This article is part of a larger study on teacher development. The main study investigated teacher development within primary school Mathematics teachers’ classrooms to determine if teaching practices could be enhanced through a didactisation-based programme. It sought to develop teachers within their own environments and classrooms. Design research (both designing the conditions for change and studying the results of those conditions) enabled the researchers to design a programme that was congruent with teachers’ own needs and experiences. The programme ran for a period of a year with regular contact between the teachers and the researcher conducting the programme (the first author). The programme set out nine didactisation practices: active students, differentiation, mathematisation, vertically aligned lessons, accessing student thinking and ideas, probing student thinking and ideas, connecting student ideas, assessing students and reflecting on practice. One practice, student activity, is the focus of this article. It was found that by initiating discussion and cognitive conflict in teachers by using modelling problems, and further allowing teachers to observe pupils working in groups with modelling problems, teachers were starting to incorporate the didactisation practices within their own classrooms. This article documents specifically the fundamental role of student mathematical activity and the importance of improving student mathematical experiences, both for teacher development and for student mathematical learning. The study may be valuable in structuring and planning further effective teacher development programmes.

Mathematics teaching and learning is a priority in many countries around the world. South Africa is no different and the latest national assessments indicate mathematics education in South Africa is in dire straits, with the Grade 9 average for Mathematics being 14% (Department of Basic Education, 2013). Although the situation is complex, and many factors contribute to this, Mathematics teachers need to be developed so that a change in performance can be realised. In this article, a teacher development programme is described that formed part of a larger study on didactisation practices (Biccard,

Teacher change in the study was catalysed through teacher exposure to mathematical modelling tasks and being involved in in-depth reflective discussions. The focus of this article is on how student activity (as one of the didactisation practices) was optimised by teachers and how this effected a change within the classroom culture and within teachers’ own development of didactisation practices. Traditional mathematics classrooms are characterised by low student-initiated mathematical activity. Lessons often follow a predictable teacher explanation session followed by routine procedures completed by students. Mathematics presented to students in these classrooms is largely deductive: a general rule is taught upfront and practised by means of various examples in what Freudenthal (

The effect of increasing student mathematical activity in mathematics classrooms needed to be gauged and documented. Mathematical activity where students are exploring, conjecturing and making connections is more difficult to incorporate in lessons than simply solving mechanical problems. The effect of broadening student mathematical activity experiences on both the nature of student learning and teacher development needed to be studied. It was hoped that the classrooms would change from being based on teacher explanations only, to being more learner centred and more problem centred. The spin-off of making classrooms more learner participatory is that the role of both the students and the teachers had to be reconsidered. It was through mediating these changes that teachers would reflect on and develop their practices. A number of questions are scrutinised in this article. Can teachers change the level of student mathematical activity in their classrooms? What is the nature of this change? How does this affect the nature of mathematical learning in these classrooms? How would an increase in student activity assist teachers in developing other didactisation practices? The level, type and nature of student activity is the focus of this article.

Realistic mathematics education (RME) provides a basis for the theoretical framework of the article. RME is a teaching and learning theory that is underpinned by the work of Freudenthal, Treffers, Gravemeijer and others. The principles of guided re-invention, self-developed models and didactical phenomenology (Gravemeijer,

Active students – what mathematics are the students working through or dealing with? Is it procedures only or do they have to engage with the material and make sense of it? Are there different ways of approaching the problem or are they simply repeating a procedure? Is there a need to discuss what they are working on? Is there more to discuss than only the solution?

Differentiation – does the problem lend itself to a variety of approaches? Can students with different understanding deal with and access the problem?

Mathematisation – this is at the heart of a mathematics lesson. What mathematics are students learning and how are they learning it? Can they bridge from a contextual problem to mathematics? How can they shorten their methods by using more elegant solution paths?

Vertically aligned lessons – this refers to the teaching and learning trajectory a teacher has set for students, whether it is a short-term trajectory through a particular section of work or a long-term yearly trajectory. In terms of mathematics, a progressive mathematisation is sought where there is a building of mathematical structures or a deeper, more complex understanding of surface features of a problem. Essentially, does the teaching trajectory allow for scaffolding or reflecting on student informal ideas to more formal mathematics?

These components formed an integral part of the didactisation practices envisaged for teacher development in the study. They encapsulate essential features of mathematics classrooms where student involvement and student understanding is central. Understanding mathematisation is fundamental to theorising about teacher actions and teacher decision-making that may lead to significant learning in a mathematics classroom.

Freudenthal (

Generalising – looking for analogies classifying and structuring.

Certainty – reflecting, justifying, proving (using a systematic approach, elaborating and testing conjectures).

Exactness – modelling, symbolising and defining.

Brevity – symbolising and schematising (developing standard procedures and notations) (p. 82).

The level of thinking in mathematics is raised from looking for analogies and structuring to reflecting and proving and finally to symbolising and schematising. This level raising, according to Gravemeijer specifically involves generalising and formalising. Formalising embraces modelling, symbolising, schematising and defining while generalising refers to a ‘construction of connections’ and not an application of general knowledge (Gravemeijer,

Teacher development programmes within mathematics education would benefit from keeping these descriptions close to their aims and focus. Teacher development needs to be centred on student mathematising and how teachers and students together can raise the level of thinking through solving contextual problems. Level raising in mathematics classrooms presupposes mathematical activity on the part of the students. Teacher development should consider teachers and students as active reflective participants within their own classrooms as suggested by Fosnot, Dolk, Zolkower, Hersch and Seignoret (

We engaged in-service teachers in experiences that involved action, reflection and conversation within the context or learning/teaching. We took the perspective that teachers need to construct new gestalts, new visions of mathematics teaching and learning. To do this they need to be learners in an environment where mathematics is taught as mathematising, where learning is seen as constructing in terms of professional development of teachers. (p. 7)

To further understand and guide the actions and decisions of Mathematics teachers, a framework that focused on the work of teachers was also needed to understand the very complex, messy domain of mathematics classrooms. Wilson and Heid (

probe mathematical ideas

access and understand the mathematical thinking of learners

know and use the curriculum

assess the mathematical knowledge of learners

reflect on the mathematics of practice.

A proficiency framework was preferred to a knowledge-only framework since it allowed closer access to identifying and understanding authentic Mathematics teacher decisions that translate into actions in classrooms. The

student activity

differentiation

mathematisation

vertically aligned lessons

access student thinking and ideas

probe student thinking and ideas

connect student ideas

assess student thinking

reflect on practice.

As the professional development programme progressed, it was found that an improvement in student activity was the first to develop. Furthermore, it was found that other didactisation practices are stimulated by, and could further be developed through, meaningful student activity and teacher reflection on this activity. This resulted in a hierarchy of didactisation practices with student activity at its base (Biccard,

Starting with a basic definition of activity allows one to place it within the mathematics learning domain. According to the Concise Oxford English Dictionary

This means that activity can be graded by the

Brousseau (

Skemp (

Treffers (

It would therefore appear that part of the conditions under which student activity becomes meaningful in mathematical learning is that of reflecting on their activity and the effect of their actions. Classrooms that are teacher-directed and teacher-presented allow very little student discussion or deep reflection. Vygotsky (

Lesh and Doerr (

The professional development programme designed in this study changed the conditions under which teachers learn by presenting them with modelling problems; firstly, they solved the problems as a group and, secondly, they observed groups of students solving the same problem. This led to teachers reflecting on these activities and making different decisions for their lessons in ways that each teacher decided according to their own goals and resources. Biccard (

The study involved five primary school Mathematics teachers. They were teaching Grade 5 and Grade 6 students (aged 10–12 years). The five teachers volunteered to be part of the programme, which lasted about 9 months over a period of one year. Teachers were briefed on the intentions and aims of the study and they signed consent documentation. They understood that all references were anonymous and they could withdraw at any time.

The study implemented the principles of design research. This means that the researcher had to create an innovative learning environment for the teachers and study the conditions that made the environment conducive to professional development and, similarly, study the effect of the professional development. Cobb, Confrey, diSessa, Lehrer and Schauble (

Design experiments

Design experiments are highly interventionist in nature.

Design experiments are both prospective and reflective. They include a hypothetical

Design experiments have an iterative design.

The theory generated by design experiments is relatively humble.

Bakker (

The main study included the following sequence of activities in the teacher development programme that was designed:

An observation of each of the volunteer teachers in practice. The didactisation principles were gauged using a variety of instruments. The pedagogy scale, use of context scale and mathematical content scale as presented by Fosnot et al. (

A number of contact sessions with all five teachers. During the first, fourth and seventh contact sessions, teachers worked through a modelling problem as a group. As concluded by Biccard (

A ‘fishbowl’ session, in which teachers observed a small group of students engaging in the same modelling task the teachers had completed in the previous session.

Reflection session and reflection instruments at various stages.

Promotion of inductive reasoning.

These activities were repeated over three cycles. The researcher also provided various resources for teachers. During the third contact session teachers were given two groups of tasks: one group were of a problem-solving nature, while the other was traditional numbers-only problems. They were asked to match the number-only problem to its contextual partner problem. Teachers were then asked to discuss the nature and value of each type of problem in classrooms. Through this discussion teachers were assisted in changing traditional problems into more context-based and collaborative problems for their classrooms. Teachers also contrasted the differences between the different types of problems. It was not a requirement that teachers were to present any particular (modelling or problem-based) lesson in any of the observed lessons. It was also clear to teachers that they did not have to present any particular content during observed lessons. Teachers were to continue covering their curriculum during observed visits. Teachers were left in control of ‘filtering’ aspects of the professional development programme through to their own classrooms and their own practices.

Part of the programme involved resourcing teachers with the type of problems that could lend themselves to more active students in classrooms. Stigler and Hiebert (

Since design research is qualitative in nature, the validity and reliability of the study was hinged around McKenney, Nieveen and Van den Akker’s (

‘Student activity’ was one of nine didactisation practices that were described and explicated in the professional development programme of the main study. The didactisation principles were gauged by the researcher during observation of lessons. The first observation lesson took place before the professional development programme had started. Teachers were asked to present a typical lesson. The second and third observation lessons took place after three (and six) professional development contact sessions while the final lesson observation took place at the very end of the programme (one year after the beginning of the programme). Although the programme took place over a year, actual contact was maintained with the teachers for a period of nine months since research is not allowed at schools during the final term of the school year when pupils and teachers are preparing for the final examinations.

This article will only focus on student activity during the four lessons each of the five teachers presented. A brief summary of the student activities for each lesson is outlined in

Student activity per lesson.

Teacher | Baseline lesson (July 2012) | Cycle 2 lesson (February 2013) | Cycle 3 lesson (April 2013) | Final lesson (June 2013) |
---|---|---|---|---|

A Grade 6 | Whole class explanation on percentage increase and decrease. |
Students worked in pairs with manipulatives to calculate equivalent fractions. | Students worked in pairs solving word problems. | Students worked in pairs on ordering decimal numbers, then reported to larger groups and finallypresented solutions to class. |

B Grade 5 | Answered teacher’s questions individually. |
Whole class discussion on inverse operations required for input/output flow diagrams. |
Individual and group work. Building of 3D shapes. |
Students worked in pairs to solve one ‘big’ (multi-step) problem involving money, profit and loss. |

C Grade 6 | Teacher-led discussion on converting fractions to percentages. |
Group work. Using blocks to show fractional grouping (e.g. 2/3 of 18). Presenting solution to rest of class. | Factor trees. Pair-work on large sheets of paper. Presenting solutions to the teacher during whole class discussion. | Modelling problem – groups solving a real-life problem that needs structuring and analysing. |

D Grade 6 | Mental calculations, teacher-led questions and group calculations. | Teacher-led question and answer lesson based on textbook presentation of fraction wall. | Whole class discussion based on walk around the school grounds to find various 3D shapes. | Pair work on mixed word problems. |

E Grade 5 | Drill lesson on multiplying by units of 10, 100 and 1000. | Teacher-led whole class discussion on number patterns followed by individual textbook work. | Sharing wholes by cutting pictures of pizzas and chocolates. Packing and pasting fractions into books. | Exploring 3D shapes. Looking, feeling shapes. Drawing own 3D net for shapes. Comparing own net to given net. |

The main shift in the cycles of student activity in these results is from teacher doing to student doing. Another shift is from individual work to pair work or group work. The type of activities moved from single solution to multi-step solutions and multi-approach solutions. The teachers were ‘devolving’ the responsibility of the mathematical work to their students as the lessons progressed over the year-long period. The type of activities changed from teacher explaining ideas and concepts to students encountering and engaging with the ideas and concepts.

This change in the activity level, and the types of activities students were involved in, was gauged as summarised in

Occurrence of student activities over five lessons.

Student activities that took place during the lessons | Number of lessons where activity took place (Baseline lessons) | Number of lessons where the activity took place (Cycle 2 lessons) | Number of lessons where the activity took place (Cycle 3 lessons) | Number of lessons where the activity took place (Final lessons) |
---|---|---|---|---|

Listen | 5 | 5 | 5 | 5 |

Anticipate | 1 | 2 | 5 | 3 |

Answer questions | 4 | 5 | 5 | 4 |

Explain | 1 | 3 | 3 | 4 |

Read | 1 | 2 | 1 | 3 |

Calculate | 4 | 5 | 2 | 4 |

Write | 1 | 3 | 3 | 4 |

Rewrite | 2 | 2 | 0 | 1 |

Organise | 1 | 2 | 3 | 4 |

Physical activity | 0 | 2 | 3 | 1 |

Mental work | 1 | 4 | 5 | 5 |

Modelling | 0 | 0 | 0 | 1 |

The five lessons were observed and the activities students were involved with were recorded and are presented in

The teacher as supervisor of the activity, in which the teacher answers questions or clarifies if students ask.

The teacher as director or manager, in which the teacher initiates discussion and controls the topic, but allows or invites input.

The teacher as facilitator, in which the teacher sets up a structure, interacts with students and students interact with each other and the materials (p. 4).

The findings are summarised in

Teacher probing of student ideas.

Teacher probing role | Teacher as supervisor | Teacher as director or manager | Teacher as facilitator |
---|---|---|---|

A Baseline | X | - | - |

B Baseline | X | - | - |

C Baseline | X | - | - |

D Baseline | X | - | - |

E Baseline | X | - | - |

A Cycle 2 | - | - | X |

B Cycle 2 | X | - | - |

C Cycle 2 | - | - | X |

D Cycle 2 | - | X | - |

E Cycle 2 | - | X | - |

A Cycle 3 | - | - | X |

B Cycle 3 | - | X | - |

C Cycle 3 | - | X | - |

D Cycle 3 | - | X | - |

E Cycle 3 | - | - | X |

A Final | - | - | X |

B Final | - | - | X |

C Final | - | - | X |

D Final | - | - | X |

E Final | - | - | X |

The teachers’ roles changed as they incorporated more mathematical activity for their students and as they varied the activities that students were involved in. The change in teacher role and student role is reciprocal and is influenced by the landscape of mathematical activities in the classroom. As the conditions and responsibilities changed due to students being more active in the lessons, so the type of mathematical thinking involved changed (see

The use of materials also changed across the lessons. From a distributed cognition point of view (Pea in 2007, p. 13), the distribution of intelligence is across a system that comprises the individual, tools and the social context in which learning takes place. Pea (

Change in the use of materials and social context per lesson.

Teacher | Baseline lesson | Cycle 2 lesson | Cycle 3 lesson | Final lesson |
---|---|---|---|---|

A | Smart board |
Chocolates to pack and sort |
Textbooks |
Worksheet |

- | +M +S | + S | +S | |

B | White board |
Chalkboard |
Chalkboard |
Chalkboard |

- | - | +M +S | +S | |

C | Overhead projector |
Wooden blocks |
Chalkboard |
Modelling problem |

- | +M +S | +M +S | +S | |

D | Overhead projector |
Textbook |
3D shapes |
Word problems on workcard |

- | - | +M +S | +S | |

E | Overhead projector |
Chalkboard |
Worksheets with printed pictures |
3D shapes |

- | +S | +M +S | +M +S |

As can be seen in

A further aspect of development within these mathematics classrooms can be dissected using Stigler and Hiebert’s (

Although lessons showed shifts along an inductive-deductive continuum, it is important to state that mathematics lessons may include a judicious mix of the two. Traditional lessons tend to be exclusively deductive in nature. It is evident in this study that teachers are trying new ways of teaching mathematics. The changing roles of both students and teachers mediated by increased student activity and ameliorated by the social interaction through activities raised the level of teacher professional activity in the classrooms. The results of this study show that teachers incorporated more active learning activities in their own way and with the content that they were required to teach. The results suggest that smaller problem-based activities are the first step in changing activity levels in classrooms while modelling problems were starting to come to the fore at the end of the programme.

Teachers’ didactisation practices in the main study showed signs of development through the programme. After careful consideration of each teacher’s personal development, the following hierarchy was formulated to show which didactisation practices developed first. As a result of increased student activity, the other practices were made visible to teachers and could evolve and develop. Effective professional development of Mathematics teachers should therefore have a solid base of how and why student activity contributes to both abstract and generalisable thinking in students and changing of teaching practices in teachers.

Didactisation practices hierarchy.

in this teacher development programme. The hierarchy also shows that the more abstract thinking required by teachers is in the form of generalisations needed for connecting student ideas and vertically aligning lessons. The increasing difficulty in these practices is as a result of abstraction and generalisation on the teacher’s part.

Teacher professional development programmes for both inservice and pre-service teachers would benefit from focusing on the level and conditions of student activity. This formed a cornerstone in this article. Although the professional development programme in the main study focused on all nine didactisation practices, student activity appears to have resonated most with the teachers. It was only once they incorporated student activity into their own teaching in their own ways and to suit their own needs, did the other didactisation practices develop. It also provided teachers with their own experiences to share and build upon. By focusing on increasing student mathematical activity in the classroom teachers were able to mediate the challenge of the changing roles for themselves and their students. An evolution of changing social contexts for the lessons took place. Students were given more responsibility in the latter lessons as they were given different activities to complete. Students were also doing more talking in the latter lessons. It was however found that tying together and connecting different student ideas and integrating different ways of thinking about concepts was still challenging for teachers by the end of the programme (Biccard,

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

P.B. (Stellenbosch University) was the researcher for the doctoral study and D.W. (Stellenbosch University) was the promoter for the study. This article was written with consistent input from both authors.