Chris Linsell and Megan Anakin are based at the University of Otago College of Education. Lynn Tozer is an independent consultant. The Teaching Algebra Conceptually project was funded by the Teaching and Learning Research Initiative (TLRI). The article is adapted from the authors' paper Teaching Algebra Conceptually in Years 9 and 10, which includes a full list of project team members.

Students in the early secondary years frequently struggle with introductory algebra, often due to the limited preparation they have received at the primary school level (Warren, 2003). Teachers need a sound knowledge of how to help students learn algebra and how to help them make up for gaps in their background knowledge of maths. In these respects, however, teachers themselves frequently need more extensive preparation (Anthony and Walshaw, 2007). Perhaps as a result, algebra in schools is often reduced to rules for transforming and solving equations (Bednarz, Kieran & Lee, 1996) and these rules often seem meaningless to students (Brekke, 2001).

The current article describes a project in New Zealand designed to address this issue: to explore and create teaching approaches that help students in years 9 and 10 to develop a conceptual understanding of algebra. The project was known as Teaching Algebra Conceptually (TAC). The article is adapted from the authors' paper Teaching Algebra Conceptually in Years 9 and 10.

The project is based on a conception of algebraic thinking as awareness of mathematical structure, rather than as a collection of rules and procedures to be learnt. It follows the New Zealand Numeracy Development Projects in designing learning experiences that address the identified needs of the students.

Implementing the project

The project took place over 2010–2011. In its first year it involved 99 year 9 students in five classes, and their five classroom teachers. The following year it involved 185 students in years 9 or 10 in eight classes, as well as a small year 11 numeracy class, along with their eight teachers.

The project team met regularly for full-day workshops four times per year, with shorter meetings about every two weeks. Guided by the researchers, the teachers designed and trialled new approaches to instruction in algebra, and examined and refined existing approaches. The teachers captured their lessons on video and shared selected excerpts with the other participants.

A key component of the workshops was analysis of student-assessment data, using a previously developed diagnostic tool (Linsell et al, 2006; Linsell, 2010). At the start of each year, students completed tests to provide baseline data about their existing knowledge of algebra and their understanding of strategies for the solution of algebraic problems. At the workshops, a great deal of time was spent poring over this information.

The diagnostic assessment revealed that many students did not have a good understanding of the arithmetic skills and knowledge of the kind that many high school teachers might assume that their students would have learnt at primary school. Many students lacked understanding of arithmetic structure, inverse operations and equivalence, and did not have instant recall of basic facts.
 

Approaches to teaching

A number of teaching ideas emerged from the workshop discussions. While teachers' approaches to instruction responded to the particular needs of each class of students, and reflected their individual teaching styles and beliefs, all participants agreed that the following approaches were effective ways to help students learn algebra.

Assessment

The diagnostic assessment used by project participants has a more general relevance for teaching purposes. It is essential to know where the students are at so that teachers can decide on next learning steps.

'Algebra everywhere'

The teachers rejected the common practice of teaching algebra as isolated units of work once a term or once a year. Instead, they integrated the teaching of algebraic thinking with other elements of mathematics throughout the year.

Rigour and vocabulary

The teachers were particularly careful with correct use of mathematical vocabulary throughout the year. Any terms that were unfamiliar or ambiguous to the students were defined and written into the students' notebooks. Teachers were careful that their board work was correctly set out, and they insisted that their students meet similar standards. They did not assume their students knew the conventions of notation, but explicitly taught the conventions. Similarly, mathematical identities and laws were made explicit and expressed as generalisations using algebraic notation. The schools ensured through discussion at mathematics-department meetings that there was consistency from teacher to teacher and from one year level to the next.

The toolbox

When algebraic skills were identified, they were presented to the students as tools to put in their 'toolboxes'. This metaphor was used to promote acceptance and understanding of the skills. These skills included, but were not limited to, substitution, manipulating terms, expanding brackets, factorising, and strategies for solving equations. Number skills related to indices, integers, order of operations, basic facts, squares, cubes, and highest common factors were also placed in the students' toolboxes. When solving problems in any context, students were encouraged to select and use tools purposefully. This approach avoided skills being taught in isolation, as students appreciated that these were tools that they would use frequently. Thus the skills became more readily transferable.

Context

It was essential that students should perceive algebra as being meaningful, and a wide variety of contexts such as taxi fares and cost of food for the school canteen were used to ensure this. Science teachers were consulted to ensure that approaches to solving equations and using notation were consistent between mathematics and science lessons. Also, some of the teachers chose to develop thematic units aimed at developing algebraic concepts in holistic ways.


Improvements in student performance

Students' baseline assessment results were compared to results of summative assessments at the end of both years. See the original paper for a more detailed discussion of these results. In summary, significant improvements were found in:

• strategies for solving equations
• most aspects of algebraic knowledge, including knowledge of arithmetic structure, inverse operations, algebraic notation and convention, acceptance of lack of closure, and understanding of equivalence
• strategies for finding relationships between variables
• strategies for expressing generality.

Conclusion

Several key findings emerged from the project. Firstly, achievement in algebra can be enhanced by taking a structural perspective that provides high-quality diagnostic-assessment information to teachers. Secondly, effective teaching approaches that have an effect on student outcomes integrate algebra into the curriculum, rather than treating it as a separate topic. Teachers should be supported to recognise and model algebra in many contexts and should not pigeonhole it to a single block of teaching time on its own. Thirdly, students should be encouraged to build a 'toolbox' of knowledge and skills that assist them in solving algebraic problems.

The TAC website has been created to share findings and to act as a platform for further development of ideas. Teachers and researchers are invited to visit and contribute.

Project contact: Chris Linsell, Senior Lecturer, University of Otago College of Education.
Phone: +64 3 479 4253
Email: chris.linsell@otago.ac.nz

References

Anthony, G., & Walshaw, M. (2007). Effective pedagogy in mathematics/pangarau Best Evidence Synthesis iteration (BES). Wellington, Ministry of Education.

Bednarz, N., Kieran, C., & Lee, L. (1996). Approaches to algebra: perspectives for research and teaching. In N. Bednarz, C. Kieran & L. Lee (eds), Approaches to Algebra: Perspectives for Research and teaching, pp 3–12. Dordrecht, Kluwer Academic.

Brekke, G. (2001). School algebra: primarily manipulations of empty symbols on a piece of paper? In H. Chick, K. Stacey, J. Vincent, & J. Vincent (eds), 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra, pp 96–102. Melbourne, University of Melbourne.

Linsell, C. (2010). Secondary numeracy project students' development of algebraic knowledge and strategies. In Ministry of Education (ed), Findings from the New Zealand Numeracy Development Projects 2009, pp 100–117. Wellington, Learning Media.

Linsell, C., Savell, J., Johnston, N., Bell, M., McAuslan, E., & Bell, J. (2006). Early Algebraic Thinking: Links to Numeracy. Retrieved from http://www.tlri.org.nz/sites/default/files/projects/9242_summaryreport.pdf

Warren, E. (2003). The role of arithmetic structure in the transition from arithmetic to algebra. Mathematics Education Research Journal, 15(2), 122–137.