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From Primitive Knowing to Formalising: The Role of Student-to-Student Questioning in the Development of Mathematical Understanding
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Aiesha explains that she is initially counting both handshakes, then dividing the second handshake out. She draws the chart (bottom of figure 6) as if she were A and Shaniqua were B. She continues by writing a 2 between two circles (which represent people) on her picture to show the two handshakes that took place between each set of two people (top of figure 6 and the top of figure 2) to answer Crystal’s question.
Figure 6: Linking representations to each other
Aiesha was able to show how her formula mapped into her original representation
involving circles and loops, the action of actually shaking hands, as well as her chart with letters. Her ability to set up a hypothetical situation about the existing problem, develop multiple representations for the same idea, connect the representations to each other, and ultimately provide a solution that is generalizable indicates that she has reached the formalising layer. Further, some six months later, Aiesha’s class was given the opportunity to investigate a task that was structurally similar to this handshake problem. Within a few minutes, Aiesha and her group moved through most of the representations they constructed six months earlier, and reconstructed the formula to generalize, using the correct symbolic notation. Interestingly enough, many students around the room also used the formula Aiesha presented six months earlier for this new task.
CONCLUSIONS
We conclude that student-to-student interactions and questions played a central role in
Aiesha’s movement from primitive knowing to formalising, as well as her movement to linking representations to each other. This ultimately led to her ability to retain and retrieve her ideas when presented with similar types of problems months later, which is a central goal of the teaching and learning process. Of course, we cannot say with complete certainty that these interactions were exclusively responsible for the development of the ideas, however, we believe that our analysis suggests that they played a key role. While it is not possible to draw overwhelming conclusions based on this limited example, we do believe that an analysis of
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| | Authors: Warner, Lisa. and Schorr, Roberta. |
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Aiesha explains that she is initially counting both handshakes, then dividing the
second handshake out. She draws the chart (bottom of figure 6) as if she were A
and Shaniqua were B. She continues by writing a 2 between two circles (which
represent people) on her picture to show the two handshakes that took place
between each set of two people (top of figure 6 and the top of figure 2) to answer
Crystal’s question.
Figure 6: Linking representations to each other
Aiesha was able to show how her formula mapped into her original representation
involving circles and loops, the action of actually shaking hands, as well as her chart with
letters. Her ability to set up a hypothetical situation about the existing problem, develop
multiple representations for the same idea, connect the representations to each other, and
ultimately provide a solution that is generalizable indicates that she has reached the
formalising layer. Further, some six months later, Aiesha’s class was given the opportunity to
investigate a task that was structurally similar to this handshake problem. Within a few
minutes, Aiesha and her group moved through most of the representations they constructed
six months earlier, and reconstructed the formula to generalize, using the correct symbolic
notation. Interestingly enough, many students around the room also used the formula Aiesha
presented six months earlier for this new task.
CONCLUSIONS
We conclude that student-to-student interactions and questions played a central role in
Aiesha’s movement from primitive knowing to formalising, as well as her movement to
linking representations to each other. This ultimately led to her ability to retain and retrieve
her ideas when presented with similar types of problems months later, which is a central goal
of the teaching and learning process. Of course, we cannot say with complete certainty that
these interactions were exclusively responsible for the development of the ideas, however, we
believe that our analysis suggests that they played a key role. While it is not possible to draw
overwhelming conclusions based on this limited example, we do believe that an analysis of
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