Historically, mathematics was often regarded as a ‘drill and practice’ subject, where learners were discouraged from talking, collaborating and perhaps even thinking. Teaching of mathematics was often very formal and learners were taught to memorise a procedure and remember a formula. Those who could were successful, those who could not, practised until they could.
In the first blog we explored what the TWM characteristics are and looked at examples to help us understand them in mathematical contexts. But how might we include these TWM characteristics as part of our mathematics teaching and learning?
How can I integrate the Thinking and Working Mathematically characteristics into a lesson?
Most importantly, the TWM characteristics should not be seen as separate to any mathematics learning or lessons. They are intended to be woven into as much teaching and learning as possible. If you are using the new Cambridge Mathematics PLS Curriculum (from September 2021) examples of the TWM characteristics are included in the Scheme of Work. If you are using the CUP resources, the TWM characteristics are integrated into teaching notes and learner activities to help you bring them to life in your lessons.
In the current curriculum, there are four (Primary) or five strands (Lower Secondary), one of which is problem solving. In the new curriculum, these strands have been re-sorted so there are now three (Primary) or four (Lower Secondary) strands, but the previous problem-solving strand no longer exists. Instead, the TWM characteristics are placed at the heart of the curriculum. Problem solving objectives have been replaced with the eight TWM characteristics.
Integrating the TWM characteristics may require some shift in your pedagogical practice, depending on how you currently teach. There are some easy ways to do this, although to begin with, these may take a little time to get used to.
Task Design
Firstly, not all lessons will (or need to) include the TWM characteristics. However, careful task selection and design will either help to enable or disable learners using the characteristics.
While teachers are the most important resource learners can have, the tasks that teachers choose to use with their learners are a powerful tool. The tasks that we give to our learners can make all the difference between enabling learners to think and be inspired or to disengage them and limit their thinking.
Choosing tasks that create the conditions that will promote characteristics such as conjecturing, or classifying is a skill.
So, how do we choose tasks that enable the TWM characteristics? If you are using Press resources, then you can be sure that activities bring in TWM skills. But if you are unsure, here is a quick (but not limited) checklist:
- Does the task allow for multiple solutions?
- Does the task allow for different methods to be used?
- Is the task accessible to a wide range of learners?
- Does the task offer opportunities for initial success?
- Does the task encourage collaboration and discussion?
- Does the task encourage learners to think for themselves and make decisions?
- Does the task allow learners to pose their own questions and conjectures?
- Does the task have the potential for learners to spot patterns and discover generalisations?
- Is it an interesting and curious task?
Not all tasks will do all of these things, but it is important that they include some of them, if we are trying to integrate the TWM characteristics into a lesson. The good news is that almost all tasks can be edited slightly to include some of these elements.
For example, a task might ask: what is 12 x 5?
Posed in this way, the task does not allow for multiple solutions, it is not particularly interesting or curious and it does not allow learners to spot patterns. If the teacher chose to, they could ask learners to offer a conjecture or encourage them to use different methods. But, without any intervention from the teacher, the task is limiting.
However, we can change the task: what two numbers can we multiply together to give a product of 60?
By making a simple change the task now includes many of the above elements. In fact, this task could now be explored for the rest of the lesson, without the need for any more tasks to be set. It also allows for several of the TWM characteristics to be used:
Specialising: I will start by trying 10 x 6.
Conjecturing: I don’t think I can use two odd numbers.
Generalising: When two odd numbers are multiplied together the result is always an odd number.
Convincing: I can explain why I can’t use two odd numbers.
So, task design can not only help to cover the content within the curriculum but also integrate the TWM characteristics at the same time.
Thinking and Working Mathematically Habits
The eight TWM characteristics alone will not be enough to sustain the level of thinking and working mathematically that we aspire our learners to be able to do. There are additional habits (or behaviours) that they need to develop, and which we need to model, encourage and enable. Learners who are aware of these habits and use them alongside the TWM characteristics will gain the confidence to make mathematical decisions, to deal with any mathematical problems they face and ultimately adopt the dispositions and behaviours of a mathematician.
1. Being stuck
Everyone gets stuck. Being stuck is a good place to be in. In fact, Mason et al (1982) would say it is ‘an honourable and positive state from which much can be learned’. When we are in this state it means that we really need to think what to do next. Should we try to specialise using specific examples, or should we look for a pattern and try and generalise? Whatever we do, we are actually working harder than if we were not stuck!
2. Working collaboratively
Working with others, as opposed to working on your own, enables many of the TWM characteristics. Working in this way offers learners the opportunity to value different approaches, so they may begin to generalise or classify. It allows learners to communicate their thinking and to listen to that of others, thus critiquing and improving
3. Working systematically
We want to encourage learners to work on problems in different ways, but they are more likely to be successful if they work systematically. While choosing examples is a good way to start (specialising), if they are to then begin to spot patterns (generalising) being systematic is an important element because patterns are more likely to be evident among related examples than with randomly chosen ones.
4. Mathematical talk
Mathematical talk is essential if we want to develop learners understanding of mathematics. Purposeful talk can clarify, refine, explain, convince, justify and improve thinking and working. Learners need regular opportunities to engage in purposeful talk. The TWM characteristics are a perfect vehicle for enabling classrooms of rich mathematics talk. Asking learners to offer conjectures, or to convince one another of their thinking or to critique others work all requires them to talk.
5. A culture and climate of thinking and working mathematically
The TWM characteristics may require a shift in how we think, plan and teach mathematics. We may need to change how we choose tasks, or how we seat learners, so they have opportunities to talk to each other. There will be many learners who have been conditioned to focus only on finding the answer (quickly) and resist recording their thinking and working. Some learners may struggle with finding more than one solution to a problem if they have already reached an answer, and others may not wish to expose their mistakes for fear of failure. Setting up a culture and climate where these things are viewed positively may take time, but it will be worth it!
Questions to Prompt Thinking
The questions we ask are critical in all mathematical learning. Here are a few questions that you may find helpful to pose to your learners and which will help to draw out the TWM characteristics.
- What do you notice?
- Can you convince me?
- What is the same? What is different?
- How do you know?
- Can you sort these by …?
- Which is the odd one out?
- Why do you think that …?
- What happens if we change …?
- Is there a different way?
- What happens in general?
- Is there a rule for that?
- How would you justify that?
- What happens if …?
- Can you give me another example?
- When would you use this approach?
- Can you spot a pattern?
- Is this a special example?
- Can you find an example that doesn’t …?
- What needs to change?
Summary
The TWM mathematics characteristics are an exciting and unique feature to the new Cambridge PLS Mathematics curriculum, which is supported by the CUP resources.
For some of us this will require a shift in how we currently teach mathematics, for others less so. However, ensuring that these TWM characteristics exist and are cultivated in our planning and teaching will result in our learners being able to make sense of ideas, realise connections, reason and generalise, and above all enjoy mathematics.
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
Blog 3: Thinking and Working Mathematically: a task
