The use of tasks is an important factor in developing early algebraic reasoning. As outlined in Chapter Two, there are a range of research studies which illustrate how early algebra can be taught through tasks involving generalised arithmetic and functional reasoning. However, research studies also highlight some general factors within task design which potentially provide greater affordances for algebraic activity. These factors will be outlined in the following section. Facilitating algebraic reasoning can be accomplished through variation of the task parameters. Here the openness of the task, achieved by extending the number of answers from a closed single answer to multiple solutions, shifts the purpose of the task from computation to examining patterns or relationships (Blanton & Kaput, 2005b; Kaput & Blanton, 2005; Smith &
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Thompson, 2008; Soares, Blanton & Kaput, 2005). For example, Blanton and Kaput (2005b) describe how the task parameters could be extended to develop what they term an ‘algebrafied’ problem by changing the single solution problem: How many telephone calls could be made among 5 friends if each person spoke with each friend exactly once on the telephone? to a sequential set of problems: How many telephone calls would there be if there were 6 friends? Seven friends? Eight friends? Twenty friends? One hundred friends? Organize your data in a table. Describe any relationship you see between the number of phone calls and the number of friends in the group. How many phone calls would there be for n friends? This set of problems provides opportunities for students to examine patterns and relationships. It also facilitates engagement with the mathematical practices of developing conjectures, justifying, using representations, and generalising.
Similarly, designing tasks which involve series of number sentences offer opportunities to engage students in algebraic reasoning (Carpenter et al., 2005b; Kaput & Blanton, 2005). Examples from research studies (e.g., Baek, 2008; Carpenter et al., 2005a; Carpenter et al., 2005b; Hunter, 2010) provided in Chapter Two showed how students could develop conjectures and generalisations from carefully chosen sequences of number sentences which illustrated patterns. When students were encouraged to examine sequences of unexecuted sums without computing answers, they began to engage in the types of structural analysis which supported early algebraic reasoning (Kaput & Blanton, 2005; Smith & Thompson, 2008).
Another method of facilitating students to attend to structure and generality is through tasks that require students to generate more problems of the same type (Mason, 2008; Soares et al., 2005). For example, students could be asked to offer solutions for the following question: Nineteen divided by two equals nine with a remainder of one, what other numbers share this property? In
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this way an arithmetic task can be developed into a task focused on generalisation. Also, asking students to generate more problems of a similar type can be used as a formative assessment task. For example, Carpenter et al. (2003) advocate asking students to generate their own true and false number sentences to provide an indication of their thinking about equality, relations, and other important mathematical ideas.
Another factor in designing or adapting tasks to facilitate algebraic reasoning is the use of contextual support within word problems to support sense-making of abstract concepts (Ding & Li, 2010; Koedinger & Nathan, 2004; Smith & Thompson, 2008). Ding and Li’s (2010) study of text-books’ presentation of the distributive property found that contextual support of word problems within USA text-books was limited when compared with Chinese text-books. For example, within the Chinese text-book the structure of the problem concerning finding the cost of 102 t-shirts priced at 32 yuan allowed students to make sense of distributive property by splitting 102 into 100 and 2 and then relating this to the context of the problem with 102 viewed as 100 t-shirts and 2 t-shirts. Careful construction of word problems allows students to solve problems through use of informal strategies and to make sense of different concepts within algebra (Carraher et al., 2008; Ding & Li, 2010; Koedinger & Nathan, 2004).
Tasks which offer students the opportunity to use multiple representations can also facilitate algebraic reasoning. Research studies (e.g., Beatty & Moss, 2006; Kaput & Blanton, 2005; McNab, 2006) demonstrate how tasks that offer opportunities for multiple representations can cultivate the practice of students using different forms of representation to communicate reasoning and to justify thinking. For example, Beatty and Moss (2006) describe the use of a problem where students were required to generate a functional rule that predicted the number of chairs that would fit around any number of tables. This required the students to use concrete
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materials, verbal explanations, and tables of data. Such tasks also support students to develop the ability to access different forms of representation including symbolic forms and move between these forms flexibly utilising the representation which provides the greatest affordance for the task (Schoenfeld, 2008).
A further design principle in developing tasks that support algebraic reasoning involves the utilisation of connections between different mathematical content areas. Tasks which exploit the connections within mathematics provide learners with opportunities to think about content in new ways. Also, a foundation is laid for the meaningful use of algebraic tools. For example, Schoenfeld (2008) describes how a rate and ratio problem could be extended through the introduction of a graphical representation. This then shifts the problem to include ideas about functional relationships. Extending tasks in this way supports students to view the connections within mathematics and algebra. It also allows students to be introduced to new representations in a way which links their interpretations with a meaningful context.
The careful design or extension of tasks within different mathematical areas to include algebra allows algebraic reasoning to become an everyday part of mathematics lessons (Kaput & Blanton, 2005; Schoenfeld, 2008). For example, Kaput and Blanton (2005) illustrate how a teacher with a class of students aged 6-years to 7-years old in adapting a combinatoric problem involving combinations of outfits of pants and shirts was able to support her students to build generalisations. The task was enacted in the classroom with students using coloured cut-outs of the pants and shirts and recording information onto chart paper. These researchers argue that many tasks can be extended and ‘algebrafied’ across the mathematics curriculum thereby allowing algebraic reasoning to permeate mathematics instruction.
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It is necessary also to consider factors which support or inhibit successful task implementation. While the use of carefully designed algebraic tasks is important, there are also a range of classroom factors which may influence engagement with the task. Research studies (e.g., Henningsen & Stein, 1997; Sullivan, Mousley, & Zevenbergen, 2006) have investigated how to successfully maintain the cognitive demand of challenging tasks such as those used to develop