K. Juuti, and J. Lavonen. NorDiNa, (2006)Construction of research based teaching sequences through Developmental research (Linsje, 1995), Educational reconstruction (Duit, Komorek & Wilbers, 1997), or Ingenierie Didactique (Artigue, 1994), can be considered very similar with design-based research. On the one hand, these approaches take into careful consideration students’ previous knowledge and emphasise basic scientific concepts and how they are related to the teaching sequence (Méhuet, 2004) and on another hand they aim to design the artefacts. For example, Andersson and Bach (2005) produced a teacher guide as an artefact describing the research-based sequence for teaching geometrical optics. However, these approaches focus on research-based design and the adoption of the innovations needs, for example, teachers’ in-service training.
(p 56).
M. Scaife, Y. Rogers, F. Aldrich, and M. Davies. CHI '97: Proceedings of the SIGCHI conference on Human factors in computing systems, page 343-350. New York, NY, USA, ACM Press, (1997)
J. Maloney, L. Burd, Y. Kafai, N. Rusk, B. Silverman, and M. Resnick. C5 '04: Proceedings of the Second International Conference on Creating, Connecting and Collaborating through Computing, page 104-109. Washington, DC, USA, IEEE Computer Society, (2004)
H. Freudenthal. Educational Studies in Mathematics, 1 (1/2):
3-8(May 1968)Systematization is a great virtue of mathematics, and if possible, the student has to learn this virtue, too. But then I mean the activity of systematizing, not its result. Its result is a system, a beautiful closed system, closed, with no entrance and no exit. In its highest perfection it can even be handled by a machine. But for what can be performed by machines, we need no humans. What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematizing reality and if possible even that of mathematizing mathematics..
P. Thompson, and L. Saldanha. Research companion to the Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA, (2003)
M. Masuch, and M. Rüger. Proceedings of the 3rd Confernece on Creating, Connecting and Collaborating through Computing, 2005. (C5 2005), page 67-74. Cambridge, MA, (2005)
J. Smith, and P. Thompson. Employing children's natural powers to build algebraic reasoning in the context of elementary mathematics, Erlbaum, New York, (2007)
P. Ernest. Why Learn Maths, London University Institute of Education, London, 1. To reproduce mathematical skill and knowledge based capability
The typical traditional reproductive mathematics curriculum has focused exclusively on this first aim, comprising a narrow reading of mathematical capability. At the highest level, not always realised, the learner learns to answer questions posed by the teacher or text. As is argued elsewhere (Ernest 1991) this serves not only to reproduce mathematical knowledge and skills in the learner, but to reproduce the social order and social injustice as well.
2. To develop creative capabilities in mathematics
The progressive mathematics teaching movement has added a second aim, to allow the learner to be creative and express herself in mathematics, via problem solving, investigational work, using a variety of representations, and so on. This allows the learner to pose mathematical questions, puzzles and problems, as well as to solve them. This notion adds the idea of creative personal development and the skills of mathematical questioning as a goal of schooling, but remains trapped in an individualistic ideology that fails to acknowledge the social and societal contexts of schooling, and thus tacitly endorses the social status quo.
3. To develop empowering mathematical capabilities and a critical appreciation of the social applications and uses of mathematics
Critical mathematics education adds in a third aim, the empowerment of the learner through the development of critical mathematical literacy capabilities and the critical appreciation of the mathematics embedded in social and political contexts. Thus the empowered learner will not only be able to pose and solve mathematical questions, but also be able to address important questions relating to the broad range of social uses (and abuses) of mathematics. This is a radical perspective and set of aims concerned with both the political and social empowerment of the learner and with the promotion of social justice, and which is realised in mainstream school education almost nowhere. However, the focus in the appreciation element developed in this perspective is on the external social contexts of mathematics. Admittedly these may include the history of mathematics and its past and present cultural contexts, but these do not represent any full treatment of mathematical appreciation.
4. To develop an inner appreciation of mathematics: its big ideas and nature
This fourth aim adds in further dimension of mathematical appreciation, namely the inner appreciation of mathematics, including the big ideas and nature of mathematics. The appreciation of mathematics as making a unique contribution to human culture with special concepts and a powerful aesthetic of its own, is an aim for school mathematics often neglected by mathematicians and users of mathematics alike. It is common for persons like these to emphasise capability at the expense of appreciation, and external applications at the expense of its inner nature and values. One mistake that may be made in this connection is the assumption that an inner appreciation of mathematics cannot be developed without capability. Thus, according to this assumption, the student cannot appreciate infinity, proof, catastrophe theory and chaos, for example, unless they have developed capability in these high level mathematical topics, which is out of the question at school. The fourth aim questions this assumption and suggests that an inner appreciation of mathematics is not only possible but desirable to some degree for all students at school..(2000)
V. Barre, C. Chaquet, and H. El-Kechaï. Proceedings of Artificial Intelligence in Education: Workshop on Usage Analysis in Learning Systems, Amsterdam, (2005)
F. Arzarello, M. Bussi, and O. Robutti. Proceedings of the 28th International Conference of the International Group for the Psychology of Mathematics Education, 4, page 89-96. (2004)
J. Carroll, G. Chin, M. Rosson, and D. Neale. Proceedings of the 3rd conference on Designing interactive systems: processes, practices, methods, and techniques, (2000)
V. Giraldo, L. Carvalho, and D. Tall. Proceedings of the 27 thAnnual Conference of the International Group for the Psychology of Mathematics Education, (2003)
L. Radford. Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, 1, page 143-145. Melbourne, Australia, University of Melbourne, (2005)
J. DeLoache. Child Psychology in Retrospect and Prospect: In Celebration of the 75th Anniversary of the Institute of Child Development, Lawrence Erlbaum Associates, Mahwah, NJ, (2002)
L. Lee. Approaches to algebra: perspectives for research and teaching, Kluwer Academic Publishers, p 102
… it is much of a challenge to demonstrate that functions, modelling, and problem solving are all types of generalizing activities, that algebra and indeed all of mathematics is about generalizing patterns.
p 103
The history of the science of algebra is the story of the growth of a technique for representing of finite patterns.
The notion of the importance of pattern is as old as civilization. Every art is founded on the study of patterns.
Mathematics is the most powerful technique for the understanding of pattern, and for the analysis of the relationships of patterns.(1996)