Encyclopedia of Mathematics Education

Living Edition
| Editors: Steve Lerman

Algebra Teaching and Learning

• Carolyn Kieran
Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-77487-9_6-5

Keywords

School algebra Research on the teaching and learning of algebra Algebraic thinking Structure Generalization Properties Algebraic procedures Algebraic concepts Functional approach Generalized arithmetic approach Symbolic algebraic thinking Technological tools in algebra learning Generational activity Transformational activity Global/meta-level activity

Definition

The learning and teaching of the area of mathematics known as school algebra has traditionally involved the secondary school student (approximately 12–18 years of age) and has focused on forming and operating on polynomial and rational expressions, representing word problems with algebraic expressions and equations containing variables and unknowns, and solving algebraic equations by means of axiomatic and equivalence properties. However, over the past several decades, changes in perspective as to what constitutes school algebra have occurred, with the result that several different conceptualizations of school algebra have emerged. For example, Arcavi et al. (2017) define the aims of school algebra as including “expressing generalizations, establishing relationships, solving problems, exploring properties, proving theorems, and calculating” (pp. 2–3). In Stacey and Chick (2004), school algebra is seen as “a way of expressing generality; a study of symbol manipulation and equation solving; a study of functions; a way to solve certain classes of problems; and a way to model real situations” (p. 16). The lack of universality regarding definitions of school algebra is emphasized by Leung et al. (2014) who provide evidence that algebra lessons around the world can vary not only from country to country, but also within country, and that this diversity can be characterized not only in terms of content but additionally as to whether the main focus is either procedural or conceptual or some combination of the two.

Some years ago, Freudenthal (1977) characterized school algebra as consisting of not only the solving of linear and quadratic equations but also algebraic thinking, which includes the ability to describe relations and solving procedures in a general way. This latter facet highlighting algebraic thinking, quite novel at the time, not only opened up additional dimensions for conceptualizing school algebra at the secondary level but also provided an avenue for developing an algebraic thread in primary school mathematics, resulting in a movement that has come to be referred to as early algebra or the algebraization of arithmetic (e.g., Cai and Knuth 2011; Kaput et al. 2007; Kieran 2018; Kieran et al. 2016). At the core of this movement at the primary school level has been a focus on mathematical relations, patterns, and arithmetical structures, with detailed attention to the reasoning processes used by young students, aged from about 5 to 12 years, as they come to construct these relations, patterns, and structures – processes such as noticing, seeking structure, conjecturing, generalizing, representing, and justifying. A notable aspect of the cultivation of algebraic thinking with the younger student is the use of alternatives to alphanumeric symbols (e.g., words, artifacts, or other mathematical signs) for the expression of generality involving indeterminate objects (see Radford 2018). To sum up, contemporary definitions of school algebra, while remarkable for their diversity, embrace on the one hand, sign-based activity involving mathematical objects and the structural relations between them and, on the other hand, the mathematical thinking processes underpinning such activity. Because the focus of this encyclopedia entry is on the teaching and learning of school algebra involving the student aged about 12 up to 18 years of age, the reader is encouraged to consult the entry on Early Algebra Teaching and Learning for material related to students younger than this.

Evolving Perspectives on School Algebra and Its Research Over the Years

Up until the second half of the twentieth century, algebra was viewed as the science of equation solving – as per its invention by Al-Khwârizmî in the ninth century. This perspective on algebra, as a tool for manipulating symbols, was reflected in school curricula as they emerged and took shape through the 1800s and into the 1900s. Accordingly, the research conducted during the first half of the twentieth century on the learning of school algebra – scant though it was – tended to focus on the relative difficulty of solving various types of equations, on the role of practice, and on students’ errors in applying equation-solving algorithms. During the 1960s, the research took a psychological turn when cognitive behaviorists used the subject area as a vehicle for studying more general questions related to skill development and the structure of memory. In the late 1970s, when algebra education researchers began to increase in number and to coalesce as a community (Wagner and Kieran 1989), research embarked on the ways in which students construct meaning for algebra, on the nature of the algebraic concepts and procedures they use during their initial attempts at algebra, and on various novel approaches for teaching algebra (e.g., Bednarz et al. 1996). While the study of students’ learning of algebra favored a cognitive orientation for some time, sociocultural considerations added another pivotal dimension to the research on school algebra from the end of the 1990s (Lerman 2000). And most recently – from the 2010s – neurocognitive research has begun to offer additional insights related to algebra learning (Kieran 2017).

The years since the late 1980s witnessed a broadening of the content of school algebra, a content that had often been referred to as that of generalized arithmetic. While functions had been considered a separate domain of mathematical study during the decades prior, the two began to co-exist at this time in school algebra curricula and research. Functions, with their graphical, tabular, and symbolic representations, gradually came to be seen as legitimate algebraic objects (Schwartz and Yerushalmy 1992). Concomitant with this evolution was the arrival of computing technology, which began to be integrated in varying degrees into the content and emphases of school algebra. A further change in perspective on school algebra was its encompassing in an explicit way what has come to be called algebraic reasoning: that is, a consideration of the thinking processes that involve indeterminate objects and that can give rise to, and also accompany, activity with algebraic symbols. This widening of perspective on algebraic activity in schools reflected a double concern aimed at making algebra more accessible to all students and at engaging primary school students in the early study of algebra.

As the vision of school algebra broadened considerably over the decades – moving from a letter-symbolic and symbol-manipulation view to one that included multiple representations, realistic problem settings, and the use of technological tools – so too did the vision of how algebra is learned. The once-held notion that students learn algebra by memorizing rules for symbol manipulation and by practicing equation solving and expression simplification was replaced by perspectives that take into account a multitude of factors and sources by which students derive meaning for algebraic objects and processes.

Researchers began to study the specific question of meaning making in school algebra (e.g., Kaput 1989; Kirshner 2001). By the early 2000s, the various ways of thinking about meaning making in algebra were considered to include a triplet of sources (see Kieran 2007): (a) meaning from within mathematics, which includes meaning from the algebraic structures and objects themselves, involving the letter-symbolic form, and meaning from other mathematical representations, such as tables and graphs; (b) meaning from the problem context; and (c) meaning derived from that which is exterior to the mathematics/problem context (e.g., linguistic activity, gestures and body language, metaphors, lived experience, and image building). Further theoretical development of this area was carried out by, for example, Radford (2006) with his conceptualization of a semiotic-cultural framework of mathematical learning, which was applied to the learning of algebra. Through words, artifacts, and mathematical signs, which are referred to as semiotic means of objectification, and in line with Radford, the cultural objects of algebra are made apparent to the student in a process by which subjective meanings are refined.

Some research studies have used the nature of algebraic activity as a lens for investigating the various components of students’ learning experiences in algebra. Several models have been proposed for describing algebra and its activities (see, e.g., Bell 1996; Mason et al. 2005; Sfard 2008). For example, a model developed by Kieran (1996) characterizes school algebra according to three types of activity: generational, transformational, and global/meta-level.

The generational activity of algebra is typically where a great deal of meaning building occurs and where situations, patterns, and relationships are interpreted and represented algebraically. Examples include equations containing an unknown that represent problem situations, expressions of generality arising from geometric patterns or numerical sequences, expressions of the rules governing numerical relationships, as well as representations of functions by means of graphs, tables, or literal symbols. This activity also includes building meaning for notions such as equality, equivalence, variable, unknown, and terms such as “equation solution.”

The transformational activity of algebra, which involves all of the various types of symbol manipulation, is considered by some to be exclusively skill-based; however, this interpretation would not reflect current thinking in the field. In line with a broader view, mathematical technique is seen as having both pragmatic and epistemic value, with its epistemic value being most prominent during the period when a technique is being learned (see Artigue 2002). In other words, the transformational activity of algebra is not just skill-based work; it includes conceptual/theoretical elements, as for example, in coming to see that if the integer exponent n in xn – 1 has several divisors, then the expression can be factored several ways and thus can be seen structurally in more than one way. However, for such conceptual aspects to develop, technical learning cannot be neglected.

Lastly, there are the global/meta-level activities, for which algebra may be used as a tool but which are not exclusive to algebra. They encompass more general mathematical processes and activities that relate to the purpose and context for using algebra, and provide a motivation for engaging in the generational and transformational activities of algebra. They include problem solving, modeling, working with generalizable patterns, justifying and proving, making predictions and conjectures, studying change in functional situations, looking for relationships and structure, and so on – activities that could indeed be engaged in without using any letter-symbolic algebra at all.

What Does Research Tell Us About the Learning of School Algebra?

There is a considerable body of research on the learning of algebra that has accumulated during the past 40 years or so. What does this research have to say? (For details of the findings synthesized herein, the reader is urged to consult the following handbooks and related resources: Arcavi et al. 2017; Kieran 1992, 2006, 2007; Stacey et al. 2004; Warren et al. 2016.)

Research on the learning of school algebra with the secondary level student can be roughly divided into studies involving the 12- to 15-year-old, when students are typically introduced formally to algebra, and those involving the 15- to 18-year-old when students move more deeply into the content. For the former age group, where the majority of studies have occurred, research has tended to focus on generational activity, with considerably fewer studies dedicated to transformational activity. The studies focusing on global/meta-level activity have been concentrated in the areas of generalizing and problem solving. While not directly research related, it may be of interest to note that, in the 2015 TIMSS (Trends in International Mathematics and Science Study) international assessment of 8th graders (Mullis et al. 2016), 14 countries showed higher achievement in comparison with the 2011 results in the content domain of algebra – an indicator that could be interpreted to suggest the improving state of algebra learning worldwide.

Studies involving the 12- to 15-year-old age range of student tended during the 1980s and 1990s to focus on student difficulties in making the transition from arithmetic to algebra and on the nature of the algebraic concepts and procedures developed and used by students during their initial attempts at algebra. Later work in the following decades evolved in three different directions: (i) a shift from meaning being based primarily on the letter-symbolic form and on the problem situation/context toward the kinds of meanings derived from the use of graphical and tabular representations; (ii) a developing emphasis on students’ attention to structure; and (iii) the use of technology as a conceptual and technical support for algebraic activity.

For example, earlier research told us that students have difficulty with conceptualizing certain aspects of school algebra, difficulties that include: (a) accepting unclosed expressions such as x + 3 or 4x + y as valid responses, thinking that they should be able to do something with them, for instance, solving for x; (b) viewing the equal sign not necessarily as a signal to compute an answer but also as a relational symbol of equivalence; (c) interpreting algebraic expressions as mathematical objects in addition to viewing them as computational processes; (d) counteracting well-established natural-language-based habits in representing certain problem situations such as the classic “there are 7 times the number of students as professors” situation; (e) moving from the solving of word problems by a series of undoing operations toward the representing and solving of these problems by transforming both sides of the equation; and (f) failing to see the power of algebra as a tool for representing the general structure of a situation.

More recent algebra research with this age range of student has been able to build upon and extend the earlier work through the use of freshly developed theoretical frameworks, tasks, and technological tools. For example, this more recent body of work tells us that: (a) students engage in the act of generalizing figural sequences in a variety of ways and that interactions involving teacher and students play a role in supporting productive generalizing in classroom contexts; (b) students’ visual imagery of inequalities and of equations that involve linear functions on both sides of the equal sign is greatly assisted by graphical representations and by digital software featuring such representations; (c) students’ difficulties with recognizing structure in algebraic expressions and equations are a reflection of the difficulties they have with recognizing structure in number and arithmetical operations; and (d) students’ representations of functional problem situations evolve from first using numbers as the only means of modeling, then to intensively working with graphs and tables, and lastly to using more symbolic representations. These more recent research findings on the learning of school algebra illustrate how the field has moved from an exclusively cognitive orientation related to the symbol-oriented activity in algebra toward one that also encompasses a sociocultural, multirepresentational, technology-supported perspective that involves a much broader view of the scope of school algebra. As will be seen in the following paragraphs, aspects of this movement are reflected in the research involving the 15- to 18-year-old, in particular with respect to the use of digital technology as a conceptual tool, but also extending the role played by this technology to that of a tool of mathematical work.

Studies involving the 15- to 18-year-old age range of student, which have always been fewer in number than for the younger secondary-level student, have tended to focus on all three types of algebraic activity, albeit at a level appropriate for the more advanced student of algebra. Research on generational activity has centered on form and structure, as well as the study of functions and parameters. Research related to the transformational activity of algebra has investigated areas that include the following: notions of equivalence and meaning building for equivalence transformations, the solving of equations and inequalities, the factoring of expressions, and the integration of graphical and symbolic representations. Studies oriented toward the global/meta-level activity of algebra have focused, for instance, on modeling and proving, with the latter devoted especially to number-theoretic problems.

This body of research has revealed, for example, that (a) few students, even in their last year of secondary school, display a sense of structure for algebraic expressions and equations and those who do are inconsistent; however, specially designed learning environments supported by digital technology can improve the ability to notice structure; (b) students’ difficulties with interpreting quadratic equations show conceptual gaps in their understanding of two-solution equations; (c) when introduced to systems of equations, students are more inclined to make sense of comparison than substitution approaches, with manipulation difficulties adding to their problems with the substitution method; (d) the transition from viewing a function as a process to treating it as an abstract mathematical object remains a serious obstacle for a majority of students; (e) activity with technological tools such as Computer Algebra Systems (CAS), in combination with paper-and-pencil work, can promote both conceptual and technical growth, as long as the learning of technique is also attended to; (f) recognizing equivalence, even in simple cases, can be a significant stumbling block for students and requires teacher intervention and the use of appropriate tasks; and (g) students’ approaches to proving tend to rely more often on numerical instantiation than on symbolic manipulation.

Recent studies in cognitive neuroscience using neuroimaging technology and algebra-related tasks, and which have involved older algebra students, are offering new perspectives on current understandings of algebra learning. Though these studies are few in number, they provide evidence for the cognitive effort involved in doing and in being successful at algebra. For example, one set of studies compared the so-called model method of problem solving with the symbolic algebraic method (Lee et al. 2007, 2010). The researchers were interested in investigating whether the model and symbolic methods draw on similar cognitive processes and impose similar cognitive demands. The young-adult research subjects who participated in both studies were equally proficient in both methods at the outset of each experiment. In the 2007 study, the researchers used fMRI (functional Magnetic Resonance Imagery) to study the differences between the model and symbolic methods in the early stages of problem solving involving the transformation from text to either the model or the symbolic representation. They found that, while both methods were associated with activation of the working memory and quantitative processing regions of the brain, the symbolic method resulted in greater activity of those parts of the brain associated with attentional requirements. The 2010 study, which used the same neuroimaging technology, focused on the second stage of algebra word problem solving, that is, the computation of the actual solution to the problem from either the given model or the given symbolic representation. The greater activation of similar areas of the brain allowed the researchers to infer that additional attentional and executive resources are required for generating a numeric solution from an algebraic equation than from a diagrammatic model representation. Thus, the findings from both of the Lee et al. studies led to the conclusion that the symbolic method is more demanding than the diagrammatic model method. That the symbolic method is more effortful than the model method, even for competent algebra-problem-solvers, is a clear challenge to the traditional belief that algebraic methods of problem solving are easier than other methods and that algebraic solving activity involves simply the execution of automatized techniques for symbol manipulation.

A second example is drawn from a study conducted by Waisman et al. (2014). The researchers investigated the mathematical area of translation from graphical to symbolic representations of functions and their cerebral activation in groups of 16- to 18-year-old participants that differed in general giftedness and excellence in school mathematics. By means of the Event-Related brain Potentials (ERP) technique, the researchers found, not surprisingly, that the students with extraordinary mathematical abilities exhibited the highest accuracy along with the shortest reaction time. What is of, perhaps more, interest is the additional finding that the students who were not generally gifted but who excelled in mathematics achieved higher accuracy by means of greater mental effort. This finding offers further evidence to dispel the popular, yet naïve, view that students who do well in mathematics are “born with a certain talent for doing mathematics” and that they achieve this excellence without a great deal of cognitive effort.

These two examples from cognitive neuroscience research studies serve to raise our awareness levels of certain cognitive constraints associated with algebraic activity, even among older more experienced algebra students. The finding that algebraic excellence requires a great deal of mindful attention and cognitive effort should sensitize teachers and researchers to the mental demands involved in doing algebra. Furthermore, the remarkable similarity in cerebral activation between the conceptual work of representing a problem with algebraic symbols (Lee et al. 2007) and the procedural work of actually computing with those symbols (Lee et al. 2010) would suggest a rethinking of the age-old conceptual-procedural dichotomy in algebra (see also Kieran 2013).

What Research Says About the Teaching of Algebra

Interwoven throughout the above-described research related to the learning of algebra at the secondary school level has been an emphasis on the role played by the teacher in orchestrating that learning by means of appropriately designed tasks and by fostering classroom communication supportive of that learning. While the research studies that have focused specifically on the teaching, as opposed to the learning, of algebra remain fewer in number, two areas are worthy of note. One concerns the initiation of students to an algebraic frame of mind and the other concerns attempts at teaching students to notice structure in algebraic expressions and equations.

Many students beginning the study of algebra in secondary school come equipped with an arithmetical frame of mind that predisposes them to think in terms of calculating an answer when faced with a mathematical problem. A considerable amount of time is required in order to shift their thinking toward a perspective where relations, ways of representing relations, and operations involving these representations are the central focus. Teaching experiments within a number of research studies have been designed to explore various approaches to developing in students an algebraic frame of mind. Approaches that have generally been found to be successful include those that (a) emphasize generalizing and expressing that generality within activity involving patterns, functions, and variables; (b) focus on thinking about equality in a relational way; (c) move beyond the goal of searching for the correct answer to taking the time to examine expressions and equations with the aim of noticing underlying properties; (d) include making explicit conceptual connections when demonstrating procedures; and (e) use problem situations that are amenable to more than one equation representation and engage pupils in comparing the resulting equation representations to determine which one is better in that it is more generalizable. Despite these instructional moves that have proved useful in developing an algebraic frame of mind in students, research also emphasizes that the transition from nonsymbolic to symbolic algebraic thinking is a long-term process that requires a certain sensitivity and ability to notice and listen on the part of teachers.

Complementary to the process of generalizing in the development of algebraic thinking is the process of seeing structure. Linchevski and Livneh (1999), who coined the phrase “structure sense,” maintain that students’ difficulties with algebraic structure are in part due to their lack of understanding of structural notions in arithmetic. These researchers thereupon suggest that instruction be designed to foster the development of structure sense by providing experience with equivalent structures of expressions and with their decomposition and recomposition. Hoch and Dreyfus (2006) have also reported that very few of the secondary-level students they observed had a sense of algebraic structure, that is, very few could: “(i) recognize a familiar structure in its simplest form, (ii) deal with a compound term as a single entity and through an appropriate substitution recognize a familiar structure in a more complex form, and (iii) choose appropriate manipulations to make best use of structure” (p. 306). Warren et al. (2016), in their review of algebra research conducted by members of the PME (Psychology of Mathematics Education) group over the 10-year period from 2005 to 2015, emphasize that there is a continuing need to teach explicitly the abilities included in structure sense and offer the following research-based, instructional suggestions: “use brackets to help students ‘see’ algebraic structure, … work with examples where analysis or classification of problems in terms of their structural properties is the goal of the activities, … ask how definitions and properties can be used, and ask students for the goal of the activity instead of the solution” (p. 95).

The importance of teaching students to notice structural aspects in algebra and how teachers might go about doing this is also one of three main recommendations formulated by Star et al. (2015) in their resource titled, Teaching strategies for improving algebra knowledge in middle and high school students. Based on findings from 15 exemplary research studies published between 1993 and 2013, Star et al. developed the following three recommendations for teaching algebra to 12- to 18-year-olds (note that the term problem as used in these recommendations refers primarily to tasks involving expressions, equations with unknowns, and functional equations):
1. 1.

Use solved problems to engage students in analyzing algebraic reasoning and strategies. The substrategies associated with this recommendation are: (i) have students discuss solved problem-structures and solutions to make connections among strategies and reasoning; (ii) select solved problems that reflect the lesson’s instructional aim, including problems that illustrate common errors; and (iii) use whole-class discussions, small-group work, and independent practice activities to introduce, elaborate on, and practice working with solved problems.

2. 2.

Teach students to utilize the structure of algebraic representations. The substrategies associated with this recommendation are: (i) promote the use of language that reflects mathematical structure; (ii) encourage students to use reflective questioning to notice structure as they solve problems; and (iii) teach students that different algebraic representations can convey different information about an algebra problem.

3. 3.

Teach students to intentionally choose from alternative algebraic strategies when solving problems. The substrategies associated with this recommendation are: (i) teach students to recognize and generate strategies for solving problems; (ii) encourage students to articulate the reasoning behind their choice of strategy and the mathematical validity of their strategy when solving problems; and (iii) have students evaluate and compare different strategies for solving problems.

For Further Study and Reflection

Numerous advances have been made over the last several decades with respect to our knowledge of the learning and teaching of algebra. These include, but are not restricted to, developing in students an algebraic frame of mind; enlarging their views on equality, equivalence, unknowns, and variables; extending the meaning being given to algebraic objects and situations by the introduction of functions and their various representations; expanding students’ awareness of the role played by generalization in algebra; and improving their technical abilities and conceptual knowledge, often with the help of digital technology. Despite these advances, more remains to be done. In particular, seeking, using, and expressing structure within algebraic expressions and equations is an area that requires the continued attention of researchers and teachers. There is indeed a dual face to algebra: one face looking toward generalizing, and, alternatively but complementarily, the other face looking in the opposite direction towards “seeing through mathematical objects” and drawing out structural aspects. This structural face is one that students need to come to see; achieving this entails focused experience and instructional guidance.

One rather promising aspect of the current research scene is the increase in researcher interest in the development of algebraic thinking at the primary school level – an interest that has the potential to enhance our understanding of the learning and teaching of algebra at the secondary level as well. Assuming that the present level of interest in the development of symbolic algebraic thinking and in the complementary processes of seeking structure and generalizing continues at the primary level to the degree at which it is currently engaged, there is every likelihood that the topic of the learning and teaching of algebra will one day encompass the entire school cursus from the beginning of primary through to the end of secondary school – a coordinated integration that would surely be beneficial for the algebraic learning of all students.

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