SAPERE trainer Rod Cunningham is running a one-day workshop on maths and P4C in London on Thursday 28 March and again on Thursday 20 June. The course will provide a range of mathematical starting points invoking creative, collaborative, critical and caring thinking and which lend themselves to exploration of ‘big’ mathematics ideas. The course is suitable for primary and secondary and examples for all ages will be used.
To find out more and book a place on the course, see the SAPERE website.
Rod has worked as a mathematics educator for over 30 years and P4C trainer for five. He is interested in developing the use of P4C approaches across the curriculum. Further details of his work can be found on his website.
Here Rod blogs about the benefits of using P4C in maths and discusses the example of the Chinese number puzzle.
An OFSTED report into promoting achievement in mathematics claims that success is linked to ‘Teaching that focuses on developing students’ understanding of mathematical concepts and enhances their critical thinking and reasoning together with a spirit of collaborative enquiry that promotes mathematical discussion and debate.’ (OFSTED 2006)
My experience of working in schools that practice P4C shows that there are many ways in which learning in mathematics can be enhanced through collaborative work which builds upon P4C experience. I believe there are two key reasons for the success of using 4Cs thinking and enquiry in mathematics. First, mathematics is built upon a network of interrelated concepts in much the same way as ordinary language. Second, the facilitation skills that teachers hone in P4C enquiry are crucial to the conduct of collaborative learning in mathematics.
The nature of mathematics and the importance of enquiry
It may sound strange to some that mathematics is a creative discipline that employs powers of the imagination as much as those of reason. However, mathematics is underpinned by a number of big conceptual ideas which develop and grow as the learner progresses towards more sophisticated understanding and competency. Like any conceptual system, such understanding develops through practical and collaborative enquiry and interaction. Brent Davies suggests that “If we consider mathematics as a human activity, as opposed to a collection of actual objects independent of our being human, then mathematics is a body of knowledge entwined with culture, including language.” (Davis et. al. 2015 p78)
The importance of facilitation in mathematical enquiry
The facilitation of enquiries in P4C is a subtle skill which takes time to acquire. It involves modelling the language of dialogue and reasoning, providing provocation whilst avoiding becoming overly interventionist; letting the dialogue take its course but at the same time encouraging the participants to dig deeply into their thinking and that of their fellow enquirers. My observation is that teachers who have facilitated communities of enquiry regularly become better able to assist pupils to build collaborative understanding of difficult mathematical concepts.
Making links between P4C and mathematical thinking
In my experience, pupils with a P4C background appear to be more confident in tackling novel problems and in working collaboratively to solve them. They demonstrate 4Cs thinking through the language they use when offered open-ended mathematics problems. The subsequent reflection and dialogue enables pupils to build connections between important mathematical ideas lending coherence to the subject. The teacher/facilitator can enhance this by helping pupils ask ‘big’ mathematics-related questions which can promote coherence within the subject through further dialogue.
Story books figure strongly in developing mathematical ideas in the Early years. This is dependent upon teachers being aware of the extent and importance of such ideas. Models, images and metaphors are important for the effective learning of mathematics across primary and secondary ages. In upper primary and secondary years, I draw heavily upon the work of Malcolm Swan (2006), who demonstrates that particular activities are particularly effective for developing deep thinking and learning in mathematics.
Below is an example of the type of activity which engages mathematical enquiry and builds on experience of P4C. This activity is not necessarily new to the mathematics classroom. The difference in a P4C-rich classroom was the language used by pupils during their work on the problems, and the type of prompts given by the facilitator and the subsequent “bigger” questions posed which are then followed up. This example is appropriate for upper primary and lower secondary classes but the principles extend across the age range.
Image: The Chinese number jigsaw
Groups of three year 6 children were given a Chinese number jigsaw. They took some time to complete these. They were asked to think about the patterns in the way the Chinese numbers are written. It was pointed out that the number representation we use has developed from several different traditions over a long time and suggested that an investigation of the history of this would be very interesting, but for another time.
Image: Children working with a Chinese number jigsaw
The facilitator asked, “Can you make 64 using the Chinese script?” which led to discussion about how the numbers were formed. The facilitator then asked, “How about the system that we use?” There was general discussion about the use of just ten symbols and with some prompting, the importance of the position of each digit. The groups were then given a poster as a stimulus and asked to come up with questions, observations, points of interest and record these, choosing one question to share with the whole class. Some groups focused on the value of the poster as a stimulus, which was in itself interesting given that they are used to evaluating the stimulus in P4C. The children are used to responding to the prompts ‘good because’ and ‘bad because’.
Image: Place value questions poster
The groups shared their observations with the whole class. Further questions came from the learners about the use of the base ten system. A list of these questions was produced for future sessions available for further discussion and investigation by the whole group. These questions included:
- Why is the base ten system useful?
- When was it ‘invented’, and is ‘invented’ the right word?
- Is it the ‘best’ system?
- How would we decide if [a number system] is good, or the best?
There was widespread interest in finding out about other number systems, as well as the history of the base ten system. This included enthusiasm for making up jigsaw puzzles in other number systems.
Davis, B. and the Spatial Reasoning Group 2015 Spatial Reasoning in the Early Years. New York: Routledge
OFSTED report 2006 (Ref HMI 2611) Evaluating mathematics provision for 14-19 year olds
Swan, M. 2006 Collaborative Learning in Mathematics: A challenge to our beliefs and practices. London: NRDC/Niace
Thanks to the teachers and pupils at Willowtown Community Primary School in Ebbw Vale where this example was recorded.