Judy Bailey, University of Waikato
Over the past year, a number of international mathematics teachers and researchers have visited New Zealand to work with teachers around the use of rich tasks and mathematical problem-solving. Most recently, Professor Jo Boaler explored the concepts of mathematical freedom and rich tasks with approximately 500 teachers in a day-long workshop in Hamilton.
There is much interest in these approaches to teaching mathematics, but there can also be real challenges. Teaching mathematics using rich tasks or mathematical problems can represent a significant shift in practice (and mindset!) for teachers who have often personally experienced mathematics as a rather dry discipline involving learning set procedures. It can be difficult to change your practice when your own experiences and beliefs about a subject are different to those being suggested (or even mandated) by a national curriculum. This blog post explores some of the key shifts involved in teaching through rich tasks and problem solving, and provides some practical suggestions and resources for teachers who are beginning or already on this journey.
What is mathematics?
If you are a teacher of mathematics (primary or secondary) and are thinking about adapting your practice, a useful starting point might be to think about what you believe about the nature of mathematics – what is mathematics really about? Ask a few people, “What is maths?”, and you are likely to get comments about maths being (a) to do with numbers and (b) something “they are no good at”. People may share their recollections about doing many, and apparently meaningless, exercises from a school textbook. Such experiences and ideas can result in mathematics being considered as a body of knowledge containing truths to do with quantity, patterns, shape, space, and chance. For teachers holding this belief, teaching and learning mathematics is likely to be about transmitting ‘rules’ and trying to convey and/or understand pre-determined, fixed ideas. Such lessons may be what Foster (2013) called “triple-x” lessons where the teacher explains and then students do examples and exercises.
A different way of thinking about mathematics is to see maths as a constructive, creative, social endeavour. When teachers hold this belief, learners are expected to do their own exploring and discovering of ideas. Few people (except, perhaps, those who have not yet reached school age!) would consider ideas of creativity, imagination and wonder in conjunction with the word ‘mathematics’. However, I think of an experience with my 22-month-old grandson. I enjoyed observing him spending almost an hour loading feijoas into the engine of his tractor (there was a small cavity to put things in), shifting stones onto the top of the tractor’s engine (feijoas were now getting a bit squashed!) and then taking the stones off and throwing them on the lawn! He listened and responded to his grandmother’s explanation that we couldn’t leave the stones on the lawn because of the potential impact on his parents’ lawnmower. The play was creative, focused, concentrated, social, and interspersed with learning to steer the tractor as Nana obligingly provided the engine power. Opportunities for learning included spatial awareness, volume, “more” (a favourite word) and social constructs such as thinking of others. Mathematical concepts (and others) were being learned in a creative, imaginative and play-based way. Wonderful!
Teaching mathematics as creative, imaginative and wonderful: The importance of genuine problems
If the idea of mathematics being creative, imaginative and wonderful for our learners (of any age) appeals, a question might be, how can we teach mathematics in this way? A powerful way to move forward is embedded (although perhaps easily overlooked) in the New Zealand Curriculum for English-medium schools within the stem of each set of mathematics achievement objectives from levels 1 to 8. At all curriculum levels, this stem reads,
“in a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems [emphasis mine] and model situations that require them to: …”.
For those who remember MiNZC, the previous mathematics curriculum, the idea of mathematical problem-solving being at the heart of mathematics is not new. However, not having ever truly experienced the joy (and challenge) of real problem-solving myself, I recall how I (as a classroom teacher) used to relegate ‘problem-solving’ to the occasional Friday afternoon or end of a unit – once the supposedly ‘real’ (read ‘triple-x’ lessons) mathematics had been done. Looking back, I can now see the issues with this approach: I was still primarily communicating to my students that mathematics is about reproducing individual, disconnected procedures, and by spending minimal time on problem solving I wasn’t giving the students sufficient opportunity to develop their real mathematical ‘muscles’.
So, if examples and exercises are not problems, what is? The central feature of a real problem is that for the problem-solver, there is no immediately known means of solving it. If a teacher has demonstrated a procedure on the board there is actually no ‘problem’. A rich task has many overlaps with mathematical problems in that the focus is on inquiry, and there are multiple methods, pathways and/or representations that relate to the task. A rich task often has a ‘low floor’ and ‘high ceiling’, meaning that the task is accessible for a wide range of learners.
Problem-based teaching in action
What does mathematical problem-solving or the use of rich tasks look like in action? It means the students are likely to not immediately know what to do, and that we, as teachers, have to learn to initially stand back and trust learners to be creative, imaginative and give the problem or task a go. At first, of course, we will have to coach our learners through this new experience and establish new norms for our mathematics classroom – for example, helping students understand that it’s okay not to immediately know how to solve a problem; that we will work together to see what progress we can make; that we will value each other’s contributions during this process; and that the teacher will not ‘rescue’ them as soon as they ask for help. There are helpful frameworks available to support teachers in learning to facilitate such lessons – e.g. see here and here.
International research, examples of good practice here in Aotearoa NZ as well as my own experiences and research across a range of NZ classrooms all show that problem-based mathematics teaching engages children. Recently, for example, I’ve witnessed this in a large modern learning environment of 60 children, all engaged in the one ‘low-floor, high ceiling’ problem. The problem that was used in that lesson is reproduced below – many teachers will recognize it as a classic maths task.
Such problems and tasks take time (at least a whole lesson or more for a single problem/task) and involve students playing with a range of ideas (some of which won’t work), working individually and in small groups, communicating with others, justifying and explaining your thinking, struggle and persistence. Opportunities to link to the NZ curriculum’s key competencies as well as the aspirations of Te Marautanga o Aotearoa are abundant.
How can teachers learn to teach using problems and rich tasks?
Changing your teaching can be difficult and challenging as well as rewarding and invigorating. So, how do we learn to teach mathematics in this way? Three things come to mind.
First, it is beneficial for teachers to personally experience problem-solving or engaging in a rich task, i.e. to actually engage in solving mathematical problems themselves. This may mean seeking out relevant professional development opportunities where you will not only talk about these pedagogies but actually have opportunities to take on the students’ role and experience them from the inside.
Second, research suggests that working collaboratively with other teachers is an effective way to learn about and reflect on one’s teaching. This suggests the next step could be to find an interested teaching colleague, locate a genuine problem for you both (try looking here, here, or here) and have a go. After successfully solving the problem (seek help if you get stuck – that’s all part of it), have a think about what this experience might mean for mathematics teaching and learning.
Third, learning to teach mathematics as a creative endeavour requires that we trust and position children as capable and competent problem-solvers. Sadly, some children are already entrenched in unhelpful beliefs about mathematics – for example, that maths is an external body of pre-determined, fixed and difficult ideas, or that maths is about being able to get the right answers quickly. Tackling these beliefs may take some intentional un-doing and persistence on our part. Recent work on developing mathematical mindsets and establishing positive beliefs about mathematics could be useful here, encouraging students to come to know themselves as capable and competent mathematicians.
Eighteen months ago, we witnessed the end of the mandatory use of national standards and freedom from the associated pressure this policy placed on learners, parents, teachers and schools. This change has provided us with an opportunity to reconsider our mathematics programmes. Teachers I’ve spoken with have expressed delight about the ending of the mandatory National Standards requirements, explaining they see this as an opportunity to offer more creative mathematical and integrated curriculum experiences for learners. Embedding mathematical problems and/or rich tasks within teaching programmes might be one way to achieve this.
This blog post is based on the following article published in Teachers and Curriculum, in compliance with a Creative Commons licence:
Bailey, J. (2018). The end of national standards: An opportunity to find creativity in primary mathematics? Teachers and Curriculum, 18(1), 85-87.
Judy Bailey is a primary mathematics teacher educator and researcher. She enjoys working alongside pre-service and in-service teachers as they investigate what the teaching and learning of mathematics can look like. Most recently she has been collaborating with beginning teachers exploring what supports are needed for beginning teachers to embed mathematical problem solving within their teaching programmes.