This article presents an interpretive analysis of three different mathematics teaching cases to establish where the bigger picture should lie in the teaching and learning of mathematics. We use pre-existing data collected through pre-observation and post-observation interviews and passive classroom observation undertaken by the third author in two different Grade 11 classes taught by two different teachers at one high school. Another set of data was collected through participant observation of the second author’s Year 2 University class. We analyse the presence or absence of the bigger picture, especially, in the teachers’ questioning strategies and their approach to content, guided by Tall’s framework of

Our major concern in this article lies with pursuing the question: where is the bigger picture in the teaching and learning of mathematics? It is prompted by the kinds of tasks, questions, classroom interactions and targeted content that ground mathematics teaching and learning within and across the different educational levels. In most cases, these teaching and learning activities seem to lack coherence, lack focus on important mathematics and lack appropriate articulation. One of the contributing factors that always affects students’ mathematics performance is the fact that a variety of teaching styles are to be found in operation in mathematics lessons, each depending on the teacher’s knowledge (skills and attitudes) of mathematics and the teacher’s knowledge about teaching mathematics. Hiebert and Grouws (

For Tall (

Mathematical understandings of a number of different big ideas are important ideas that students need to learn because they contribute to an understanding of the bigger picture of mathematics. This is explained well by Charles (

The purpose of the study was to pursue the question: where is the bigger picture in the teaching and learning of Mathematics?

Over the years, researchers have generally defined good mathematics teaching implicitly, with focus on various processes, such as reasoning and problem-solving (Wilson, Cooney & Stinson,

Tall’s (

Pirie and Kieren’s (

Within the interpretive paradigm, this study pursued the question: where is the bigger picture in the teaching and learning of mathematics? We use pre-existing data collected through pre-observation and post-observation interviews and passive classroom observation by the third author in two different Grade 11 classes taught by two different teachers, Thabiso and Lerato (pseudonyms), at one high school. Pre-observation interviews were conducted to gather information on the teachers’ beliefs about the nature of mathematics and how students learn and should be taught, given the current demand for curriculum reforms. Classroom observations, which commenced a week after a pre-observation interview, exposed the teachers’ teaching approaches and classroom interactions, which offered an opportunity to establish consistency between the interview responses and the classroom practices. Observations also offered an opportunity to search for alignment between the teachers’ teaching philosophy and classroom practice. A post-observation interview sought to get some clarity on issues that emerged during classroom observation. Another set of data was collected through participant observation in the second author’s Year 2 University class. In this article, we focus particularly on the teachers’ questioning strategies and the approach to content.

We analyse these three different mathematics’ teaching cases, guided by Tall’s (

Permission was granted by the Education Department, the school and the two teachers who participated. The teachers were made aware of the fact that they were free to voice their opinions, give advice and withdraw, if they felt so inclined. Because the research was not directly focused on the learners, we requested the teacher-participants to explain the research to the learners and their parents, via the school principal. Participation was voluntary and pseudonyms have been used to identify the two school teachers (Thabiso and Lerato), ensuring anonymity and confidentiality.

Approval for data collection was obtained from the department in which the study was located, the students and the relevant university structures beyond the department. In both locations, the nature and purpose of the study were declared, inclusive of potential audiences and substantive foci. Erickson (

It was possible for the three different mathematics’ teaching cases to involve prolonged engagement, persistent observation, peer debriefing and member checks with the teachers because the third author was teaching at the same school. With respect to the data collected at the university, the second author was teaching the class and the first author was the internal moderator for the module. We thus had enough opportunity to hear the teachers’ voices, which contributed to establishing credibility of this study (Bitsch,

We organise our results and discussions of the three cases (two cases of high school mathematics teaching and one case from a Year 2 university mathematics education class) below, commencing in each case with a brief biography, followed by two excerpts from the teachers’ mathematics lessons, our analysis of the lessons using the identified theoretical framework and a reflection on where the bigger picture is in relation to the teaching and learning of mathematics concepts being addressed.

Thabiso, a male teacher aged 43 years, held a Bachelor in Science Honours degree, majored in Mathematics. He had 18 years of teaching experience and had taught Mathematics in the Further Education and Training band (FET, that is, Grade 10, 11 and 12 classes). He attended Dinaledi Project workshops as part of his professional development.

Thabiso’s two teaching excerpts.

In both excerpts, learners were expected to operate in the symbolic world, beginning with processes or actions that are symbolised and coordinated for calculation and manipulation (Tall,

In excerpt 1 it appears that the teacher was confident that the learners understood what quadratic equations are and how they are derived. For him, it was not necessary to explain to the learners when to use a quadratic formula to solve the quadratic equation. The teacher assumed that learners could work with symbolic representations thus introduced to use the quadratic formula ^{2} – 3

Question 1c) deviated a little from the pattern as it required learners to first realise that they needed to simplify the given expression to its standard form before they could follow the routine procedure. It is doubtful whether the learners comprehended why ± should be separated and what the meaning of the equal sign in the context of quadratics was. It would also be interesting to find out whether learners would know how the values of

In excerpt 2 the teacher expected the learners to build on their experiences with working with triangles. The learners were expected to recall that they could only use the sine rule if two angles and a side are given or two sides and the non-included angle are given, that is, their met-befores. They had to connect that experience with a concept of the sine rule to find the unknowns. It could have been more empowering if learners were exposed to the given triangle to solve for the unknowns without being reminded of which formula to use. Meaningful learning occurs when learners succeed in choosing effective mathematical strategies to solve given problems.

Reflecting on excerpt 1, we wonder whether learners would know, before even attempting any given quadratic equation to solve, that the equation might have two solutions, or one solution, or perhaps no solution. The bigger picture with regard to quadratic equations lies in their origins. Egyptian, Babylonian and Chinese mathematicians dealt with areas of quadrilaterals and were interested in finding the length and breadth of a rectangle with known area (Gandz, ^{2} + ^{2} +

A similar observation is made with regard to excerpt 2 regarding the solution of triangles. The sine rule is one of the many strategies that is used to solve the triangles. Once the rule is derived, it is important that it is adequately analysed to establish the conditions under which it applies and the opportunities it gives us in solving triangles. In other words, in the bigger picture of solution of triangles, when is it appropriate or more ideal to use the sine rule? Determining whether the sine rule is appropriate is more important than its actual application. Thabiso’s approach was to ask the learners to simply plug in the values to calculate the missing values. Once again, the technical approach was not placed in the appropriate context within which the sine rule would have been seen as a particular strategy more suitable for a particular situation.

When approached in Thabiso’s way, mathematics is viewed as a collection of rules or formulae that learners must memorise, often out of context. Its role as a way of observing and interpreting our daily experiences is stripped away. In his defence, one might argue that at this stage Thabiso was simply helping learners to develop the skill of using the formulae or the rules and that the context would be brought in at a later stage when all different skills had been acquired. This is where we differ from Tall (

Lerato, a female teacher aged 51 years, held a Primary Teachers Diploma (PTD), Advanced Certificate (ACE) in Mathematics Education (majored in FET Mathematics teaching). She had 26 years teaching experience and taught mathematics in the FET band (that is, Grade 10, 11 and 12 classes). She attended in-service training workshops at the Mathematics, Science and Technology College (MASTEC) in 2008 as part of her professional development.

Lerato’s two teaching excerpts.

In both excerpts, learners operated in the symbolic world but were subjected to a different questioning strategy as compared to Thabiso’s case. Lerato, like Thabiso, led the learners to perform a procedure to find the correct answer. It is, however, doubtful whether with her questioning, learners would have managed to build up the symbolic mental imagery that is the basis of true understanding (Tarlow,

In excerpt 1, the expected learners’ met-befores were to know the standard form of quadratic equations and to factorise. The questions that required learners to share their challenges in factorising, and in making

In excerpt 2, Lerato reminded the learners about the expected met-befores: to use the horizontal reduction and special angles, to locate the given angle and to check the sign of the ratio. Learners were also expected to know that angles are conventionally measured counter-clockwise from the right hand horizontal axis and that angles measured in a clockwise direction are considered negative. Although an opportunity was provided to

The bigger picture with regard to quadratic equations was outlined in Thabiso’s case. In this section the focus is on the questioning strategy that Lerato used to facilitate learning. The questions that Lerato raised were generic and required learners to solve for

The question “when can we use the quadratic formula?” captures another aspect of the bigger picture. This prompts the question that, if the general formula method works so well, why would we ever use factoring? If pursued, it would reveal that the bigger picture involves a realisation on the part of learners that general formulas exist only for polynomials with degree less than 5, as proved by the French mathematician Galois (Moore, ^{2} +

The reduction formulae are meant to take advantage of the circularity of the angles, something that is rarely acknowledged. Lerato’s questioning in excerpt 2 prevented the learners from seeing fundamental relationships between lengths, angles and areas of triangles in a broader sense. The use of ‘degree’ as the unit for measuring angles (the Babylonian astronomers’ unit, Emerson,

The teacher, a male aged 48 years, held a doctoral degree in mathematics education. He majored in Mathematics and Applied Mathematics for his Bachelor of Science (BSc) degree, did a BSc Honours degree in Applied Mathematics and a Higher Education Diploma as a teaching qualification.

Kwena’s two teaching excerpts.

While the learning in the two excerpts started from different worlds, the

In excerpt 1, just like in Thabiso’s case, the learning was approached from the point of view of orientating learners on one procedure to solve a system of linear equations represented by

Matrices emanated from the study of solutions of a system of linear equations. The idea was to find different strategies that could be used to solve the unknowns efficiently and reliably. However, as mathematical objects, their properties are also amenable to the four mathematical operations addition, subtraction, division and multiplication. At the core of all these operations, is the solution of the matrix equation

The same scenario applies to the limits of functions. Given the function

Both excerpts require students to analyse the strategies used in resolving the problems at hand. While the first excerpt is wide open in terms of the strategies around the bigger picture, the second excerpt is anchored around a particular strategy: the mean value theorem. This makes the first excerpt relatively stronger in addressing the bigger picture. On the other side, the application is closed in the first excerpt, as a particular scenario is given, whereas it is open in the second excerpt, as students are expected to come up with their own problems. This makes the second excerpt relatively stronger with regard to the bigger picture. The insistence on reflections on own experiences and the generation of reflective questions in both excerpts makes the realisation of the bigger picture in both cases more likely. The approach in both cases, if well developed, has the potential to contribute to an integrated development of the bigger picture with regard to the concepts at hand.

In this article we pursued the question: where is the bigger picture in the teaching and learning of mathematics? We used three teaching cases to analyse the presence or absence of the bigger picture, especially, in the teachers’ questioning strategies and their approach to content. In both Tall’s (

In the three teaching cases that we have used in the study, we found that all the topics covered had potential for the incorporation of a bigger picture of mathematics. However, the analysis of the content and questioning strategies in those lessons revealed that either the influence of a bigger picture was non-existent or it had minimal influence. In the case of Thabiso’s approach, the influence of the bigger picture was non-existent on both the content structure and the questioning strategy with regard to quadratic equations and the solution of triangles. In both Lerato’s and Kwena’s cases, engagement with the content was framed by their questioning strategies. In Lerato’s case, the questioning strategy had glimpses of influences of the bigger picture of quadratic equations and that of the solution of trigonometric equations. In Kwena’s case, the questioning strategy revealed a concerted effort to incorporate the bigger picture of matrices and that of limits in the lessons. However, a lot is still needed to improve on that practice.

Generally, all three teachers can benefit from exposure to the influence of the bigger picture in the teaching and learning of mathematics. Both Tall’s (

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

S.M. was the project leader and moderator of K. Masha’s assessment instruments. K. Masha facilitated lessons in the Year 2 university class and captured data from that activity. K. Maphutha collected data from the two teachers’ mathematics classes and assisted in peer debriefing and member-checks with the teachers involved. S.M. and K. Masha conceptualised the article and thereafter S.M. produced the first draft. The two then proceeded to work on the final draft.