Examples that teachers choose and use are fundamental to what mathematics is taught and learned, and what opportunities for learning are created in mathematics classrooms. This qualitative multiple case study, using Sfard’s commognitive theory, draws attention to mathematics teachers’ classroom practices during functions lessons which is unexamined in the South African context. In this article, data sets include unstructured non-participant classroom observations on functions, which were videorecorded. Sfard’s commognitive theory served as an appropriate lens in interpreting and analysing teachers’ discourses and giving meaning to teachers’ classroom practices during functions lessons. The findings demonstrate that the example selection and sequences teachers used during functions lessons either constrained or enabled the development of endorsed narratives about the effect of parameters on the different families of functions.

The concept of functions has received attention within the field of mathematics education (Kabael,

Within the functions concept in school mathematics, both its importance and problems relating to its learning have been researched and documented in mathematics education research (Moalosi,

‘for the motivation of mathematical concepts by using concrete examples in the teaching of mathematics stems from the commonly accepted notion that, nowadays, students are interested in the study of the subject matter if they are confident in the applicability of the material they are about to learn.’ (Abramovich & Leonov,

The interest and confidence are influenced by the quality of a teacher because learners observe and do what they observe the interlocutor does during teaching and learning in the classroom (Sfard,

How do teachers select and use examples while teaching functions?

How does the selection and use of examples facilitate or limit the development of learners’ knowledge of functions?

My contention is that for learners to understand the different properties of the concept of functions, teachers should ensure that learners are taught to undertake the following actions: interpretation and construction of functions to help them to comprehend them. Interpretation refers to ‘action by which student makes sense or gains meaning from a graph (or a portion of a graph), a functional equation, or a situation’ (Leinhardt et al.,

Earlier, Anderson (

Firstly, the world of changes entails an identification of ‘what’ is changing in given relationships and ‘how’ the change is taking place. In this sense, teachers should teach learners how to work with the idea of ‘transformation’ in functions, and pay attention to the appearance, displacement and orientation of functions (Chimhande,

Secondly, teachers should teach the learners how to observe change between the given variables and identify the relationships between them.

Accordingly, the nature of teachers’ exemplification during functions lessons plays a crucial role in promoting or hindering learners’ understanding of the topic. In this article, a critical examination of the examples that five participating teachers selected and used while introducing functions, as well as their sequencing of such examples, enables me to unearth the effectiveness of their teaching of the topic.

In addition to the above discussion, for Sierpinska (

‘One quantity changes in a predictable or recognisable pattern, the other also changes, typically in a differing pattern. Thus, if one can describe how _{1} changes to _{2} and how _{1} changes to _{2} then one has described a functional relationship between

This is not sufficient to constitute a function. According to Bazzoni (

Thirdly, a function is considered a rule that governs the relationship between variables (Sierpinska,

Sfard’s (

According to Sfard (

The word ‘discourse’ implies the use of words and symbols in a way that is generally endorsed by members of a community (Sfard,

According to Sfard (

In this article, I use the commognitive theory to describe discourses of functions presented by five teachers in the study, to reveal the opportunities as well as constraints for learning offered by the examples that the five teachers selected and used during teaching. The data from which this article emanates revealed that the dominant narratives were the presentations of functions as formulas, while there were limited opportunities for learners to make conjectures and engage in proof activities. The prevalent routines included the sketching of graphs from functions presented in symbolic form. The following section details the research methodology for the study, to highlight the nature of data generation I used.

The empirical data in the current article consist mainly of videotaped lessons presented by five mathematics teachers at five different school sites in Mpumalanga province of South Africa, representing multiple cases. As reported in this article, a qualitative research approach was espoused (Creswell,

The current study used a multiple case study design. This design enabled me to understand the nature of mathematics teaching, specifically the teaching of functions within a bounded context and bounded activity (Creswell,

Teachers’ biographical information.

Pseudonym | Gender | Mathematics education qualifications | Number of years teaching | Institution trained at to become a teacher |
---|---|---|---|---|

Zelda | Female | Bachelor of Education | 5 years | North-West University, South Africa |

Mafada | Male | Honours in Mathematics Education | 20 years | Giyani College of Education, South Africa |

Tinyiko | Female | Bachelor of Education | 5 years | University of Venda, South Africa |

Mutsakisi | Female | Bachelor of Education | 30 years | University of Zimbabwe |

Jaden | Male | Bachelor of Education | 17 years | College of Education, India |

The empirical data in the current study were generated by means of unstructured non-participatory classroom videotaped observations. Johnson and Christensen (

According to Nieuwenhuis (

In addition to the above discussion, analysing teachers’ classroom discourses while teaching the topic under study required an approach that allowed me to look within and across the different teachers’ lessons, to create a picture of the quality of each teacher’s teaching. Thus, I overlaid into the tenets of the commognitive theory for both structure and generality about the teachers’ discourses during the observed lessons. I initially chunked lessons into episodes based on what activities were set and their related examples for each lesson. Within the episodes, I then noted the nature of teachers’ mathematical discourse as framed by the four components of commognitive mathematical discourse.

Discourses in Mafada’s teaching episodes.

Sfard’s commognitive theory |
||||
---|---|---|---|---|

Episodes and observable actions | Visual mediator | Words used | Endorsed narratives | Routines |

1. Introducing the four families of functions and showing learners what a coordinate is in the form of coordinate pairs ( |
Symbolic mediators: written functions are:^{2}; ^{2} |
Functions; parabola; hyperbola; straight-line graph; variable; coordinates; dependent variable; independent variable; point; |
Object-level narrative: “ |
Clarifying. |

2. Introduction of the parabolic function in the form ^{2}. Demonstrating the change in representations, from algebraic representation of the parabolic function ^{2} to the table of values. Substituting specific values of ^{2} is a function. |
Symbolic: using the function ^{2} to compute the table of values |
Function; |
Object-level narrative: “ |
Rituals to determine the output values for given input values, completing the table of values and plotting and drawing the parabola. |

3. Summarising the steps needed to draw graphs of functions. Illustrating to the learners that the shape of the parabolic function does not always give the shape given by the function ^{2.} |
Iconic: graph of a function ^{2} and sketches of ‘other’ parabolic functions |
Plot; hyperbola; subject of the formula; sign of a function; face up | Meta-level narrative: “ |
Memorisation ritual on how to draw the graphs of parabolic functions. |

4. Demonstrating to learners how to determine the intercepts for ^{2} – 3 and ^{2} – 1 and in turn sketching their graphs. |
Symbolic mediators: written functions are:^{2} – 1; ^{2} ^{2} – 3^{2} – 3 and ^{2} – 1 |
Value of |
Object-level narrative about the effect of parameter |
Rituals to determine the |

Mafada viewed the equations as merely producing a result of calculating, resulting in seeing the different functions as recipes to apply to numbers, then remaining unchanged across numbers.

Discourses in Mutsakisi’s teaching episodes.

Sfard’s commognitive theory |
||||
---|---|---|---|---|

Episodes and observable actions | Visual mediator | Words used | Endorsed narratives | Routines |

1. Using the general equation for linear functions in the form |
Symbolic mediators: written functions are: |
Variable; dependent variable; independent variable; |
Object-level narratives, identifying |
Clarifying. |

2. Using the notion of intercepts to draw the graph of the function in the form |
Symbolic: using the function |
Intercepts; positive; |
Describing the notion of intercepts: “ |
Ritual to complete the table of values from an algebraic equation. Ritual to find the |

3. Introduction of a new family of functions (parabolic) to juxtapose the structural differences between linear and parabolic functions in terms of their symbolic appearances. | Symbolic mediators: written functions are: ^{2}; ^{2} + 2; ^{2} + 1. |
Quadratic functions; straight line; linear functions; output values; domain; input values; range; |
Distinguishing linear and parabolic functions in terms of their symbolic representations: “ |
Ritual to complete the table of values from an algebraic equation. |

4. Using the parabolic function in the form ^{2} + 1 to show learners how to complete the table of values and in turn draw the graph. From the graph, she identified the turning point and the |
Symbolic mediators: written functions are: ^{2} + ^{2} + 1, ^{2}; ^{2} and ^{2} |
Turning points; greater than; positive; smile; faces up | Narrative about turning point |
Ritual to complete the table of values from an algebraic equation. |

The observable action that is prevalent across Mutsakisi’s lessons was drawing the graphs of functions, which was characterised by Mutsakisi demonstrating the drawing of graphs of functions. That is, functions given in symbolic form were represented in tables of values, whereby the teacher demonstrated to the learners the algebraic calculations to find

Discourses in Tinyiko’s teaching episodes.

Sfard’s commognitive theory |
||||
---|---|---|---|---|

Episodes and observable actions | Visual mediator | Words used | Endorsed narratives | Routines |

1. Recapping on the features of linear functions to set the scene for parabolic functions. Substitution and calculations and completion of table of values to draw the parabolas of given functions. | Symbolic mediators:^{2}; ^{2} + 1;^{2} – 1^{2} depicted in the table of values. |
Parabola; linear; functions; straight; linear graphs; dual-intercept method; intercepts; gradient; turning point; |
The effect of parameter ^{2} + 1 |
Clarifying. |

2. Showing learners how to use the dual-intercept method and table method to determine the output values for chosen inputs. |
Symbolic syntactic mediators:^{2} – 1; ^{2}; ^{2} + 1; ^{2} ^{2} to perform mathematical calculations^{2} and ^{2} |
Substitute; formula; stealing; |
The object-level narrative about calculations: “ |
Rituals to use the dual-intercept method and table method to determine the output values and drawing graphs of functions. |

3. The teacher intended to introduce hyperbolic functions, but the example she introduced was for exponential functions. Although she realised this after engaging in mathematical calculations, she continued performing rituals to demonstrate to the learners how to substitute and calculate for output values. | Symbolic syntactic mediators: ^{2}; ^{2} + 1 |
General formula; hyperbola graph; |
Describing the notion of asymptote: “ |
Rituals to use the dual-intercept method and table method to determine the output values and drawing the graph of the function. |

4. Substitution and calculation of intercepts for |
Symbolic syntactic mediators: |
General equation; linear graphs; parabola; cup; cave; concave; gradient; table method; |
The object-level narratives about intercepts: “ |
Rituals to use the dual-intercept method to determine the output values. |

5. Using two examples |
Symbolic syntactic mediators: |
Arrow; continuing; inputs; domain; range; points; facing up; output; positive; negative | The effect of parameter |
Memorisation about the effect of parameter |

Tinyiko’s pedagogical actions in all the episodes reveal the prominent use of the rituals to draw or sketch the graphs of given functions in symbolic form. This makes the observable action of drawing or sketching the graphs the end goal of Tinyiko’s teaching of functions. Focusing on the how of the routines resulted in Tinyiko’s discourse of rituals rather than explorations, as she emphasised the following of rules without explication and understanding their applicability.

Discourses in Zelda’s teaching episodes.

Sfard’s commognitive theory |
||||
---|---|---|---|---|

Episodes and observable actions | Visual mediator | Words used | Endorsed narratives | Routines |

1. Introducing the functions: ^{2}; ^{2} + 1 and ^{2} – 1 in the table of values and engaging in the process of substituting and calculating the output values and completing the table of values. |
Iconic visual mediators: functions are: ^{2}; ^{2} + 1 and ^{2} – 1 depicted in the same table of values |
Same axes; compare; values of |
(none) | Ritual to substitute and calculate the values of |

2. Exploration of the effect of varying the values of parameter |
Iconic: graphs of the three functions ^{2}; ^{2} + 1 and ^{2} – 1 on the same set of axes ^{2}; ^{2} ^{2}; ^{2} to juxtapose the direction of the graphs because of the sign of the value of ^{2}; ^{2} + 1 and ^{2} – 1 in the form (0, 0); (0, 1) and (0, –1) respectively |
Plot; highest value; |
The effects of parameters |
Exploration of the effect of parameter |

3. Introduction of a new family of functions (hyperbolic functions) using two examples: |
Iconic visual mediators: functions are: |
Undefined; Cartesian plane; increasing; decreasing | Describing the notion of asymptote: “ |
Exploration of the effect of parameter |

What is starkly evident in Zelda’s teaching is that she engaged her learners in interpretations of the given functions using an interactive communicative approach, highlighting the critical global features for the families of functions she focused on. In terms of discursive routines, Zelda’s teaching appeals to the use of explorative routines to help learners to observe some critical features for each family of functions focusing mainly on applicability routines.

Discourses in Jaden’s teaching episodes.

Episodes and observable actions | Visual mediator | Words used | Endorsed narratives | Routines |
---|---|---|---|---|

1. Using the function machine approach to demonstrate to the learners the ritual to substitute and calculate output values. Determining the ‘rules’ for given relations using the patterns-oriented approach. | Iconic visual mediators: Using the function machine to calculate the output values |
Output values; input values; function; difference; values of |
Meta-level narrative about the rule underpinning the relation: “ |
Ritual to substitute and calculate the values of |

2. Teaching learners how to substitute and calculate the output values using the function notation in the form |
Symbolic: Using the examples of functions: ^{2} – 1; |
Meta-level narrative about how the function notation is used: “ |
Clarifying ritual to demonstrate to the learners how to use the function machine to determine the output values. | |

3. Using the examples of linear functions |
Iconic visual mediators: functions |
Graph; linear function; exponent; line; substitute; increasing; steeper; gradient; |
Object-level narrative about the relationship between the symbolic mediators and the graphical mediator: “ |
Exploration of the effect of parameter |

Jaden’s teaching were rituals to translate the functions presented in symbolic form into the table of values and drawing of graphs. Across all Jaden’s episodes, the teaching was dominated by his explanatory talk without providing learners with the learning opportunities to create mathematical meanings for themselves during the lessons.

This section addresses participating teachers’ selection of examples during algebraic function lessons, and whether and how they facilitated or constrained the learning of functions’ critical features during teaching. According to Renkl (

Development of parameters discourse within the function concept depends on the content teachers make available for learners to learn, the teaching approach they use to convey the notion of parameters to the learners as well as how they select and vary examples to develop learners’ thinking about the effect of different parameters on the behaviour of the functions. Two of the five teachers in this study struggled to offer explanatory talk that would enable learners to make conjectures, prove them and make generalisations about the effect of the different parameters for the different families of functions they worked with in the classrooms. Analysing how the teachers selected and sequenced a set of examples in each lesson enabled a view of whether and how the examples accumulate to bring the object of learning in different lessons into focus for learners, and whether there is movement to achieve generality which is one of the curriculum objectives for Grade 10 level in South Africa (Adler & Ronda,

For the two sub-themes, the above statement means that teachers’ systems of examples and their sequencing reveal whether there was movement towards generality relating to the parameters of functions. This relates to curriculum statement 3 for functions which expects learners to ‘investigate the effect of ^{x}

It is discernible that in the examples that Mafada and Tinyiko used in their lessons, they did not use patterns in which they vary one parameter while keeping the other one invariant.

Sequences of examples in Mafada’s and Tinyiko’s lessons.

Teacher | Examples and their sequence |
---|---|

Mafada | ^{2}; ^{2} – 5; ^{2} + 3; ^{2}; ^{2}; ^{2} – 1; ^{2} – 3 |

Tinyiko | ^{2}; ^{2}; ^{2}; ^{x}^{x} |

Mafada starts with ^{2} and moves on to change both

While Mafada’s and Tinyiko’s sequencing of the examples moved from simplicity to complexity, as presented in ^{2} where simple example is taken to complex ones’ (p. 61). For Mafada and Tinyiko, the lack of an invariance-variance relationship to bring the world of changes to the fore in the example sets did not allow for systematic comparison of the different families of functions in terms of the effect of changing the values of

I argue that Mafada’s and Tinyiko’s examples across their lessons have constrained the discernment of the meaning and structure of the parameters of functions, because there was no systematicity in terms of what varies and what remains the same between two parameters. That is, the set of examples the teachers used did not demonstrate knowledge of what changes, what stays invariant and what the underlying meanings behind varying parameters

In addition, the patterns of variation in Mafada’s and Tinyiko’s examples are contrary to Leung’s (

‘invariants are critical features that define or generalise a phenomenon … for a major aim of mathematical activity is to separate out invariant patterns while different mathematical entities are varying, and subsequently to generalise.’ (p. 434)

The ways Mafada and Tinyiko varied the parameters during teaching did not bring about the discernment of structure in working with the different families of functions as well as generality about the effect of the parameters

This lack of interpretive elaborations and intellectual discussions with the learners about the effect of the parameters indicates that the teachers did not create a teaching and learning environment that facilitates learners’ deep understanding of functions. The following extract exemplifies the ritualistic routines in Tinyiko’s teaching:

1 | Tinyiko: | If for instance you are given |

4 | Learners: | ( |

5 | Tinyiko: | The |

7 | Learner: | We let |

8 | Tinyiko: | We said let |

9 | Learners: | ( |

10 | Tinyiko: | And let’s remember that the |

12 | Learners: | The outputs. |

13 | Tinyiko: | Good, after you get all the values, all you need to do is draw the graphs. |

The questions ‘^{1}

The above discussion resonates with Mason’s (

‘depends on discerning common and differing features among examples and experiences, generalising from these according to the scope of examples that are presented, and fusing these features into a concept.’ (p. 8)

It is arguable that varying the two parameters simultaneously without first varying one while the other remains invariant makes it difficult for learners to experience the difference of their effect on the functions.

Sieving out invariants in the parameters during the teaching of functions is an essence of experiencing the depth of the topic, and in turn developing conceptual understanding as this facilitates symbolic mediation for different functions (Chimhande,

Sequences of examples in Zelda’s, Jaden’s and Mutsakisi’s lessons.

Teacher | Examples and their sequence |
---|---|

Zelda | ^{2}; ^{2} + 1; ^{2} – 1; ^{2} |

Jaden | |

Mutsakisi | ^{2}; ^{2} + 2; ^{2} + 1 |

Zelda’s, Jaden’s and Mutsakisi’s patterns of variation in the selected lessons as presented in ^{2}, ^{2} + 1 and ^{2} – 1. Zelda’s fourth example (^{2}) was introduced to mediate learners’ thinking about the effect of changing the value of parameter

For Mutsakisi, the pattern of variation also focused on varying the values of parameter ^{2}, ^{2} + 2 and ^{2} + 1 while parameter

According to Lo and Marton (

‘

‘^{2} ^{2}

‘The graphs of

Zelda, Jaden and Mutsakisi created opportunities through the system of variation and sequencing of examples to bring the idea of ‘transformation’ in functions, attention on the appearance (structure), displacement and orientation of functions, to the fore (Chimhande,

In this study, the teachers who did not vary one parameter while keeping the other invariant in the examples did not engage in interpretive actions about the effects of the parameters on the different families of functions, relating to the lack of explorative routines. Accordingly, this results in lack of formal word use and endorsed narratives related to quadratics and linear functions. This demonstrates that systematic variation, selection and sequencing of examples in symbolic form are the preconditions of productive communication about the behaviour of different parameters for families of functions in terms of formal words and endorsed narratives. That is, without systematic and sequential variation of parameters, teachers’ communication becomes limited to rote steps to draw graphs, and nothing is revealed to the learners about the effect of the parameters. The teachers who selected and sequenced the examples showing the variance-invariance patterns of working with parameters for different families of functions engaged in interpretive actions about the effect of the parameters; as such, the variation patterns mediated both their communication about the effect of the parameters and created opportunities for learners to learn about the notion of parameters. Word use and endorsed narratives are enabled or constrained by the availability and systematicity of patterned variation or lack thereof.

The participating teachers from the five Acornhoek Schools in Mpumalanga are hereby acknowledged.

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

H.W.M. is the sole author of this article.

Before the study could commence, ethical clearance was granted by the University of the Witwatersrand and access to the schools was permitted by the Mpumalanga Department of Education (certificate number 2018ECE006D). All teachers were informed of the purpose, confidentiality and voluntary nature of participation in the study before any data generation processes commenced and all participating teachers signed informed consents. I also adhered to the importance of ensuring that the identity of participants is protected, both in terms of keeping the information they provided confidential and by using pseudonyms to conceal their true identities as well as those of their respective schools. The assurances for confidentiality and anonymity in this study extended beyond protecting the teachers’ names and those of their schools to also include the avoidance of using self-identifying statements and information.

The financial assistance of the National Institute for the Humanities and Social Sciences (NIHSS), in collaboration with the South African Humanities Deans Association (SAHUDA), towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the NIHSS and SAHUDA.

Data sharing is not applicable to this article as no new data were created or analysed in this study.

The views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author.

This observable action was frequent also across the other episodes in this lesson and in other observed lessons for the same teacher.