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Abstract of the Working Seminar (16-18 years old)

Surface and deep approach in the mathematics classroom
Lambrecht Spijkerboer,

STA-international, the Netherlands

As a trainer/consultant in the educational field my current work is to organise training for in-service teacher training. Because of visiting classrooms weekly, I can connect new ideas for education direct to daily practice and making it profitable. In the different working seminars we will try to find opportunities for making motivational lessons for gifted and non-gifted students in our classes and work on topics like:
  • Surface and deep approach in learning mathematics. In an everyday math lesson, besides the explanation of the knowledge, mostly making exercises (of the same kind as explained the instruction) is also done part of the time. The idea behind this lessondesign is: having enough practice is the way to master the theory, and be able to pass exams with equal exercises as were proposed during the education before. Important is to distinguish the differences between routine exercises where students are supposed to give back what was learned by the instruction of the teacher (reproduction), and challenging motivational questions to invite students to think themselves and solve problems by using their mathematical knowledge (insight). The differences between the learning activities in dealing with exercises are explained more extended in the “OBIT model” (Maréchal & Spijkerboer, 2017).
  • Different approaches in building the concept of relations. I.e. The discovery of relations is an important part of mathematics education. Relations are described by stories, tables, formulas and graphs. We will work on different problems to invite students to understand the meaning of a formula, a graph, an algebraic solution, … Does it make sense to start with the description in words, doing manipulations in the field and go from there to the more numeric side of the problem? By using the OBIT-model, you see different approaches in building the concept of relations. Mostly math and science books, start with a formula, to make a table and graph of it. The relation is imagined by the graph, several questions are asked, to read the graph properly. All this kind of activities, are mostly done by learning activities like Remembering and Understanding. (surface approach), in order to come to deep approach later. After a lot of rehearsals. There is another way to start: with Integration and Application. It is a didactical choice to start with Integration and Application (deep approach) and to go to Understanding later. The need for doing calculations and computation is more obvious after students have learned the key of the concept and about the application of the concept because that motivates the learning.
  • Ways of working to deal with differences. Different ways of working, for exploring mathematical skills, for rehearsal of routine tasks and for explanation of the concepts behind the posed problems can be discussed. Nevertheless it always turns out to have an active participation of students. A lot of research done about cooperative learning is useful for exploring different ways of working in the mathematics classroom to invite students for deep approach (Hill, S. & Hill, T., 1990; Bellanca J. & Fogarty R., 1994). The use of different ways of working not only makes the learning more effective, but also has impact on the way students perform in their mathematical development, besides learning to cooperate with person you didn’t choose yourself. This has implications to the role of the teacher to provide students with questions instead of answers, process-help instead of product-help only and tasks instead of exercises. The focus is on motivational lessons with active participation of students.
Colby, S.A. & Smith, T.W. (2007). Teaching for deep learning: Washington: Heldref Publications.
Hill, S. & Hill, T. (1990). The Collaborative Classroom: A Guide to Cooperative Learning. South Yarra, Victoria: Eleanor Curtain.
Maréchal, J. & Spijkerboer, L (2017) Leerlingen AANzetten tot leren, Pica Huizen, the Netherlands
Spijkerboer, L. & Santos, L. (2015). Organising dialogue and Enquiry in Gellert, U e.o Educational Paths to Mathematics, Springer, Swiss.
Spijkerboer, L., Math that matters in Sabena, C. (a cura di) (2015). Teaching and Learning mathematics: resources and obstacles. Actes/Proceedings CIEAEM 67. Palermo: Università degli Studi di Palermo - GRIM.

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