Figure 2- Aiesha’s initial strategy for finding the number of handshakes when 11 people
are at the party
Bea: When you did the demonstration you did up there, I didn’t get it. [She is referring
to Aiesha presenting figure 1 to the class a few days earlier.]
Aiesha:
What do you mean?
Bea: All of these lines [pointing to the loops on figure 2]. What about these people
[pointing to all of the circles on the right]?
Aiesha: I’m going to show you all. I am multiplying [writing 11 x 10 = 110].
Bea:
This person [pointing to the circle all the way on the left] is shaking hands with all
of these people, and this is all of his shakes. Right, and how many handshakes is
right here [pointing to the first circle to the left]?
Aiesha:
Ten.
Bea:
And then this one (pointing to the second circle) is going to be nine, right?
Aiesha:
And then eight, seven, six, five, four, three, two, one.
Aiesha then begins drawing the chart in figure 3 and explains it to her peers.
Figure 3 - Aiesha’s chart for finding the number of handshakes when 11 people are at the
party
Aiesha’s explanations indicate that she has moved to the image having layer. “At the
level of image having a person can use a mental construct about a topic without having to do
the particular activities that brought it about.” (Pirie & Kieren, 1994, p.66) Aiesha now
appears to have an “image” of the handshakes, and is no longer tied to the action of showing
each handshake.
Convention