteristics as part of your mathematics teaching and learning.
As the previous two blogs have highlighted, the new TWM characteristics are designed to be integrated into mathematics lessons where possible. Not all lessons are required to include the TWM characteristics but as you become more familiar and confident with them, you will find that they become a natural part of planning and teaching. Some tasks (as blog 2 explored) are more suited to including the TWM characteristics, others less so.
As a rule, the more open the task, the more TWM characteristics you can include. This blog takes one task and shows you how it is possible to include all eight TWM characteristics.
Thinking and Working Mathematically task
This task is from the NRICH. NRICH is a unique collaboration between the Faculties of Mathematics and Education at the University of Cambridge. Its aim is to focus on problem solving for students to learn mathematics through exploration and discussion. It provides thousands of free online mathematics resources for ages 3 to 18, covering all stages of early years, primary and secondary school education: all completely free and available to all.
Magic Vs
– Place each of the numbers 1 to 5 in the V shape so that the two arms have the same total
– How many different possibilities are there?
– What do you notice about all the solutions you find?
As with any mathematical problem, there are many different ways to use this task in the classroom. This is just one example.
Step 1
Draw two Vs on the board; one which is ‘magic’ (i.e. whose arms have the same total) and one which is not. Ask learners ‘what do you notice?’ or ‘what is the same and what is different?’ Clarify why the Magic V is magic! Ask learners what they could suggest about Magic Vs. For example, how many possibilities are there using the numbers 1 to 5? Write down their conjectures.
- What were the learners’ initial thoughts? Did they notice the ‘magic’?
- What conjectures did they offer? Are they confident at giving conjectures or do they need more practice?
Step 2
Ask learners to work in pairs on the task. Remind them to record their thinking. You may want to give learners digit cards so they can move the digits around quickly, but this is not essential; a piece of paper and a pencil will also suffice.
- Notice how they are recording. Is it systematic or random?
- Are they quick/eager to begin the task?
- As they get into the task are they resilient?
Step 3
Bring the class back together and invite them to comment on what they notice.
- What patterns have they spotted?
- Did they improve their recording as they got further into the task? Did this help or hinder?
Step 4
If time, challenge the learners to extend the task in a different way.
Can they apply their thinking using different numbers in the V? (e.g. 2 to 6? 12 to 16? 37 to 41? 103 to 107?)
- Would the same rules apply if the V had an arm of length 4?
- If you were to change the task to use different numbers what do you think would happen? Would you still find a magic V?
- Ask learners if they liked the task.
In addition to the suggested teaching sequence above, when working with learners, asking the following questions will also help to include the TWM characteristics into this task.
Summary
Thinking and Working Mathematically is a process by which learners can increase their understanding, competence, and hopefully enjoyment, with solving problems. These characteristics show learners that mathematical thinking requires more than just getting answers to questions.
They allow learners to become aware of what it is they are doing, what they should draw attention to and what they should ignore. They will ultimately enable learners to become better mathematical problem solvers and to be able to solve those problems they have yet to encounter.
If you are looking for more support with TWM, the Cambridge University Press revised Primary and Lower Secondary mathematics series supports the 8 characteristics. Just take a look at the maths resources on our Primary and Lower Secondary hub page.
Other bogs in this series:
Blog 1: Thinking and Working Mathematically: definitions and examples
