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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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determined by student-to-student questions and/or interactions.
In the sections that follow, we examine the development of a particular student, Aiesha,
by identifying student-to-student questions and/or interactions in the context of the Pirie/Kieren model for mathematical understanding.
RESULTS
MOVING FROM PRIMITIVE KNOWING TO IMAGE MAKING
Primitive knowing is the starting place for the growth of any particular mathematical
understanding, what the student can do initially, with the exception of the knowledge of the topic. In this case, Aiesha begins by shaking hands with a member of her group and then moves to a picture and number representation for her idea (figure 1). She moves to the image making layer (doing something to get the idea of what the concept is), using a picture representation to construct an idea of multiplying the number of people by one less than the number of people to arrive at the number of handshakes. Every time she multiplies, however, she arrives at double the number of actual handshakes in the correct solution. At first, she doesn’t notice this mistake and becomes frustrated, explaining that there is no answer. After another student shares his solution, she realizes that the answer is eight and divides her answer to an eight person party by 2. She is working in the image making layer because she is “tied to the action or doing”.
Figure 1- Aiesha’s move to image making
MOVING FROM IMAGE MAKING TO IMAGE HAVING
Two weeks later the students were challenged to begin a new task involving an extension
of the original task. In this episode, another group member, Bea, questions Aiesha about her initial representation. Aiesha then restructures her knowledge to generate a representation that is more understandable to her peers. In doing so, she has developed a new and ultimately more useful representation.
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| | Authors: Warner, Lisa. and Schorr, Roberta. |
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determined by student-to-student questions and/or interactions.
In the sections that follow, we examine the development of a particular student, Aiesha,
by identifying student-to-student questions and/or interactions in the context of the
Pirie/Kieren model for mathematical understanding.
RESULTS
MOVING FROM PRIMITIVE KNOWING TO IMAGE MAKING
Primitive knowing is the starting place for the growth of any particular mathematical
understanding, what the student can do initially, with the exception of the knowledge of the
topic. In this case, Aiesha begins by shaking hands with a member of her group and then
moves to a picture and number representation for her idea (figure 1). She moves to the image
making layer (doing something to get the idea of what the concept is), using a picture
representation to construct an idea of multiplying the number of people by one less than the
number of people to arrive at the number of handshakes. Every time she multiplies, however,
she arrives at double the number of actual handshakes in the correct solution. At first, she
doesn’t notice this mistake and becomes frustrated, explaining that there is no answer. After
another student shares his solution, she realizes that the answer is eight and divides her
answer to an eight person party by 2. She is working in the image making layer because she
is “tied to the action or doing”.
Figure 1- Aiesha’s move to image making
MOVING FROM IMAGE MAKING TO IMAGE HAVING
Two weeks later the students were challenged to begin a new task involving an extension
of the original task. In this episode, another group member, Bea, questions Aiesha about her
initial representation. Aiesha then restructures her knowledge to generate a representation
that is more understandable to her peers. In doing so, she has developed a new and ultimately
more useful representation.
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