The National
Council for Teachers of Mathematics (2000) states that reasoning and proof
should be a standard in every classroom. Similarly, the Common Core State Standards for Mathematics (CCSSM, 2010)
emphasizes justification as one of its key mathematical practices. Teaching
with justification, however, is challenging. Previous studies have documented low
levels of student justifications in U.S. classrooms (e.g., Jacobs et al., 2006)
and specific challenges of enactment (Bieda, 2010).
The lack of positive examples makes it imperative that we better understand
pedagogical moves that do support
student justification. Therefore, this study was guided by the question: ÒWhat
teacher practices are associated with high levels of mathematical justification?Ó
Theoretical
Perspective
Teaching to
support studentsÕ justification is a complex and multifaceted activity. In
seeking to understand teachersÕ pedagogy of justification, we draw on sociocultural
theories as presented by Vygotsky (2002), Wertsch (1998), and others, and focus on teachersÕ
discursive acts as a mediating tool and critical resource that teachers use as
they make efforts to support, prompt and shape studentsÕ engagement with
mathematics. This perspective attunes us to the moment-to-moment interactions,
as these specific discursive moves can create space for studentsÕ justification
or can close down productive discussions.
Methodology
This study is a
collective longitudinal case study; each of the seven selected teachers
operated as an intrinsic case study. We analyzed the implementation of the Number Trick task, a task requiring the
use of the distributive property or equivalent reasoning. Data included video
transcriptions, teacher reflections and student work.
Before coding,
transcripts were split into episodes which included all discussion centered on
a single question. Then, each student utterance and the episodes as a whole
were coded, using two different coding schemes which emphasized the level of
justification present. Interrater reliability was 93%
when agreement was counted as two out of three researchers agreeing.
Summary of Findings
We will offer
preliminary findings by reporting three factors that were identified as
supporting studentsÕ use of justification: (1) attending to the mathematics of
the students, (2) supporting students in articulating their mathematical ideas,
and (3) creating a critical classroom culture where students support each otherÕs
justifications. Only the third factor is addressed here. All three factors, and
others identified through our continuing analyses, will be reported in the
session.
Supporting a Critical
Classroom Culture
A critical
classroom culture is one where students question each other about their ideas. The
excerpt in Figure 1 shows how students negotiated how to represent the first
part of the task with an algebraic expression.
01. Dan: So since this problem (references n+4+9) is the same as this
problem (points to 5 + 4 + 5 + 4), because since n=5, the ns are basically counting (inaudible)
like this problem so (pauses and puts
in 5s for the 2 ns right underneath) it could basically just be 5+4 and
5+4 again, and n+4 and, this could be n+4 plus any number. 02. Teacher: Any number, right? 03. S1: These two problems are
together. 04. Teacher: Alright, so I did
this you guys, I actually wrote up a rule for rule number 1, I wrote these
over there, can you guys see them over there? IÕm going to leave them over
there, thatÕs our rule. Questions about that? 05. S: WouldnÕt the second one (n+4+9) not work because if your
variable was something different than 5 then it
would not be 9. 06. Teacher: Ooh what do you
guys think about that? Ooh Viv what makes you think
that, come up here and show us what you mean because thatÕs a good point. 07. S2: I didnÕt even catch it. 08. Teacher: And you guys, ask
any questions on this, but explain it to us Viv. 09. Viv: So wouldnÕt you have to
add 4 every time, if this was 6 then thatÕs going to be 10 not 9 (referring to n=6 +4 = 10). 10. Teacher: Questions for her,
questions Viv, you got people. 11. S1: (inaudible) 12. Teacher: Okay do you have
more questions? 13. Viv: [calling on a peer] Dan. 14. Dan: What um, didnÕt she write
9n=5. So wouldnÕt she know that n is supposed to be 5? 15. Viv: Yeah but like if it was
any other number, for every number. 16. Dan: So are you saying that
if that n=5 and was not there that couldnÕt be any other number but is that
what you are saying? So you are saying that if see how she wrote n=5, so you
are saying since you will always know what n equals if that wasnÕt there then
it could be any number, except wouldnÕt 9 minus 4 equal 5? 17. Teacher: Yeah, like 9
wouldnÕt work in that spot for any number (inaudible). 18. Shelby: I agree with that
because of you wanted to do the n+4+ 9 you would have to put like you would
almost have to put like parentheses in and more like variables and it could
become like really confusing. 19. Teacher: So then what I am
hearing you guys say is that this wonÕt work every time, so I have to get rid
of it, right. |
Figure 1. An example of a supporting a critical classroom
culture.
In turn 1, Dan suggested
that the first part of the number trick could be represented by n+5+9. This
works when n=5, but not when n is any other number. The students negotiated
this, with strategic guidance from the teacher, eventually concluding that this
proposed representation of the trick will not work for all values of n.
Though we
examined each factor in isolation, and can see how they are distinct, in
reality a teacher uses more than one practice in a given moment. This study
provides evidence that particular teaching practices may support studentsÕ use
of mathematical justification. Further analyses will refine additional
pedagogical strategies that are associated with high levels of student
justification and focus on those pedagogical moves that support student justification, and not only student participation in classroom
discourse.
References
Bieda,
K. (2010). Enacting proof-related tasks in middle school mathematics:
Challenges and opportunities. Journal for Research in Mathematics Education, 41, 351-382.
Council of Chief
State School Officers & National Governors Association Center for Best
Practices. (2010). Common core state standards. District of Columbia: Council
of Chief State School Officers & National Governors Association Center for
Best Practices.
Jacobs,
J., Hiebert, J., Givvin,
K., Hollingsworth, H., Garnier, H. & Wearne, D.
(2006). Does
eighth-grade mathematics teaching in the United States align with the NCTM Standards? Results from the TIMSS 1995
and 1999 video studies. Journal
for Research in Mathematics Education, 36, 5-32.
National
Council of Teachers of Mathematics (2000). Principles
and standards of schools mathematics. Reston, VA: NCTM.
Vygotsky, L. S. (2002). Thought
and language (13 ed.). Cambridge, MA: MIT Press.
Wertsch, J. V. (1998). Mind as action.
New York: Oxford University Press.
Though the mathematics education community values students’ use of justification, teachers have often found it difficult to support. The speakers will discuss strategies associated with mathematically acceptable argumentation that emerged from a study of teachers participating in a research-and-development program on justification.
Session Type: Poster Session