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This section outlines the potential use of spreadsheets for the teaching and learning of school algebra. This is achieved by first providing an overview of how spreadsheets may be function as cognitive tool for teaching and learning of school algebra. This is followed by a description of how spreadsheets may be suitable for use to enhance learning of some concepts (concept of variable; simplifying expressions; solving equations, graphs) within the Grade 9 algebra curriculum.

Arganbright (1984, p.185) gives the following description of a spreadsheets program:

A spreadsheets program uses a large matrix whose rows are identified by positive integers and columns by letters. Locations, or cells, in the matrix are identified by row and column. Each cell can contain a descriptive label, a number, or an algebraic expression that refers to other cells in the spreadsheets. The program calculates values for the expressions using the values of the cells to which the expressions refer and displays the evaluated spreadsheets on the computer screen. (Arganbright, 1984, p.

185).

Figure 2.3is an illustration of the features of spreadsheets.

Figure 2.3 Part of Excel Spreadsheetsss

Spreadsheets, as cognitive tools, may be used for amplifying and reorganizing mental functioning. Spreadsheets programs were mainly used to create budgets and manage accounts. They affected ways of dealing with accounting process in handling large sets of data. They can be used to perform various mathematical calculations more quickly and accurately. Spreadsheets can also be used to store and modify data quite easily. When data used in a calculation is altered, the rest of the data resulting from the calculation is automatically modified. Spreadsheets can also be used to perform multiple calculations and allow the user to handle large sets of numerical data. They can also convert numerical data

that reflects relationships, into charts and graphs.

It is on the bases of these affordances that spreadsheets were then investigated for potential use in the teaching of school algebra. Research studies (Rojano, 1996; Berdnarz, et al, 1996; Heugl, et al, 1997; Yerushalmy, 2005; Tabach et al, 2008) indicate that spreadsheets like

Excel, offer opportunities for learners to develop algebraic skills such as generalization,

modeling and shifting between different forms of representations (graphical, numerical and symbolization).

Spreadsheets have the potential to bridge the gap between arithmetic and algebra. They provide learners with opportunity to progress from their arithmetical intuitive methods to more algebraic ones (Haspekian, 2005, p. 114). Use of spreadsheets involves rule making, as such it requires learners to be able to formulate rules connecting data presented in the cells. The rules represent some form of modeling of the mathematical situation dealt with. Identifying values and developing rules that show relationship between them enhances understanding of algorithms that are used to compare them. In this way, learners would be working on numerical representations while at the same time working on generalizations and patterns. (Jonassen & Reeves, 2001, p. 712).

Spreadsheets also function as cognitive tools in that they may be seen as learners’ intellectual partners sharing the cognitive burden of performing long calculations. Learners are left with cognitive processing of information while the technology does that which it performs best. They “allow students to handle, observe and generate large sets of numerical instances” (Tabachet al, 2008, p. 50) offering them opportunity to conjecture, compare and identify patterns and relationships in data.

Chazan (1999, p. 123) noted that the capacity of this technology to carry out many calculations rapidly supports the transition from examination of single cases towards the examination of multiple cases at once. With their “dragging” power, spreadsheets can enable learners to extend any given sequences of numbers, slowly realize need for generalization, and finally appreciate the power of symbolism (Tabach & Friedlander, 2004, p. 428).

Spreadsheets can thus be seen as tools that “allow children to appreciate the need for an algebra-like notation and provide new ways for children to be introduced to it” Ainley in (Haspekian, 2005, p. 114). From analysis of the learners’ work in the Anglo/Mexican Project, Rojano (1996, p.144) also affirms, “spreadsheets environment supported pupils in moving from thinking with the specific to the general, both in terms of the unknown and of the mathematical relationships expressed in the problem”. Spreadsheets may thus be seen as tools that may enable learners to generalize, mathematize and communicate mathematical ideas.

The relationship that has been identified and expressed using spreadsheets notation may then be modeled mathematically, using rules to describe relationships in the model. Spreadsheets offer opportunities for learners to model the same situation in multiple ways, at the same time, all on the same display. The models can be symbolic (equation), tabular and graphical. Multiple representations afforded by the spreadsheets help learners to get deeper understanding of the situation at hand. Constructing spreadsheets representation of situations requires abstract reasoning by the user, which is consistent with one of the goals of using cognitive tools. (Jonassen & Reeves, 2001, p. 716).

Spreadsheets may function as cognitive tools supporting problem solving. In the spreadsheets environment, learners engage in more pupil-centered experimental learning where they explore mathematical ideas through supposing, testing, correcting their suppositions, testing again, applying results, proving their results, etc. Working in spreadsheets offers learners opportunity to construct their own conceptualization of the organization of the content domain. In this way the spreadsheets do not make the work easier, but requires from the learner higher thinking skills, processing the information at hand, generating thoughts that would otherwise not be possible without the technology. (Jonassen & Reeves, 2001, p. 697). The immediate feedback that learners get while working on a formula allows them to experiment, conjecture, realize, and correct their errors (Haspekian, 2005, p. 114). Problem solving within spreadsheets reveals all steps taken, showing the progression of calculations as they are carried out. This models all the logic implied by the calculations and making these

obvious to the learners, helps them understand the interrelationships and the procedures involved.

Spreadsheets may also be seen as tools that support reflective thinking. They provide for learners’ flexibility to quickly rearrange information and re-engage with the activities from fresh perspectives. Here learners can test and reflect on their output. In this way, spreadsheets allow and foster risk-taking and experimentation in an attempt to make sense of a situation at hand (Calder, 2010, p. 2). Heugl et al indicate that problem solving in the age of CAS involves modeling, operating and interpreting versus the earlier approaches to problem solving which viewed it as involving problem formulation and looking for solutions, with calculations as the main activity (Heugl et al, 1997, pp. 37-38).

The next section provides a discussion of how spreadsheets may be used to enhance some basic algebraic concepts and mathematical skills at Grade 9 level.

i) Concept of variable and algebra notation

The concept of variable as a placeholder for a general number may be more visible to learners through spreadsheets. A cell in spreadsheets can be viewed as a container into which numbers may be placed, and a cell address or cell reference describes the physical location of the cell. This cell address may also be used to refer to the particular number in a cell or any number that may be entered into that cell. According to Haspekian (2005, p. 121), the cell argument A1, highlighted in Figure 2.4, may be seen as:

 An abstract, general reference: it represents the variable (indeed, the formula does refer to it, making it play the role of variable);

 A particular concrete reference: it is here a number (in case nothing is edited there, some spreadsheets attribute the value 0);

 A geographic reference (it is a spatial address on the sheet);

 A material reference (it is a compartment of the grid; some pupils can see it as a box). (Haspekian, 2005, p. 121).

Figure 2.4. Cell argument A1 in Spreadsheetsss

The ambiguity with which a cell reference is used, “offers a strong visual image of the cell as a number container whose contents may be changed” (Bills, Ainley, & Wilson, 2006, p. 42). In this way, cell addresses are concrete representations of variables. The “fill down” command on a column using a formula again supports the idea of “variable as a range of numbers in functional relationships” (Bills et al, 2006, p. 42). On clicking on the individual values that have been generated by filling down, learners may realize that each one of them was produced by substituting entries in adjacent cells in place of the cell reference used in the formula, hence the concept of a “variable column” also emerges (Haspekian, 2003, p. 6). Using a letter to name the variable column and consequently asking them to write a formula for the column with generated values, learners may be able to form a link from the spreadsheets notation to the formal algebraic symbolism and thus appreciate the role of the letters in this case(Ainley, Bills & Wilson, 2005, p. 190).

ii) Simplifying expressions

The process of simplifying expressions through expansion of brackets is also one of the problem areas in algebra learning. Learners tend to give, for example, 2x + 3 as a simplified form of 2(x + 3). When sequences are generated from already formulated expressions in a spreadsheets environment, learners may be led into identifying those that yield similar results and thus appreciate how the distributive property is applied. The enablement that spreadsheets permit working with actual values can be used to develop the concept of equivalent expressions, involving factorization and expansion of brackets. (Tabach &

Friedlander, 2008, p. 29). The two expressions may also be treated as functions, say, f(x) = 2x + 3, f(x) = 2x + 6 and f(x) = 2(x+ 3), and comparing the graphs of the three drawn on the same axes would help learners recognize the differences and similarities.

iii) Solving equations (linear and quadratic)

Solutions of equations may also be done by using tables of values generated through the “dragging” or “fill down” command of spreadsheets. Solving equations in this case may be viewed as identifying the value of the variable for which the output for the expression is a given number. This case may also be seen as comparing two functions, where the task is to identify the value of the independent variable for which the two functions or expressions are equal.

For example, Figure 2.5 shows how spreadsheets may be used in solving for x in 2x + 3 = 3x.

Figure 2.5: Solving Linear Equations in Spreadsheets

Solution of equations may also be done through graphs. In this case, an equation is viewed as representation two expressions that are connected together by an equal sign. Creating separate graphs, on the same pair of axes, for functions formulated from each expression, and identifying the point of intersection, provide the solution for the original linear equation. This is illustrated in Figure 2.5.

iv) Multiple representations

Three main functions served by use of multiple representations in the teaching and learning of mathematics have been identified. Multiple representations may be used to complement each other, that is, one representation may complement information displayed in another form or they may support complementary processes. Using multiple representations for this purpose means that one form of representation would not be sufficient to carry or reveal all information about a situation. Information displayed in these cases may bear redundant features as well as those that are unique to each form, or each may encode different aspects of the domain. (Ainsworth, 1999, p. 137).

Another function of multiple representations is seen when one form of representation may also be used to constrain possible (mis)interpretations from use of the other. In this case, a familiar form of representation is used to help learners make correct interpretations of the less familiar form. The familiar form is used to support learners’ cognitive processes as they extend or construct and reconstruct meaning and understanding on the unfamiliar form. (Ainsworth, 1999, p. 139).

Multiple representations can also be used to help learners construct deeper understandings of a situation represented (Ainsworth, 1999, p.134). Kaput (1989) in Ainsworth (1999, p. 141) notes “the cognitive linking of representations creates a whole that is more than the sum of its parts…it enables us to “see” complex ideas in a new way and apply them more effectively”. This includes situations where multiple representations may be introduced simultaneously to learners so that they (learners) may be able to establish relationships between them. Integration of information from different representations of a situation permits learners to gain deeper insight into the situation that would rather be more difficult with one representation. (Ainsworth & Van Labeke, 2004, p. 250).

Figure 2.6 shows the different functions for which multiple representations may serve, as viewed by Ainsworth (1999, p.134).

Figure 2.6: A Functional Taxonomy of Multiple representations

The multiple representations (graphical, tabular and symbolic) that spreadsheets permit on the same screen, helps learners to make connections between all data displayed. Even though there are suggestions that translations between different forms of representations are difficult (Ainsworth, 1999, p.132), the design of spreadsheets allows for automatic translation, “dyna- linking”. When learners can see the immediate change on a graph, as a result of altering values in a table, this can help them establish the connection between the two forms of representation of the same relationship (Calder, 2010, p. 2). These multiple representations enable the learners to visualize those aspects of a problem that may not be apparent in one form and thus provide for better interpretations and understanding of the mathematical situation at hand.

Within spreadsheets environment, learners can choose to use any of the forms of representation depending on their preference or may use them in parallel as may be determined by the needs of the problem tackled. Depending on their levels of experience and expertise with varying representations, an appropriate combination of representations leaves

each learner with freedom to select and exploit that which they feel most comfortable with. (Ainsworth, 1999, p. 136).

Multiple representations enable learners to conjecture critique and justify their solutions. They enable learners to prove correctness or reasonableness of their responses to problem situations (Tabach et al, 2008, p. 49). According to constructivism, best learning results from thinking in meaningful, mindful ways. On working in spreadsheets, learners may represent what they know in multiple ways and this requires them to think mindfully about what the spreadsheets can afford them. Empowering learners to design and produce representations of what they know and share it with each other is a powerful learning experience. (Jonassen & Reeves, 2001, p. 695). In addition, using more than one form of representation is more likely to have a motivational effect on the learners.

v) Exploring properties of graphs

The fact that spreadsheets also allow for specifying a formula for performing a calculation is very important in constructing tables as well as graphs for a range of values for any given relationship. Spreadsheets promote experimentation with data. Learners may be challenged to make extensions to problems and investigate what happens when certain variations are made. This would thus make it possible for learners to explore behavior of graphs of linear equations, under varying values for the gradient and y-intercept. Learners would therefore be able to generalize about the properties of graphs of linear functions. According to Tabach and Friedlander (2008, p. 28), learners’ investigations of variation processes can be considered as generational and global /meta-level mathematical activities.

Having reviewed the potentialities that spreadsheets may provide for the teaching and learning of Grade 9 algebra, I now get into the discussion of what teaching and learning algebra through spreadsheets requires, paying particular attention to the kind of cognition involved.