Abstract
Before taking the reader into the classroom, we need to introduce the children who will be found there. Other chapters introduce the school in which the classroom was located and the teachers who carried out the lessons, but here we are focusing on the students in a particular classroom. Who were they? The first key piece of information is that since the school was an essential element of the COREM (Center for Observation for Mathematics Education Research) admissions were emphatically not selective. The school was the public school for a blue collar neighborhood, and its students were the ones who lived around it. Parents were kept informed about the unusual aspects of the teaching, but there were no special requirements or requests of them. On the other hand, the lessons we visit took place in the fifth grade with students most of whom had been at the school since age three or four, so all of their expectations for what would happen in a mathematics class were built around the kind of activity and responsibility we see in action. They needed no persuasion to involve themselves.
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Notes
- 1.
This use of the equal sign is incorrect. The teacher will mention it during the discussion time.
- 2.
Some expressions used spontaneously by the children to make themselves understood – either by the teacher or by the other children – are adopted by the whole class and accepted by the teacher. Some terms are thus used that are not necessarily “mathematical” but only serve temporarily for dealing with particular situations. They are not institutionalized, and are therefore later forgotten. Examples: “trap” and “in-between number”
- 3.
The drawing should be prepared before class, either on the board or on paper.
- 4.
“The more you pedal less hard, the less you go forward”, as a child once explained to a flabbergasted psychologist.
- 5.
As well as the English one!
- 6.
We have shown that if the teaching of “problem-solving methods” follows the classic conceptions relative to knowledge and learning, the teaching will lead to uncontrollable metadidactical slippage and to failure. [Metadidactical slippage is discussed in Chap. 5.]
- 7.
Connaissances and savoir both translate to “knowledge”, but they are used very distinctly. A good working definition is that connaissance is general knowledge and savoir is reference knowledge. For a more nuanced definition, see Chap. 5.
- 8.
A problematic is something that constitutes a problem or an area of difficulty in a particular field of study [Oxford English Dictionary] The French use problematics more specifically to refer the set of questions posed in a science or philosophy with respect to some particular domain.
- 9.
This activity can take place in the context of a Social Studies class – a drawing, for example.
- 10.
The teacher’s pantographs are also set to scale factors of 3 and 1.5.
Bibliography
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Brousseau, G., Brousseau, N., & Warfield, V. (2004). Rationals and decimals as required in the school curriculum Part 1: Rationals as measurement. Journal of Mathematical Behavior, 23, 1–20. Elsevier.
Brousseau, G., Brousseau, N., & Warfield, V. (2007). Rationals and decimals as required in the school curriculum Part 2: From rationals to decimals. Journal of Mathematical Behavior, 26, 281–300. Elsevier.
Brousseau, G., Brousseau, N., & Warfield, V. (2008). Rationals and decimals as required in the school curriculum. Part 3: Rationals and decimals as linear functions. Journal of Mathematical Behavior, 27, 153–176, Elsevier.
Brousseau, G., Brousseau, N., & Warfield, V. (2009). Rationals and decimals as required in the school curriculum. Part 4: Problem-solving, Composed Mapping and Division. Journal of Mathematical Behavior, 28, 79–118, Elsevier.
Warfield, Virginia McShane. (2007). Invitation to Didactique, Xlibris.
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Brousseau, G., Brousseau, N., Warfield, V. (2014). The Adventure as Experienced by the Students. In: Teaching Fractions through Situations: A Fundamental Experiment. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2715-1_2
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