Case 5: Use mathematical tools to explore students' thinking about mathematics

This case explains how the effective use of three Numeracy Project tools (the number framework, diagnostic interview, and teaching model) enables teachers to design rich mathematical tasks for diverse (all) students. The tools support teachers to explore students' thinking about mathematics, their own expectations of students, and their beliefs about teaching mathematics.

Recent international studies show that in-school variation in mathematical achievement is greater than the variation between schools. This case describes some of the processes schools can use to build teachers' mathematical knowledge and pedagogical practices so that they can employ the tools effectively to improve mathematical outcomes for all students.

See also BES Exemplar 1 "Developing communities of mathematical inquiry" and BES Case 26 "Strengthen teacher pedagogical knowledge".

Download: BES Case 5 [PDF 368kB]

Case 9: Use everyday rhymes and waiata to develop spatial skills and awareness

Children's informal mathematical knowledge originates with their everyday activities.

This short case illustrates how a young child uses the various spatial words in a waiata to direct her actions with a poi.

See also BES Case 19 "Use fun games to build mathematical knowledge and confidence in young learners."

Download: BES Case 9 [PDF 893kB]

Case 19: Use fun games to build mathematical knowledge and confidence in young learners

Playing fun mathematical games is one of the most powerful and positive ways in which families can support mathematical learning.

This case demonstrates how adults can support children to develop deeper understanding of fractions by playing informal language games. In this case, the game was played for a few minutes several times a week. The difficulty of the tasks and problem-solving approaches varied, but all were based around the familiar context of sharing cookies amongst a number of children. This case will be useful for parents, caregivers, and educators in early childhood centres and schools.

Download: BES Case 19 [PDF 893kB]

Case 20: Use a participation framework to support students to discuss their problem-solving strategies

This case illustrates the importance of positioning students so that they are able to contribute to mathematical discussions - as problem-solvers, solution-reporters, and claim-defenders. Above all, it highlights the importance of ensuring that students who find fractions difficult are treated as competent, with legitimate contributions to make. The teacher in this case carefully supported her students by making use of their prior knowledge and providing them with multiple opportunities to learn.

See also BES Exemplar 1 "Developing communities of mathematical inquiry" and BES Case 29 "Use pedagogical leadership to enable more equitable and effective teaching for all learners".

Download: BES Case 20 [PDF 1.1mB]

Case 21: Develop tasks that provide high-level challenge and high-level involvement

The teacher in this case selects mathematically rich learning activities that are responsive to students' knowledge and interests, invite exploration and discussion, and develop understanding of fractions.

The case describes how a "pizza fractions" kit provided the impetus for a "fraction flags" activity when the teacher saw a student making a "flag" using pieces from the pizza fractions kit and then exploring fraction-related questions using this different representation.

Download: BES Case 21 [PDF 1.5mB]

Case 22: Support students to make their own choice of tools and representations when solving problems

This case illustrates how cognitive conflict can be a resource for learning. When an apparent contradiction surfaced during problem solving, instead of circumventing discussion by providing an explanation, the teacher in the case ensured that all students were challenged to explore their own thinking, and encouraged to use a range of tools to solve the problem. In this way, the teacher allowed the students to focus on justifying their methods, based on the logic of mathematics.

Download: BES Case 22 [PDF 709kB]

Case 23: Scaffold learning through the careful selection of tasks and problems

Teachers need to be careful that their contribution to mathematical conversations does not take from their students valuable opportunities to do some thinking.

This case describes how a secondary school teacher maintained high levels of involvement after noticing that the students were all using variations of the same incorrect strategy to tackle a particular task. The teacher introduced a different but parallel task to demonstrate the shortcomings of the strategy. Once the students saw the issue, the teacher supported them to think about another strategy.

Download: BES Case 23 [PDF 636kB]

Case 24: Expect students to be accountable for thinking through the mathematics involved in a problem

The mathematics BES emphasises the importance of basing pedagogical practices on socio-mathematical norms that collectively establish expectations about what mathematical thinking is and is not.

Through the use of contrasting examples from four different teachers, this case illustrates important differences between effective and less effective teaching. Above all, effective teaching emphasises conceptual thinking, not superficial sharing of ideas and strategies, and mathematical argumentation, not procedural description/summary.

While this case is located in grades 4-5, it is relevant to all teachers of mathematics, including secondary school teachers.

See also BES Exemplar 1 "Developing communities of mathematical inquiry"

Download: BES Case 24 [PDF 1.4mB]

Case 25: Develop a mathematical community of practice

In a mathematical community of practice, students learn to articulate their thinking and engage in exchange of ideas in an environment that is both challenging and safe. The safety is particularly important because, to develop mathematical understanding, students need to be able to get things "wrong" and learn from their "mistakes" without being embarrassed or defensive.

This case provides a window into the relationship between a teacher and two students as the teacher challenges one of the students to clarify his explanations.

See also BES Exemplar 1 "Developing communities of mathematical inquiry"

Download: BES Case 25 [PDF 722kB]

Case 26: Strengthen teacher pedagogical knowledge

Using fractions as a context, these two cases highlight the importance of teachers being able to create conceptually correct representations of mathematical ideas. Without the knowledge to do this, attempts to connect mathematics with students' lives are always going to fall short.

The examples in these cases provide a springboard for teachers to discuss their own mathematical understandings with colleagues or in a professional learning community. This process could highlight areas that need addressing.

See also BES Case 5 "Use mathematical tools to explore students' thinking about mathematics."

Download: BES Case 26 [PDF 1.0mB]

Case 29: Use pedagogical leadership to enable more equitable and effective teaching for all learners

Teaching approaches influence student identity and social outcomes as well as academic outcomes. Research shows fixed-ability grouping or streaming can have negative outcomes on all three types of outcome, especially for underachievers.

In this case, a primary principal encourages his teachers to review their beliefs about mixed-ability grouping by giving them the experience of working on a mathematical problem with a heterogeneous group of colleagues. Although the context for the case is primary mathematics, it has relevance across all levels and areas of the curriculum.

The case highlights the importance of leaders' knowledge of teaching and learning and provides a number of useful thinking tools and processes.

See also BES Exemplar 1 "Developing communities of mathematical inquiry"

Download: BES Case 29 [PDF 328kB]