explanation in that book [Reflections on Number] with the explanation in [another book
being used in the course], but there was no explanation at all that I could find in that
(pointing at the Reflections on Number book on her table).
After the class had shifted attention away from the tutor and back to the mathematical idea of “6
divided by 0,” Jessica made some comments that turned the discussion again to Beth’s idea that
the tutor should have told Harvey that he had forgotten to include himself.
Jessica: To put myself in a kid’s point of view, I would think that 6 divided by 0 is 6. You
have 6, you divide it by nothing, you have 6. I think Harvey’s explanation for his age
made perfect sense and I was like, well how do you say it’s not? And the tutor used a
calculator. That’s not a great learning tool. It still didn’t tell him [Harvey] why. He [the
tutor] is just like, “Oh, it doesn’t work. The calculator says it doesn’t work so it doesn’t
work.”
Cathy: 6 times 0 would be 0 and not 6. I mean, it only checks one way, which may confuse
them, like she [Jessica] said. Explaining would help!
Beth: She [the tutor] should have said, “Well, you’re splitting it by no one but you have it
yourself, so it’s really 6 divided by 1.”
These teachers’ comments indicate concern about the lack of direct explanation by the tutor in
the investigation.
The comments made above are consistent with teachers’ more general views of the
curriculum materials they were using in the course. Nicole’s description of the lack of
explanation in the Reflections on Number investigation is suggestive of many more extensive
comments and complaints about the lack of explanations in the curriculum materials. For
instance, Meg described Reflections on Number as “more of a pamphlet than a textbook” because
. . . it goes through different topics as a textbook would, like the number zero or Pascal’s
triangle, but a typical textbook has a lesson and a group of questions. What we have is
similar but it doesn’t necessarily teach what we’re learning.
In Meg’s view, the curriculum materials “show different methods and problems, but not as much
of what you would see in an instructional textbook. It’s more questions and a lot of explanations
required. They’ll ask questions and then ‘Why? Why this? or What could be different?’” In other
words, although the curriculum materials sometimes presented mathematical information, that
information was usually followed by questions and problems that demanded extensive analysis
or use of the information.
For many of the preservice teachers, the unfamiliar format of the Standards-based materials
was problematic. As Jessica explained, examples and explanations are important elements of
textbooks for her and for children learning mathematics:
I like to have an explanation and then maybe an example or two and then some problems.
And I think that’s beneficial for elementary school children, or any student, because if there’s
any confusion when they get to the problems, they can always convert back to the examples
and see how the example was done, which gives them an idea of how to do it.
For Jessica, the materials’ lack of explanation contributed to feelings of frustration at times:
“Some of the books just say ‘do this problem’ and you don’t really have any basis for what
you’re doing, or where to start.” Her desire for the curriculum materials to provide the
explanations and examples to which she was accustomed was a determining factor in her general
preference for CMP materials over MiC materials. In her view, the CMP materials provided
more information to the student. For instance, comparing CMP’s Prime Time (Lappan, Fey,
Fitzgerald, Friel, & Phillips, 1998) and MiC’s (1998) Reflections on Number, Jessica explained:
Convention