Title: Are Their Arguments Convincing?
Topic: Mathematics Education
Presentation Format: Paper session
Paper Authors:
Robert Sigley
Robert B. Davis Institute for Learning
Rutgers University Graduate School of Education, USA
robert.sigley@gse.rutgers.edu
Carolyn A. Maher
Department of Learning and Teaching
Rutgers University Graduate School of Education, USA
carolyn.maher@gse.rutgers.edu
Marjory F. Palius
Robert B. Davis Institute for Learning
Rutgers University Graduate School of Education, USA
marjory.palius@gse.rutgers.edu
Cindy E. Hmelo-Silver
Department of Educational Psychology
Rutgers University Graduate School of Education, USA
cindy.hmelo-silver@gse.rutgers.edu
Are Their Arguments Convincing?
Abstract
This study examines the conversations provided by teachers studying videos of children’s
mathematical reasoning in an online environment over a two-week period in a hybrid,
graduate mathematics education course. In particular, it investigates whether the
justification of a solution to a problem posed to a group of five students in an informal
problem-solving environment, as presented in a video clip of that group’s problem
solving, was convincing. Analysis of the teachers’ discourse about students’ reasoning
revealed that teachers made reference to their own problem solving in referring to the
forms of reasoning presented by the students in the video. In comparing their own
arguments with the students in the video, they indicated that studying the video offered a
context to discuss the reasoning of the students. Further, they indicated that observing the
reasoning of the students resulted in reflecting about their own problem solving.
Theoretical Perspective
Opportunities to reflect and revisit ideas, and discuss them within a community of
learners, have shown to be especially powerful in the development of mathematical
reasoning (Maher, Powell & Uptegrove, 2010). Based on longitudinal and crosssectional studies of the development of reasoning in learners, a unique collection of
videos and related metadata are being prepared for storage on the Video Mosaic
Collaborative (VMC), a repository at Rutgers University1 (Agnew et. al 2010).
The repository provides access not only to videos but also to related metadata, such as
descriptions, transcripts, and written work of students (Agnew, et al, 2010; Palius &
Maher, 2011). The VMC also makes available innovative tools for using the materials in
teacher education contexts, both in face-to-face situations and in collaborative online
environments (Maher, Landis, & Palius, 2010; Maher, Hmelo-Silver, Palius, Sigley,
2011).
The research presented here is part of larger project of design research in teacher
education that is being conducted in association with the development of the video
repository and its tools2. The research is grounded in constructivist views on the learning
and teaching of mathematics (Davis & Maher, 1990; Maher & Davis, 1990; Davis, 1990),
and is premised on the concept that videos of student learning can serve as powerful
pedagogical tools (Maher, 2008). Making use of the repository resources, a hybrid course
was designed to offer a learning environment that blended in-class problem solving,
asynchronous online discussion, access to videos, and reading assignments of relevant
literature. A goal for the instructional intervention is that teachers learn to recognize
students’ reasoning as presented on video clips selected from the VMC repository.
1
The video mosaic is located at: http://www.videomosaic.org
Research supported by the National Science Foundation grant DRL-0822204, directed
by C. A. Maher with G. Agnew, C. E. Hmelo-Silver, and M. F. Palius. The views
expressed in this paper are those of the authors and not necessarily those of the funding
agency.
2
2
Setting
The intervention occurred in a graduate-level, hybrid mathematics education course in
which teachers worked on challenging mathematical tasks in the classroom environment,
and then studied videos of students who had been working on the same tasks. For the
online component, the teachers were organized into four discussion groups of 6 to 7
teachers per group. The format involved posing questions for discussion about the
reasoning of the students in the video and whether the teachers found the reasoning
convincing, and if so, why or why not.
Data Source
The data were collected within a course management system at a major public research
university and consist of online discussions among twenty-five teacher participants in the
course setting described above. Data were used to conduct an analysis of the threaded
discussions in a course unit that focused on a particular task, which included problem
solving, video clip viewing, and related reading (Maher & Muter, 2010). The unit took
place in the fifth week of the course, over a two and a half week period.
The Task
During the in-class problem solving, the teachers worked in small groups on the
following problem:
How many different block towers can be built selecting from three colors of blocks such
that the towers have at least one of each color?
They then presented their solutions to the entire class. Their follow-up assignment was to
study a video, Romina’s Proof to Ankur’s Challenge3, of five 10th grade students who
worked on the same problem-solving activity. Specifically, the posted assignment was:
This week's assignment for online work involves a video and two
readings, with threaded discussion, that follows class work on problem
solving for the Ankur's Challenge task.
The following questions are intended to guide discussion in your
small groups (and will also be posted in the introduction to group
discussion threads).
(1) Describe Romina’s strategy for solving the Ankur’s Challenge problem.
(2) In your opinion, is this solution a convincing one? Why or why not?
Research Questions
The research questions that guided this study are: (1) How, if at all, does teachers
studying the video of Romina’s Proof contribute to understanding their own problem
3
See http://hdl.rutgers.edu/1782.1/rucore00000001201.Video.000062055
3
solving? And (2) To what extent, if at all, do the teachers find the students’ reasoning
convincing?
Results
Pooling the data analyzed across the four groups of teachers, a total of 71 posts were
made in the discussion threads of the course unit we studied. Individual posts tended to
be rich in the scope of commentary that the teacher offered in their respective group’s
discussion thread. That is, a single post often commented on more than one aspect of
teacher reflection on problem solving within context of this study. Five themes recurred
with high frequency in our analysis of the teachers’ postings, which give compelling
evidence that studying the video proved to be a strong catalyst in teachers’ analysis of the
students’ reasoning. Teacher comments reflected acknowledgement of Romina’s proof
as correct, clever, convincing, elegant, and impressive. The logic of Romina’s proof was
mentioned repeatedly across the postings. Teachers also made reference to whether the
proof was similar and/or different than that of their own or other class members. This
included references such as similarity in notation, cleverness in notation, personally
insightful, brilliantly represented, and so forth. What is particularly noteworthy is that the
study of the students’ reasoning triggered in the teachers a reflection about their own
problem solving. The following table summarizes the recurrence of top five themes that
appeared from our coding of the teacher discussion threads.
Theme of reflective commentary
Related video to own problem solving
Related video to others’ problem solving
Acknowledged positive qualities of Romina’s proof
Mentioned specifically the logic of Romina’s proof
Referred to arguments / justification as a social activity
Frequency
23
14
20
17
21
Percentage
92%
56%
75%
68%
84%
The fifth theme in above table is especially interesting, as it refers to the notion of doing
mathematics as discussion among a community of learners through the process of making
and evaluating arguments to justify solutions to problems. The teachers expressed this
idea in various ways, and their commentary reflected what they think is important and
necessary about engaging in mathematics as a social activity. We share some examples of
teacher postings taken verbatim from the course management system, using pseudonyms
in lieu of actual names.
For instance, Michael addressed student-to-student discussion of mathematical
arguments, and connected his idea to a comment made by classmate Beth, in his post:
I think asking the students to convince their peers is what makes this
study special, the solution is not very important. When trying to
convince someone the students really deepen their understanding and
as (Beth) says reformulate, reorganize, rethink, restate their argument.
By considering the video episode in its broader context of the research study from which
it came, Michael is noticing how certain features of the learning environment, namely
peer evaluation of arguments, contribute to learning mathematics with understanding.
4
Teachers also reflected on the need for a student to justify a solution. Sam asserted a
connection between convincing mathematical explanations and the deep thinking that
leads to understanding and learning.
The need for justification is what leads students to think deeper
about a topic and fully understand the specific problem. Being able
to explain and justify their solutions and convince others of their
solution is a very powerful tool and perhaps is where the most
learning takes place. Conceptual learning and understanding arises
when there is reason to support it. This is what made Romina's proof
so convincing. In fact, her proof became more convincing and
with each time she explained it, gaining little insight and
knowledge each time. In accordance with Glaserfeld's view, I think
Romina was the knower who came to know more. Romina had the initial
thoughts and idea of her process but through reasoning and explaining,
her proof became clearer because she became clearer on her own method.
Classmate Jennifer responded to Sam’s post to elaborate further on the link between
justification and deeper understanding.
By justifying her answer, she not only learned new math but also learned how to
articulate her ideas effectively, leading to a deeper understanding… It is important for
students not only to understand and be able to justify their own solutions, but understand
other solutions as well.
The same theme arose in a different discussion group, where Tom responded to one of his
classmates by citing a particular idea and giving further evidence from the video about
how notations created by the student became more refined along with re-writing the
solution more convincingly.
I agree with you that "explanation and justification help the student rethink about his/her
ideas and push him to make sense of their findings before making them public". This is
why Romina was successful in her proof. Every time she explained herself, she was able
to make more sense of it and therefore justify her findings better. At first, she had 2's
next to each of the 6 towers and then got confused herself why they put them there, then
erased them and rewrote her solution in a more clear, convincing way. If Romina did not
have to explain herself and justify her solution, she may have never came up with the
nice, elegant solution that she did.
The last example comes from Maria, who moved beyond the specifics of the assignment
and reflected more generally on how the studying of videos has transformed her view of
what mathematics is.
From watching numerous videos throughout this semester, I have learned that
mathematics is more than just finding an answer (correct or not). Math is about being
able to explain your thinking and justify your solution to others. I believe one truly
understands a concept when they are able to explain it in such a way that others can now
understand. By explaining your solution, you are contributing to the learning of others,
but, more importantly, you are contributing to your own learning. When one justifies
their solution, they are using higher order thinking skills. They must be able to present a
5
complete argument that leaves the reader with no doubt or questions that the argument is
correct. That can be a hard task to accomplish.
Conclusions and Implications
As our study has shown, the teachers compared their own and each others’ problem
solving with the problem solving of the students in the video. They found the arguments
provided by the students in the video to be logical and convincing, and they backed their
assessment with detailed explanations. Understanding how teachers are stimulated to
reflect on their own and each others’ problem solving, as well as on the problem solving
of students, can help us design learning environments that afford opportunities for this
reflection. This study suggests that giving teachers the opportunity to work
collaboratively on a challenging mathematical task, share and discuss their and the
students’ solutions with each other, can provide a powerful incentive for deep reflection
about what constitutes a convincing argument in a mathematics task. Analysis of the
detailed conversations suggest how carefully selected videos can enrich collaborative
teacher learning about mathematical reasoning. Attention to student reasoning is an
important aspect of mathematics teaching. Further research in other contexts and with
other problem solving and video resources is suggested in order to examine how
reflection can be enhanced by collaborative mathematics learning and through studying
videos of students learning in other contexts.
References
Agnew, G., Mills, C. M., & Maher, C. A. (2010). VMCAnalytic: Developing a
collaborative video analysis tool for education faculty and practicing educators. In
R. H. Sprague, Jr. (Ed.), Proceedings of the 43rd Annual Hawaii International
Conference on System Sciences (HICCS-43): Abstracts and CD-ROM of Full
Papers. IEEE Computer Society, Conference Publishing Services: Los Alamitos,
CA.
Davis, R. B. (1990). Discovery learning and constructivism. In R. B. Davis, C. A.
Maher, & N. Noddings (Eds.), Constructivist Views on the Teaching and
Learning of Mathematics, Journal for Research in Mathematics Education
Monograph No. 4, 93-106. Reston, VA: National Council of Teachers of
Mathematics.
Davis, R. B. & Maher, C. A. (1990). What do we do when we do mathematics? In R. B.
Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist Views on the Teaching
and Learning of Mathematics, Journal for Research in Mathematics Education
Monograph No. 4, 65-78. Reston, VA: National Council of Teachers of
Mathematics.
Maher, C. A. (2008).Video recordings as pedagogical tools in mathematics teacher
education. In D. Tirosh and T. Wood (Eds.), International Handbook of
Mathematics Teacher Education: Vol. 2: Tools and Processes in Mathematics
Teacher Education (pp. 65-83). Rotterdam, The Netherlands: Sense Publishers.
6
Maher, C. A. & Davis, R. B. (1990). Building representations of children's meanings. In
R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist Views on the
Teaching and Learning of Mathematics: Journal for Research in Mathematics
Education Monograph No. 4, 79-90. Reston, VA: National Council of Teachers
of Mathematics.
Maher, C. A., Hmelo-Silver, C. E., Palius, M. F. & Sigley, R. (2011). Reflecting on
Reasoning by Studying Videos. Paper presented at EARLI Conference, August
2011, Exeter.
Maher, C. A., Landis, J. H. & Palius, M. F. (2010). Teachers attending to students’
reasoning: Using videos as tools. Journal of Mathematics Education 3(2), 1-24
Maher, C. A. & Muter, E. (2010). Responding to Ankur’s Challenge: Co-construction of
Argument Leading to Proof. In C. A. Maher, A. B. Powell, & E. B. Uptegrove
(Eds.), Combinatorics and Reasoning: Representing, Justifying, and Building
Isomorphisms (pp. 89-96). Springer: New York, NY.
Maher, C. A., Powell A. B. & Uptegrove, E. B. (Eds.), (2010). Combinatorics and
reasoning: Representing, justifying and building isomorphisms. New York:
Springer.
Palius, M. F. & Maher, C. A. (2011). Teacher education models for promoting
mathematical thinking. In B. Ubuz (Ed.), Proceedings of 35th Conference of the
International Group for the Psychology of Mathematics Education 3, 321-328.
Ankara, Turkey: PME.
7