Students’ solution of arrangement problems and their connection to Cartesian product problems

ABSTRACT

Two-hour long developmental teaching interviews were conducted with each of 14 sixth grade students, ages 11–12. The purposes of the interviews were to investigate how students solved arrangement problems (APs), and how their solutions of these problems differed from their solution of Cartesian product problems (CPPs). The 14 students represented a balanced mix of students operating with each of three different multiplicative concepts that have been identified in prior research. This paper reports on the 11 students who were using the first or second multiplicative concept. Students operating with different multiplicative concepts all experienced similar perturbing elements in their solution of APs relative to their solution of CPPs, but they operated differently to resolve these perturbing elements. These differences are identified and their significance discussed in relation to other research findings on students’ combinatorial and multiplicative reasoning.

KEYWORDS:

• Combinatorial reasoning
• learning
• second order models
• middle grades
• multiplicative reasoning

Notes

1. “G” represents the color green; “Y” represents the color yellow; “R” represents the color red; “Pnk” represents the color pink; “Bk” represents the color black; “O” represents the color orange; “P” represents the color purple; “B” represents the color blue.

2. Elementary units include discrete ones that can be used to create more complicated units like outfits.

3. An abstract unit of one is a unit that has been stripped of specific sensory material (e.g., dots that a student imagines to be in a particular configuration); the process of stripping specific sensory material is what produces a unit of one that is abstract.

4. In figures like Figure 3, I represent units that a student creates in activity with dashed parenthesis, and units that are interiorized (i.e., not made in activity) with solid parenthesis.

5. Given school schedules, five students participated in three 40-minute interviews.

6. He used the letters “Bl” for blue, “B” for black, “P” for purple, “Pnk” for pink, “G” or “Gr” for green, “R” or “r” for red, “Y” for yellow, and “O” for orange.

7. In the Data Excerpts, I is for the interview-researcher, W is for a witness-researcher, and all other letters stand for the student. Comments enclosed in brackets [] describe students’ nonverbal action or interaction from the interviewer’s perspective. Ellipses (…) indicate a sentence or idea that seems to trail off; four periods (….) denote omitted dialogue or interaction; and parenthesis () are the researcher’s interpretations of a word meaning.

8. Portions of this data excerpt occurred prior to the data that was shared in analyzing issue 1 and issue 2.

9. This problem has similarities to the Extension of the Outfits Problem outlined in the Theoretical Framework. However, this problem is different in two ways: (a) no new colors are added; and (b) a single color, rather than one shirt and one pant, creates the row and column structure. Recall that some MC2 students could solve the Extension of the Outfits Problem in the absence of using a list or an array while others could not.

10. We note that Terionna uses an array in her solution to the problem. Students’ transition to arrays entailed some changes in their reasoning. We do not analyze those changes here because the reasoning that Terionna produced largely occurred prior to her consideration of the implications it had for her array representation.

11. This array is a replica of her array. Her array had too many markings on it to reproduce it. Note that she labeled the vertical axis as representing the color in the first position in a pair and the horizontal axis as representing the color in the second position in the pair, which reverses the conventional way that ordered pairs are read.

Additional information

Funding

The research reported in this manuscript was supported by the IUPUI Research Support Funds Grant. The findings and statements in the paper do not represent the views of IUPUI.

Notes on contributors

Erik S. Tillema

Erik S. Tillema is a former middle and high school teacher. Currently, he is faculty at Indiana University where he teaches course to pre-service teachers, and conducts research on learning. His research specifically focuses on how students' combinatorial reasoning can support their quantitative and algebraic reasoning.