I have just had the opportunity to spend the weekend learning alongside a large group of mathematics teachers. After a day and a half of talking and thinking about mathematics teaching, I am excited to get back to school and try out some new ideas. There were also some key takeaways for me that I share below. These are the questions or wonderings that my mind wandered to while listening to the numerous talented speakers over the past two days. A provocative presentation creates space for the audience to reflect on the ideas shared and how the new ideas which emerge from the experience challenge our existing practice. The presenters at the MANSW Annual conference certainly gave their audience plenty to think about.
It was nice to spend time surrounded by people who believe that mathematics is a creative endeavour. Being around people who understand this aspect of mathematics, people who see the beauty of its patterns and operations makes you question why we feel the need to add Art to STEM. Not that Art and Mathematics do not make excellent companions but that the mix is too often advocated for by those hoping to add a creative element where they imagine it is missing. There is a need to be aware of an individual’s history with mathematical thinking before we launch into lessons requiring a creative approach. If a student has not been exposed to creative approaches to mathematics early in their learning journey, such methods are likely to cause confusion. Students trained to believe mathematics is about learning a set of procedures to be applied as taught, will find requests for creative solutions are like asking a non-swimmer to ‘just dive in and see what happens’. The result is likely to be an increased level of maths anxiety and an even greater desire to be shown what to do and when to do it. The answer is to ensure that creativity is a part of mathematical explorations from the earliest days of a child’s learning. As this learning is formalised, opportunities to play, explore, create, and wonder need to be maintained. A playful exploration of mathematical ideas should be enhanced when an experienced mathematical thinker joins the child on their learning journey. It should not be replaced by a forced march to the expert’s drum.
Students require mathematical freedom. Kirsty Thorpe beautifully shared this idea. Teachers like to be organised. We like to have our resources ready to go. Our lesson planning includes notes on how the lesson should evolve and what the students are going to need. With this in mind, the well-organised teacher prepares the resources that students are going to need and has them ready for easy distribution. Except when we do this, sometimes we are limiting how our students engage with their learning. As Kirsty noted, if we supply them with counters, don’t be surprised if all of their solutions involve manipulating sets of counters. Allowing students to select the materials they will use to visualise their mathematical thinking gives them mathematical freedom. It also allows more opportunities for students to demonstrate their reasoning skills. When students select the materials they use to visualise their thinking; they are making choices which they should then have to defend. Working with my colleague Stellina Sim, we have created Number Talk Bags. Each bag contains an assortment of commonly used mathematical manipulative and students make choices about which ones allow them to share their thinking with their classmates. Having the Number Talk Bags ready to go maintains a smooth flow to the lesson while enabling the students to make choices.
There are, of course, times when a limited set of resources can produce better thinking. Having to work with limitations can be a part of a challenge and is typical of problems we encounter in the real world. Limiting students to a particular set of manipulative can also guide them towards the thinking that is required for the learning objective. When we started exploring patterns with kindergarten, we limited the resources they had access to. Early on, it was almost impossible for the students not to arrange the objects we offered them into a pattern. Yes, we were limiting their freedom, but we ensured that they were forming, seeing and describing patterns. Once the foundation was laid, we offered other possibilities from an expanded palette.
Mathematical confidence requires a culture that supports mathematical exploration and risk-taking. Students who believe mathematics is about right answers are going to struggle in open classrooms. For teachers who are discovering new approaches to mathematics and exploring pedagogies where learning from mistakes is celebrated a word of caution. Some students will find this very confronting. If your entire mathematical career has led you to believe that good mathematicians don’t make mistakes, and now your mistake is shared on the whiteboard, you are going to want to dig a hole and climb in. As we change our mindset as teachers, as we explore the use of challenging tasks and mathematical inquiries, we need to take our students with us on the journey.
And it’s not just changing how we perceive mistakes. Some of our students love mathematics because they believe that there is just one right answer and one best way to find it. They imagine themselves to be confident and competent mathematicians because they can find that one correct answer and do so using the right method and quickly too. Now we are asking them to find another way and telling them that there is not one right answer. In essence, we are taking away the rock upon which their mathematical confidence rests. Some students will love this and see mathematics in a whole new light. Others will feel that their teacher has turned off the light and hidden the switch.
Less is more. When we teach less, we make time for tasks to go as long as they need to. We need to understand that a challenging task taught in ways that require students to genuinely engage and explore in multiple ways does not fit in an hour block. The advantage of teaching in this way is that the learning should stick. If we get it right, we shouldn’t have to start each year with an introduction to fractions. We also streamline our teaching when we allow students to make use of their mathematical knowledge in contexts which span multiple strands of the syllabus. Instead of teaching two weeks of multiplication and then two weeks of area, rich tasks allow us to to do a better job of teaching both concepts together and in less time. Maybe three weeks full of challenging inquiry into area fuelled by meaningful opportunities to build an understanding of multiplication.
The best learning can occur when students have to share their thinking, and this is more important than solving the problem. In many classes, the focus of problem-solving is on finding a solution, once the problem is solved, we move on and the learning stops. If you just solve the problem, you are not creating opportunities for communicating and reasoning and students only see one solution. In this way, opportunities for flexibility are missed. When you include ample time for students to share their solutions, explain the choices they made and compare options, you create opportunities for mathematical reasoning and communication. Problem-solving is great, but it is only one part of working mathematically.
And, Learning Outcomes written on the board for all to see might not be the panacea we are all looking for. Yes, the teacher needs to have an understanding of what they hope their children will learn, but it should not be assumed that publishing this enhances learning. Such a reading of the research is easily shown to be flawed. We should also consider the sequence of learning rather than just the endpoint. We need to ask more questions about the path our learners might be taking towards an objective. What are the waypoints on our journey towards the learning objective? How do we know where we are and where we are going next? What paths are learners likely to take, what mistakes might that make, what do their attempts reveal about their emerging understanding?
Fiona Foley shared with us the Five Practices for Productive Mathematics Discussion. This is a process which allows teachers to understand better the impact that their planned teaching might have and to shape their plan to maximise this. It evolved from the research of Margaret Smith and May Stein working for the National Council for Teachers of Mathematics (NCTM). It includes anticipating how students might respond, monitoring how they respond, selecting which students share solutions, sequencing the solutions shared to support learning and helping students to make connections. The Sequencing component is a particularly useful tool for teachers hoping to clarify the learning journeys students are likely to take. A group of teachers begin by anticipating the solutions and methods students are likely to use, both successful ones and those which are flawed. The teachers then consider which of these they will unpack with the class first and why. The intent is to understand how the mathematical conversation that results from a challenging task can lead the learners towards the desired learning. The result is a discussion where students are best able to learn from the methods attempted and the mistakes which were made. When undertaken as a collaborative process, it extends the mathematical understanding of the teachers significantly and is well worth the time it takes.
So the conference is over, but the learning will continue. It is reassuring to know that there are many teachers with a great depth of knowledge to share and a genuine passion for teaching mathematics. Thanks to organisations like Mathematics Association of New South Wales (MANSW) for providing ways to connect teachers and to share resources. A rising tide lifts all ship as they say and for mathematics, the tide is coming in.
By Nigel Coutts