This article sets out a professional development programme for primary school mathematics teachers. Clark and Hollingsworth’s model of teacher change provided the theoretical framework necessary to understand teacher change. A design study allowed for increased programme flexibility and participator involvement. Five volunteer primary school teachers teaching at South African state schools were involved in the programme for a period of one year and their pedagogy, use of mathematical content and context developed during the programme. Twenty lessons were observed over the year-long period. An observation rubric that specifically focused on mathematical pedagogy, use of context and mathematical content scale guided the researcher to gauge global changing teacher practices. Teacher growth was evident through their professional experimentation and changes in their personal domain. The design features emanating from the study are that teachers be given opportunities to experience reform tasks (e.g. model-eliciting tasks) in the role of learners themselves and teachers should be encouraged to use contextual problems to initiate concept development. More mathematical detail in lesson planning is also necessary. Furthermore, teachers need appropriately designed resource materials to teach in new ways. It is recommended that professional development includes teachers engaging collaboratively in solving rich tasks. This study adds to the growing body of knowledge regarding teacher development programmes that focus on how teachers change their own classroom practices.

The need to develop mathematics teacher practices is known worldwide (Borko,

Professional development programmes that have been successful include those that focus on: representations, explanation and communication (Hill & Ball,

The teachers involved in this study were Grade 5 and Grade 6 teachers in South African primary schools. All teachers had at least three years of teaching experience and were teaching in well-resourced suburban schools. All schools and classes were mixed in terms of cultural background, ethnicity, gender and ability. The language of teaching and learning in all schools was English. However, for about one-third of the learners in the classes, English was not their home language. Only one teacher specifically trained to be a mathematics teacher. The challenges and economic effects of poor-quality mathematics education in South Africa are well documented (Spaull,

Schoenfeld (

In a review essay that critically examined mathematics teacher change, Goos and Geiger (

Teacher change should be conceptualised holistically through interventions, resources and change processes within the classrooms. However, teacher development programmes should empower teachers to determine their own growth and change paths (Frid & Sparrow,

In this study, Clark and Hollingsworth’s (

The research question guiding this design study was:

This leads to the sub-question:

The idea around change and growth in teachers is complex. The word change may denote any change (positive or negative) while growth suggests improvement in teaching practices. In this study the term ‘change’ was used since I did not necessarily pre-empt a positive change in teachers due to the professional development intervention. In the next section, a model to understand change from teacher professional development is set out.

Model of teacher change.

Clark (1998 in Clark & Hollingsworth,

Clark and Hollingsworth’s (

Clark and Hollingsworth (

A guiding principle in this study was to acknowledge that teachers learn in ways that they find most useful (Clark & Hollingsworth,

The methodology for the study is an intervention study, which can be seen as a type of design study, which is a common type of design study (Cobb, Jackson, & Dunlap,

Five mathematics teachers volunteered to be part of the PD programme. The teachers indicated that they were interested in learning about new ways of teaching mathematics.

These teachers were teaching Grade 5 and Grade 6 learners (10 to 12 years old). They were teaching at three schools that formed a convenience sample. The schools were suburban government primary schools that were well resourced. The classes were of mixed ability, cultural background and gender with around 33 to 35 learners in each class. The study ran for a period of one year, although actual contact with the teachers was for a period of nine months. The relevant stakeholders and gatekeepers gave their permission for the study to take place.

The PD ran in three cycles (see

An example of a modelling task.

Design-based research cycles.

During Session 2 of each cycle, the teachers would observe small groups of learners solve the same problem. The teachers were asked to become critical reflectors during this session. During Session 3 of each cycle, I considered the discussions in Session 1 and Session 2 and provided support and scaffolding that the teachers indicated they required. PD through DBR requires both engaging teachers in one setting (the PD setting) while trying to reorganise their practices in another setting (the classroom) (Cobb, Zhao, & Dean,

Observing and recording holistic changes in a mathematics teacher’s lessons is not a simple task. For this purpose, the rubric created by Fosnot, Dolk, Zolkhower, Hersch and Seignoret (

we engaged in-service teachers in experiences that involved action, reflection, and conversation within the context of 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 mathematizing, where learning is seen as constructing. (p. 7)

For validity purposes, an existing observation framework was used to understand the observed lessons and to document the teacher’s development through the programme. The rubric is too extensive to repeat here, so a summary is presented in

Classroom observation rubric.

These formulations allowed me to consider teacher development across three interrelated domains that could capture some elements of teacher change in a mathematics classroom. The rubric is detailed enough to ensure consistency in its interpretation and use. The rubric focuses on teacher actions within the classroom and the repercussions for learning through these actions.

This PD followed a similar approach to that of Cobb et al. (

The research was granted ethical clearance by the overseeing university (Reference no. DESC_Biccard2012). Permission to conduct the research was also given by the overseeing provincial department of education. Principals of the schools also granted permission for the study to take place. Participants took part voluntarily in the study and were assured confidentiality. Participants signed informed consent documents and were allowed to withdraw from the research at any point.

The findings of some of the other areas of this study have been partially disseminated (see Biccard,

Twenty lessons were observed through the programme and the development of the teachers gauged using the rubric in each case by the same researcher. A brief summary of the lessons is presented (see

Summary of baseline lessons.

Baseline lessons | Rubric rating |
---|---|

Word problems on percentage increase and decrease |
P1 |

C1 | |

M1 | |

This was a revision lesson. Learners were generating mind maps on the concepts of Area and Perimeter. The teacher reminded learners of the various formulae for calculations. Learners worked individually on their mind maps. | P1 |

C1 | |

M1 | |

The teacher presented methods to convert fractions to decimal numbers and percentages through using area representations. The lesson was concluded with an individual worksheet. | P1 |

C1 | |

M1 | |

The teacher presented learners with ‘sum search’ sheets where they had to find addition and subtraction problems. Learners worked in groups but worked individually to find the ‘sums’. Groups had to calculate the average number of sums they found. | P2 |

C2 | |

M2 | |

The teacher asked learners to write out tables multiplying by 10, 100 or 1000. The teacher explained that this was holding learners up in their multiplication methods, so she spent the lesson trying to get them to identify patterns such as: |
P1 |

C1 | |

M1 |

C, context scale; M, mathematical content scale; P, pedagogy scale.

Teaching by telling was observed as the dominant pedagogy during the baseline lessons. The learners were mostly seated and silent during these lessons. Only one lesson involved pupils working in groups to calculate averages followed by a whole class discussion. However, this was to present answers and not to facilitate learner constructions. This lesson was identified as showing ‘signs of change’ (P2).

During the weeks that followed the baseline observations, teachers were involved in Cycle 1 of the PD programme (see

Example of card matching activity.

Teachers discussed how two different types of questions on the same concepts could either alienate students or be more inclusive:

These [

And if I don’t remember?

Then it’s over.

But that [

They don’t have to remember that you have to do that, and then that…

You see, I am still telling them…I have to go to where they work it out for themselves…that is where I am struggling. (Biccard,

What is evident here is that teacher change must be approached holistically to include teachers’ personal domain since this is a strong enabler of changing a teachers’ domain of practice to develop improved classroom outcomes.

I visited the teachers at the beginning of the next school term;

Summary of Cycle 2 lessons.

Cycle 2 lesson | Rubric rating |
---|---|

Creating equivalent fractions. |
P2 |

C3 | |

M2 | |

The teacher explained ‘inverse’ operations using a flow diagram drawn on the board. An extensive teacher-led whole class discussion took place before individual seatwork. | P1 |

C1 | |

M2 | |

The teacher asked learners to work in groups. She provided each group with small wooden blocks and a big sheet of paper. She asked learners to calculate 2/3 of 18 and to show their working (by packing blocks on the projector) to the rest of the class using the blocks. Learners could use the paper for calculations to verify their answers if necessary. | P3 |

C2 | |

M2 | |

The teacher led a whole class discussion on presenting equivalent fractions in area models. She held an extensive question and answer sessions and wanted to show connections between learner understandings. | P2 |

C1 | |

M1 | |

The teacher led a whole class discussion on different number patterns that she presented on the board. She wanted learners to give the next number in each pattern and explain the pattern. She used both additive and multiplicative patterns. Learners were encouraged to show various methods of working out the next number. The teacher focused on learners discussing how + 2; + 4; + 8 is the same as × 2 in the pattern: 2; 4; 8 … |
P3 |

C1 | |

M2 |

C, context scale; M, mathematical content scale; P, pedagogy scale.

During the follow-up lessons in Cycle 2, two lessons demonstrated ‘signs of change’ where teachers allowed ideas from learners to guide the discussion while the question and answer sessions were more in-depth and focused on connecting the mathematical ideas.

Two lessons facilitated learners’ own constructions, which presents some evidence of professional experimentation in the domain of practice. These lessons saw learners working in groups trying to construct meaning and not working individually and silently as in the first series of lessons. In the one lesson, groups had to use blocks to explain to the rest of the class what of 18 ‘looked like’ while the second lesson involved learners working in pairs to calculate equivalent fractions of area models. The inverse operations lesson was still traditional in nature. Two lessons were still dominated by teachers specifying methods while three lessons had learners think about methods. The time allocated for student interaction (Teacher A and Teacher C) in the lessons allowed for students to think reflectively about their working with the manipulatives.

The first session of PD that followed these lessons had teachers solve the second modelling task in groups. Some teachers struggled with calculating the ratio for the task (e.g. how to increase a 4 cm length to 7 cm) and once again felt that learners would need more guidance. Teachers also worked through a variety of other proportional reasoning problems in a variety of contexts. The shift from additive reasoning to multiplicative reasoning was the main discussion of this session. The summary of proportional research (Van De Walle et al.,

During the lesson observation, I also noted that teachers’ lesson planning was not very detailed. The lesson plans were mostly populated with dates and topics but no further elaboration of concept development. I, therefore, presented a session on thinking about the mathematical goal of the lesson and hypothetical learning trajectories as an important aspect of lesson thinking and lesson planning. I wanted teachers to see that the mathematical goal of the lesson not as the lesson title, for example adding fractions, but rather as

I visited the teachers again and the lessons are summarised in

Summary of Cycle 3 lessons.

Cycle 3 lesson | Rubric rating |
---|---|

Multiplication and division word problems. |
P2 |

C2 | |

M2 | |

Learners sat in groups and had to observe and point out the features of a given 3D shape. They were then asked to create the same 3D shape using toothpicks and jelly sweets. The teacher moved around looking at the constructions and asking questions without explicitly telling learners what to do. The learners worked on traditional textbook work on 3D shapes afterwards. | P3 |

C3 | |

M2 | |

The teacher asked learners to work in groups. She provided each group with a large sheet of paper. Groups had to make as many factor trees as possible and present their solutions to the rest of the class. | P2 |

C2 | |

M3 | |

The teacher revised the names and types of 3D shapes the learners had learnt the year before. She discussed their features. She then took learners on a walk around the school grounds and asked them to identify as many of the 3D shapes as they could. She stopped at strategic places on the school grounds. The lesson concluded with a whole class discussion on the various shapes they identified. | P2 |

C2 | |

M2 | |

The teacher provided each learner with a page where some pizzas and some chocolate bars were printed. She then asked them to share the pizzas (3 pizzas shared between 4 children) or 4 chocolates shared between 12 children. Learners worked individually but were in discussion with their peers. She then led a whole class discussion on the different ways that learners used to share out the pizzas or chocolates. | P3 |

C3 | |

M3 |

C, context scale; M, mathematical content scale; P, pedagogy scale.

The third cycle lesson observation showed a move away from teachers’ teaching by telling toward a greater focus on asking learners to work collaboratively. Lessons now included questions that stimulated learner thinking rather than seeking specific answers. The external domain appeared to have activated a change cycle in the other three domains (domain of practice, domain of consequence and the personal domain).

Three lessons during Cycle 3 showed ‘signs of change’ while two allowed for learners’ own constructions to guide the lesson. In the three that showed signs of change, learners worked in groups and discussed their ideas regarding multiplication and division word problems, factor trees and 3D shapes that they identified around the school grounds. In the two lessons coded as P3, one lesson saw learners cutting paper pizzas and chocolates to share between numbers of learners and in the other lesson, learners had to construct their own 3D prism before moving onto formal terminology, diagrams and concepts. Learners were spending more time working with each other and verifying their work with each other. Teacher B’s and Teacher E’s lessons included questions that stimulated student thinking rather asking for specific answers. Lessons in this cycle also reflected many more mathematics moments due to the increase in student activity. The professional experimentation results in a change to the domain of consequence (outcomes) and changes to the personal domain. It supports Guskey’s (

The final sessions of PD once again started with the teachers solving a modelling task (see

Teachers indicated throughout the PD that they needed assistance in finding contextual problems that covered the curriculum topics at the correct level. This is consistent with Borromeo Ferri and Blum’s (

At the end of this session, I asked teachers to reflect on their own teaching. Their responses show that they are starting to consider their roles as facilitators more critically. They stated the following when asked which aspects of their teaching they still wanted to improve. These responses indicate changes in their personal domain.

To have the patience not to give groups who are struggling the answer, but to guide them patience with their methods/ideas – not to tell them how to work out the answer. Talk less and listen more to facilitate more and control less. (Biccard,

I visited the teachers at the end of the term and the summary of lessons is presented in

Summary of final lessons.

Final lessons | Rubric rating | |
---|---|---|

Learners worked in groups on lists of decimal numbers that had to be ordered. The groups had to agree on the order and one person from each group had to present their solutions. A whole class discussion with teacher connecting ideas concluded the lesson. | P2 | |

C1 | ||

M1 | ||

The teacher presented learners with a complex problem on calculating profit. It involved repackaging a large packet of peanuts into smaller quantities. Learners worked in pairs to solve it. The teacher gave them big blank sheets of paper to work on and not their usual workbooks. |
P3 | |

C3 | ||

M2 | ||

The teacher presented learners with a model-eliciting problem. Learners were given the results of athletics events and the list of ‘winners’. Some events are won because of the larger number (e.g. long jump) while other events are won because of a smaller number (e.g. a race). Learners worked in groups to determine if the winners’ list provided to them was correct. | P3 | |

C3 | ||

M3 | ||

Learners worked in pairs on multiplication and division revision problems. The teacher created two sets of worksheets with word problems using similar mathematical concepts but in different contexts. Learners presented their solutions. She led a discussion on how the problems on the two different worksheets were similar in structure and solution path. | P1 | |

C1 | ||

M2 | ||

The teacher provided each group of learners with a 3D shape. She asked them to look carefully at the edges, corners, etc. She provided each learner with a large sheet of paper. Learners had to draw a net of the shape. Thereafter she provided learners with a pre-printed net and asked them to compare their net to the pre-printed one. She asked them to look at what was the same and what was different. She then asked them to construct both nets. | P3 | |

C3 | ||

M3 |

C, context scale; M, mathematical content scale; P, pedagogy scale.

In the final lessons, two showed signs of change while three focused on learners’ own constructions. The signs-of-change lessons included pair work on word problems with a follow-up whole class discussion while the second lesson involved groups of learners working collaboratively on ordering decimal numbers. For the three learner construction lessons, one involved a model-eliciting problem and two involved learners’ own constructions on profit models and learners constructing their own nets for 3D shapes. Teacher A and Teacher D appeared to have folded back to more traditional teaching for this lesson.

The baseline lessons and Cycle 2 lessons were predominantly calculation-based lessons. During Cycle 3 lesson observations, three teachers used some other contexts as starting points (e.g. word problems) while two teachers presented truly problematic situations where no known procedures could simply be applied to the problems (sharing pizzas and constructing 3D objects). In the final lesson observations (almost a year after the start of the programme) three teachers presented truly problematic situations but only two of these teachers guided the classroom discussion to connect learner understandings. It was during this cycle that one of the teachers presented the model-eliciting problem based on finding the winner of an athletics event. The teacher noticed that this problem (presented before moving onto typical textbook problems) assisted her learners in understanding concepts related to ordering decimal numbers. This teacher tried a mini-experiment in her classes (professional experimentation). She had one of her classes work on the textbook problem first and then complete the model-eliciting problem. She found that this class struggled with the textbook problem while those that did the model-eliciting problem first did not.

Teachers’ initial lessons were typical mathematics lessons in terms of focusing on bare numbers, procedures or skills. During Cycle 2 observations, some of the teachers started exploring mathematical moments in their classes by allowing learners to enter the mathematical discussion through self-directed activities and group work. By Cycle 3 observations, two teachers had their classes explore mathematical ideas through their own constructions (e.g. pizza sharing lesson). By exposing mathematical ideas through contexts, teachers in this study were able to capitalise on the ‘mathematics moments’ in their classrooms. Of the 20 lessons observed, only 4 reached the third level on the pedagogy scale. Changing teacher practices is a complex endeavour and the time needed for paradigm shifts in mathematics teaching cannot be underestimated (Guskey,

The lesson observation rubric focused on three big areas (pedagogy, context and content) where the potential for mathematisation can be gauged in a classroom. The three elements of the rubric are interrelated in terms of classroom practice. ‘Teaching by telling’ was often associated with bare numbers emphasising procedural mathematics (with the teacher sticking to a predesigned script) while a pedagogy of facilitating learner constructions necessitated the use of truly problematic situations and a focus on the underlying structure of the problem. In the latter lessons, teachers were using learner involvement to a greater extent, although not always making explicit connections between learner ideas. Teachers’ evolving pedagogy involved using more contexts and more focus on learners’ own constructions during the lessons. This is consistent with the findings of Cobb et al. (

Teachers are the most ‘critical layer of the school system in terms of efforts to change what happens in schools’ (Smith & Southerland,

This study sought to contribute to the micro theory level by proposing activities such as modelling tasks for teacher professional development and did not necessarily seek replication of tasks in the classroom. In an attempt to move away from deficit-type narratives regarding teachers, the DBR approach in this study did not prescribe which practices teachers

This study was limited to five teachers in one region of South Africa, teaching Grade 5 and Grade 6 mathematics that is congruent with one of the known limitations of DBR – designing to scale (Cobb et al.,

The author declares that she has no conflicting interests in producing or publishing this article.

I declare that I am the sole author of this article.

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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.