Teaching and Learning Research Papers

Please click on the links below to read the abstracts of the papers, alternatively you may wish to scroll through them.

Lapp D, Berry J and Nyman M (2010) Student Connections of Linear Algebra Concepts: A Temporal Analysis of Concept Maps
International Journal of Mathematical Education in Science & Technology, 41 No 1, 1 – 18
This article examines the connections of linear algebra concepts in a first course at the undergraduate level. The theoretical
underpinnings of this study are grounded in the constructivist perspective (including social constructivism), Vernaud’s theory of conceptual fields and Pirie and Kieren’s model for the growth of mathematical understanding. In addition to the existing techniques for analysing concept maps, two new techniques are developed for analysing qualitative data based on student-constructed concept maps: (1) temporal clumping of concepts and (2) the use of adjacency matrices of an undirected graph representation of the concept map. Findings suggest that students may find it more difficult to make connections between concepts like eigenvalues and eigenvectors and
concepts from other parts of the conceptual field such as basis and dimension. In fact, eigenvalues and eigenvectors seemed to be the most disconnected concepts within all of the students’ concept maps. In addition, the relationships between link types and certain clumps are suggested as well as directions for future study and curriculum design.

Pratt, N. (forthcoming, 2010) Learning mathematics outside the classroom. In Waite, S. (Ed.) Learning outside the classroom: birth to eleven. London, Sage Publications.

Pratt, N. & Back, J. (2009) ‘Spaces to discuss mathematics: communities of practice on an online discussion board’, Research in Mathematics Education, 11(2), pp. 115 – 130.

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Kelly P, Berry J and Battersby D (2007), Developing teacher expertise: teachers and students doing mathematics together.
Journal of In-service Education, 33 No 1, 41-65
This article provides an exploration and evaluation of teachers’ experiences of participating in a Postgraduate Professional Development opportunity where they learnt alongside their students, thus allowing teacher expertise to grow within a complex web of distributed knowing and collaborative learning. A group of thirteen teachers, each accompanied by a small group of their students, attended a mathematics masterclass at their local university led by a professor of mathematics education. Both students and teachers were supported in working as mathematicians together, but were debriefed separately. The students reflected together on what they had learned about how mathematicians see the world, whilst the teachers critically reflected on how the expert mathematician supported the learning of novices, comparing this to learning in their classrooms and devising implications for their future practice. The teachers’ views of this process were recorded in an ongoing reflective journal, as were their contributions to an end of course evaluation session. Teachers completed a questionnaire and a sample was interviewed three months after the final session. These data are analysed raising implications for teacher learning and future research.

Pratt, N. & Berry, J. (2007) ‘The joy of mathematics’, in D. Hayes (Ed.) Joyful teaching and learning in the primary school, Exeter, Learning Matters.

Pratt, N. (2006) Interactive maths teaching in the primary school, London, Paul Chapman Publishing.

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Pratt, N. (2006) ‘’Interactive’ teaching in numeracy lessons: what do children have to say?’, Cambridge Journal of Education, 36(2), pp. 221 – 235.

BerryJ and Sahlberg P (2006) Accountability Affects the Use of Small Group Learning in School Mathematics
Nordic Studies in Mathematics Education, 11 No 1, 5-32
This study investigates the perspectives of a sample of teachers on the use of cooperative small groups in the teaching and learning of mathematics. We asked teachers (N = 18) in England and Finland about their experiences and ideas of small group learning in mathematics. The research tool used the ordering by each teacher of eight mathematics tasks into a hierarchy from those tasks that are best for small group working to those tasks that are best for individual working as a frame for in-depth interviews. We conclude that the role of small group learning as seen by most of the teachers is for doing mathematics, introducing social skills and discussion rather than learning mathematical knowledge and skills. Furthermore we report on the barriers to using small group learning caused by the accountability structures inherent in the educational systems of both countries.

Picker, S.H. (2005). Mathematical Reasoning Using Bipartite Graphs and Graph Colouring. Mathematics in School, 34(1).

MacBean J, Graham E and Sangwin C (2004), Group Work in Mathematics: A Survey of Students' Experiences and Attitudes, Teaching Mathematics and its Applications, Vol. 23, 49-68.
This study investigates students’ experiences of group work at both school and at university, along with their current attitudes towards group work, in mathematics. Three groups of students from differing universities were involved, and data was gathered using a questionnaire and semi-structured interviews. Results revealed some significant differences between the universities in the students’ experiences of, and attitudes towards, group work. Overall their attitudes were positive, but mainly utilitarian, and the most common form of group work they experienced was of an informal nature. No difference was found in their school experience of group work.

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Berry J and Nyman M (2003) Promoting students’ graphical understanding of the calculus. Journal of Mathematical Behaviour,
22, 481-497
Our purpose in this paper is to report on an observational study to show how students think about the links between the graph of a derived function and the original function from which it was formed. The participants were asked to perform the following task: they were presented with four graphs that represented derived functions and from these graphs they were asked to construct the original functions from which they were formed. The students then had to walk these graphs as if they were displacement-time graphs. Their discussions were audio tape recorded and their walks were captured using data logging equipment and these were analysed together with their pencil and paper notes. From these three sources of data we were able to construct a picture of the students' graphical understanding of connections in calculus. The results confirm that at the start of the activity the students demonstrate an algebraic symbolic view of calculus and find it difficult to make connections between the graphs of a derived function and the function itself. By being able to 'walk' an associated displacement time graph we propose that the students are extending their understanding of calculus concepts from symbolic representation to a graphical representation and to what we term a 'physical feel'.

Sahlberg P. and Berry J., 2003, Small Group Learning in Mathematics: Teachers’ and pupils’ ideas about groupwork in school. Finnish Educational Research Association, Volume 13, ISBN 952-5401-12-XS; ISSN 1458-1094, Turku Finland
The style of teaching and learning mathematics and the type of tasks used affect the conceptions, attitudes and views of mathematics held by teachers and pupils. Our purpose in this research study is to investigate the occurrence of small group learning and the methods of co-operative learning in school mathematics in Finland and in England. Furthermore, we wish to draw more educators’ and researchers’ attention to the role of these methods of learning and to the importance of task design in implementing them into school mathematics.
We conclude that small group learning is not common in the teaching and learning of mathematics and that the tasks used in school mathematics determine the teaching and learning styles and these are not conducive to the methods of contemporary co-operative learning.

Sahlberg P. and Berry J., 2002, One and One is Sometimes Three in Small Group Mathematics Learning, Asia Pacific Journal of Education, Vol 22 No 1, pp 83 - 94
In recent years, Mathematics teaching has been confronted by demands for higher standards and better pupil achievement in several parts of the world. Researchers have suggested a shift from teacher-centred instruction towards more active participatory learning methods as one way to improve the quality of the learning process. The tension between whole class teaching versus small group learning in Mathematics has been particularly apparent in many education systems. This article analyses the development of Mathematics teaching by asking whether small group learning is an effective arrangement in teaching school Mathematics. We conclude that although there is no unanimity about the effects of small group learning on student achievement in school Mathematics, it seems that it produces at least equal academic outcomes among all students compared to more traditional methods of instruction. Working in pairs is a particularly effective form of learning Mathematics and that small groups are beneficial for developing mathematical problem-solving skills. We also conclude that the present educational policies and increased quality assurance structures in many countries conflict, or are not consistent with scientific-professional thinking and research on the teaching of mathematics.

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Berry J.and Nyman M., 2002, Small Group Assessment Methods in Mathematics, The International Journal of Mathematics Education in Science and Technology, Vol 33 No 5, pp 641-649
Assessment of students’ attainment in courses is often driven by the method of instruction. When mathematics is taught in the traditional style of lectures on theory coordinated with homework on standard problems, the testing is often oriented to reproducing the skills demonstrated by the instructor. If a more collaborative teaching method is used, how does one assess the students’ acquisition of problem-solving and mathematical thinking skills. In this paper we discuss a team-oriented formal testing method used in a mathematical modelling course taught during the Alma College intensive spring term.

Nyman M. and Berry J., 2002, Developing Transferable Skills in Undergraduate Mathematics Students through Mathematical Modelling, Journal of Teaching Mathematics and Its Applications, 21 No 1, 29-46, 2002, ISSN 0268-3679
Undergraduate mathematics students should possess certain transferable skills upon completion of a degree. The Mathematical Association of America’s recently published CUPM Discussion Papers about Mathematics and the Mathematical Sciences in 2010: What Should Students Know and the NCTM’s Principles and Standards for School Mathematics both suggest the importance of transferable skills for success in student’s later careers. These skills and a strategy for developing them via an intensive short course on mathematical modelling are discussed. Student reactions and comments on the course are included. The evaluation suggests that this course on mathematical modelling developed modelling as well as generic skills through the teaching, learning and assessment style.

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