limitations to each argument, and a

limitations to each argument, and a more convincing
argument would be a combination of the two. The merits
and limitations they mentioned were similar to reasons
other students provided for choosing one argument over
another. What is clear is that even as seventh graders,
these students are reading arguments beyond surface fea-
tures they might have generalized from classroom work,
such as that writing complete sentences is good or pro-
viding examples is bad, and they are looking for whether
the arguments provide enough information to help the
reader understand why the statement is true. Similarly,
mathematicians expect visual arguments to be accompa-
nied by verbal explanations (Inglis & Mejia-Ramos,
2009), and these results suggest that students begin, at the
middle-school level, to adopt conceptions aligned with
the discipline that arguments must be clear and illustrate
why a statement to be mathematically convincing. Table 1
provides a summary of the reasons students provided for
their choice in question 1.
Show and Tell: Improving Arguments to Be More
Convincing
During the interview, after students chose either the EB
or G response as more convincing and justified their
choice, students were asked to describe how they would
improve the response not chosen to make it more convinc-
ing (question 3 of the protocol, see Methods section).
Analyses of students’ responses to this third question
offered an opportunity to determine whether students
would revise an argument in ways that corresponded to
features they previously identified for a convincing argu-
ment. We found that students tended to revise arguments
using a “show and tell” strategy; that is, if they chose the
EB argument, students indicated that including examples
would improve the G argument by showing how the state-
ment worked with examples. On the other hand, if a
student chose the G argument, they indicated that provid-
ing facts or an explanation would improve the EB argu-
ment. In some cases, students who chose the G response
because it described why the rule worked suggested
adding examples to show how the case worked; as one
student stated, “If you mixed these two up a little bit, they
would pretty much be, like, really good.”
Not surprisingly, each of the 15 students who chose the
EB argument recommended improving the G argument by
including examples. In justifying their modifications, stu-
dents’ reasons were coded into one of four categories: (a)
sense-making, (b) show the mathematics, (c) visualize the
mathematics, and (d) symbolic preference. Excerpts to
illustrate responses coded to each of these four categories,
respectively, are as follows: