Nigel Calder by Processing Mathematics Through Digital Technologies

Author:Processing Mathematics Through Digital Technologies
Language: fra
Format: azw3
Published: 2022-09-20T10:09:11.196000+00:00

CHAPTER 6

NEGOTIATING SHARED MEANINGS

The following data was interesting for the way in which the two pupils focussed on (Jo & Sam) drew different interpretations of the same data, when it was engaged through the same pedagogical medium. It illustrates how approaching it from varying individual conceptual positions differentiated their interpretation. They disagreed with each other’s generalisation, but through further iterations of the hermeneutic circle, each interpretation was folded into the other’s evolving perspective. Through investigation with the spreadsheet, and the subsequent discourse, they found a common appropriate interpretation. Their disagreement, the tension generated by the other’s approach, followed by the moderation of their personal perspective, led to the accommodation of a shared interpretation of the generalisation, that was facilitated by the exploratory medium.

They had already negotiated the sense of the task through initially entering a formula to represent the situation, then interpreting the table of values that was generated. They drew on the preconceptions borne of their prevailing discourses.

At this stage they had used the table to make generalisations regarding multiplying two-digit numbers by 101, and were now investigating multiplying three digit numbers by 101. They had adapted their approach to entering a three-digit number, then interpreting the output through their existing frame.

95243

They had tried 943, which generated:

99283

Then 983, which generated:

The pupils’ initial generalisations centred about viewing their data through a visual lens. They looked at the situation of the digits and changes to particular digits in those positions. They both interpreted the situation from their personal perspectives, but as they oscillated between these broader perspectives and engaging with the task, their interpretations diverged.

Jo:

First digit of the 3-digit number is the starting number of the final number.

Sam: First digit equals first digit; second digit equals second digit plus one. Get it? Third digit equals third digit minus one.

Jo:

Let’s try another. 18584.

Sam: What’s your other number?

Jo:

184. The middle one is the last one added to the first, or is it plus one.

Jo seemed intent on building a more rigorous generalisation by exploring other inputs, whereas Sam had found a pattern that fits the first two outputs generated and was keen to formalise that in some way. She seemed to anticipate that there 82

AN INFLUENCE ON THE LEARNING PROCESS

was a generalisation that would give her a methodology to predict. She was motivated by what she saw in front of her in a visual sense, her experience with the table from the two-digit exploration, and the direction in which the medium and its underlying discourses led her initial conjecture. Once again, their personal perspectives influenced their interpretation of the situation in diverging ways, as evidenced by what they said and what they did, by their dialogue and actions.

Sam:

My rule is first digit equals first digit; second digit equals second digit plus one. Third digit equals third digit minus one. Fourth digit equals second digit; fifth digit equals third digit.

Jo:

Let’s try… no, 3 digits now.

Sam: Let’s try 175.

1 7 6 7 5

was generated on screen.

Sam: That’ll be 18475 (Using her version of the rule:

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