GENERALIZING AND PROGRESSIVELY FORMALIZING IN A THIRD-GRADE MATHEMATICS CLASSROOM: CONVERSATIONS ABOUT

EVEN AND ODD NUMBERS

Maria L. Blanton

James J. Kaput

University of Massachusetts Dartmouth University of Massachusetts Dartmouth

mblanton@umassd.edu jkaput@umassd.edu

Abstract: This study explored third-grade students’ capacity for generalizing and formalizing their mathematical thinking in classroom conversations about even and odd numbers. Data were collected during a 90-minute mathematics class in an urban school district multiple times per week over the course of an academic year. Analysis showed that these students have a capacity for making robust generalizations in progressively formal ways and with intuitive supporting arguments. In particular, their activity of generalizing about even and odd numbers was based on diverse forms of reasoning, involved the algebraic treatment of the terms ‘even’ and ‘odd’ as placeholders, or variables, in generalized expressions, and extended across mathematical operations. Moreover, the process of argumentation and justification was a factor in how and what students came to know about even and odd numbers.

Perspective and Purpose of the Study

There is a growing recognition that algebraic reasoning takes several forms and can simultaneously emerge from and enhance elementary school mathematics (NCTM, 1998). Kaput (1998) has characterized these forms of reasoning as including (a) the use of arithmetic as a domain for expressing and formalizing generalizations; (b) generalizing numerical patterns to describe functional relationships; (c) modeling as a domain for expressing and formalizing generalizations; and (d) generalizing about mathematical systems abstracted from computations and relations. From this characterization, it is apparent that the processes of generalizing and progressively formalizing a generality are at the heart of the development of algebraic reasoning (see also Mason, 1996). As such, these processes represent the kind of mathematical thinking necessary to extend children’s elementary school experiences beyond arithmetic proficiency to include habits of mind that can support the more complex and abstract mathematics that will become increasingly important in the new century (Kaput, 1999; Romberg & Kaput, 1998).

This perspective on the algebraic potential of elementary school mathematics raises important questions about students’ understanding of and capacity for generalizing and progressively formalizing mathematical thought in the social context of argumentation and justification. The purpose of this study was to examine these processes within a third-grade classroom by mapping the evolution of students’ thinking in conversations about even and odd numbers. In particular, this study explored

(a) third-grade students’ capacity for building generalizations and the idiosyncratic ways in which those generalizations were expressed;

(b) the extent to which students were able to formalize their thinking; and

(c) the role of purposeful argumentation and justification in these processes.

Methods/Data Source

The study took place in a third-grade mathematics classroom in an urban, under-achieving school district, with eighteen students representing diverse socioeconomic and ethnic backgrounds participating in the research. The classroom teacher (Jan-pseudonym) was concurrently a 2^nd-year participant in a district–based project, led by the authors, designed to develop elementary teachers’ ability to identify and strategically build upon students’ attempts to generalize and formalize their thinking and to engineer viable classroom instructional activities that would support this (Kaput & Blanton, 1999). Jan’s classroom was subsequently selected as the research site for this study because of her commitment to the development of students’ algebraic reasoning.

Data collection occurred during Jan’s 90-minute mathematics class approximately three days per week for one academic year. The data consisted of audio recordings, Jan’s reflections and field notes that focused on students’ verbal discourse. While various other mathematical themes reflecting children’s activities of generalizing and formalizing have emerged from these data (e.g., the evolution of children’s symbol sense), classroom conversations about even and odd numbers were selected here as the unit of analysis, with the class as the grain-size for analysis. (Ultimately, we will look across other themes in order to get a more complete picture of students’ capacity for algebraic thinking.) Viewing learning as a nonlinear series of transformations, we expected that students would be at different points in a continuum of development. As such, the goal of our analysis was not to quantify students’ activities of generalizing and formalizing, but instead to use qualitative methods to explore the nature and progression of students’ generalizations and how these came to be established within the classroom.

Results and Conclusions

Analysis of the data showed a progression in the complexity of generalizations and supporting justifications that students were able to construct about even and odd numbers. At the beginning of the academic year, students could only identify the parity of a number (Jan had a chart in the classroom that categorized a set of numbers as even or odd). They were unable to argue why a particular number was odd or even and thus were unable to build generalizations based on these properties. During the academic year, we observed both planned instruction (see Jan’s reflections) as well as impromptu discussions about even and odd numbers. Through this, students were eventually able to build generalizations and construct spontaneous arguments, even when making such arguments were an abstraction from the task at hand. Classroom protocols and Jan’s reflections (italicized) are included here to illustrate this progression.

Jan. I asked the class what would happen if I added 2 even numbers together. Most of them said that I would get an even number. When I asked what would happen if I added 2 odd numbers together, most of them said that I would get an odd number. When asked about odd and even together, the answers were mixed. In the past I would have told them the answers by giving them some examples (e.g., 5+5=10). But…I wanted them to see how it really works, so that they could see that it would [generalize to all cases]. We did [an] activity combining (square) grid-paper shapes to model adding even and odd numbers. I asked the same questions again. This time they answered with more certainty.

One student later explained the generalization that ‘the sum of any two odd numbers is even’ using the idea of adding square shapes: “If you have two odd numbers it makes it even because if you have leftovers the two leftovers go together.”

Jan. The only confusion came when [Sarah] said that odd + even was odd and even + odd was even. [Stephen] responded that that couldn’t be. He used numbers in place of odd and even and said that it (using ‘odd’ and ‘even’) was the same as [using letters instead of numbers].

Sarah explained to the class, “I thought that all the time when odd is the first one it was supposed to be odd and when even was first it was going to be even. [But then I figured that that wasn’t correct] because once you start turning them around, then it’s the same thing. It doesn’t make a difference.”

This vignette suggests that students initially used the type (e.g., all odd) or, in Sarah’s case, the position (e.g., even number first) of the terms ‘even’ and ‘odd’, not their intrinsic mathematical properties, to make a generalization about their sums. At this point, Sarah was unable to see the term ‘even’ or ‘odd’ in the algebraic way that Stephen did, as a placeholder or variable. Sarah was eventually able to construct a commutativity argument that disproved her initial generalization (based on the position of an odd or even number in a sum), and it was through peer argumentation with Stephen that her generalization was challenged and refined. In this sense, the social context of argumentation and justification ultimately provided the impetus for building a valid generalization and thus became a critical factor in how and what students came to know about even and odd numbers.

Jan extended this activity with students. She wrote, “I asked them, ‘If we added odd + odd + odd, what would the sum be?’ They figured out that the sum would have to be odd because 2 odds make an even and when you add odd + even, you get odd.” In this case, students reasoned that they could associate two numbers at a time and thereby iteratively reduce the task of adding 3 odd numbers. Additionally, they were able to achieve a level of abstraction in which they could reason with a generalization to produce a generalization. That is, they were able to reason with the general referent of ‘odd’ in the expression ‘odd+odd+odd’ (vs. specific sums of odd numbers, such as 1+5+7) to produce the generalization ‘the sum of any 3 odd numbers is odd’.

Several months after the events described above, students extended their generalizations from sums to products of even and odd numbers during a spontaneous whole-class discussion that grew out of a separate mathematical task. After working through different numerical combinations, students conjectured that ‘even times even = even’, ‘odd times odd = odd’ and ‘even times odd = even’. One student explained that multiplying any number by an even number would always result in an even number because it requires adding pairs of numbers. This episode further speaks to these students’ emerging capacity for generalizing. For example, it grew out of students’ observations about the task at hand (in particular, a multiplication sentence kept producing even number results), and it illustrates students’ potential to construct justifications using sophisticated arguments (e.g., the idea of multiplication as repeated addition).

The analysis suggests that these “average” students have a capacity far beyond arithmetic proficiency for making robust generalizations with intuitive supporting arguments. We found that, in their activity of generalizing, students

(a) used multiple forms of reasoning, including representational reasoning (e.g., using graphical or figural objects to model even/odd numbers), numerical reasoning (breaking numbers apart to identify their properties) and pattern-based reasoning (“zero is even because it is in the pattern that includes 2, 4, 6, 8,…”);

(b) used the terms ‘even’ and ‘odd’ algebraically, that is, as placeholders or variables ;

(c) were able to reason with a generalized referent (e.g., ‘odd’) to produce a generalization;

(d) were able to extend their generalizations about even and odd numbers across mathematical operations in a spontaneous way and over a sustained period of time;

(e) mediated the validity of their generalizations through peer argumentation; and

(f) were able to construct sophisticated justifications for their conjectures.

Although students were (expectedly) not at a symbolic level of formalism in their generalizations, there was evidence that some students saw even numbers as multiples of 2 (a precursor to the formalism 2n, where n is an integer) and, more significantly, could construct arguments based on this idea.

References

Kaput, J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. In S. Fennel (Ed.), The nature and role of algebra in the K-14 curriculum: Proceedings of a national symposium (pp. 25–26). Washington, DC: National Research Council, National Academy Press.

Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema, & T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133–155). Mahwah, NJ: Erlbaum.

Kaput, J. & Blanton, M. (1999). Algebraic reasoning in the context of elementary mathematics: Making it implementable on a massive scale. Paper presented at the American Educational Research Association, Montreal, Canada.

Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65–86). Dordrecht, Netherlands: Kluwer.

National Council of Teachers of Mathematics. (1998). Principles and standards for school mathematics: Standards 2000 draft. Reston, VA: Author.

Romberg, T., & Kaput, J. (1999). Mathematics worth teaching, mathematics worth understanding. In E. Fennema, & T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 3–32). Mahwah, NJ: Erlbaum.