The research reported in this paper is design research: its purpose is to develop a ‘toolkit’ of resources, consisting of classroom mathematics tasks, lesson plans and guidance. This toolkit will support teachers in using formative assessment effectively in their classrooms.

Design research in education, also called design-based research or design experiments, is essentially an intervention in which something, such as, for example, a lesson plan, is designed drawing on previously developed theory. It is tried out and iteratively improved in the light of what happened (Carraher & Schliemann, 2002; Cobb, Confrey, Disessa, Lehrer, & Schauble, 2003; Collins, Joseph, & Bielaczyc, 2009). In the case of schools and classrooms, although improving the design is the focus of the intervention, it is also hoped that, in the course of the intervention, students will learn some mathematics and teachers will learn something about teaching mathematics.

In the research reported here, the intervention was the classroom implementation of pre-written lessons or ‘learning experiences’ (Swan, 2008). Teachers were given mathematics tasks to use with their students and asked to adopt teaching approaches which emphasise formative assessment (Black & Wiliam, 2009). In particular they were asked to have the students work in small groups or pairs, and to discuss the work, thus providing teachers with opportunities to gather information about their students’ current levels of understanding. It was suggested that they should not intervene in the students’ discussions or tell the students what to do, but rather respond to the information they were gathering by asking questions to move the students’ thinking on. The teachers were given lesson plans and guidance notes, which explained what the teacher should do and why, provided examples of questions they might ask and described some of the approaches the students might take. The findings from the classroom interventions will inform the development of the lessons, and accompanying guidance, which will eventually have a place in the toolkit. This paper discusses one such lesson.

The lesson was trialled with three teachers and their classes in a college for Technical and Vocational Education and Training (TVET). In addition, one secondary school teacher trialled the lesson with his class. All the students were following a mathematics literacy course. Mathematics literacy aims to prepare students for real-world situations, and one approach is to provide them with tasks similar to those they could encounter in the workplace. When the research took place, they were working on scale drawings, ground plans and related ideas. The teachers said they would like to teach a lesson within this topic area and the researchers made a suggestion.

The lesson they were asked to teach was a problem-solving task (Schoenfeld, 2013), which allowed multiple different approaches and outcomes. The lesson was written for an American audience, and one of the research questions was how it should be adapted for a South African audience.

In the lesson, the students are asked to take the role of a garden designer and are given an email from Mandy, the client. Mandy has a set of requirements for the garden design: she says she has ordered a shed (and gives the dimensions), she wants decking for the barbecue area, with enough space to seat six people, a circular pond of about 7m^{2} and some paths and borders. It was highly unlikely that any of the students would live in a house with a garden such as Mandy’s, but it was likely that they would have seen such houses and gardens. The students are given a sheet of paper with the plan of a rectangular garden drawn on it: on one of the shorter sides is the house, and in one of the longer sides is a gate. The plan indicates that the longer side measures 10m. Students are also given a ruler, so that they can measure the long side of the garden, which is 20cm. They calculate the scaled size of the garden features, and place them in the garden in line with the constraints Mandy has set out (e.g. the barbecue area should be near the patio doors). Clearly there is no one correct design, and students have some freedom to choose their own design, but they should also be able to explain their choices. To finish the lesson, the teachers decided that they wanted some of the small groups to present their design decisions to the class.

Almost all the students in all the classes were Xhosa speaking although in the TVET college there were also some Cape Coloureds, who spoke Afrikaans and English fluently. The language of instruction is English in all cases. However, in the light of the language of the students and their cultural background, the teachers and researchers changed some of the language in Mandy’s email. (e.g. ‘paving for the braai area’ rather than ‘decking for the barbecue area’). It was also decided to provide the students with coloured paper (brown for the shed, blue for the pond), scissors and glue, in case they wanted to measure and cut out shapes to represent the features of the garden. These could then be moved around until the students were happy with their design.

The three teachers at the TVET college taught the lesson on the same day, but at different times. The teacher at the secondary school taught the lesson about two months later. After each lesson the researchers and teachers discussed what had happened in the lesson and made some adaptations to the lesson plan in the light of these discussions.

This paragraph describes examples of adaptations. In the first lesson, it seemed that cutting the shapes out of paper confused the students so much that they could not get started. The second teacher decided not to use cut out paper but rather to ask students to sketch the shapes. Some students said that they did not understand the word ‘shed’ and the teacher explained that it was a Wendy House. Some students asked what a pond is, and the teacher explained that it was ‘like a small dam’. The second and third teachers explained these terms at the start of the lesson and for the secondary school, Mandy’s email was changed to include the words ‘Wendy House’. In the TVET college, almost all students attempted to calculate the radius of the pond using the area formula, but most needed help from the teacher, who told them what to do. Very few of the students completed their garden design in one class period (an hour). For the secondary school Mandy’s email was further changed to leave out the borders, to change the round pond to a flower bed and to give the diameter of the flower bed.

The revised version of the lesson uses simplified language, and vocabulary suitable for the South African context. It has fewer requirements from Mandy, as outlined above. The

As explained above, although the ultimate focus of the research was the design of the lesson and accompanying supporting material, it was also important to the researchers and the teachers that the students benefitted from the experience. Overall, it seems that the lesson was well received by teachers and students. At the TVET college, students filled in a post-lesson questionnaire and the results suggest that for these students, the most important things that stood out were group work and, related to this, the fact that the task encouraged them to discuss their ideas. It also seems that the task forced them to think, and it was relatively exciting, but perhaps a little confusing. Only one said it was boring. All four teachers said that they had liked the task, and the teaching approach, and would try it again with other classes.

References

Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. *Educational Assessment, Evaluation and Accountability*, *21*(1), 5–31.

Carraher, D., & Schliemann, A. (2002). Design Research: What We Learn When We Engage in Design. *Journal of the Learning Sciences*, *11*(1), 1–24.

Cobb, P., Confrey, J., Disessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. *Educational Researcher*, *32*(1), 9–13.

Collins, A., Joseph, D., & Bielaczyc, K. (2009). Design Research: Theoretical and Methodological Issues. *Journal of the Learning Sciences*, *13*(August 2014), 15–42.

Schoenfeld, A. (2013). Reflections on Problem Solving Theory and Practice. *The Mathematics Enthusiast*, *10*(1), 9–34.

Swan, M. (2008). Designing a Multiple Representation Learning Experience in Secondary Algebra. *Educational Designer*, *1*(1), 1–17.

Acknowledgement

This project was funded by the European Union under the Framework 7 programme.