Textbook content has the ability to influence mathematical learning. This study compares how linear functions are presented in two textbooks, one of South African and the other of German origin. These two textbooks are used in different language-based streams in a school in Gauteng, South Africa. A qualitative content analysis on how the topic of linear functions is presented in these two textbooks was done. The interplay between procedural and conceptual knowledge, the integration of the multiple representations of functions, and the links created to other mathematical content areas and the real world were considered. It was found that the German textbook included a higher percentage of content that promoted the development of conceptual knowledge. This was especially due to the level of cognitive demand of tasks included in the analysed textbook chapters. Also, while the South African textbook presented a wider range of opportunities to interact with the different representations of functions, the German textbook, on the other hand, included more links to the real world. Both textbooks linked ‘functions’ to other mathematical content areas, although the German textbook included a wider range of linked topics. It was concluded that learners from the two streams are thus exposed to different affordances to learn mathematics by their textbooks.

Mathematics textbooks have been found to have a strong influence on classroom practice (Fan,

The topic of functions was selected as the focus for the textbook analysis because it is generally agreed that functions form one of the most important unifying ideas in mathematics education (Knuth,

The overall purpose of this study was therefore to analyse and compare the manner in which linear functions are presented in the South African and German mathematics textbooks used at the observed school by learners en route to Grade 10. Due to variations between the South African and the German mathematics curricula adhered to in the two language-based streams, similar mathematical topics are not necessarily taught at the same grade level across the two streams. The topic of linear functions is introduced and taught as an area of focus in Grade 7 in the German stream while this occurs in Grade 9 in the English stream at the school. In order to compare textbook content on linear functions pitched at a similar level for purposes of this comparison, the English stream Grade 9 textbook,

How are linear functions presented in

What affordances for learning about linear functions are made available in

It must be noted that this investigation did not aim to endorse either textbook’s approach as superior. Furthermore, this study was not aimed at making any value judgements with regard to the academic strategy of using different language-based streams in the research school. Rather, the study intended to provide a thorough comparison of both textbooks’ content originating in two different education systems, with the aim of understanding the affordances for learning made available to the two groups before progressing to Grade 10.

It has been argued that the concept of functions is one of the most important in all mathematics (Dubinsky & Harel,

When selecting criteria by which to compare the textbook content on linear functions and the affordances for learning made available to learners in the two textbooks, this study therefore investigated whether this fundamental topic is presented in a manner promoting procedural or conceptual knowledge, as well as how these types of knowledge are sequenced. Against this background, the integration of the multiple representations of functions, and the presence of links established between linear functions and the real world, as well as to other mathematical content areas, were also considered.

The debate as to whether procedural or conceptual knowledge should be predominantly promoted in mathematics instruction has been well documented over the past century (Hiebert & Lefevre,

A distinctive aspect of the study of functions is its integration of different types of representations. In mathematics, representations include verbal, numerical, graphical and symbolic descriptions of a concept, which allow for it to be interpreted, communicated and discussed (Tripathi,

A further criterion that contributes to the mathematical learning process is the use of contexts that link mathematics to the real world. The realistic mathematics education movement is a prime example of how using tasks that relate to the real world can be utilised as ‘a source of learning mathematics’ (Van den Heuvel-Panhuizen,

As indicated previously, textbooks have been shown to have a large influence on classroom practice (Stylianides,

Mathematical tasks, in many ways the building blocks of mathematics textbooks, have also been a focus of much international research in past years. Watson and Ohtani (

This study was framed theoretically and analytically by an adapted version of Stein et al.’s (

As shown in

A summary of task analysis guide.

This task analysis guide was for us the springboard from which to evaluate the opportunities for learning in this study’s analysed textbook content. In order to engage with this study’s research questions specifically, the guide was adapted using the literature on procedural and conceptual mathematical knowledge, while also considering literature on the multiple representations of functions. Firstly, the task analysis guide was used to investigate the interplay of procedural and conceptual knowledge in the two textbooks in the following manner: textbook elements that resonate with the description of ‘lower-level demands’ were described as promoting procedural knowledge, while those that resonate with the description of ‘higher-level demands’ were described as promoting conceptual knowledge. This use of the task analysis guide was based on the language used to describe procedural and conceptual knowledge by key theorists on the topic, particularly Hiebert and Lefevre (

In addition, the Trends in International Mathematics and Science Study (TIMSS) textbook investigation (Valverde et al.,

The approach used in the larger study (Mellor,

The different purposes of textbook content used to partition the selected chapters.

Purpose | Description |
---|---|

Instructional narrative | Written language (that may incorporate mathematical symbols) that presents or explains mathematical content |

Graphic | A picture, diagram or graph that is incorporated to complement the instructional narrative |

Exercise | Unsolved mathematical tasks that the learner should attempt |

Worked example | Solved mathematical tasks that aim to provide instruction or explanation |

Other | Any content that does not fit the above criteria |

After the chapters had been partitioned into ‘blocks’, each individual block was then coded so that the content could be accurately described, allowing for analysis to take place thereafter. Making use of the adapted task analysis guide, as described above, each block was first coded as promoting conceptual or procedural knowledge of linear functions. During this coding process, we realised that Stein et al.’s (

Examples of the types of content ‘blocks’ found in

Sample ‘block’ delineated by purpose | Category | Procedural/Conceptual knowledge | Representations | Link to the real world | Link to other mathematical content area |
---|---|---|---|---|---|

Procedures without connections | Procedural | Ordered pairs | No | No | |

Set up a table for input values Sketch the graph Describe a real-world situation that could be described with this equation. |
Procedures with connections | Conceptual | Equation |
Yes | No |

Doing mathematics | Conceptual | Equation | No | Yes: geometry | |

Reading for understanding | Conceptual | Graph | Yes | No |

Valverde et al. (

The two textbooks differ markedly in terms of their macrostructures.

Of CM’s 78 content blocks, just over half were coded as promoting procedural knowledge (54%) while only about a third of EDM’s 166 blocks were coded as procedural (37%). As is visible in

Comparison of blocks that promote procedural and conceptual knowledge in

With the aim of gaining a more holistic understanding of how the respective textbooks utilise conceptual and procedural knowledge, the linear sequencing of the content was also analysed. As indicated earlier, the chapters on linear functions are arranged differently in the two textbooks: CM has three chapters inserted at different locations of the textbook while EDM has only one chapter with six subsections. Examining the linear sequencing of the coded content blocks in the separate sections highlighted the interplay between the individual blocks (Mellor,

Analysis of coded content blocks, sequenced as in the respective chapters, in terms of procedural and conceptual knowledge.

Analysis of the three chapters of CM did not indicate a consistent method of arranging content that promotes procedural and conceptual knowledge. The chapters ‘Functions and Relationships 1’ and ‘Functions and Relationships 2’ can be viewed as a unit as they cover the same content. Mellor (

It therefore appears that in these two chapters the content is arranged predominantly according to the ‘concepts-first’ view. This view posits that learners first acquire knowledge of concepts and use this as a foundation to develop procedural knowledge (Rittle-Johnson & Schneider,

In EDM, there appears to be a more consistent pattern of content blocks supporting concept or procedure. Like the ‘Graphs’ chapter in CM, the subsections of the Linear Functions chapter also indicate a predominantly ‘iterative’ view, with procedural and conceptual knowledge being more consistently integrated. There is, however, also a trend of the subsections beginning with conceptual content. As a whole, the content of EDM appears to combine conceptual and procedural content more consistently than CM across the analysed chapters (Mellor,

Further points of interest regarding the pattern of EDM’s content in

Analysis of individual content blocks provided further explanation of the different approaches to conceptual and procedural knowledge found in the two textbooks.

Comparison of each category used in the coding process for procedural and conceptual knowledge.

Comparing specifically selected tasks from the two textbooks indicated that although the two textbooks include similar tasks with similar desired outcomes, the level of cognitive demand of the tasks is generally higher in EDM than in CM. Consider the following comparison between

Translating between function representations in

Task from

The tasks in

The use of graded questions also differs in the two textbooks’ linear functions content. Both books include a highly similar task on isosceles triangles that requires learners to describe the functional relationship between the base angles and the third angle. These two tasks are provided in

Task from

Task from

The comparison between the exercises in

The comparisons between

Both the task analysis guide (Stein et al.,

Percentage of content blocks that utilise one or more function representations (flow diagram, table, equation, graph, written words).

The two textbooks have different foci in terms of multiple representations. About half of CM’s content blocks make use of only one representation of functions. Of this, the notable contributors are ‘graph only’ questions (28% of total blocks) while ‘equation only’ questions make up about 15% of all content blocks. The next most frequent category is shared between blocks that integrate two and three representations, each making up about a fifth of total blocks. Finally, just over 10% of blocks incorporate either four or five representations. This indicates that although there is a clear focus on content that involves either graphs or equations, the book does place importance on integrating other representations into its content. In comparison, only about a fifth of EDM’s content blocks utilise just one representation. Over half of its content blocks integrate two representations of functions. This modal category is composed overwhelmingly of content blocks that combine graphs and equations (45%). Blocks that utilise three representations make up its second highest category (just over 20% of the data) with only 2 out of the 166 coded blocks using four representations. Thus, although CM focuses on individual skills related to linear functions more than EDM does, the book also emphasises translating between three or more representations to a higher degree than the German book: a third of CM’s coded content integrates three or more representations while only a quarter of EDM’s content encourages this.

As indicated previously, the ability to translate between multiple representations in mathematics is seen to contribute toward conceptual understanding (Kilpatrick et al.,

Without procedural fluency, students have trouble deepening their understanding of mathematical ideas or solving mathematical problems. The attention they devote to working out results they should … compute easily prevents them from seeing important relationships. (p. 122)

Translating between the multiple representations of functions is an acknowledged weakness in high school mathematics (Leinhardt et al.,

Although the content of EDM does not promote the translation between representations to the point that CM does, the book focuses in depth on the relationship between linear functions and solving linear equations. Consider, for example, the task in

(a) Task from

Such a task has the potential to develop strong connections and deep understanding of the relationship between linear equations and linear graphs. Although EDM may not develop the procedural fluency that CM does in translating between representations of functions, tasks such as this present the opportunity for procedural steps to be understood conceptually in a manner that was not evident in CM. Thus, overall, EDM can be seen as promoting deep, interconnected knowledge of linear functions in equation, graphical and tabular form, while CM promotes procedural fluency in a broader range of representations.

Hiebert and Lefevre (

However, although these types of links have the potential to develop learners’ conceptual understanding of mathematical content, they do not necessarily contribute to the learners’ understanding of the value of linear functions, nor how functional relationships manifest themselves in the real world. While perhaps not originally meant in this sense, Hiebert and Lefevre’s (

In terms of including content that demonstrates the use of linear functions in the real world, EDM and CM use different approaches. EDM includes more blocks linked to the real world than CM – 32% of content blocks in EDM compared to 24% in CM. Furthermore, their placement of these blocks is not the same, particularly in terms of how the chapters are introduced.

This study has argued that CM and EDM create different affordances for learning about linear functions. Both textbooks provide opportunity to learn about linear functions procedurally and conceptually, but with different emphases.

This study also found that CM includes a broader range of function representations than EDM. However, CM also includes a higher percentage of tasks that only deal with one type of representation. In comparison, over half of EDM’s coded content links two types of representations, predominantly linear equations to graphical representations of functions. In general, the findings indicated that although the German book covers fewer representations, those included are studied more in depth and conceptually than is apparent in CM.

Lastly, EDM creates more links between linear functions and other mathematical content areas as well as to the real world than CM does. According to Hiebert and Lefevre (

Overall, this study has exhibited that two different textbooks can present the topic of linear functions in different ways, and thus create different affordances for this content to be learnt. At the school where these textbooks were used in two language-based streams, learners entering a combined class in Grade 10 were thus exposed to different methods of making sense of mathematics. What do these findings mean for a teacher who is tasked with teaching Grade 10 learners in such a school? We argue that the key to successfully teaching learners in a combined class is the awareness and acknowledgement of the different affordances learners have been exposed to in terms of their mathematics content prior to Grade 10. This awareness may mean tailoring lesson plans by using the types of examples from the German book that would benefit the English stream learners, and vice versa.

K.M. thanks the University of the Witwatersrand for the Postgraduate Merit Award that provided financial support during the course of this research. The article is based on the B.Sc. Honours research report of K.M. which was jointly supervised by R.C. and A.A.E.

The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.

K.M. wrote the majority of the article. All three authors contributed to the conceptualisation of the methodological approach used in the study. Data analysis was carried out by K.M., R.C. and A.A.E. based on the jointly developed analytical framework.