the magic pot (Russell et al. 2008b). When I walked past Kristen, I saw her
usual series of tally marks on her paper, along with the equation, 9 32 36.
She also had a collection of cubes scattered on her desk. I checked in to try to
make sense of what she was thinking and soon discovered that there was no ap-
parent connection between the cubes on her desk and the tally marks on her
paper.
TEACHER: What was your plan for using the cubes? [silence] Can you just go
ahead and count and tell me how you counted?
KRISTEN:[touches and counts each cube] 1,... 37.
TEACHER: Then what did you do, once you knew that you had 37 cubes?
KRISTEN: Then I counted these 1s and it made 36.
TEACHER: Show me how it made 36.
KRISTEN: Count them?
TEACHER: Can you show me what you did? [Kristen counts the same cubes
again and gets 37.]
TEACHER: Then what?
KRISTEN:[continues counting some other cubes on her desk] 38, 39, 30... 40,
41, 42, 43, 44, 45, 46, 47.
TEACHER:[I still could not see the connection between the cubes and the problem.
Perhaps she didn’t understand the story context.] What’s happening in this
problem? What’s it about?
KRISTEN: There were people going in the magic pot and then they all come
out and then there’s 9.
TEACHER:[In retrospect, I wish I had asked Kristen where the 9 came from.
Instead, I focused on her understanding of what the problem was asking.] Let’s
see if you remember how the magic pot works. What happened when
Mrs. Haktak put a hairpin in the magic pot?
KRISTEN: One other 1.
TEACHER: Then when they put 2 bags in the magic pot, how many bags
came out of the magic pot?
KRISTEN: Two more.
TEACHER: What would happen if we put our class into the magic pot? How
many more people would come out? How many people are in our class?
KRISTEN: I forgot.
TEACHER: You wrote it on your paper.
KRISTEN: 18.
TEACHER: So 18 people would come out of the magic pot. Then how many
more people would come out?
KRISTEN: 18.Assessing and Developing Early Number Concepts