Engaging learners with mathematics
Engaging learners with mathematics is one of a set of professional development modules designed to support school leaders as they lead professional learning about the National Standards for years 1–8 within the New Zealand Curriculum. The modules are suitable for use during the cycles of professional inquiry that leaders and teachers engage in to improve outcomes for their students.
See Engaging learners with texts, which is an equivalent module for literacy.
Introduction to the module
The focus of this module
This module focuses on deepening understandings of how to engage learners with mathematics in relation to the mathematics standards for years 1–8. The module builds on the understandings developed in the module Knowledge of mathematics learning.
Like the New Zealand Curriculum, the mathematics standards emphasise the ability to solve problems and model situations in a range of meaningful contexts by selecting and applying appropriate knowledge, skills, and strategies.
This content of this module is closely related to discussions in Effective Pedagogy in Mathematics/Pāngarau: Best Evidence Synthesis Iteration (BES) and to the dimensions of quality teaching described in Numeracy Development Projects, Book 3: Getting Started.
The structure of this module
This module has three main sections.
Key outcome of the module, which:
 states what the module aims to help school leaders and teachers achieve
 lists indicators that describe what to look for as evidence that they have achieved the outcome
 provides a rationale for the key outcome.
Reflective questions for school leaders and teachers, which:
 helps determine the professional learning needs of the whole staff, syndicates, or individual teachers or leaders
 can be used within activities for leaders and teachers (see next section).
Leading shifts in practice through focused activities, which:
 outlines some professional development activities that relate to the reflective questions
 can be used flexibly to help meet identified needs
 draws on existing resources and professional development opportunities.
A final section, Resources and references, lists texts cited and quoted in the module along with resources that include useful information about mathematics teaching and learning.
How to use this module
School leaders can use this module to identify and explore shifts in practice that might be needed as their school works with the National Standards.
Teachers can use the reflective questions and/or activities to guide them through any changes they might need to make as they work with the National Standards.
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Key outcome of the module
The key outcome of this module is that school leaders and teachers develop their understandings of how to effectively engage students with mathematics and statistics.
Indicators
Indicators that this outcome is being achieved include the following:
 Teachers use problem solving in a variety of contexts to help students to explore mathematical and statistical ideas.
 School leaders and teachers reflect on their mathematics programmes and the extent to which they enable their students to:
 apply skills and knowledge in solving problems and modelling situations
 communicate to offer and justify ideas
 investigate problems that have multiple lines of inquiry and solutions
 use tools and representations to develop mathematical and statistical ideas.
Rationale for key outcome
By studying mathematics and statistics, students develop the ability to think creatively, critically, strategically and logically. They learn to structure and to organise, to carry out procedures flexibly and accurately, to process and communicate information, and to enjoy intellectual challenge.
NZC, page 26
Without a problem, there is no mathematics.
Holton et al. (1999)
A problem presents a situation that requires action; it may be realistic or imaginary. The mathematics standards are based on problem solving.
The expectations defined by the standards include how a student solves a given problem, not only the student’s ability to solve it.
The standards acknowledge and accept variability in the ways in which students solve problems and communicate their findings.
The challenge for school leaders and teachers is to select problems that extend students but are achievable for them. Together, the curriculum levels and the mathematics standards provide helpful guidance for addressing the challenge of problem selection.
A second challenge is to consider the opportunities that a problem offers. This means being aware of the mathematical and statistical ideas that could emerge from a student’s attempts to solve the problem. It also involves considering the “path of ideas” that students could travel as they engage with the problem and how teachers can support their engagement through their questions and suggestions.
The following problems illustrate these two challenges.
Problem A
You are preparing your July roll return for the Ministry of Education. Before you have saved your work, your computer crashes – and with it, your student management system.
Without your system, you must work from the incomplete data set you have, shown in the table below.
Roll return Room
 1
 2
 3
 4
 5
 6
 ...
 17
 18
 19
 20
 21
 22

Years
 1
 1
 12
 2
 2
 2
 ...
 5
 5
 5
 6
 6
 6

Roll
 21
 19
 24
 25
 26
 23
 ...
 28
 31
 29
 32
 27
 30

How could you count or estimate the total number of students in the school? What other strategies or approaches could you use to solve the problem?
Problem B
Watch a short Youtube clip from the TV show Numbers.
Compare what you did to solve problem A with what Charlie does to solve his problem of intercepting the assassin (problem B). Problems A and B are clearly different, but they share many similarities.
Both problems:
 involve data (not all data are numbers)
 are based on a contextual need
 require the creation of models
 require people to use their knowledge
 require the use of tools, for example, diagrams, symbols, words, and physical objects.
The problems differ in that one has a reasonably precise solution (the number of students) and the other involves probability: its solution has an element of uncertainty.
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Reflective questions for school leaders and teachers
The following questions are designed to help school leaders and teachers understand their school’s current practice in relation to engaging learners with mathematics. They can then make comparisons with the practices embedded within the mathematics standards.
Use one or more of the reflective questions to identify areas for further exploration through the activities that follow.
These reflective questions are drawn from Effective Pedagogy in Mathematics/Pāngarau: Best Evidence Synthesis Iteration (BES). Refer to chapter 5 for discussion about the mathematics tasks, activities, and tools that students need to engage with.
1. How confident are we that mathematics learning in classrooms is promoted through problem solving and mathematical thinking?
 Do teachers understand the opportunities that mathematical problems offer?
 Do teachers select problems to promote connections between mathematical concepts?
2. To what extent do teachers consider students’ prior knowledge and experience when selecting tasks?
 Are tasks interesting and engaging to students? How do we know?
 How do teachers determine which contexts will be familiar or interesting to students?
 Do tasks provide appropriate challenge? How do we know?
3. Do the mathematical tasks that teachers use provide opportunities for cognitive engagement and a 'press for understanding'? (The press for understanding is the insistence by the teacher that students engage cognitively. For further discussion on press for understanding, see Effective Pedagogy in Mathematics/Pāngarau: Best Evidence Synthesis Iteration (BES), page 120.)
 In what ways is teaching oriented to developing mathematical understanding and thinking processes (as opposed to achieving correctness or completeness)?
 In what ways do teachers use formative assessment to help students to justify and explain their thinking?
4. In what ways, and how effectively, are teachers supporting mathematics learning for diverse learners?
 What tools (for example, diagrams, equipment) are most commonly used as learning supports?
 How do teachers select and use these tools to support the diverse learners in our school?
Use the understandings gained from discussing the reflective questions above to identify the shifts in practice and/or professional learning that may be required in the school. Select from the following activities to support these shifts as part of professional inquiry.
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Leading shifts in practice through focused activities
Select activities to help the school community deepen its understandings about engaging students with mathematics and statistics. Further exploration may be needed to reach the outcome for this module. For example, discussions may reveal a need to explore real or perceived barriers to learning for particular groups of students in greater depth.
The activities can be used in a variety of ways for wholestaff, syndicate, group, or individual inquiry. For example, teachers working with years 1–3 may use activities differently from those working with years 4–8.
The activities in all of the modules, including this one, are based on the core resources listed in the Overview. Refer to these as appropriate when exploring practice through the activities.
These activities use either real or imaginary mathematical and statistical problems. The problems aim to trigger discussion about the nature of mathematics learning and the ways in which students can engage with mathematics.
Activity 1: Learning opportunities offered by problems
How confident are we that mathematics learning in classrooms is promoted through problem solving and mathematical thinking?
Teachers can use a single problem to teach mathematical understandings at many different year levels. From problems A, B, and C below, choose the one that best suits the level you are teaching at or focusing on.
Consider the following questions:
 What mathematical learning does the problem offer?
 How might students respond to the problem? Is there a progression in students’ likely responses?
 What expectations from the standards does this problem relate to?
 How might we organise the responses of students to provide evidence of their achievement?
Problem A
Toby has 6 marbles. Rewa has 10 marbles. How many more marbles does Rewa have than Toby? How many marbles does Rewa need to give Toby so that they have the same number?
Problem B
5 cows have 12 bales of hay to share equally. 3 horses have 8 bales of hay to share equally. Which animals get the biggest share each: the cows or the horses?
Problem C
Starting from your home, how far could you travel over a period of 4 hours?How much would the journey cost?
The open version of problem A is, 'Rewa has x more marbles than Toby. How many marbles should Rewa give Toby so that they have the same number?' This is open because there are endless possibilities: it is likely to lead to a richer task. As students experiment with different numbers, they will note that the excess number of marbles must be even if Rewa and Toby are to have the same number of marbles.
With colleagues, discuss your responses to the problem(s). Note that learning opportunities may involve exploring different 'paths of ideas', depending on how students respond. For example, problem A could explore the path 'how to determine half of a set', or it could explore 'what happens when two people have unequal shares'. Teachers must know the learning opportunities that a problem offers so that they can help their students to explore different paths.
Problems A and B are 'little problems': we might select these problems to develop a particular set of mathematical or statistical ideas with students. By contrast, some 'big problems' develop multiple mathematical ideas and are sometimes referred to as rich mathematical tasks. We might select these types of problems to develop connections between mathematical ideas and to engage students in extended work over many lessons.
What makes a rich mathematical task?
 It must be accessible to everyone at the start
 It needs to allow further challenges and be extendable
 It should invite learners to make decisions
 It should involve learners in speculating, hypothesis making and testing, proving and explaining, reflecting, interpreting
 It should not restrict learners from searching in other directions
 It should promote discussion and communication
 It should encourage originality/invention
 It should encourage 'what if' and 'what if not' questions
 It should have an element of surprise
 It should be enjoyable
Ahmed (1987), page 20
Consider problem C and then consider Ahmed’s criteria for a rich mathematical task. Which of Ahmed’s criteria does problem C meet?
By attempting the problem, which expectations from the standards could students meet?
Activity 2: Linking to students' prior knowledge of experiences
To what extent do teachers consider students’ prior knowledge and experience when selecting tasks?
Share an example of a problem that you gave to your students and that you considered interesting and relevant to them. With colleagues, consider the following questions:
 Why would your students have found this problem interesting?
 How did the problem link to students’ prior knowledge and experiences?
 What are the learning opportunities offered by the problem?
 What curriculum objectives and expectations from the standards could the problem help students to meet?
Activity 3: Providing opportunities for cognitive engagement
Do the mathematical tasks that teachers use provide opportunities for cognitive engagement and a press for understanding?
In a discussion with two or more classroom teachers, chart the tasks or activities they use for each class in a specified week. Identify the learning outcomes, the standard(s) to which the tasks relate, and the ways in which mathematical understanding is developed through each task.
 How 'open' are the learning outcomes in terms of the ways in which they might be achieved? To carry out the tasks, what kind of thinking would students need to engage in?
 How will we know when or if the students have reached each outcome? Can we use this as evidence towards making an overall teacher judgment?
 How do we use the tasks and activities to help students to justify and explain their thinking?
Select one task and discuss what “a press for understanding” might mean for that task (see Effective Pedagogy in Mathematics/Pāngarau: Best Evidence Synthesis Iteration (BES), pages 120 and 184).
Activity 4: Supporting diverse learners
In what ways and how effectively are teachers supporting mathematics learning for diverse learners?
 Carry out an informal survey of tools that are used as learning supports for mathematics and statistics teaching across the school. Check resource rooms, skim records of teachers’ planning and assessment, and use a brainstorm to generate a list.
 Categorise the tools using criteria that your colleagues suggest, for example, models, concrete materials, stories, and computer programmes.
 Focusing on one category at a time, explore the ways in which we can use the tools to support understanding at different year levels and with different groups of students.
Consider the following questions:
 Are some tools more useful than others for different groups of students? Why might this be?
 What preconceived ideas might influence the use of some tools? For example, do we favour pencil and paper or mental calculation over the use of calculators? Is this always justifiable?
 To what extent are students involved in choosing tools to use?
 Which tools (or ways of using tools) do we want to learn more about? How can we do this?
 What pedagogical content knowledge do we need to make better use of tools to support understanding?
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Resources and references
This section includes details of texts that are cited or quoted in the module and/or that will be helpful to users of this particular module. The full list of core resources is available in the Overview.
 Ahmed, A. (1987). Better Mathematics: A Curriculum Development Study. London: HMSO.
 Ministry of Education (1999–). Figure It Out (Mathematics curriculum support books). Wellington: Learning Media.
 Holton, D., Anderson, J., Thomas, B., and Fletcher, D. (1999). “Mathematical Problem Solving in Support of the Curriculum”. International Journal of Mathematical Education in Science and Technology, vol. 30, no. 3, pp. 351–371.
 Ministry of Education (2008). Numeracy Development Projects, Book 3: Getting Started. Wellington: Ministry of Education.
Published on: 19 Feb 2010
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