The research presented in this article reports on the results of a case study examining the classroom practice of one early career Grade 4 teacher (Nox, pseudonym) as she teaches equivalent fractions. The goal was to explore the ways and extent to which her instruction reflected a dialogical teaching approach, defined as a pedagogical approach underpinned by five specific principles that can be enacted through a range of possible talk strategies to achieve sustained participation of learners and thus enhance meaningful learning. I provide a pedagogical activity to illustrate to teachers how, by instigating and developing classroom talk in the primary classes, a dialogic teaching sequence may be implemented. However, the majority of the existing literature on dialogic teaching stems from studies conducted in Asian, European, and North American countries, whereas systematic research on dialogic teaching across international contexts remains limited. Nox was interviewed after obtaining observational data to seek clarity on some of the observed instructional practices. Analysis of transcripts using the notion of dialogic teaching as a theoretical lens revealed that there was little evidence of Nox’s attempts to use talk to make learning of equivalent fractions a cumulative process. In addition, time constraint was the most significant factor in Nox’s teaching of equivalent fractions: she considered the curriculum too congested. Implications are drawn for evaluating dialogic teaching in primary mathematics classrooms. Future, larger studies may shed light on the extent of these results and, if need be, a significant investment on initial teacher training may be necessary to underscore the value of dialogic teaching in enhancing meaningful learning of, at least, equivalent fractions.

Fractions is not only one of the most important topics in mathematics, but also one of the most multifaceted (Pedersen & Bjerre,

Meaningful learning of fraction equivalence is crucial for learners’ success in algebra, a gatekeeper to post-school education ‘and the careers such education affords’ (Kilpatrick & Izsák,

In a study conducted by Jigyel and Afamasaga-Fuata’i (

Although direct teaching,^{1}

As the name suggests, dialogic teaching is described as teacher-led interactions with learners – one of the two main kinds of interaction, the other being learner-learner interactions (Mercer & Sams, ^{2}

Prior to providing a review of literature, it will be helpful to define the notions of ‘meaningful learning’ and ‘dialogic teaching’, which are particularly clearer if they are contrasted with ‘rote learning’. According to Ausubel (^{3}

In short, given the complexity of educational contexts, it would be simplistic to assume that dialogic teaching would be suitable in all classrooms. The results in this article should broaden our knowledge of early career teachers’ uptake of dialogic teaching as a framework that prioritises learners’ ideas in the development of mathematical concepts. By doing so, the results should transform the links of dialogic teaching to equivalent fractions and thus improve teaching of equivalent fractions content.

Previous research suggests that teaching approaches^{4}

The remainder of this article is structured as follows. Having provided a brief background to the study, I elaborate on the literature pertaining to dialogic teaching and exemplify the kind of strategies that may partly characterise instruction on equivalent fractions. Then, I provide a framework guiding this study. Next, an analysis of the results is undertaken followed by a discussion in which I interweave the literature with the findings. On the basis of the findings, I end the article with implications for classroom practice and recommendations for future research.

In recent years, many researchers in the field of education have shown interest in the work of Robin Alexander, David Ausubel, and Mikhail Bakhtin. These scholars have conceptualised terms like ‘dialogic’ and ‘meaningful learning’, which have gained increased attention in both mathematics and science education studies on classroom interactions. The term ‘dialogic’ has gained increased attention and classroom talk has become a key topic in educational sciences (Arend & Sunnen,

Teaching and learning is primarily concerned with the acquisition, retention, and use of information such as facts, propositions, principles, and vocabulary in the various disciplines (Ausubel,

What is the capital of Peru? (I)

Lima. (R)

Yes, quite correct (F)

The IRF structure involves ‘closed’ questions in which there is only one answer, which is already known to the teacher. Although this structure does not typify the pattern of talk in all classroom activities (learners may ask questions of teachers, or of other learners), IRFs have been observed as a common feature in classrooms the world over (Mercer,

In a study of Danish and Indonesian teachers’ pedagogical approach to equivalent fractions, Putra and Winsløw (

Learners should understand why a procedure works prior to using it. During interviews with two preservice teachers, Shongwe (

Hogan, Nastasi and Pressley (

Different fractions that name the same whole (

Most preferably, instead, treatment of equivalent fractions using concrete models such as fraction bars to compare, for example, thirds and sixths and see that

Meaningful learning of fraction equivalence includes having an integrated knowledge, which can be displayed and articulated by means of the following five attributes:

A fraction represents a quantity being measured in relation to a

A fraction quantity can be represented using manipulatives or pictorially by

Equivalent fractions can be constructed from manipulatives or pictorial representations by repartitioning or

Equivalent fractions can be constructed using symbolic notation.

A fraction quantity is a member of an equivalence class in which all fraction numerals represent the same quantity (Wong,

Physical quantity, that is, the size or amount of a physical characteristic of an object, can be measured. For example, physical quantities encountered by young learners are: length, perimeter, area, volume, mass, etc. However, determining whether an answer about a physical quantity of an object, arrived at by using a calculator or paper-and-pencil method, is reasonable requires estimation ability. Estimation is the mental process of arriving at an approximate measurement without the aid of measuring instruments.

This then requires a learner to make a judgement of the size of a physical quantity relative to some specified unit. The notions of

A referent unit is a non-standard or standard unit that can be used to estimate a quantity. For example, if area is to be measured a two-dimensional unit like a playing card can be used as a referent unit or if the length of a pencil is to be measured matchsticks can be used as referent unit. Thus, the choice of an appropriate unit for measurement is a mathematical skill that is fundamental in learning to measure.

Partitioning means engaging in an intuitive activity that generates quantity to build knowledge about fair sharing. For example, in determining an equivalent fraction for

Division with partitioning.

The teaching approach adopted by teachers in relation to equivalence may manifest itself in learners’ work. Kerslake (

^{5}

Development of a general statement about equivalent fractions.

An alternative generalisation approach, provided by Putra and Winsløw (

For example, for nd

However, the ‘paper-folding’ activity is a superior method in promoting meaningful learning through dialogic teaching. As Gattegno (

The study reported in this article employed Alexander’s (

Dialogic teaching combines four repertoires (and their subcategories): talk for everyday life, learning talk, teaching talk, and classroom organisation. These repertoires are used flexibly, on the basis of fitness for purpose. Additional to these repertoires, Alexander (

structuring of questions so as to provoke thoughtful responses

individual teacher-learner exchanges are coherent, connected lines of enquiry that do not leave learners stranded.

In this article, I analysed Nox’s teaching of equivalent fractions. The purpose of this article was neither to praise nor be critical of her pedagogy. Similarly, I did not intend to advocate or critique the use of dialogic teaching approach in equivalent fractions. Rather, the purpose of this article was to present an analysis of her instruction itself and thus offer a more precise description of what occurred in her classroom. A second purpose was to discuss why Nox chose to teach equivalent fractions in the way that she did.

Using Alexander’s (

How does Nox structure her questions to learners so as to provoke thoughtful responses in the learning and teaching of equivalent fractions?

Why did Nox teach equivalent fractions in the way she did?

The study reported in this article is part of a larger project looking at the classroom practices and professional development of an early career primary mathematics teacher. A qualitative case study design was adopted. In a qualitative inquiry the researcher studies meanings constructed by participants on a phenomenon in their natural setting (Denzin & Lincoln,

Ethical approval was granted by the Ethics in Research Committee of the University of KwaZulu-Natal.

This research is part of a larger project designed to follow the experiences an early career teacher in the pedagogy of mathematics and ethical approval was granted by the Ethics in Research Committee of at a university in south-eastern South Africa with a protocol number HSSREC/00001902/2020. Informed consent was obtained from all participants.

The empirical research reported in the present study involved data from a main project designed to follow the professional development of an early career teacher, Nox (pseudonym), in a Grade 4 class (nine- and ten-year-olds) who were conveniently sampled to participate in the study. Nox had a four-year professional teaching degree in primary mathematics. At the time of the study (2020), she had 2 years of teaching experience. She taught at a rural combined primary school in the Eastern Cape, South Africa, which I shall call Fundisanani. The school is situated in a village (rural area) and serves learners from a low socio-economic background, in an isiXhosa-speaking African community with high absenteeism rate for both learners and teachers. Like most rural schools in the area, it was under-resourced, had no laboratory or media centre, nor sports field. The names of all participants and schools have been changed to preserve anonymity.

The context of the study was kept as normal as possible, in three ways. First, Nox was observed teaching equivalent fractions in her Grade 4 classroom under the general topic ‘Fractions’. Second, the lessons on equivalent fractions that were audio recorded were selected from the typical, prescribed primary school mathematics curriculum. Third, the researcher was a spectator observer (Patton,

Qualitative data were collected by employing two methods: classroom observation and semi-structured interviews. The purpose of the observation was to understand how they orchestrate dialogic strategies in their context-specific settings. As already mentioned, Nox was observed in her natural setting (i.e. the classroom). Data were audio recorded and transcribed verbatim. An audio recorder was strapped to Nox’s waist to capture all the teaching proceedings including interactions in group work and any non-verbal communication. Given that the purpose of the observation was to understand how Nox orchestrated dialogic strategies in her context-specific settings – that is, I was particularly interested in the teacher talk – I concentrated on recording solely Nox. Another reason for recording only her talk was to minimise interference with learners’ behaviour so that their talk would be natural.

The classroom observation protocol was used to answer the first two research questions by documenting what Nox said in telling her story on equivalence. The tool also included background information, contextual background and activities, and 10 items on a four-point Likert-type scale. The items are on a continuum ranging from ‘

Additionally, Nox was interviewed after the lesson observations to establish reasons for the observed teaching sequences. The interview also helped in triangulation of observation data with interview data. Sample questions on the interview schedule included ‘Why do you ask learners to clarify their homework activities?’, ‘Describe your teaching of equivalent fractions’, and ‘Why do you teach equivalent fractions the way you did in the lesson?’. The interview lasted for 45 minutes.

The purpose of the proposed study is to investigate how practising Grade 4 teachers use dialogic teaching to orchestrate classroom interaction in the context of equivalent fractions. The data were subjected to thematic analysis since this method offers ‘a more accessible form of analysis, particularly for those early in a qualitative research career’ (Braun & Clarke,

In particular, transcribed excerpts from episodes of audio recorded lessons of Nox’s dialogic strategies were used to examine how she drew on dialogic teaching principles proposed by Alexander (2004) to make understanding of equivalent fractions available on the social plane, at their different schools. That is, Alexander’s predetermined categories were used to assess how the participating teachers make opportunities for understanding equivalent fractions available for learners in their classrooms, particularly probing learners’ answers with the ‘Why do you think that?’ question. Nox’s talk, rather than learner-learner interactions, was analysed. It is worth emphasising that the microphone was sensitive enough to record learner-learner talk, as well.

On the whole, to investigate the two research questions underpinning this study, the analysis mainly focused on the identification of interactions using Alexander’s (

The observation transcript was also coded by a doctoral student whose work was in the area of dialogic teaching. A very high inter-rater reliability – where researchers are expected to independently identify the same codes in a transcript and the codings compared for agreements – was reached (Creswell,

Collection of classroom observation data ceased when saturation was reached. Data saturation is a methodological principle in qualitative research that is commonly taken to indicate that, on the basis of the data that have been analysed hitherto, further data analysis is unnecessary (Fusch & Ness,

To provide answers to the first research question meant analysing ways Nox structured her classroom talk, especially her questions. This was also a way to search for evidence of dialogic teaching of equivalent fractions as her lesson unfolded. As already mentioned, the analysis focused solely on Nox’s talk structure given the consensus that, as a teacher, she was the single most important school variable influencing learner achievement. The question remains whether her instructional practice showed traces of the principles of dialogic teaching – the lens through which the data in this study was analysed – that would have enhanced learning of equivalent fractions.

To highlight the principles, we consider Nox’s lesson on equivalent fractions in her Grade 4 class. Primary school teachers are encouraged to engage learners in mental work prior to beginning a mathematics lesson. However, interactions in this respect, though interesting, were not analysed as they fell outside the scope of the study. The results are presented in terms of the two research questions.

The first research question was formulated to assess how Nox probed her learners to say more about their responses. In the classroom, Nox was standing in the front of the learners facing the chalkboard. The learners were sitting at desks arranged in rows. She wrote the topic ‘Equivalent fractions’ on the chalkboard. The excerpt below shows how Nox introduced her lesson, proceeding from the previous lesson’s activity; perhaps some patterns of dialogic teaching could be observed. Nox checked learners’ thinking on a previous activity involving different ways of representing fractions (using fractions to describe pictures). Noticeable was that the work was void of context; I expected a word problem, at least.

Routines in Nox’s lessons.

Turn | Speaker | Transcription |
---|---|---|

1 | Nox | Ok … [with a loud voice trying to quiet down the learners and get their attention], good morning Grade 4! |

2 | Class [Chorus] | Good morning, Mam! |

3 | Nox | How are you? |

4 | Class [Chorus] | We are fine. Thank you, teacher and you? |

5 | Nox | I am fine. You can sit down. Ok, the last time we were dealing with … [inaudible], isn’t it? Let us mark the homework. Who can write two-fifths in symbols? |

6 | Class Chorus] | Me, me, me, … |

7 | Nox | Yes, Xola! |

8 | L1 | [Moving to the board and writes] |

9 | Nox | Hmmm … Let’s see [looking at the fraction diagram on the wall]. How do you know it’s correct? |

10 | L1 | Ahhh … Mam. I take … hmm … two things in 5 things. |

11 | Nox | Oh, I see. Class is he correct? [interruption by knock at the door] Yes, come in, Nonhle [another learner arriving 10 minutes into the class]. Yes, Nosipho, tell us! |

12 | L2 | Yes, Xola … have the right things. |

13 | Nox | Good. Now, let’s go to the second problem. |

Moving now to the gist of the lesson – that is, the focus on how she orchestrated the interactions with her learners to tell the story of equivalence – we find Nox trying to engage her learners in more talk on the topic, equivalent fractions (see

Nox’s instruction moves from conviction to seeking generalisation.

Turn | Speaker | Transcription |
---|---|---|

1 | Nox | Using the fraction wall, in how many different ways can make a |

2 | L3 | There’s 5, Mam! |

3 | Nox | Why do you say 5? Can you mention just one? |

4 | L3 | |

5 | Nox | Ok. How do you know that? |

6 | L3 | You see, hmmm … if you cut the … |

7 | L4 | |

8 | Nox | No [Seeking more ideas]. What do you think, Lunwabo? |

9 | L5 | Mam, |

10 | Nox | |

11 | L5 | If we look at the two shades [pointing at 2nd and 4th rows], the one half and the four eights … they take the same space [area]. |

12 | Nox | Class, do you see what they are saying? |

13 | Class [Chorus] | Yes, Mam! |

As an early career teacher, Nox deferred the same question to other learners for their ideas, using phrases such as ‘Sipho thinks that half is the same as two quarters, do you agree with him?’ Put another way, I sought to examine whether Nox chained learners’ responses to follow coherent lines of thinking about the same idea to enhance meaningful learning of equivalent fractions. Her phrasing of the question suggested that she wanted to provoke thoughtful responses and see interanimation of ideas in her classroom talk. Seemingly convinced that the learning of equivalent fraction had taken place, Nox proceeded to seek a general rule to check if two fractions are indeed equivalent, as shown in Turn 16 (see

Learners show difficulty with making a generalisation.

Turn | Speaker | Transcription |
---|---|---|

14 | Nox | Now, we need to find a rule for working out equivalent fractions? Alipheli, what can we do? |

15 | L7 | Eish, Eish, Mam, |

16 | Nox | If |

17 | L2 | I don’t see this thing, Mam. |

18 | L7 | Me, too. |

19 | Nox | We can say that … Hmmm … two fractions are equivalent or equal if the product of the numerator ( |

20 | L6 | Yes, what is a? Yeah, what is the use … [of the] letters? |

21 | Nox | Look, if |

22 | L3 | Yes, Mam, … Hmmm … I find 12 is 12. |

23 | Nox | Good, if you see that the two sides are equal, it means that the fractions are equivalent. If they are not, the two sides will not be equal. Let’s look at this! |

24 | L7 | 2 × 4 |

25 | Nox | What do you say, |

26 | L9 | Oh, yeah, |

27 | L8 | It’s looking difficult. |

28 | Nox | Try to practise using your own examples. Hmmm … you’ll understand, |

29 | Class [Chorus] | Yes … |

30 | Nox | Do these exercises in your Bluebook! [Calling out the page number of the practice exercises in the workbook (commonly referred to as the “Bluebook” and used as a textbook in South Africa from reception year to Grade 6)]. |

What is happening in the extract above is that Nox seems to have involved many different learners in seeking an understanding of the rule. However, a close examination reveals that the voices she sought were on different aspects of the generalisation rule (i.e.

Although Nox asked a question that appeared ‘dialogic’ in that it contained the ‘why’ cue, she did so from the perspective of a teacher with an authoritative stance (as she sought to maintain school mathematics position; pressing for the three principles of dialogic teaching was not explicit in the lessons observed). This need not necessarily be construed as an indictment on Nox’s practice. She had to teach her learners in an environment that was not conducive to the principles of dialogic teaching; this could have limited her enactment of dialogic teaching in the classroom. Learners’ responses to such a question functioned to respond to mathematics content because Nox utilised cues such as ‘Why …’ and ‘How do you …’ only to understand how closely the learners’ responses aligned with some school-sanctioned or predetermined disciplinary stance.

Nox’s conceptual framework in teaching equivalent fractions.

Although Nox mobilised learners’ ideas, she did not anchor her questions and comments in learners’ contributions. In other words, despite utilising cues such as ‘why’ and ‘do you believe’, she only truly cared about how closely the learners’ response aligned with disciplinary knowledge. This talk structure does not, in any way visible to an analyst, embody any ‘scaffolding’ intended for meaningful meaning of equivalent fractions. Nox engaged her learners in a mathematical activity of talking about equivalent fractions. Accordingly, she showed part of mathematical practice in her use of the ‘why’ cues to seek confirmation or further development of an idea.

In her 29 turns of talk, Nox asked a total of 16 questions that could be designated as ‘closed’ exchanges. During the dialogue, learner participation can be characterised as providing ‘correct’ answers intended to demonstrate to Nox that her learners were indeed recalling the knowledge as transmitted in the previous lesson. Nox implicitly indicated that a learner’s answer was incorrect by not recording it on the chalkboard or by ignoring it and continuing to ask other learners for the ‘answer’ (Turn 4 and Turn 7 of the second excerpt). Worthy to mention here is that including other episodes would have served no purpose in that data saturation was reached at this point. That is, further analysis beyond this point did not yield new results. Thus, further coding of observational data was no longer feasible.

Dialogic teaching is an approach whose success is affected by contextual factors. Hence, it was necessary to conduct an interview to understand why Nox approached the teaching of equivalent fractions the way she did. School factors such as the routine of holding assembly in the morning prior to commencement of lessons, the movement of learners as they change classes, the pressure to complete the curriculum, and late arrival played a role in her teaching of equivalent fractions. Although cross-national research studies have shown that a significant number of teachers exit their initial (preservice) teacher preparation programmes with inadequate knowledge of mathematics (eds. Tatto, Rodriguez, Smith, & Reckase,

Asked to reflect on her teaching in the lesson, she pointed out that collecting learners’ ideas ‘makes the lesson messy; these kids get involved in arguments’. Interested to understand better what she meant by ‘arguments’, I probed (see

Nox’s reflections on her lessons.

Variable | Reflections |
---|---|

Researcher | Why do you ask the ‘why’ questions and make no further attempt to build on it to sustain learners’ participation? |

Nox | Hmmm … I have to remember that I have a duty to complete the curriculum … I wish I has sufficient time. But, I have other subjects to move to. |

Researcher | I hear you. I hear you, well. But, don’t you think that it’s better to not finish the curriculum but have meaningful learning of equivalent fractions? |

Nox | Yeah, yes, honestly that will be a great thing to do. But, if I do it, yeah … I’ll [be] behind the schedule. I’ll be called to the principal’s office to explain. No one will listen to my reasons. Hmmm no time for consolidation of work. |

Researcher | What do you mean by ‘consolidation of work?’ |

Nox | There’s too much content. Eish, yo, yo, yo. Don’t get me wrong. I enjoy my maths but the learners only know the surface of maths … to pass the grade. |

During the interview, Nox also raised time as a scarce resource, as captured in the following excerpt.

As can be found in pedagogical exchanges in most classrooms the world over, Nox clearly indicated that time limited her disposition to engage in dialogic teaching because she considered the curriculum too heavy. This suggestion brings to the fore the need for mentoring programmes or an improvement in inductive programmes for early career teachers so that they can enact reform-oriented teaching such as dialogic teaching. Most probably time can be found by looking into the activities that seemed to consume teaching time. For instance, Nox’s lesson was scheduled to take 40 minutes yet half of it was taken by the morning assembly lasting beyond the allocated time on the school’s timetable. There is truth in what Nox is saying: she is in a daily grind in which she is facing challenges that come with being an early career teacher (Lortie,

Another limiting factor was learners’ late arrival which disturbs the flow of teaching. Nox has had to recap to bring the late learners trickling into the classroom up to speed with the ideas already entertained in the class. Her efforts support the notion that teachers generally want all learners to understand the contents of their lessons (Van de Walle, Bay-Williams, Lovin, & Karp,

The purpose of this study was to explore possible instances of dialogic teaching in Nox’s mathematics classroom. In particular, the exploration involved examining how she uses dialogic strategies to tell the story of equivalent fractions, that is, how different ratios can have the same value. To remind the reader, dialogical teaching, in a weaker sense, refers to classroom interactions in which multiple speakers take extended turns which take account of others’ ideas (Scott, Mortimer, & Aguiar,

Teacher’s questions in dialogic teaching approach are structured so as to provoke thoughtful responses, which refers to questions that provoke further questions to create a coherent line of enquiry in relation to equivalent fractions. Nox’s interest in collecting learners’ ideas was to give learners an opportunity to relate them to their existing knowledge, in this case knowledge of equivalent fractions. This practice was in fulfilment of Ausubel’s (

In addition, Nox-learner exchanges were not, in Alexander’s (

Classroom observation of and interview with Nox provided insight into the contextual factors that influenced her teaching of equivalent fractions. The actual time spent observing Nox’s lessons was not commensurate with that allotted in the timetable. The effect of this factor was corroborated in Nox’s own words. During the interview, she highlighted that ideally she would like to see her learners make sense of equivalent fractions. In other words, she indicated that she would like to see her learners build on what has been said by the previous speaker to increase the coherence of the exchanges. However, such exchanges were few and far between, because ‘if I do it, yeah … I’ll [

Nox pointed out that she had to resort to teaching methods that offer little in terms of meaningful learning of equivalent fractions. She acknowledged that learning equivalent fractions in arbitrary fashions undermines the connected nature of mathematics concepts. Interview results sustained the inference made in the observation data that Nox’s teaching approach mimicked transmission teaching. However, what emerged from the interview was that Nox’s choice of teaching approach was not made with reckless disregard for meaningful learning; school environmental circumstances contributed.

In sum, analysis of both classroom observation data and interview transcript call for teaching approaches that position learners’ ideas at the centre of instruction in line with current reform initiatives in curriculum documents (Common Core State Standards Initiative [CCSSI],

As is often the case in any research, one limitation should be borne in mind when interpreting the results of this study. Learner-learner interactions were not analysed. This was not designed to discount learners’ voices; learners’ talk is a crucial part of a pedagogical event. This limitation suggests an opportunity for further research. Future research may incorporate this component of classroom interactions to better understand how learners view the teaching approach adopted by early career teachers in equivalent fractions.

In this article, I described how the teaching approach adopted by the focal teacher, Nox, made use of talk to guide learners to think and talk about equivalent fractions. The study was framed by the concept of ‘dialogic teaching’, described here as a pedagogic approach underpinned by specific features enacted through a range of possible talk strategies. Analysis of both observational and interview transcripts revealed that her teaching approach could be classified as following the IRF format, despite her attempts to elicit learners’ ideas on equivalent fractions.

Interview responses confirmed her allegiance to this format. However, her reasons for adopting this approach were found to be reasonable, given the work environment of a daily grind, in which she practised her craft. Professional development programmes can mitigate these obstacles by capacitating early career teachers with skills as they navigate their way in a teaching and learning environment such as that in which Nox found herself. The complexity of teaching notwithstanding, future studies must investigate the impact of the kind of environment in which early career teachers work as they form their styles and strategies of teaching. In particular, such studies may focus on the effect of the environment on early career teachers’ ability to engage learners in meaningful learning, on a large scale (i.e. using survey methods).

I am particularly indebted to the Grade 4 early career teacher and her learners for opening their classroom and being generous with their time.

The author declares that he has no financial or personal relationships that may have inappropriately influenced him in writing this article.

The corresponding author worked the manuscript from concept right up to its conclusion.

This work was supported in part by the grant received from the University Capacity Development Programme (UCDP) at UKZN.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Any opinions, findings, and conclusions or recommendations expressed in this article are those of the author and do not necessarily reflect the views of the UCDP.

I follow Hiebert and Grouws (

By mathematical practice is meant mathematical activity involving plausible (inductive) reasoning, through which conjectures are generated, and demonstrative (deductive) reasoning that is formalised in proof (Reid,

Paraphrasing Watson and Mason (

I join Arbaugh and Benbunan-Fich (