Oliver: Diofantus also did not use algebraic symbols Adriana: How can a student of that age construct such a proof? Although the class seemed to come to a consensus, one prospective teacher reminded the audience that the mathematics curriculum suggested a formal approach to problems of that kind.
This creates some tensions again: has been taught. Following a problem solving method mechanically is not good for his future mathematical development. It is more important to encourage him to make nd meeting Through the above comment, Oliver brought to the foreground the critical role of symbols and to view it as an integral pa development.
In the third, fourth and fifth meeting the prospective teachers commented on critical incidents which they had noticed in the classroom observations. A variety of issues difficulties to make connections between different representations, the relation nd prior knowledge in learning and the effectiveness of specific teaching examples and tasks e.
In the discussions a central issue was the construction of mathematical meaning. Initially However, during the discussion the prospective teachers started to link this phenomenon to t findings. We will try to indicate these shifts by using examples from our data. Some examples were the transformation of a fraction to an equivalent one, the solution of a first degree equation, arithmetic or algebraic computations.
The first student completed the transformation by multiplying both terms of the fraction by 6. Then the second student wondered why he did not use a faster common technique nominator.
The prospective teacher interpreted the phenomenon by considering this conceptual understanding: rd method works, the essence of the method. Kostas stated that he from his fieldwork observation how another teacher managed a similar situation. The prospective teacher found this approach original as it was beyond his own experiences: was emphasizing that we do the inverse operations.
I find this approach very different, more advanced Lefteris, 3rd class meeting In a subsequent class meeting, the prospective teachers themselves started to build connections between learning, teaching and research. They had been asked to find and read a research paper that would help them to develop adequate explanations of the fact that students often do things at an operational level without deeper managed to give deeper interpretations by realising the meaning of the variable, the double meaning of the equal sign and the transition that the students needed to make from arithmetic to algebra.
In terms of teaching they identified tasks such as a tree diagram that could help students to understand the priority of operations and use it for solving equations or they talked about the emphasis needed to be given on algebraic structures in arithmetic.
Finally, they started to focus on the role of teaching practices to the development of learning and identify fine elements of teaching. In the last three class meetings the prospective teachers presented critical incidents or unexpected responses, to the appropriateness of the designed tasks, to epistemological aspects and to classroom management. Almost all the prospective teachers participated in the discussion by presenting and justifying their critical phase, the main part o representations; the difference between procedural and conceptual understanding; cognitive and affective needs; and the epistemological characteristics of geometry.
Moreover, they often seemed to overtly recognise the critical role of tasks in In the 7th class meeting, Katerina talked about what she learned from her 8 th grade students while working on a task she had designed for comparing the areas of three irregular polygonal areas: regular shapes.
We let the students to work on their own. I had expected them to develop three or four different strategies but when I analysed them afterwards I discovered that they were twelve! What I have understood is that when you let the students to work on themselves they have a lot of different ideas. We can also see how they are thinking Moreover, she acknowledged the importance to provide space to all students to think mathematically during the teaching.
In the discussion that followed the other prospective teachers also commented on the added think that it has to do with the nature of the tasks. A very specific task does not allow for Spirithoula, 7th class meeting In the last class meeting Aggeliki and Maria presented a critical incident from their teaching in a 9th grade class.
One student who was engaged in calculating areas in the geometrical context he recalled the formula without connecting it to the problem. The two prospective teachers did not expect this response and they interpreted that the student did not make any connection to the problem but he only recalled the relation without understanding: th class meeting was at school.
The teacher was telling me something and when I th did not know it I was giving him the formula I knew. An Se Thimitho - Giannis Tassios Monos Arga Mesanyxta - Giannis Ploutarxos Krisi - Nikos Makropoulos Osoi Antres Agapane - Giorgos Tsalikis Mipos Eimai Trelos - Notis Sfakianakis O Aetos - Notis Sfakianakis Feugw Xorizw - Pasxalis Terzis Triti Porta - Triantafillos Xamenoi Politeia - Vassilis Karras Oi Nyxtes Tou Trelou - Antypas Enas Katharos Egoismos - Triantafillos Sou Vazo Diskola - Natasa Theodoridou Add image 5 more photos.
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