Description of Project
Forms of Mathematical Reasoning and Encouraging Student Justification
Dina Yankelewitz, Assistant Professor of Developmental Mathematics
Aims and Objectives: I am applying for RP&D funding to prepare three manuscripts for submission to
journals in the fiscal year 2012-2013. Based on my dissertation research, these manuscripts will report on
important foundational work that will assist me as I prepare to work on a large project for which I am
applying for external funding this year.
There are two goals of the manuscript preparation that I will be undertaking. First, I would like to wrap up
the loose ends of my current research before beginning the next large project. Second, I will be using the
results from my research on students’ reasoning while working on fraction tasks to inform my new
research, and the preparation and publication of this research will further this end.
My dissertation research focused on identifying the forms of reasoning and the nature of the
argumentation that was elicited as fourth grade students worked on a strand of fraction tasks. These forms
of reasoning (e.g. direct reasoning, indirect reasoning, recursive reasoning) are not identical to formal
proofs, but echo aspects of formal proof that, if further developed throughout the grades, may better equip
students to confront formal proof in more advanced classes. Given the importance of proof in advanced
mathematics, it is particularly important to encourage the development of informal mathematical
reasoning in this content area (National Council of Teachers of Mathematics, 2000; Ball, Hoyles, Jahnke
& Movshovitz-Hadar, 2002). In addition to tracing the reasoning and argumentation of these students, I
looked for trends in the students’ reasoning in an attempt to identify ways in which the development of
the more uncommon forms can be encouraged in the classroom. I noted several patterns in the reasoning
that occurred, and I also noticed trends in the kinds of tasks and classroom conditions that elicited the
more sophisticated reasoning.
In the past three years, I have published or submitted for publication over a dozen journal articles, the
majority of which were directly connected with the research on students’ mathematical reasoning that I
am currently completing. The manuscripts that I hope to prepare using the RP&D funding will report on
what I consider to be the most significant findings that resulted from my dissertation research, including
the identification of an important pattern in the elicitation of indirect reasoning, which is a challenge for
many students of mathematics (Epp, 1998; Leron, 1985; Thompson, 1996; Antonini, 2003). My report on
this trend will be submitted to The Journal for Research in Mathematics Education, the leading
mathematics education in the country, as I believe that the findings are significant and of wide interest. In
addition, in a second article I will report on the influence of group writing tasks in promoting generic
reasoning, which is a form of reasoning that has not been studied by many researchers but is pivotal to the
development of mathematical reasoning and proof in advanced mathematics (Rowland, 2002; Selden,
2010). Lastly, a third article will present a new framework for tracking students’ reasoning, the
argumentation chart, which I hope will be useful for researchers as well as teacher educators, and can
further ground the new project that I will be undertaking. This last report will also draw on educational
psychology literature (e.g. Chinn & Anderson, 1998 )
In July 2010, I submitted an NSF CAREER proposal. Although not funded, the proposal was deemed
competitive and reviewers suggested that I conduct a pilot of my study to provide evidence of the
viability of the project. This was accomplished in 2011-2012 via RP&D funding, and I will submit a
proposal (due July 1st) to the Spencer Foundation to fund the larger study. This larger study will build on
the research that I’ve done in the area of mathematical reasoning, and my current plans to conclude my
analyses and dissemination of my previous research will help to inform my new research plans.
Background work already accomplished: My dissertation research (described in the third paragraph
above) will provide the skeleton of the content of the manuscripts that I will prepare and submit.
Although completing three manuscripts is a tall order for the coming year, this goal is within reach in
light of the work that I’ve already completed. I’ve already completed the literature review for the first
manuscript (that focuses on indirect reasoning), and the main findings were already reported (in brief) in
my dissertation. The literature review for the second manuscript (focusing on generic reasoning) has also
been undertaken, and I have already highlighted, transcribed, and described the video data that will be
used as the basis for the article. Lastly, I have made two conference presentations that were related to the
topic of the third manuscript (argumentation charts), one directed toward an audience of researchers, and
the other for teachers and teacher educators. I hope to build on the work that I’ve already completed for
those two presentations as I prepare this last manuscript for submission.
Procedures/Methodology:
I will use the following timeline to pace my work:
July ’12: I will spend the month of July preparing the first manuscript. I will spend the first two weeks
writing the important ideas gleaned from the literature review as well as the theoretical framework.
During the third week, I will write the results section of the manuscript, and the fourth week will be
devoted to the discussion and conclusions. After completing the first draft of the manuscript, I will send
the draft to several colleagues for comments. This will be especially important for the first manuscript, as
it will be submitted to an A* journal.
August ‘12: During the month of August, I will follow the same procedure for the second manuscript. If
time permits, I will also try to complete the literature review for the third manuscript in this time.
Sept.-Jan. ‘13: I will revise the first two manuscripts based on my own editing and the comments of my
colleagues. I will also complete a first draft of the third manuscript.
Feb.-June ‘13: I will revise the third manuscript, and then work on formatting the manuscripts according
to each journal’s specifications. I hope to submit all three manuscripts by the end of June.
Importance or Value: In addition to informing my new research project for which I will be applying for
external funding this spring, the manuscripts will be of value to mathematics educators and researchers
and will further the knowledge of how students’ reasoning can be developed and traced in elementary
school classrooms. I hope that the publication of the manuscripts in prestigious journals will allow a wide
audience to gain from the research, enabling other scholars and teachers to use my work to further the
improvement of students’ mathematical learning. In addition, I find that my research is useful in my own
teaching. One of my goals in my own teaching of mathematics is that of training my students to think in
mathematically sound ways, and my involvement in research of this focus has allowed me to incorporate
my knowledge in my own pedagogy.
Further Research or Study: As discussed above, the research that I will be reporting will inform and
assist me as I begin to work on the research project that is the subject of my Spencer grant proposal,
which will be submitted by July 1, 2012. This larger study will use the trends identified in the students’
reasoning about fraction ideas to develop geometry tasks for grades 3-7 which will be designed to elicit
varied forms of reasoning in students. The project will include the development of the tasks, alignment
with the Common Core State Standards for Mathematics, professional development of teachers, and the
collection and analysis of students’ work and reasoning about the tasks.
Outcome: The results of the proposed project will be the publication of the three manuscripts in
mathematics education research journals. As mentioned earlier, the first manuscript will be submitted to
The Journal for Research in Mathematics Education, the leading journal in the field. The second and third
journals may be submitted to Mathematical Thinking and Learning and Mathematics Education Research
Journal, both A ranked journals in the field of mathematics education
References
Antonini, S. (2003). Non-examples and proof by contradiction. In N. A. Pateman, B. J. Dougherty, & J.
Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PMENA (Vol. 2, pp. 49–55).
Honolulu.
Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In L. I.
Tatsien (Ed.), Proceedings of the International Congress of Mathematicians, (Vol. III, pp. 907920). Beijing: Higher Education Press.
Chinn, C. A., & Anderson, R. C. (1998). The structure of discussions that promote reasoning. Teachers
College Record, 100, 315–368.
Epp, S.S. (1998), A Unified Framework for Proof and Disproof, The Mathematics Teacher, v. 91(8) pp.
708-713.
Leron, U. (1985), A Direct Approach to Indirect Proofs, Educational Studies in Mathematics, 16,821-325.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.
Reston, VA: National Council of Teachers of Mathematics.
Rowland, T. (2002). Proofs in number theory: History and heresy. In Proceedings of the Twenty-Sixth
Annual Meeting of the International Group for the Psychology of Mathematics Education, (Vol. I,
pp. 230-235). Norwich, England.
Selden, A. (2010). Transitions and proof and proving at tertiary level (Technical Report No. 2010-3).
Cookville, Tennessee: Tennessee Technological University.
Thompson, D.R. (1996), Learning and Teaching Indirect Proof, The Mathematics Teacher v. 89(6) pp.
474-82.