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«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»

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Although much has been learned about children’s reasoning in the whole-number domain, children’s reasoning about integers has received relatively little attention (Kilpatrick, Swafford, & Findell, 2001). The introduction of the integers poses conceptual challenges for students, as they are required to expand their mathematical worlds to include negative numbers (Bruno & Martinón, 1999; Janvier, 1983; Vlassis, 2004). This extension contradicts students’ previous conceptions, which often involve overgeneralizations of their experiences with the natural numbers, e.g., that addition makes larger and subtraction makes smaller. Children’s difficulties with integers can be appreciated in light of the history of mathematics, wherein famous and accomplished mathematicians struggled with counterintuitive notions associated with negative numbers (Gallardo, 2002; Hefendehl-Hebeker, 1991; Henley, 1999; Thomaidis & Tzanakis, 2007). At the same time, researchers have reported on cases in which children reasoned productively about integers, even in the lower elementary grades (Behrend & Mohs, 2006; Bishop, Lamb, Philipp, Schappelle, & Whitacre, 2011; Hativa & Cohen, 1995; Wilcox, 2008). In contrast with the literature concerning children’s reasoning about whole numbers, the literature concerning children’s reasoning about integers is sparse. There is not an established framework for integer reasoning, and developmental trajectories have yet to be identified.

Methodology In 2010, we conducted more than 90 interviews with K-12 students, as we piloted a variety of integer-related tasks. We refined our interview protocol on the basis of these. In 2011, we conducted 160 interviews at seven school sites across three districts – 40 each with children at grades 2, 4, 7, and 11. We used a range of tasks in these interviews, but this report focuses on open number sentences (such as 5 –  = 8), which were used extensively with students at each grade level. We developed codes for children’s strategies via a process of constant comparative analysis (Strauss & Corbin, 1998), and we have organized these strategies into a framework based on the underlying number conceptions that they suggest. We have also used descriptive statistics to compare the relative difficulty of problems both within and across grade levels. Our ongoing analysis involves relating students’ strategies and conceptions to problem difficulty.

Results We present examples of various ways of reasoning from elementary, middle school, and high school students. In grades K-4, we identify ways of reasoning about integers prior to formal instruction. Some of these children were entirely unfamiliar with the notion of negative numbers.

Their responses reveal the counterintuitive nature of ideas related to negative numbers for children who live in a whole-number world (e.g., 3 – 5 is impossible). Other elementary children were familiar with negatives. Many of them were able to engage productively with our tasks, although they had received no formal instruction in integer arithmetic. At the middle-school level, we see the influence of instruction on children’s reasoning. Many responses are indicative of attempts to follow school-learned procedures. Often the reasoning of these children is in contrast to the sense making approaches of their younger counterparts. At the high-school level, some students’ approaches remain very procedural, while others employ a variety of productive ways of reasoning about integers. In this short proposal, we offer a few specific examples.

James, a first grader, had heard of negative numbers and could solve some of the open number sentences that were posed to him. For example, he was given the problem -5 + -2 = .

James wrote -7 in the blank and explained his thinking as follows: “Because like five plus two equals seven. So, like, if you’re doing negatives, it’s like the same as regulars.” James applied an 15TH Annual Conference on Research in Undergraduate Mathematics Education 599 analogy between negative numbers and “regular” numbers, which enabled him to obtain some correct answers. Essentially, if the given numbers were both negative, he thought about the problem the same way as he would if the given numbers were both “regulars,” and then he simply wrote a minus sign in front of his answer and called the number “negative.” James could not solve problems that involved both a positive and a negative number. He said that the numbers behaved like magnets that would repel one another. Thus, his reasoning about integer arithmetic was rather limited. On the other hand, James’s way of reasoning enabled him to solve problems such as -5 – -3 = , which were difficult for some seventh and eleventh graders.

Roland was in fourth grade. He had heard of negative numbers, and he knew the ordinal relationship between these and positive numbers. Although Roland was not familiar with addition or subtraction involving negatives, he was able to solve many of our tasks. For example,

Roland solved -5 + -1 =  by employing an analogy between negative and positive numbers:

Since 5 plus 1 equals 6, -5 plus -1 equals -6. In contrast with James, however, Roland had a meaningful justification for his approach. He reasoned that combining two negative numbers would give a result “farther from the positive numbers,” so that -6 made sense. In several instances, Roland reasoned productively by deducing whether the given operation should result in moving in the direction of the positives or away from them. He even solved -5 – -3 =  by reasoning in this way. (The reader is encouraged to imagine the details of Roland’s solution.) Jane, a fifth grader, had received instruction in integer addition and subtraction. When she was given the problem -12 + 7 = , Jane changed it to read -12 – +7 = . She came up with two possible answers, +5 and -19, and she decided that -19 was correct. A song that Jane’s teacher had taught her informed her thinking about the problem. Jane mentioned this song and recited it: “Same signs, add and keep. Different signs, subtract. Take the sign of the higher number. Then you’ll be exact.” Jane’s reasoning contrasts starkly with Roland’s. Whereas he made sense of problems on the basis of the ordinal relationship between positive and negative numbers, Jane attempted to apply an arbitrary and unclear rule. She did not make explicit any specific relationship between the song and her solution to this problem. It seemed that she thought she should change something before computing, when in fact this was unnecessary.





Implications One compelling finding from this study is that children are capable of reasoning productively about integers prior to formal instruction. Research has shown that children are capable of inventing their own mental calculative strategies, when given the opportunity to do so (Carpenter, Franke, Jacobs, Fennema, & Empson, 1997). However, in order to support students’ invention, teachers need knowledge of relative problem difficulty. Understanding children’s ways of reasoning affords teachers models of student thinking, and therefore the ability to anticipate problem difficulty (Carpenter, Fennema, Peterson, & Carey, 1988; Fennema et al., 1996). As an example of integer problem difficulty, only 58% of seventh graders correctly solved 6 – -2 = , while 75% correctly solved -5 – -3 = . Procedurally, these problems look similar. If anything, -5 – -3 might appear more difficult. However, reasoning like that of James and Roland helps to explain why this problem was actually less difficult for some students.

The findings that we will present can be used instructively with preservice teachers in two ways: (1) to engage them in thinking about integers themselves, and (2) to introduce them to children’s ways of reasoning, which may be very different than their own. Knowledge of children’s integer reasoning is relevant to preservice elementary teachers, even if they will not be teaching about integers as such, because children’s experiences in the early elementary grades will influence their preparedness for integer instruction.

600 15TH Annual Conference on Research in Undergraduate Mathematics Education Questions Attendees will be engaged in interpreting children’s thinking and considering how the video clips could be used with preservice teachers. The specific questions will be tied to the examples of children’s thinking. Questions like the following will be posed: How would you solve this problem yourself? Do you have another way of solving it? How might a second grader think about this problem? Which of these tasks would you expect to be more difficult for a child?

How might preservice teachers think about this problem? How might you use this clip with preservice teachers? What would you hope they would take away from it?

References

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90, 449-466.

Behrend, J., & Mohs, C. (2006). From simple questions to powerful connections: A two-year conversation about negative numbers. Teaching Children Mathematics, 12 (5), 260–264.

Bishop, J. P., Lamb, L. L. C., Philipp, R. A., Schappelle, B. P., & Whitacre, I. (2011). First graders outwit a famous mathematician. Teaching Children Mathematics, 17, 350–358.

Bruno, A., & Martinón, A. (1999). The teaching of numerical extensions: The case of negative numbers. International Journal of Mathematical Education in Science and Technology, 30, 789–809.

Carpenter, T. P., Fennema, E., Peterson, P. L., & Carey, D. A. (1988). Teachers’ pedagogical content knowledge of students’ problem solving in elementary arithmetic. Journal for Research in Mathematics Education, 19, 385-401.

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann.

Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1997). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29, 3-20.

Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction.

Journal for Research in Mathematics Education, 27, 403-434.

Gallardo, A. (2002). The extension of the natural-number domain to the integers in the transition from arithmetic to algebra. Educational Studies in Mathematics, 49, 171–192.

Hativa, N., & Cohen, D. (1995). Self learning of negative number concepts by lower division elementary students through solving computer-provided numerical problems.

Educational Studies in Mathematics, 28, 401-431.

Hefendehl-Hebeker, L. (1991). Negative numbers: Obstacles in their evolution from intuitive to intellectual constructs. For the Learning of Mathematics, 11, 26–32.

Henley, A. T. (1999). The history of negative numbers. Unpublished dissertation. South Bank University.

Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41, 169-202.

Janvier, C. (1983). The understanding of directed number. In J. C. Bergeron & N. Herscovics (Eds.), Proceedings of the Fifth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 295Montreal: Universite de Montreal, Faculte de Sciences de l’Education.

15TH Annual Conference on Research in Undergraduate Mathematics Education 601 Kilpatrick, J., Swafford, J., & Findell, B. (2001). Developing proficiency with other numbers. In J. Kilpatrick, J. Swafford, & B. Findell (Eds.), Adding it up; Helping children learn mathematics (pp. 231–254). Washington, DC: National Academy Press.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. New Jersey: Erlbaum.

Philipp, R. A. (2008). Motivating prospective elementary teachers to learn mathematics by focusing upon children’s mathematical thinking. Issues in Teacher Education, 17, 7-26.

Strauss, A. & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks, CA: Sage Publications.

Thomaidis, Y., & Tzanakis, C. (2007). The notion of historical “parallelism” revisited: historical evolution and students’ conception of the order relation on the number line. Educational Studies in Mathematics, 66, 165-183.

Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in ‘negativity.’ Learning and Instruction, 14, 469-484.

Wilcox, V. B. (2008). Questioning zero and negative numbers. Teaching children mathematics, 15, 202-206.

602 15TH Annual Conference on Research in Undergraduate Mathematics Education Articulating Students’ Intellectual Needs: A Case of Axiomatizing

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Abstract This study uses qualitative methods to investigate how students’ intellectual needs were articulated in an inquiry-based mathematics bridge course. One of the primary goals of this bridge course was to orient students toward more advanced mathematics by engaging them with an RMEinspired curriculum for learning abstract algebra (Larsen, 2004). Although intellectual need was not the initial object of analysis in this term-long teaching experiment, the teacher-researcher was curious about the nature of some of the student discussions that had taken place during the term. In a retrospective analysis of the teaching experiment, students’ acts of mathematizing were examined and correlated with Harel’s (2011) categories of intellectual need. A preliminary analysis of the data suggests that the act of axiomatizing a mathematical system—in this case, a group—can provide students with many opportunities to articulate and address a variety of intellectual needs.



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