Proceedings of the IConSSE FSM SWCU (2015), pp. AZ.23–25 ISBN: 978-602-1047-21-7
SWUP AZ.23
Problem solving and reasoning in the learning of mathematics
Colleen Vale*
Deakin University, Melbourne, Australia
Abstract
In Australia we have been concerned to ensure that our citizens are numerate and able to use mathematics in their daily personal and working lives. More recently mathematics curriculum writers and educators have acknowledged the need for students to be creative thinkers and problem solvers, not only when using mathematics in our daily lives, but also to contribute to improving our society and economy and enhance the employment opportunities for young people. These two examples provide illustrations of tasks that can be used to develop students understanding of algebraic concepts in both primary and secondary curriculum and to engage learners in problem solving and reasoning, especially justifying and generalizing.
Keywords algebraic concepts, problem solving, reasoning
1. Introduction
In Australia we have been concerned to ensure that our citizens are numerate and able to use mathematics in their daily personal and working lives. More recently mathematics curriculum writers and educators have acknowledged the need for students to be creative thinkers and problem solvers, not only when using mathematics in our daily lives, but also to contribute to improving our society and economy and enhance the employment opportunities for young people. In my research and engagement with teachers and schools I ve evaluated school change and development, and contributed to professional learning of coaches and teachers including through demonstration lessons and Japanese Lesson Study. Schools that are making a difference are implementing student-centred learning by planning coherent lessons that focus on a specific learning goal and address diversity of students prior learning and learning needs (Vale, et al., 2010) and working to improve students proficiency to problem solve and reason mathematically (Roche et al., 2013; Sullivan et al., 2015). In this presentation I will talk about some of my research with a focus on problem solving and reasoning related to algebra.
2. Problem solving and reasoning
When working with primary teachers in our Mathematical Reasoning Professional Learning Research Project we discovered that primary teachers struggled to define reasoning. Some believed it was thinking and some confused it with problem solving (Loong et al., 2013). TheAustralian Curriculum: Mathematicsdefines the problem solving proficiency as: the ability to make choices, interpret, formulate, model and investigate problem
C. Vale
SWUP AZ.24
situations, and communicate solutions effectively (ACARA, 2012) and reasoning as an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising (ACARA, 2012). The primary teachers in our study tended to associate interpreting the problem and making choices about how to solve it or explaining how to solve a problem as reasoning (Loong et al., 2013). Few referred to explaining why or justifying as reasoning. Other researchers have noticed that secondary teachers also tend to refer only to explaining when reporting on students reasoning.
3. Teaching and learning algebra through problem solving and reasoning
Two main topics in secondary algebra are equations and functions, and concern the big ideas of equivalence and variable. They provide opportunity to engage in relational thinking and use logical argument associated with analyzing, inferring and justifying. As part of a professional learning program I worked with primary teachers to consider how they might address students misconceptions about the meaning of the equal sign (=) and facilitate their students reasoning. Primary teachers of various year levels conducted a formative assessment using equations such as 7 + 21 = + 11 to identify misconceptions, effective solution strategies and to plan tasks (Vale, 2012). They found that children, and more commonly children in the lower grades held a misconception that the = sign means find the answer, a misconception also common in Year 12 exam papers. Teachers analysed the successful strategies and found that their students used the balance strategy, transformation strategy or a relational thinking strategy; all of which are methods explicitly taught in secondary curriculum. Teachers then worked on selecting and designing open-ended problems and justification tasks that they could use in their classrooms to address misconceptions and enable students to generate one of more of these strategies. These tasks included open-ended number sentence and justification true or false tasks. Using supportive and challenging prompts (Roche et al., 2013; Sullivan et al., 2015) are important for eliciting logical argument and justification when discussing students solutions.
Problem solving and reasoning in the learning of mathematics
SWUP AZ.25
4. Conclusion and remarks
These two examples provide illustrations of tasks that can be used to develop students understanding of algebraic concepts in both primary and secondary curriculum and to engage learners in problem solving and reasoning, especially justifying and generalising.
References
Australian Curriculum and Reporting Authority (ACARA) (2012).Australian Curriculum: Mathematics. Loong, E., Vale, C., Bragg, L., & Herbert, S. (2013). Primary school teachers perceptions of
mathematical reasoning. In Mathematics Education: Yesterday, Today and Tomorrow (pp. 466 473). Melbourne: MERGA.
Radford, L. (2000).Signs and meanings in students emergent algebraic thinking: A semiotic analysis, Educational Studies in Mathematics, 42, 237 268.
Roche, A., Clarke, D.J., Sullivan, P. & Cheeseman, J. (2013). Strategies for encouraging students to persist with challenging tasks,Australian Primary Mathematics Classroom,18(4), 27 32.
Sullivan, P., Walker, N., Borcek, C. & Rennie M. (2015). Exploring a structure for mathematics lessons that foster problem solving and reasoning. InMathematics Education in the Margins(pp. 41 58). Qld: MERGA.
Vale, C. (2013) Primary teachers algebraic thinking: Example from Lesson Study. In Mathematics Education: Yesterday,Today and Tomorrow(pp. 719 722). Melbourne: MERGA.
Vale, C. (2013). Equivalence and relational thinking: Opportunities for professional learning.Australian Primary Mathematics Classroom,18(2), 34 40.
