Engaging All Students in Mathematics
Dr Jodie Hunter leads the Developing Mathematical Inquiry Communities (DMIC) professional learning and development and research work. In this presentation, she shares how we can engage all students in mathematics.
Hunter is redefining teacher education, hoping to help raise the academic achievement for children of all cultures in New Zealand. She believes the New Zealand curriculum can build on the richness the children bring to school, especially for Māori and Pasifika pupils. Although her research is focused on maths, the tools created for teachers will be applicable to any subject in the New Zealand curriculum.
This talk was a part of Growing from Strong Foundations in Auckland on 5, June, 2018
- New Zealand Curriculum
- Te Marautanga o Aotearoa
- Te Marautanga o Aotearoa (English translation, pdf, 0.39MB)
- Te Whāriki (pdf, 4.4MB)
Best Evidence Syntheses
- School Leadership and Student Outcomes Best Evidence Synthesis: Chapter Seven (pdf, 0.7mb)
- Iterative Best Evidence Synthesis (BES) Hei Kete Raukura Resources
Developing Mathematical Inquiry Communities
- School leadership for improvement in primary mathematics education: Russell School best evidence in action implementation exemplar
- Improvement in Mathematics Education: Evidence in Action Hangaia te Urupounamu Pāngarau Mō Tātou
- Developing Mathematical Inquiry Communities
Hei Oranga Tika: Wellbeing Matters
- Hei Oranga Tika: Wellbeing Matters Emerging evidence of the impact of the Christchurch earthquakes on young children
Reading Together Te Pānui Ngātahi
- Reading Together Te Pānui Ngātahi: Implementation for impact and enduring, reciprocal high trust relationships between families, whānau and schools
- Poutama Pounamu
- Evaluation of Te Kotahitanga Phase 5 (2010-2012)
- Disciplined innovation for equity and excellence in education: Learning from Māori and Pasifika change expertise
- 'Walking the talk' matters in the use of evidence for transformative education (pdf, 0.7mb)
Writers in Schools
As you watch the presentations, think about these questions:
- What is something that has challenged or confirmed your thinking about the curricula?
- What can I do as a result?
Dr Jodie Hunter: So just to begin by introducing myself. My name is Jodie Hunter – thank you for the lovely introduction. I come from a large Cook Island family, so I am half Cook Island and half Pākehā and I also have the pleasure of being able to work very closely with my mother, Bobbie Hunter. We both lead the developing mathematical enquiry communities professional learning and development and research work, and what I will be talking about today is things that we have learnt really through this work about how we can engage all students in mathematics.
So just to begin with, we really have structured the talk that we are giving at these hui around the proposed government workplan.
The video show a diagram of the proposed government’s work plan.
So, we were really interested with the new government to have a look at what they were proposing to do, and if you look at this diagram, this was one of the documents that came out in thinking about what can we do in regards to education and changes in New Zealand.
And when we looked at that, we though that the work that we do in schools and have been doing on an ongoing basis for a number of years now, really fits into this nicely. So, what I will be talking about today is how the work and the research and development work that we do with schools aligns with each of these areas.
To begin with I wanted to just talk about where does this work come from and why are we doing this? So really our work at the heart of it is about challenging the status quo. For a long time, mathematics has been a subject which I think is about exclusion, rather than inclusion. So, mathematics has been used as a tool to exclude particular learners from career pathways, from opportunities in life but even from access in the classroom. So really our work is about thinking, how can we change what it means to be successful in mathematics and also a big part of this is who can be successful in mathematics.
I think of myself and my history of my family and I think if I look at myself and my history in my Cook Island family, I can say that I come from a long history of mathematicians. And every person who is Pasifika and Māori comes from a long history of mathematics and yet that has been discounted – both in our curriculum and our schools and by learners themselves – and teachers themselves I would say. Why I say I come from a long history of mathematics is because our ancestors use navigation to travel all around the Pacific and there is nothing more mathematical than navigation. That is pure mathematics at the heart.
But I also look at the tivaevae quilts and I will show you one of those that my grandmother makes and that is pure mathematics as well. Architecture, building, our craft work – all of that is mathematical and mathematics. Even in the ways that we work can be aligned in terms of collaboration with the ways that mathematicians work. So, all of that we can draw together to say who can be successful at mathematics. Not necessarily the people who have been positioned as typically successful.
Part of challenging the status quo is also thinking about how we can widen what success means. So, I think with national standards, my feelings about that and what I have seen happen in a lot of cases is that there has become a very narrow definition of what it means to be successful. So, when a child starts school, being successful meant that you could count to ten or count to 100 in English. But all the other aspects of mathematics were discounted as irrelevant and I think that has come through in some of the comments that we hear unfortunately from teachers when talking about children and mathematics.
One of the common comments that I hear a lot of the time when we talk about using challenging tasks and getting children to work at the top of where they can possibly work is that you don’t understand that these children come to school with nothing. They have no maths. And we really challenge that because we say that is completely impossible; nobody can have no mathematics, in fact it is about opening our eyes up as educators to see what mathematics is in the home and in the community that we can draw out and build upon.
And the unfortunate things about this as well is that children themselves deficit their eyes about who they are and we have got lots of evidence from interviews that we have done with children. So, we have children who talk about … we ask the question – how do you feel being whatever culture you are in the mathematics classroom – so we have a whole group of children who just say, well maths has nothing to do with my culture. To me that is concerning, but even more concerning is that when we have done interviews with children at particular clusters of schools we have had 20-30% of children say things like, ‘I am Tokelauan and Tokelauans don’t do maths’. Or ‘I am Samoan and in the maths classroom I have to become a Palagi and then when maths finishes I can go back to being Samoan again.’
So, it is also about challenging these perceptions from the children themselves and showing them that in fact they are mathematical, they come from a culture that is mathematical and they can be great mathematicians as well.
So how do we do that?
Firstly, we think about learners at the centre of the work that we do. So, this means that we need to use what we can call both culturally responsive and culturally sustaining approaches. Instead of viewing language and culture of students as a deficit, we can see them as a strength to support, nurture and empower. That means for everybody, stepping back and thinking about your cultural identity and what do you know about culture? How well do you know the learners in your classroom and who is sitting in front of you? And I mean truly know – not just at a surface level. That’s hard work and it really means that you have to build relationships both with the children who are in your classroom, but you also have to build relationships with parents and the community and be able to use those relationships to work together so that we can have success for all of our learners.
When we talk about culture with New Zealand teachers, often teachers only see culturally responsive teaching as a tool really to get children to buy-in to problems. If I use a problem that involves the children’s culture then they will want to do the problem. But I think we need to start seeing it as so much more than that because really culturally responsive, culturally sustaining teaching is really a holistic principle of teaching
And this benefits all children because if we think about our classrooms, they are a microcosm of society and we want all of our children to walk away from our classrooms realising that other people’s cultures have strengths in them. Being able to positively relate to each other and to be able to have understanding and empathy for different people and understand different people’s values. And if we think about culturally responsive teaching or culturally sustaining teaching in that sense – then we can think about this as a way of doing good for everybody and for our society.
So, I would just like to give you a moment to think about yourselves and what are your core values and beliefs that you grew up with. So just have a minute to think about that when you grew up, what were the core values that you were taught? Maybe by your parents or at home or through your classroom interactions? And then stop and think about how may these affect your interactions within the classroom?
So, when I think about some of the values that we hear people talking about, people will say things like ‘I was brought up with the value that if you work hard, you will succeed.’ But then I stopped to think about how does that affect our interactions with students in the classroom? Because essentially what I think we are saying there is that if everybody works hard, then everybody will succeed. But the way our classrooms and our curriculum and even our teaching at times is set up doesn’t work that way.
Other kinds of things that we hear people talking about are – you have got to be a winner in life. So that then becomes a competitive focus and if we start thinking about the interactions in our classroom, that can lead to an emphasis on speed, getting something right quickly – so correct answers. But your values also might impact ways that you think you learn – so do you learn by listening or do you learn through speaking or a bit of both? Do you learn by explaining? Do you learn by asking questions?
So, if we start thinking about this, I think it is really at a core of all our work that our values and beliefs have a huge impact in the classroom and we need to start unpacking what our own personal values and beliefs are and how they might affect the interactions that are developed in the mathematics classroom. How they might affect who has more opportunities to participate and learn within the interactions. But also, who has less opportunities to participate.
So, if we think about values and beliefs in both ourselves and also particular groups of learners in the classroom, we can think about a continuum. So, there is really a continuum of communalism and collectivism at one end and individualism and competitiveness at the other end and then other values in between those as well. If we think about Māori, Cook Island, Tokelauan, Samoan and all the Pacific island nations – often the values and beliefs of these learners are commonalities which shape the interactions and how you are as a Pacific person. And often they sit at the end of communalism and collectivism.
But, assessment processes that we have used in schools have had a huge impact I think on achievement in this sense because often we are looking at assessment as an individual process and as a type of competitive process. So that puts a whole lot of people outside of achieving and being who they are and able to achieve in that kind of a way. So, what we need to do is map out our own beliefs and that offers us an opportunity to see how they collide with a more communal view.
A key challenge for us as teachers to interrogate our own values and beliefs as I said before and how that is shaped by our upbringing and our past experiences. Think about whether you identify things such as being the best, being the neatest, hard work makes you a winner – how does that fit in with these types of values in this continuum? Or whether it is about nobody is successful unless everybody understands. Or with multiple strengths we can achieve so much more.
Our professional development activity really looks at how we can engage teachers and looking at how values shape classroom interactions with both Pasifika and Māori students. And opportunities to look at how individualistic beliefs can collide with communal beliefs and ways of participating. So, what we look to do is to really critically reflect on what perceptions we have of students because for some groups of students, they are often positioned as lower status in the classroom and this means that they have less opportunities to participate and interact. Instead what we need to do is to start thinking about why these students are reluctant to interact and to question and challenge and to think about ways that we can make this safe in the classroom or culturally acceptable ways to ask questions.
So, to do that we look at barrier-free access. In our work we use a smart tool and the smart tool we use is a communication and participation framework. So, we work on the basis that it is not natural for all students to come into the classroom and know how to communicate and participate in a productive way, so as teachers we need to offer students opportunities to be able to communicate and participate. And how can we do this? Through particular teacher actions that we set up in the classroom – so our communication and participation framework offer teachers opportunities to reflect on your practice and to see why particular students may not be participating and communicating and to look at what actions you could try and use and change to ensure that all students are participating and communicating.
The video shows a list of techniques to create barrier-free access.
What else do we use? I have mixed ability grouping up here, but I would really … I put mixed ability because I think that is what people can identify with, but I think that really in mathematics we need to start getting – well, across everything – we need to start getting rid of the notion of ability because that as been a huge part of what excludes learners in the mathematics classroom. Because we talk about high ability, low ability students, stage four, stage five, over-curriculum national standards – what was it? My overs and my unders – and all of those labels are what means that children don’t have opportunities to do rich mathematics.
So instead of thinking about high ability or low ability, think about what exactly are you talking about when you are talking about that. Are you talking just about number, are you talking about students who can ask questions, who can support each other’s learning, are you talking about students who can use multiple representations, who can come up with a conjecture? When we start thinking about all the different multiple abilities that you can have in mathematics that means that there is no such thing as high ability and low ability.
What else do we talk about using? Group-worthy problems. So, a group worthy problem is a problematic task with a low floor and a high ceiling. So, in the classrooms and the schools that we work in, we think that every child should have access to rich mathematics and to challenging complex problems. Nobody is excluded from this. But the children work together in a collaborative, supportive environment on these types of tasks, which we say are low floor, high ceiling.
So, the notion of low floor is that everybody can enter the task and the notion of high ceiling is that everybody exits that task with a different understanding and a different piece of mathematical learning. We are not saying that everybody would leave that task having developed the same understanding but that there are multiple levels of understanding and all of those understandings are valuable.
And the last part of that is using culturally responsive problems. So, problems that link in or tap into the children’s cultural identity to highlight how mathematics is part of everybody’s culture. I am just going to show you an example of one of the tasks that we designed to show this.
The video shows a photograph of a tivaevae.
Sorry, I forgot to mention that the tivaevae back on that other page is one of my grandmother’s designs, okay? And if you looked at my grandmother you would probably position her as somebody who was not mathematical as such because she came from a tiny island – Manahiki in the Cook Islands – she came to New Zealand when she was 15 on a boat and didn’t do school past probably 10 years old basically. But what she does is mathematical in terms of the tivaevae and not just in geometry. I think often we look at these and say ooh, there is lots of geometry and shape in this, but I look at it with my eyes looking for algebra and I can say that there is a heap of algebra in this as well.
The video shows a mathematical exercise about tivaevae.
I apologise because I know this is quite small.
We see this as low floor, high ceiling task. The reason being that we use this in a year 2 classroom. I don’t like to think the curriculum is limiting as such so even though growing patterns like this don’t come into the curriculum until about year 4 or year 5, I think that we can get children to be doing these much earlier. So, this was used in a year 2 classroom at a school in Mangere. So, Koru school if anybody knows that school with a little class of year 2 students. We chose this task particularly because we knew that had Cook Island children in the classroom and we wanted to highlight the mathematics that comes from Cook Island culture and also for those children to be able to have a little bit of status and expertise in explaining about the tivaevae process and how they had seen their mothers making tivaevae.
Everybody got together and sewed it together and it was a communal activity. But we used this as a growing pattern because I was looking at the tivaevae my grandmother was making and that particular design at the top – I saw it and though, ooh wow that is a really interesting growing pattern. So, I will explain it to you because it is probably a little bit hard to see.
So, you have got four leaves in the middle of it and then each time it adds on it grows by eight leaves. So, you have a constant of four in the middle and then each time it is growing by eight leaves. We see this as a low floor-high ceiling task because when we used this with this class of year 2’s we had students who were working at all kinds of levels. There were children in there who had high special needs, who were working at probably a three or four-year-old level and practicing one to one counting. This task gave them opportunities to practice one to one counting but it also gave them opportunities to listen to their peer’s explanations who had gone into other generalisations.
So, the children after working on this task for two or three days came up with generalisations for the bottom question which is ‘can you describe what the pattern would look like for the 76th position?’ So, we had children talking about how it would look like four in the middle and then eight, 76 times. We had other children talking about how you would have four in the middle and then you would have two 76 times here and two 76 times here and so on like that. And we had other children talking about how you would have four in the middle and you would have 76 leaves, 76, 76, 76 – essentially 76 eight times. Okay?
So, I can start thinking about all the multiple opportunities that the children had from this task here. So, they had opportunities for counting, they had opportunities for practicing skip counting, they had opportunities to practice multiplication, they were able to generalise, they were able to prove what they were talking about – all of that from this one task.
So, part of why we use tasks like this is that we think that we need to challenge perceptions. It is really hard to shift our own views and to shift teacher’s views of student capability unless we see them engaging in different activity. If we keep just the focus on number and counting and low-level activities, it is really hard to show what the children can do. So, in doing this and in challenging teacher perceptions of student capability and children’s perceptions themselves, we can redefine what it means to do mathematics. But we also provide multiple opportunities for students to demonstrate their strengths in multiple ways, so doing mathematics and being good at mathematics is not only about number, it is about a whole lot of content areas. It is also about a whole lot of different practices – questioning, agreeing, disagreeing, supporting others, building on somebody’s ideas, justifying your thinking.
And the final part of that is let’s remove all notions of ability and instead say that all students can learn mathematics. Everybody can do mathematics, not just a certain select group.
The video shows another mathematical exercise.
I just want you to have a think about your own classroom and have a look at this as a different type of a problem. What are the strengths that different students in your class might use to solve this collaboratively? So just quickly talk to the person sitting next to you about that.
Great to hear you talking about that. I would also make the comment here that I think in our mathematics, what we want to do is widen all the time what children … the opportunities that children have. So, I think for a long time we have been quite stuck into a narrow focus, particularly in primary schools of looking at number and we have stuck to these rules, really artificial rules I think that we were given about how much you should be doing number and how much you should be doing strand. So, you should be doing number 80% of the time and strand 20% of the time and I think we need to kind of move away from all of those kind of rules and dichotomies and start thinking about how number is really integrated across the curriculum.
So, we can do much richer tasks by looking at what we call strand but I just call mathematics. So, tasks like this and think about how number then is part of this anyway, so we can be doing number all the time as part of what else we are doing. So, the other part and another aspect of what our work is about is quality teaching. So, we use a frame of ‘ambitious teaching’. So, we talk about ambitious mathematics teaching and this is really hard work. So, I know when I go in and work in a school and with teachers that this is hard intellectual work for us to teach mathematics in an ambitious way. It is much easier to tell children that you solve the problem like this – now everybody practices.
But turning the teaching on its head and giving children complex problems and letting them come up with the solutions, and then thinking about when we step in and when we step out and what mathematical idea are we going to follow and what are different children saying and what does that mean about their reasoning… that is really hard work.
So ambitious teaching really requires us to reconstruct our practice. It is not about a slight tweaking, it is about thinking and really challenging what we mean by mathematics and what it means to do mathematics and how we teach mathematics. So significant reconstruction of practice. But it is also an opportunity to develop more sophisticated forms of maths knowledge for teaching. Where any teacher’s starting point is, it is no different to children; we can all do maths, we can all learn maths. And when I walk into a school and teachers say to me, ‘I am not really good at maths’, I go everyone can be good at maths. And actually, teaching in this way gives you multiple opportunities to develop your own mathematical understanding.
It also means that we need to have new perspectives on teaching and learning and what is our role in teaching and learning? And finally, we need to have new forms of teaching in which student thinking is elicited and built from. So really what this means is that we need to be hardworking and energetic mathematics teachers.
The final part of this is thinking about quality inclusive public education. So, in the work that we do, we are an intensive professional learning and development and research project. So, teachers in the school that we are working in have access to sustained quality professional learning through professional learning days. So, we work with schools and take staff meetings and PLD days, but a huge part of our work is also our team coming in and working in the classroom and co-constructing mathematics lessons. We take a different form of professional learning and development, so we don’t model and we don’t observe, but we work alongside teachers to redevelop what mathematics teaching looks like.
We also have access to University courses which are focussed on this work and a process called Lesson Study which is a form of collaborative professional development that we work with schools in their third or fourth year of working with us.
And the final part of his is teachers learning from parents about mathematics used in the home setting, so we advocate for different forms of parent meetings. Rather than parents coming along and being told what you need to do to support your child in maths, we say have parents come along and talk to teachers about what mathematics is already happening. Parents might not recognise it as mathematics and teachers also need to reconstruct what they see as mathematics.
And the very last thing is thinking about 21st century learning. So, the key components of this are giving students a voice, learner led enquiry, giving children flexibility and mathematics for life and I just want to finish off – because I know I am running over time – with some quotes from teachers and from the students.
The video shows two quotes.
So here is one teacher talking about a little boy in her classroom from a Pacific Island nation “who would not say anything in maths ever. Now it is getting him to shut up is the trick.”
And another teacher from one of our schools in Porirua talking about a girl in her class. “She is an amazing mathematician and she said to me that the biggest thing she has learnt this year is that she needs to listen to other people and she doesn’t have all the answers and she is not the best at maths.”
And finally, just the words of … and I think this is very interesting because I talked about the Tokelauan boy who said Tokelauan’s don’t do maths. So, this is from a school in Porirua and this is from a year 3 Tokelauan boy. We started working with these schools in 2015, so a year 3 who has come through all his schooling like this and when the teacher asked him how you feel being Tokelauan in the mathematics classroom – this was his response. “My culture is in everything I do, it is in everything I do and it is always supposed to be in everything I do.”
More from this series
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