Number Theory Concepts and Problem-Solving Strategies 1

Number Theory Concepts and Problem-Solving Strategies: A Coordinated Analysis of Teaching and Learning in One Sixth Grade Classroom

Andrew Izsák, Bradford Findell, Chandra H. Orrill

The University of Georgia

Supporting classroom instruction in which students construct connections among multiple, related ideas in a given discipline remains a central challenge for educational research and practice. New, standards-based mathematics curricula that combine content, such as numbers and operations, with processes, such as problem solving, are one attempt to meet this challenge. Tasks contained in such curricula often afford opportunities for teachers and students to work with sets of related mathematical ideas, rather than on isolated skills. This observation leads to the broad question that motivated the present study: When teachers and students work together on tasks that contain multiple related ideas, on which of those ideas do they focus and why?

The present study investigated teaching and learning in one sixth-grade classroom around a sequence of lessons from the Prime Time unit in the standards-based Connected Mathematics Project materials (CMP; Lappan, Fey, Fitzgerald, Friel, & Phillips, 2002). The CMP tasks at the center of this study are games that afford opportunities to develop simultaneously number theory concepts and problem-solving strategies. The number theory concepts include factors, multiples, prime numbers, and composite numbers. The strategies include methods of systematic exhaustion, which are useful in many problem-solving situations, as well as strategies tied to particular features of the games and that do not directly prepare students for further situations. One example of systematic exhaustion central to the present study is searching for all factors of a given number by dividing that number by 2, 3, 4, and so on, in order. A second is determining all products that can be made from pairs in a given set of numbers.

The set of concepts and strategies that can be addressed when teaching and learning with the CMP games provided a reasonable context for pursuing an instance of the research question presented above. In particular, we asked: On which of the number theory concepts and problem-solving strategies did the case-study teacher and her students focus and why? The present case is of interest because the teacher and her students focused primarily on the number theory concepts, yet the data suggested they had resources for developing methods of systematic exhaustion that they did not use. Thus, in our analysis, the instruction made good progress toward developing the number theory concepts and could have made more progress toward developing important problem-solving strategies—if the teacher and her students had made more extensive use of understandings to which they already had access. To arrive at our results, we constructed methods for accessing teachers’ and students’ understandings of the same lessons. Those methods are described in detail later in the article.

Background

We frame our research by drawing from Cohen and Ball’s (1999, 2001) perspective on instruction, adopting a stance towards teachers similar to that articulated by Simon and Tzur (1999), and using a perspective on mathematical thinking and problem solving summarized by Schoenfeld (1992). Cohen and Ball (1999, 2001) emphasized that instruction is a function of interactions among teachers, students, and content as mediated by instructional materials (see Figure 1). Instructional materials shape what teachers and students do through sequences of problems, pathways toward mathematical ideas, and external representations. Teachers use their knowledge of the content and experience with students to interpret the materials and mediate students’ opportunities to learn with those materials. Students use their prior knowledge to apprehend, interpret, and respond to materials and teachers. Moreover, students’ prior knowledge and responses to the materials and content help determine what teachers can accomplish. Thus, each of the three elements shown in Figure 1 shapes instruction through interaction with the remaining two. Cohen and Ball have used this framework to explain why simply adopting new instructional materials has repeatedly failed to improve capacity for worthwhile, substantial learning in classrooms. We will examine interactions among the case-study teacher, her students, and the Prime Time materials to understand why instruction focused much more explicitly on number theory concepts than on systematic exhaustion. To the best of our knowledge, other studies have not tracked interactions among the elements shown in Figure 1 over a sequence of several lessons in a single classroom, as we do here.[BRF1] In particular, our analysis will demonstrate that relative contributions to instruction made by each of the three components in Figure 1 can be fluid.

Figure 1. Adapted from Cohen and Ball 1999, 2001.[CO2]

Simon and Tzur (1999) accounted for teachers’ practices by explaining teachers’ perspectives from the researchers’ perspective. The researchers’ lens on practice may not be shared by teachers with whom they work because that lens is shaped by theoretical perspectives from research. Thus, researchers’ explanations for how teachers perceive, make sense of, and respond to situations may emphasize aspects of practice that are not foci of teachers’ attention. Our lens on teachers, students, and the CMP materials derives from a perspective on mathematical thinking and problem solving around which there is considerable agreement (e.g., Pólya, 1957; Schoenfeld, 1985, 1992; Silver, 1985). Schoenfeld (1992) summarized five aspects: the knowledge base, problem-solving strategies, monitoring and control, beliefs and affects, and practices. This perspective is apt for analyzing lessons we observed because the CMP materials embed mathematical ideas in problem situations. Our focus on systematic exhaustion, in addition to whole-number multiplication and number theory concepts, was shared by the case-study teacher sometimes, but not always.

Past research has shown that systematic exhaustion is a reasonable instructional goal for middle grades students. English [AI3](1993) tracked emerging strategies in students ages 7 –12 for systematically exhausting combinatorial [BRF4]problems XXXX[CO5].

Pierce Middle School and Ms. Moseley

The present study was a pilot for a project entitled Coordinating Students’ and Teachers’ Algebraic Reasoning (CoSTAR[1]) that conducts ongoing research on mathematics teaching and learning in Pierce Middle School. The school recently replaced traditional instructional materials focused on skill development with the CMP materials. The project works with teachers who have begun to use standards-based materials because teachers’ efforts to use the new materials can facilitate access to their cognition. For example, teachers’ comparisons of experiences using more traditional and reform-oriented materials can reveal aspects of their knowledge, as can teachers’ responses to new aspects of mathematics content and students’ thinking. Although the primary purpose of the project is neither to evaluate the CMP materials, nor to study the implementation of mathematics education reform, the present case study does examine how the CMP materials supported activities consistent with the Number and Operations and Problem Solving strands found in Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000).

Pierce Middle School is located in a rural community outside of a large southeastern city. Students in the district are racially and economically diverse: 35% are African American, 63% are white, and there is a growing population of Hispanics. Students are also economically diverse: 42% qualify for free lunch, while others have well-educated, middle-class parents who commute to the urban center for work. The district has ranked in the middle on state-wide mathematics tests and is making a concerted effort to improve its mathematics education program. As part of this effort, teachers and district administrators began working together to transition from traditional to standards-based materials across grades K-12 starting in the 2001-2002 school year. For the middle school, the district adopted the CMP materials.

Because CMP is an ambitious program that embeds mathematical ideas in problem situations, teachers often need significant support when first using the materials in their classrooms. Prior to the present study, which took place in the fall of 2002, teachers at Pierce Middle School had limited professional development opportunities to support their transition. The district hired a consultant to help each grade level prioritize the various units for the first year and to prepare the first units teachers would use. The consultant worked with teachers by grade level a second time at the end of the first nine weeks of instruction to prepare the second units teachers would use. The district agreed to bring the consultant a third time, but the teachers decided that they did not need further support at that time. During the year before the present study, a member of the research team not listed as author demonstrated for the teachers a few seventh- and eighth-grade lessons that used calculators.

Ms. Moseley[2] was using the Prime Time unit for the second year when we conducted this study. She held a middle grades certification with an emphasis in social studies but had been a mathematics teacher for the last 11 years. She explained that initially she liked teaching mathematics because it was straightforward and the rules and procedures made it less argumentative than social studies. As a result of recent professional development and experience pursuing a Master’s degree in middle grades education, however, Ms. Moseley no longer viewed mathematics as straightforward. Although Ms. Moseley was looking for ways to include more attention to student thinking and more use of drawn representations in her practice, she acknowledged that she often knew only one approach to a concept and was hesitant to pursue students’ solutions because she could not always understand their explanations.

Methods

Researchers in education and psychology have gained significant insights into both teachers and students in a range of content areas, but separate subfields exist for research on teachers’ knowledge and teaching and for research on students’ cognition and learning. One consequence of this separation is that educational research does not have well-established methods for accessing the interplay between teachers’ and students’ understandings of content and of each other during lessons. Our methods described below are one attempt to reduce the degree of inference required when applying the theoretical frame summarized in Figure 1 to investigate classroom teaching and learning.

We combined classroom and interview data to gain access to Ms. Moseley and her students’ understandings of the implemented Prime Time activities. First, we used two cameras to videotape Ms. Moseley’s instruction every day during the same class period. One researcher set the first camera in the back of the classroom and recorded the entire class, adjusting the levels of microphones to hear all students during whole-class discussion and to capture conversations between Ms. Moseley and individual students during group work. A second researcher used the second camera to record written work, staying at the back of the classroom to record the whiteboard during whole-class discussion and shadowing Ms. Moseley to record written work as she worked with students. We combined the video and audio using an audiovisual mixer to create a restored view (Hall, 2000) that captured much of what Ms. Moseley and her students said and looked at during the lesson.

To gain access to students’ understandings of the PrimeTime tasks and Ms. Moseley’s instruction, the first author interviewed two pairs of students from the same class in which we gathered lesson videos. From Ms. Moseley’s point of view, one pair consisted of a high-achieving boy and a high-achieving girl (Ben and Kelly), and the other consisted of a mid-achieving girl and a low-achieving girl (Maria and Jennifer).Both pairs were interviewed once a week. Based on our experiences during the student interviews, we agreed with most of Ms. Moseley’s judgments but judged the mid-achieving girl (Maria) to be high-achieving.

During the semistructured interviews (Bernard, 1994, Chapter 10), the interviewer asked students to play and discuss the two games from the Prime Time unit and used a laptop computer to show lesson video excerpts containing classroom discussions about the same games. Watching students play the games and questioning them about their reasoning provided greater access to the understandings they used than the lesson video alone typically afforded. Questioning students about lesson excerpts provided access to their interpretation of instruction related to the games. Moving back and forth between playing the games and discussing the matched lesson excerpts provided access to interactions between the students’ understandings of the lessons and their emerging understandings of the mathematics embedded in the games. The interviewer used two video cameras, one to record the students and one to record their written work and the viewed lesson clips. As with the lessons, we combined video from the two cameras to create a restored view that captured most of what the students said, wrote, and watched.

Finally, the second and third authors conducted weekly interviews with Ms. Moseley to gain access to her understandings of the Prime Time activities, how she used the materials, her understandings of her students, and the pedagogical decisions she made. The interviewers[AI6][BRF7][CO8]used a combination of lesson and student interview video excerpts selected in consultation with the first author and played on a laptop computer. When playing the lesson excerpts, the interviewers asked Ms. Moseley to discuss the mathematical content and to reconstruct her thinking as she supported student learning, asked students questions, and worked with students who were having difficulties. When playing the student interview excerpts, the interviewers asked Ms. Moseley to comment on her students’ approaches to the games and any difficulties they were having, interpret student explanations, and react to students’ interpretations of lesson excerpts. Student interview excerpts provided Ms. Moseley more detailed access to her students’ mathematical thinking and understanding of lessons than she often could gain when teaching a whole class. Ms. Moseley’s reactions to the mathematical thinking that she saw in the interviews provided us with further access to her understanding of her students and the mathematics in Prime Time. The interviewers used two video cameras, one to record Ms. Moseley and one to record the viewed lesson clips. We again combined video from the two cameras to create a restored view that captured most of what Ms. Moseley and the interviewers said and watched. Figure 2 summarizes the lesson and interview dates.

Topic / Lesson / Ben
Kelly / Jennifer
Maria / Ms. Moseley
Playing the Factor Game / 9-30-02
10-01-02
Playing Factor Game to Win / 10-02-02
10-03-02
10-04-02
Review & quiz / 10-07-02
Playing the Product Game / 10-08-02 / 10-08-02
10-09-02 / 10-09-02
Homework / 10-10-02
Making Your Own Product Game / 10-15-02 / 10-15-02
10-16-02 / 10-16-02
10-18-03
10-22-03
10-28-03
10-30-02

Figure 2. Summary of lesson and interview dates.

Analysis and Results

We organize our analysis into two main sections. The first analyzes teaching and learning with two tasks, Playing the Factor Game [AI9]and Playing to Win the Factor Game. The second section analyzes teaching and learning with two more tasks, Playing the Product Game and Making Your Own Product Game. These tasks engage elementary number theory concepts, including factors, multiples, prime numbers, composite numbers, and perfect numbers. The materials draw attention to strategies for playing the games, but do not discuss systematic exhaustion explicitly. We tracked instruction to understand on which of the number theory concepts and problem-solving strategies Ms. Moseley and her students focused and why.

Analysis in both sections will demonstrate that interactions among the three components shown Figure 1 helped maintain Ms. Moseley’s and her students’ attention on the number theory concepts across the tasks. The analysis will also demonstrate that Ms. Moseley discussed several strategies during her lessons, some tied to particular features of the games and others related to systematic exhaustion. Specific results about systematic exhaustion include:

(1)Playing to Win the Factor Game and Playing the Product Game afforded clearer opportunities for Ms. Moseley and her students to focus on systematic exhaustion than did the remaining two tasks.

(2)Ms. Moseley’s interpretation of all four tasks and her responses to student difficulties shaped students’ opportunities to focus on systematic exhaustion.