Progressions in young learners’ understandings of parity arguments

ABSTRACT

Understanding how young learners come to construct viable mathematical arguments about general claims is a critical objective in early algebra research. The qualitative study reported here characterizes empirically developed progressions in Grades K–1 students’ thinking about parity arguments for sums of evens and odds, as well as underlying concepts of pair and parity of a number. Data are drawn from classroom lessons of a Grades K–1 early algebra instructional sequence, as well as task-based interviews conducted at four timepoints during the implementation of the sequence. While most students at the beginning of the study (Grade K) did not know the concepts of even or odd and could not make any viable arguments regarding parity, by the end of Grade 1 students were largely constructing representation-based arguments to justify number parity and claims about sums of evens and odds. Results of this study align with other research that shows young learners can develop viable arguments to justify mathematical generalizations. Results also provide preliminary evidence that the instructional sequence used here can foster students’ practice of argumentation from the start of formal schooling.

KEYWORDS:

• Algebraic thinking
• early algebra
• argumentation and proof
• elementary grades
• learning progressions
• qualitative research

Acknowledgments

We would like to acknowledge Dr. Michael Battista, The Ohio State University, for helpful conversations and feedback in preparing this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

This study included human research participants. Permission to use human subjects was obtained through the Institutional Review Board at TERC, Inc., the first author’s institution (approval numbers are not given with IRB approval)

All participants in the research provided written informed consent/assent to participate in the study. Participating students and their parents or guardians were given consent/assent forms with information describing the study. Students who wished to voluntarily participate provided consent forms signed by the student and parent/guardian.

Notes

1. The term “parity” is used here to refer to whether a number (or sum) is even or odd.

2. For more on our underlying approach and the development and effectiveness of our Grades 3–5 instructional sequence, now completed, see Blanton, Brizuela et al. (2018b), Blanton et al. (2019), and Fonger et al. (2018).

3. See the section Characterizing Good Mathematical Arguments. General arguments such as representation-based proofs (Schifter, 2009) were considered valid. Empirical arguments were not.

4. We generally use “justifying” in the way that “proving” is used in the literature on argumentation in the elementary grades. Similarly, we use “arguments” or “justifications” (rather than “proofs”) to refer to the products of elementary students’ reasoning about general claims. While either terminology is acceptable in elementary grades if defined in context, our main goal is to avoid conflating terms such as “proof” with the more particular idea of “mathematical proof” in our work with teachers.

5. The study reported here focuses on data collected from Grades K–1 during the implementation of the K–2 instructional sequence. Due to school closures associated with COVID-19, final Grade 2 posttest data could not be collected. As such, we focus on the grades for which we have a full data set (K–1). For completeness, however, in this section we discuss the Grades K–2 instructional sequence overall and summarize in Table 2 all lessons across Grades K–2 that address argumentation.

6. At pretest, some students indicated they did not know what a pair was, confused it with the fruit (pear), or gave a response that did not meaningfully address the question posed. These types of responses are not accounted for in the levels identified here.

7. Task 3 was at the end of the interview. On a few occasions, some students were unable to complete this last item due to the length of the interview. These are noted in the discussion.

8. Elsewhere, we have reported on how particular tools used in our sequence mediated students’ thinking (Strachota et al., in preparation).

Additional information

Funding

The research reported here was supported by the US Department of Education under IES Award # R305A170378. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the US Department of Education.

Notes on contributors

Maria Blanton

Maria Blanton is a Senior Scientist at TERC, Inc., in Cambridge, MA. Her research focuses on the learning progressions in children’s algebraic thinking and the impacts of early algebra education on children’s algebra-readiness for middle grades.

Angela Murphy Gardiner

Angela Murphy Gardiner is a Senior Research Associate at TERC, Inc., in Cambridge, MA. Her primary research interests include students’ algebraic thinking and understanding of functions, methods for delivering effective teacher professional development, and understanding the relationship between characteristics of professional development and student outcomes.

Ingrid Ristroph

Ingrid Ristroph is a doctoral student in Mathematics Education at the University of Texas at Austin. She is interested in teacher practices fostering algebraic reasoning. She is especially interested in issues of equity that concern authority in argumentation in mathematics classrooms.

Ana Stephens

Ana Stephens is a Researcher at the Wisconsin Center for Education Research at the University of Wisconsin-Madison. Her research addresses the development of elementary students’ algebraic reasoning. She is particularly interested in students’ understanding of mathematical equivalence and the tasks that support the development of this understanding.

Eric Knuth

Eric Knuth is a Professor and Director of the STEM Center at the University of Texas at Austin. His research concerns students’ engagement in mathematical practices, particularly practices related to algebraic reasoning and mathematical argumentation (justifying and proving).

Rena Stroud

Rena Stroud is an Assistant Professor in the Department of Education at Merrimack College. She is a quantitative methodologist with research interests in teaching and learning in early mathematics.