A framework for teachers’ knowledge of mathematical reasoning

Exploring and developing primary teachers’ understanding of mathematical reasoning was the focus of the Mathematical Reasoning Professional Learning Research Program. Twenty-four primary teachers were interviewed after engagement in the first stage of the program incorporating demonstration lessons focused on reasoning conducted in their schools. Phenomenographic analysis of interview transcripts exploring variations in primary teachers’ perceptions of mathematical reasoning revealed seven categories of description based on four dimensions of variation, establishing a framework to evaluate development in understanding of reasoning.

A primary teacher's developing understanding of mathematical reasoning

To support teachers in their quest to incorporate reasoning as a mathematical proficiency as espoused in the Australian Curriculum: Mathematics, a professional learning research project using demonstration lessons was carried out. This paper reports on the impact of demonstration lessons on one participating teacher’s pedagogical knowledge about reasoning. The growth in this teacher’s knowledge was analysed using a phenomenographic framework established to evaluate teachers’ development in mathematical reasoning. The results show that demonstration and subsequent trial lessons contributed to her growth.

A framework for primary teachers’ perceptions of mathematical reasoning

Mathematical reasoning has been emphasised as one of the key proficiencies for mathematics in the Australian curriculum since 2011 and in the Canadian curriculum since 2007. This study explores primary teachers’ perceptions of mathematical reasoning at a time of further curriculum change. Twenty-four primary teachers from Canada and Australia were interviewed after engagement in the first stage of the Mathematical Reasoning Professional Learning Program incorporating demonstration lessons focused on reasoning conducted in their schools. Phenomenographic analysis of interview transcripts exploring variation in the perceptions of mathematical reasoning held by these teachers revealed seven categories of description based on four dimensions of variation. The categories delineate the different perceptions of mathematical reasoning expressed by the participants of this study. The resulting outcome space establishes a framework that facilitates tracking of growth in primary teachers’ awareness of aspects of mathematical reasoning.

Mapping variation in children’s mathematical reasoning: the case of ‘what else belongs?’

Explaining appears to dominate primary teachers’ understanding of mathematical reasoning when it is not confused with problem solving. Drawing on previous literature of mathematical reasoning, we generate a view of the critical aspects of reasoning that may assist primary teachers when designing and enacting tasks to elicit and develop mathematical reasoning. The task used in this study of children’s reasoning is a number commonality problem. We analysed written and verbal samples of reasoning gathered from children in grades 3 and 4 from three primary schools in Australia and one elementary school in Canada to map the variation in their reasoning. We found that comparing and contrasting was a critical aspect of forming conjectures when generalising in this context, an action not specified in frameworks for generalising in early algebra. The variance in children’s reasoning elicited through this task also illuminated the difference between explaining and justifying.

Primary teachers notice the impact of language on children’s mathematical reasoning

Mathematical reasoning is now featured in the mathematics curriculum documents of many nations, but this necessitates changes to teaching practice and hence a need for professional learning. The development of children’s mathematical reasoning requires appropriate encouragement and feedback from their teacher who can only do this if they recognise mathematical reasoning in children’s actions and words. As part of a larger study, we explored whether observation of educators conducting mathematics lessons can develop teachers’ sensitivity in noticing children’s reasoning and consideration of how to support reasoning. In the Mathematical Reasoning Professional Learning Research Program, demonstration lessons were conducted in Australian and Canadian primary classrooms. Data sources included post-lesson group discussions. Observation of demonstration lessons and engagement in post-lesson discussions proved to be effective vehicles for developing a professional eye for noticing children’s individual and whole-class reasoning. In particular, the teachers noticed that children struggled to employ mathematical language to communicate their reasoning and viewed limitations in language as a major barrier to increasing the use of mathematical reasoning in their classrooms. Given the focus of teachers’ noticing of the limitations in some types of mathematical language, it seems that targeted support is required for teachers to facilitate classroom discourse for reasoning.

Professional learning in mathematical reasoning: reflections of a primary teacher

Reasoning is an important aspect in the understanding and learning of mathematics. This paper reports on a case study presenting one Australian primary teacher’s reflections regarding the role played by a professional learning program in her developing understanding of mathematical reasoning. Examination of the transcripts of two interviews identified changes in her perceptions of mathematical reasoning by mapping interview responses against the Mathematical Reasoning Framework (Herbert et al., 2015). This change indicates that a well planned program of professional learning based on a demsonstration is efficacious in developing teachers’ understanding of mathematical reasoning.

Developing noticing of reasoning through demonstration lessons

Observation of fellow educators conducting demonstration lessons is one avenue for teachers to develop sensitivity to noticing students’ reasoning. We examined teachers’ noticing of children’s learning behaviours in one demonstration lesson of the Mathematical Reasoning Professional Learning Research Program (MRPLRP). The observations of teachers evident in the audio-taped post-lesson group interviews conducted at one school are reported in this paper. The teachers noticed that the children struggled to employ mathematical language to communicate their reasoning and expressed concern about gaps in children’s understanding of key mathematical concepts. The teachers viewed limitations in language and mathematical conceptual understandings as a barrier to effective reasoning.

Progressive discourse in mathematics classes - the task of the teacher

This paper uses data from two mathematics lessons to explore the nature of progressive discourse and examine critical features of teacher actions that contribute to mathematics classrooms functioning as communities of inquiry. Features found to promote progressive discourse include a focus on the conceptual elements of the curriculum and the use of complex, challenging tasks that problematised the curriculum; the orchestration of student reporting to allow all students to contribute to progress towards the community's solution to the problem; and a focus on seeking, recognizing, and drawing attention to mathematical reasoning and justification, and using this as a basis for learning.

Teaching secondary school mathematics : research and practice for the 21st century

For the last decade or so, educational policy makers and researchers in many countries have been calling for significant changes to the way mathematics is taught in secondary schools. Australian mathematics curriculum documents now promote learning goals that go beyond mastery of a pre-determined body of knowledge and procedures - the traditional emphasis on facts, skills, formulae - to include mathematical reasoning and problem solving, communication, and real world applications. There is also pressure to move away from over-reliance on teacher-centred practices such as exposition and individual seatwork, towards activities that promote learners' involvement in constructing, applying, and evaluating mathematical ideas. Further impetus for reform comes from research recommending that if learners are to develop mathematically powerful forms of thinking and habits of mind, then classrooms should immerse them in the authentic practices of the discipline by supporting a culture of collaboration and sense-making. Teaching Secondary School Mathematics - incorporates recent developments in research and practice and applications to teaching mathematics in Australian secondary schools. Covering such areas as curriculum, pedagogy and assessment; teaching mathematical content; equity and diversity in the classroom; and professional and community engagement, it is an invaluable resource for all practising and pre-service mathematics teachers.

Year 3/4 children's forms of justification

Engaging children in justifying, forming conjectures and generalising is critical to develop their mathematical reasoning. Previous studies have revealed limited opportunities for primary school children to justify their thinking, form conjectures and generalise in mathematics lessons. Forms of justification of Year3/4 children from three schools in Victoria, Australia will be examined. Evidence from children's written explanations and their verbal explanations captured in video recordings revealed that some children employed sophisticated mathamatical ideas in their justifications. The value of making children's reasoning explicit through written explanations and verbal communications is highlighted.

Children's mathematical thinking in different classroom cultures

The relationship between normative patterns of social interaction and children's mathematical thinking was investigated in 5 classes (4 reform and 1 conventional) of 7- to 8-year-olds. In earlier studies, lessons from these classes had been analyzed for the nature of interaction broadly defined; the results indicated the existence of 4 types of classroom cultures (conventional textbook, conventional problem solving, strategy reporting, and inquiry/argument). In the current study, 42 lessons from this data resource were analyzed for children's mathematical thinking as verbalized in class discussions and for interaction patterns. These analyses were then combined to explore the relationship between interaction types and  expressed mathematical thinking. The results suggest that increased complexity in children's expressed mathematical thinking was closely related to the types of interaction patterns that differentiated class discussions among the 4 classroom cultures.

Spatial reasoning via active observation

This paper presents a model for space in which an autonomous agent acquires information about its environment. The agent uses a predefined exploration strategy to build a map allowing it to navigate and deduce relationships between points in space. The shapes of objects in the environment are represented qualitatively. This shape information is deduced from the agent's motion. Normally, in a qualitative model, directional information degrades under transitive deduction. By reasoning about the shape of the environment, the agent can match visual events to points on the objects. This strengthens the model by allowing further relationships to be deduced. In particular, points that are separated by long distances, or complex surfaces, can be related by line-of-sight. These relationships are deduced without incorporating any metric information into the model. Examples are given to demonstrate the use of the model.