If going to an encyclopaedia looking for the word algebra, a long list of topics within the area of algebra would appear. However, if one were to ask students in secondary school: What is algebra? my guess is that most of them would answer: calculation with letters instead of numbers. For the students in this study the aim when introduced to algebra in grade 8, was that they should have “opportunity to work to develop their understanding of the use of letters and brackets in simple mathematical expressions and formulae …” (Hagness & Veiteberg, 1999, p. 179). This makes it likely that they were introduced to algebra as extended arithmetic; calculating with letters as substitute for numbers.
Charbonneau (1996) claims that history of mathematics provides a warning against this view, algebra is much more, and he proposes further that algebraic symbols behave like many other mathematical objects not only numbers, and he points especially to geometry where the mathemati- cal objects, are closer to algebraic symbols than numbers. He exemplifies this by addition of two numbers, then their sum is represented by a quite new number, and the original numbers have vanished. In contrast, when two segments of a line are put together one after another, the new segment includes both the two original ones. This is similar to algebraic symbols. However, when using letters, “then the letter representing the new seg- ment does not have a meaning by itself and therefore cannot be interpret- ed without specific references to the original segments” (ibid p. 34).
Sfard (1995) presents different views about what algebra is. She pro- poses that writers mostly have agreed about algebra as the ‘science of generalized computations’, but she asserts that some authors suggest that algebra as such started with the introduction of the modern algebraic sym- bol system, and thus a definition of algebra must imply this algebraic symbol system. For others this symbol system is not required. Sfard her- self states that she uses “the term algebra with respect to any kind of mathematical endeavor concerned with generalized computational pro-
cesses, whatever the tools used to convey this generality” (ibid, p. 18). This definition, emphasises the operational origin of algebraic thinking. By computational processes she refers to the theory of reification (Sfard, 1991). Mathematics is hierarchical and reification occurs on any level, therefore it is possible to talk about computational processes also in ab- stract algebra, for example as group operations related to plane symmetry. In this way Sfard, as also Charbonneau above, relates algebra to a wider scope than only to numbers.
Although traditional school curricula emphasise the letter-symbolic aspect of algebra, Stacey and MacGregor (2001) say it has been important for educators and researchers also to let other characteristics of algebra come to the forefront. This view has been promoted by recent research done with children in preschool and in lower grades, which shows that small children are able to generalise and to recognise patterns, and even to work with variables (Carraher, Schliemann, & Brizuela, 2001;
Schliemann, Carraher, Goodrow, Caddle, & Porter, 2013). Algebraic thinking or algebraic reasoning are the notions used to include these as- pects of algebra rather than the letter-symbolic one.
To define algebra is “fraught with difficulty” according to Lins and Kaput (2004, p.48), however, they say that a group of researchers in mathematics education had come to an agreement on two characteristics of algebraic thinking. The first characteristic is based on generalisation, both the acts of generalisations and the expressions of generality. The second involves “reasoning based on the forms of syntactically-structured generalisations, including syntactically and semantically guided actions” (ibid p. 48). As I interpret these characteristics, the first can be referred to as generalisation in the way children explore patterns, and the argumenta- tion made to justify their generalisation. This argumentation may be ex- pressed by normal language or by use of more formal mathematical nota- tion. The latter characteristic is based on formal algebraic notation with emphasis on transformational processes. It can be exemplified by the al- gebraic expression 3(a + 2). A syntactically guided action would be to remove the brackets and get 3a + 6. In a semantically guided action the whole expression a + 2 has to be multiplied by 3, that means that a has to be multiplied by 3 and also 2 has to be multiplied by 3. The result is the same in both occasions; what guides the actions is different. The seman- tics here is semantics within the syntactics. Semantically the expression can also be interpreted based not on the syntactical form, but for example it may be interpreted as the area of a rectangle with one side of length 3 and the other of length a + 2.
Mason (1996) says that what he calls algebraic thinking, is an aware- ness of “detecting sameness and difference, making distinctions, repeating
and ordering, classifying and labelling” (ibid p. 83). He also emphasises the importance of the processes in mathematics. According to him what he includes in algebraic thinking, is the core of algebra. And further he claims that generalisation is the “heartbeat of mathematics” (ibid p.65).
That generalisation is at the core of mathematics is not denied by Wheeler (1996). He, however, pointing to Masons’ definition of algebraic thinking, proposes that to break algebra down to its basic components leads to a list of actions, “in which all the mathematical content appears to have been stripped away” (ibid p. 319). Wheeler asserts that the list pro- posed by Mason could be a list which could also fit into other subjects, and he mentions biology as an example. Further he suggests that the prob- lem to define algebra may be solved in two different ways. One is to fol- low Mason in his use of the notion of algebraic thinking; without a de- mand for mathematical symbols. The other way is to acknowledge these mental operations of very general character, but not to define them to be algebraic before they are linked to an algebraic symbol system.
Kieran (2004a) offers a definition of algebraic thinking in the early grades of school which is aimed to make a bridge from these early grades to the algebraic activity in later grades. She had already suggested a model of algebra built on the different activities students are engaged in in
school algebra (Kieran, 2004b). This model involves three categories of activities. Students are involved in the first category, the generational ac- tivity, when forming algebraic expressions and equations. Kieran com- ments that “much of the meaning-building for algebraic objects occurs within the generational activity of algebra” (2004a, p.142). The next cate- gory she calls the transformational activity. This involves the rule-based activity of transforming expressions and equations, always on the basis of maintaining equivalence. The last category is the global, meta-level math- ematical activities. Within these activities algebra is used as a tool. Kieran mentions “problem solving, modelling, noticing structure, studying
change, generalizing, analyzing relationships, proving and predicting” (ibid p. 142). These last activities she claims students can be engaged in without using any algebra. This claim seems to be in accordance with Wheeler’s (1996) critique of Mason’s (1996) definition of algebraic think- ing.
Kieran brought in a definition of algebraic thinking, which is integrat- ing algebraic thinking from lower to higher grades:
Algebraic thinking in the early grades involves the development of ways of think- ing within activities for which letter-symbolic algebra can be used as a tool but which are not exclusive to algebra and which could be engaged in without using any letter-symbolic algebra at all, such as, analyzing relationships between quanti-
ties, noticing structure, studying change, generalizing, problem solving, modeling, justifying, proving, and predicting (Kieran, 2004a, p.149).
This definition of algebraic thinking is closely connected to what she has called the ‘global meta-level mathematical activities’ (see the paragraph above) in which algebra might be used as a tool.
As the above reveals, algebraic thinking, or reasoning is discussed within the research community of mathematics education. Research done on early algebra’ shows that “young students benefit from opportunities to start from their own intuitive representations and gradually adopt conven- tional representations, including the use of letters to represent variables as tools for representing and for understanding mathematical relations” (Carraher, Martinez, & Schliemann, 2008, p. 4). In the latest Norwegian curriculum for lower grades one competence to be attained is to find and recognise patterns. Implicitly this is part of algebraic reasoning.
I think it is appropriate to talk about algebraic reasoning in the way Blanton and Kaput (2005) have formulated a definition: “We take alge- braic reasoning to be a process in which students generalize mathematical ideas from a set of particular instances, establish those generalizations through the discourse of argumentation, and express them in increasingly formal and age-appropriate ways” (ibid p. 413).
Traditionally school algebra has been, as Pimm (1995) expresses it, about “form and transformation” (ibid p. 88), what Kieran calls the trans- formational activity (Kieran, 2004a, 2004b, 2007a). This Pimm claims to be at the core of algebra. The result has been, that many students seem to manipulate algebraic expressions based only on rules without giving meaning to the symbols they act upon, and without any consideration about why the rules work.
Lee (1997b) interviewed mathematicians, researchers, teachers, and students about algebraic understanding. She exposed seven metaphors for what algebra is: “algebra is a tool, an activity, a way of thinking, a culture, a generalised arithmetic, a language, and a school subject” (ibid p. 92). Algebra as a language and algebra as an activity were, according to Lee, the richest metaphors she found. Syntactic and semantic were words con- nected to the view of algebra as a language. These words were also used by interviewees not holding algebra as language in the foreground. The syntax in algebra is related to the rules for manipulation of algebraic ex- pressions and fluency in performing these rules. It was discussed if the semantic had to be associated with the meanings of its referents outside algebra, however, it turned out that several interviewees meant that stu- dents had to concentrate on the “semantic of the syntax” (ibid p. 96). And therefore the traditional view of students’ mindless work with symbols
and rules perhaps has to be reconsidered. Meaning may be gained from the algebra itself.
What is then algebra and what is to be taught? As seen in the literature above, algebra is not easy to define. Anderson (1978) also posed this question. He writes that some have called algebra the “study of structure” others “the language of mathematics” in the meaning of syntax and
grammar, others again “the codification of mathematical laws”. None of these alone define algebra, however all of them are part of what algebra is.
For this study algebra is taken as synonymous with school algebra, meaning algebra taught in school. Caspi and Sfard (2012) define algebra in school to be formalised meta-arithmetic. Building on the view that mathematics is a discourse creating its own objects, algebra is seen as a sub-discourse. This view is building on Sfard’s (2008) theory of com- mognition (Thinking as communicating), which will not be the basis for my research, however, I find the definition to be useful. There are accord- ing to Caspi and Sfard (2012) two basic types of meta-arithmetical tasks giving rise to algebra. These tasks are questions of numerical patterns, normally described by equalities, and tasks about unknown quantities de- scribed as equations.
3.1.1 Approaches to algebra
The students in my study in grade 11, are not expected to have been ex- plicitly exposed to teaching that emphasises algebraic reasoning from the early grades. According to the syllabi they have been following, algebra was introduced in grade 8.
It seems as if it is easier to distinguish between different ways of ap- proaching algebra, and to describe the content that is typical for the differ- ent approaches, than to define it. Although the approaches are different, and the content which is emphasised in the different approaches is not quite the same, it is not a talk about disjoint sets of content, rather differ- ent aspects and emphases.
As early as 1978, Anderson came up with four different ways of ap- proaching algebra:
Algebra as generalised arithmetic or classical algebra, Algebra as a study of patterns and structure,
Algebra as an axiomatic study, and
Algebra through concrete situations (Anderson, 1978, p. 9).
In 1996 different perspectives on algebra were proposed to reflect upon, and the result was the book: “Approaches to algebra; perspectives for re- search and teaching” (Lee, Bednarz, & Kieran, 1996). The approaches for introduction of algebra were:
A generalisation perspective A problem-solving perspective
A modelling perspective A functional perspective.
Both lists complement each other and include the generalisation perspec- tive. Anderson (1978) has generalised arithmetic, however, algebra as a study of patterns and structure also involves a generalising perspective. He mentions that an awareness of structures can be reached from the source of children’s search for patterns. The problem-solving perspective is implicit in classical algebra, although Anderson when referring to clas- sical algebra sees the problem of too much emphasis on manipulative skills.
These examples only illustrate that whatever approach is chosen, it is one out of several possibilities. What is to be noticed, is that the way stu- dents have been introduced to algebra will make impact on their further work with the subject.
The functional approach is a fairly new way of approaching algebra. Historically the concept of function came rather late in the development of mathematics. Because of new technology with opportunities for students to easily produce functions and make tables, this approach has been intro- duced to give meaning to the concept of variable. Some researchers have argued against this approach from different points of view; Pimm (1995) while he meant there was a redefinition of algebra going on in the USA “ triggered I feel more by the potentialities of these new systems and the drawbacks of an over-fragmented mathematics curriculum than by any novel epistemological insight” (ibid p.104). Others like Lee (1997a) criti- cise the functional approach while functions can be expressed without al- gebra and also algebraic expressions do not need to describe functions.
Kieran (2007a) describes how the influence of new technology has made impact on the curricula in many countries. Approaches to school algebra and content of school algebra have changed from being a topic where form and transformations were the main characteristics to a topic with a new emphasis on the study of families of functions, for example in Israel (Sfard & Linchevski, 1994b).
In Norway it seems as if the functional approach to algebra has not been in focus for curriculum makers, although one of the competencies to promote, is digital competence as a competence on all levels and related to all subjects in the last curriculum (Ministry of Education and Research, 2006).
The students in this study met algebra (letter symbols) in grade 8. This might indicate that the students have met algebra as generalised arithmetic with emphasis on the transformational activity.
In the next section the focus will be on this shift from arithmetic to al- gebra.