PROS Colleen Vale Problem solving fulltext
Proceedings of the IConSSE FSM SWCU (2015), pp. AZ.23–25
AZ.23
ISBN: 978-602-1047-21-7
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). The Australian Curriculum: Mathematics defines the problem solving proficiency
as: the ability to make choices, interpret, formulate, model and investigate problem
*
Corresponding author. E-mail address: [email protected]
SWUP
C. Vale
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.
The second example concerns generalising from spatial and growing patterns. This
introduction to algebra gained prominence toward the end of last century and is prominent
in research on early algebra (eg. Radford, 2000). During 2012-13 my colleagues and I
introduced primary teachers to Japanese Lesson Study and worked with them to plan and
conduct a research lesson involving early algebra (Vale, 2013). The match stick problem was
used for this lesson with Year 3 and Year 4 primary students. Students solved the problem to
find the number of matches needed to construct 5, 8 or 100 squares connected and arranged
in one row by counting all, creating repeated addition number sentences and creating a
quasi-generality that is, a rule for finding the solution for a particular term in the pattern.
When orchestrating the whole class discussion the teacher sequenced the presentation of
solutions to draw children s attention the source of each symbol in the number sentences
for two students to identify the variables in their rule. Secondary teachers can build on
children s learning arising from such problems and use other scenarios and representations
(such as, input-output tables) to further explore relations and use digital tools to investigate
the meaning of parameters in linear functions (y=m + c) and quadratics.
SWUP
Problem solving and reasoning in the learning of mathematics
4.
AZ.25
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. In Mathematics 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.
Vale, C., Weaven, M., Davies, A. & Hooley, N. (2010). Student-centred approaches: Teachers learning
and practice. In Shaping the Future of Mathematics Education (pp. 571 578). Freemantle:
MERGA.
SWUP
AZ.23
ISBN: 978-602-1047-21-7
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). The Australian Curriculum: Mathematics defines the problem solving proficiency
as: the ability to make choices, interpret, formulate, model and investigate problem
*
Corresponding author. E-mail address: [email protected]
SWUP
C. Vale
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.
The second example concerns generalising from spatial and growing patterns. This
introduction to algebra gained prominence toward the end of last century and is prominent
in research on early algebra (eg. Radford, 2000). During 2012-13 my colleagues and I
introduced primary teachers to Japanese Lesson Study and worked with them to plan and
conduct a research lesson involving early algebra (Vale, 2013). The match stick problem was
used for this lesson with Year 3 and Year 4 primary students. Students solved the problem to
find the number of matches needed to construct 5, 8 or 100 squares connected and arranged
in one row by counting all, creating repeated addition number sentences and creating a
quasi-generality that is, a rule for finding the solution for a particular term in the pattern.
When orchestrating the whole class discussion the teacher sequenced the presentation of
solutions to draw children s attention the source of each symbol in the number sentences
for two students to identify the variables in their rule. Secondary teachers can build on
children s learning arising from such problems and use other scenarios and representations
(such as, input-output tables) to further explore relations and use digital tools to investigate
the meaning of parameters in linear functions (y=m + c) and quadratics.
SWUP
Problem solving and reasoning in the learning of mathematics
4.
AZ.25
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. In Mathematics 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.
Vale, C., Weaven, M., Davies, A. & Hooley, N. (2010). Student-centred approaches: Teachers learning
and practice. In Shaping the Future of Mathematics Education (pp. 571 578). Freemantle:
MERGA.
SWUP