## Presentations and Articles

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This material is based upon work supported by the National Science Foundation under grant number DRL-0918780. Any opinions, findings, conclusions, and recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF.

__PRESENTATIONS__

Bishop, J. P., Lamb, L. C., Philipp, R. A., & Schappelle, B. (2014, April 9). *Opportunities for algebraic reasoning in the context of integers.* Paper to be presented at the NCTM Research Conference, New Orleans, LA.

Some students can leverage fundamental principles of algebraic reasoning in problem-solving strategies for integer tasks. In this presentation we considerlogical necessityand the use ofnonequivalent transformations—two ways in which students engaged, both successfully and unsuccessfully, with algebraic structures and invariant transformations of equations while solving integer problems.

**Bishop,** J. P., & Langrall, C. (2014, April 8). *Writing and responding to review s.* Research symposium at the NCTM Research Conference, New Orleans, LA. Andrew Izsak, Discussant.

The manuscript-review process will be examined from the reviewers' and the author's points of view. Using an article recently published in JRME as an example, the JRME editor and an author will examine constructive criticisms in the initial reviews and the author's response.

Philipp, R., & Bishop, J. P. (2014, February 14). *A CGI-approach to integers: Helping teachers structure their intuitive knowledge about children's understandings of negative numbers*. Plenary address presented at the Designing and Implementing High Quality Mathematics Professional Learning to Achieve the Common Core Standards for Mathematical Practice and Content, Teachers Development Group Leadership Seminar, Portland, OR.

Philipp & Bishop CGI Approach to Integers Plenary Talk (PDF)

Philipp & Bishop Handout 1 Number Sentences (PDF)

Philipp & Bishop Handout 2 Ways Of Integer Reasoning (PDF)

Philipp & Bishop Handout 3 Children & Mathmeticians Agree (PDF)

Lamb, L., Bishop, J., Whitacre, I., & Bagley, S. (2014, February 6). *Understanding students’ pre- and post-instructional conceptions of integers and the implications for teacher educators.* Presentation at the 18th annual conference of the Association of Mathematics Teacher Educators, Irvine, CA.

AMTE 2014 Lamb et al Integers Presentation (PDF)

AMTE 2014 Lamb et al Handout 1 (PDF)

AMTE 2014 Lamb et al Handout 2 (PDF)

We drew upon analyses of 160 clinical interviews to share students’ conceptions of integers. Our goal was to engage participants in discussing how to use this information to support work with practicing and prospective teachers.

Siegfried, J., Philipp, R., Jacobs, V., Lamb, L., Bishop, J., Nanna, R. J., & Hawthorne, C. (2014, February 6). *An analysis of mathematical content knowledge for teaching.* Individual presentation at the 18th annual conference of the Association of Mathematics Teacher Educators, Irvine, CA.

Content Knowledge AMTE 2014 Presentation (PDF)

Content Knowledge AMTE 2014 Handout (PDF)

Discussed were Common, Specialized, and Pedagogical Content Knowledge. Analysis of place-value and integer items used in two large NSF-funded research projects enabled us to consider boundaries among types of knowledge while illuminating distinctions. Implications for teaching and research were considered.

Lamb, L., Bishop, J., Philipp, R., Whitacre, I., Stephan, M., Bofferding, L., Lewis, J., Brickwedde, J., Bagley, S., & Schappelle, B. (2013, November 14–17). *Building on the emerging knowledge base** for teaching and learning in relation to integers. *Working Group presentations at the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 2013), Chicago, IL.

Bishop, J. P.

Students’ Integer Reasoning: Results From a Large-Scale Study

Bagley, S.Project Z Literature Overview: What clusters of research on integers would you identify?

Students’ understanding of integers has major implications for their success in algebra. However, relative to other content areas, research about issues related to the teaching and learning of integers is limited. Themembers of this new working group seek to explore what is known in the field in relation to integers’ teaching and learning and to identify next steps for continuing to build the research base on this topic. During the first two sessions, participants will draw from facilitators’ sharing of their work in order to identify existing research in the field and collectively categorize research strands. During the third session, the group will discuss directions for future research, seek to build consensus aboutwhat constitutes a robust understanding of integers, and plan next steps for continued engagement.

**Bishop, J. P. **(2013, June 4). *Opportunities for algebraic reasoning in the context of integers*. Presented at the Epistemic Algebraic Student Conference, University of Georgia, Athens, GA.

**Hawthorne, C.**, & Philipp, R. (2013, February 1). *Making sense of the counterintuit**iveness of integers without simply relying on rules**. * Presentation at the annual meeting of the Greater San Diego Mathematics Council, San Diego, CA.

Studentsnaturallyuse positive numbers, but negative integers tend to be counterintuitive—for a multitude of reasons. For example, why does adding suddenly make smaller, the opposite of what students expect? In this session we discuss many issues about integers that confuse students and consider approaches to help students make sense of integers, contexts, the associated operations, and integer reasoning.

**Bishop, J. P.**, Philipp, R., Whitacre, I., Stephan, M. (discussant), & Jacobs, V. (discussant). (2013, January 24). *Using integers to rethink the role of context in school mathematics. * Presented in the Pedagogical Content Knowledge symposium at the 17th annual conference of the Association of Mathematics Teacher Educators, Orlando, FL.

Teachers may leave preparation programs believing that certain practices are good or bad: For example, a good teaching practice could be using contexts. In this symposium, we rethink our stance on contexts drawing from video of 160 integer-based, student interviews.

**Whitacre, I.**, Pierson, J., Lamb, L., Philipp, R., Schappelle, B., & Lewis, M. (2012, November 2). *What sense do children make of negative dollars?* Research report to be presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 2012), Kalamazoo, MI.

We interviewed 40 students in Grade 7 to investigate their integer reasoning. In one task, children were asked to write and interpret equations related to a story problem about borrowing money from a friend. Their responses reflect different perspectives concerning the relationship between this real-world situation and various numerical representations. We identify distinct ways in which integers were used and interpreted. All the students solved the story problem correctly. Few thought about the story as involving negative numbers. When asked to interpret an equation involving negative numbers in relation to the story, about half related it to the story in an unconventional fashion, which contrasts with typical textbook approaches. These findings raise questions about the role of money and other contexts in integer instruction.

**Bishop, J. P. **(2012, September 4). *Doctoral-student seminar. * University of Georgia, Athens, GA.__ __

**Lamb, L.,** Bishop, J. P., & Philipp, R. A. (2012, June 14). *Mapping developmental trajectories of students' conceptions of integers, Year 3*. Poster presented at DR K–12 Principal Investigator Conference, Washington, DC.

In this interactive session, we share research findings from 80 hour-long interviews with Grades 2 and 4 students about their ways of reasoning about integers. The session is centered on video of children solving integer problems. Audience members react to our findings and will help us generate implications for teaching and research.

**Lamb, L., Pierson, J., Philipp, R.,** Schappelle, B., Whitacre, I., & Lewis, M. (2012, 25 April). *Children's informal conceptions of integers.* Work Session presented at the National Council of Teachers of Mathematics (NCTM) Research Presession, Philadelphia, PA.

**Bishop, J. P.,** Lamb, L. L. C., Philipp, R. A., Schappelle, B. P., Whitacre, I., & Lewis, M. L. (2012, April 17). *Ways of reasoning about integers: Order, magnitude, and formalisms.* Poster presented at the American Educational Research Association meeting, Vancouver, British Columbia, Canada.

**Bishop, J. P. **(2012, April 5 and 6). *Witches, astrology, and negative numbers.* Invited presentations at Texas State University–San Marcos, TX and University of Texas, Austin, TX.

**Whitacre, I.**, Bishop, J. P., Lewis, M., Lamb, L., Philipp, R., & Schappelle, B. (2012, February 24). *Children's reasoning about integers: Video clips to share with preservice teachers.* Preliminary research report presented at the Research in Undergraduate Mathematics Education Conference (RUME XV), Portland, OR.

We have conducted interviews with children using integer-related tasks, and we have identified various ways of reasoning that children bring to bear on these tasks. One product of this work is a collection of compelling video clips. We will share examples of children's reasoning, and the audience will be engaged in discussions of children's reasoning and use of video in instruction. Attendees will receive a free DVD with video clips that can be used with preservice teachers.

**Lamb, L., & Philipp, R.** (2012, February 10). *Supporting prospective and practicing teachers: Sharing middle and high school students' conceptions of integers.* Presentation at the annual meeting of Association of Mathematics Teacher Educators (AMTE). Fort Worth, TX.

**Lewis, M., Pierson, J., & Lamb, L.** (2012, February 3). Children's conceptions of and strategies with integers. Presentation at the annual meeting of the Greater San Diego Mathematics Council, San Diego, CA.

**Lamb, L., & Bishop, J. P.** (2011, December 9). *Witches, astrology, and negative numbers*. Colloquium presented for the Center for Research in Mathematics and Science Education and the School of Teacher Education at San Diego State University, San Diego, CA.

**Philipp, R., & Lewis, M**. (2011, November 4). *Children’s surprising understandings of negative numbers.* Presentation at the annual meeting of the California Mathematics Council Southern Section, Palm Springs, CA.

**Whitacre, I**., Bishop, J. P., Lamb, L. L. C., Philipp, R. A., Schappelle, B. P., & Lewis, M. (2011, October 22). *Integers: History, textbook approaches, and children's productive mathematical intuitions. *Brief research report presented at the meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Reno, NV.

**Whitacre, I., **Bishop, J. P., Lamb, L. C., Philipp, R., Schappelle, B. P., & Lewis, M. (2011, October 2).* Integers: History, textbooks, and children’s productive mathematical intuitions.* Presentation at the History and Pedagogy of Mathematics Conference, Point Loma Nazarene University, San Diego, CA.

**Bishop, J. P.,** Lamb, L., Philipp, R., Whitacre, I., Schappelle, B., & Lewis, M. (2011, October 1). *Viewing students’ integer reasoning through the lens of history.* Presentation at the History and Pedagogy of Mathematics Conference, Point Loma Nazarene University, San Diego, CA.

**Whitacre, I.** (2011, July 21). *Preliminary findings from Project Z: Researching children’s integer reasoning*. Presentation to Project Algebraic—a Ca-Math Science Partnership grant involving Lake Elsinore Unified School District and Temecula Valley Unified School District, Temecula, CA.

**Philipp, R. A.** (2011, June). *Understanding students’ conceptions of integers. *Presentation at the Cognitively Guided Instruction Conference. Little Rock, AK.

**Philipp, R. A.** (2011, May 13). *How a focus on children's mathematical thinking supports the professional development of elementary school teachers.* Presentation at the Critical Issues in Mathematics Education Series, Mathematical Sciences Research Institute, Berkeley, CA.

Bishop, J. P., Lamb, L. L., Philipp, R. A., Schappelle, B., & **Whitacre, I.** (2011, April). *An investigation of negative-number reasoning: The case of Violet.* Presentation for the roundtable session, Student Reasoning in Mathematics, at the 2011 annual meeting of American Educational Research Association, New Orleans, LA.

**Lamb, L., Pierson, J.**, Philipp, R., Whitacre, I., Schappelle, B., & Lewis, M. (2011, January). *Understanding students’ conceptions of integers and implications for teacher educators*. Preconference session at the 11th annual conference of the Association of Mathematics Teacher Educators, Irvine, CA.

**Lamb, L.**, Bishop, J. P., & Philipp, R. A. (2010, December). Mapping Developmental Trajectories of Students' Conceptions of Integers, Year 2. Poster presented at Principal Investigator Conference, Washington, DC.

**Bishop, J. P.**, Lamb, L., Philipp, R., Schappelle, B., & Whitacre, I. (2010, October).* A developing framework for children’s reasoning about integers.* Research report on Young Students’ Reasoning About Number at the 32nd annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH.

**Lamb, L., Pierson, J.**, Philipp, R. A., & Whitacre, I. (2010, April). *Students’ conceptions of integers*. Presentation at the National Council of Teachers of Mathematics research presession, San Diego, CA.

**Lamb, L., Pierson, J.**, & Philipp, R. A. (2009, December). Mapping Developmental Trajectories of Students' Conceptions of Integers, Year 1. Poster presented at Principal Investigator Conference, Washington, DC.

__ARTICLES__

Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., & Schappelle, B. (in press). Leveraging structure: Logical necessity in the context of integer arithmetic. *Mathematical Thinking and Learning*.

Looking for, recognizing, and using underlying mathematical structure is an important aspect of mathematical reasoning. We explore the use of mathematical structure in children’s integer strategies by developing and exemplifying the construct of logical necessity. Students in our study used logical necessity to approach and use numbers in a formal, algebraic way, leveraging key mathematical ideas about inverses, the structure of our number system, and fundamental properties. We identified the use of carefully chosen comparisons as a key feature of logical necessity and documented three types of comparisons students made when solving integer tasks. We believe that logical necessity can be applied in various mathematical domains to support students to successfully engage with mathematical structure across the K–12 curriculum.

Lamb, L., Bishop, J., Philipp, R., Whitacre, I., & Schappelle, B. (in press). The relationship between flexibility and student performance on open number sentences with integers, *Proceedings of the 38th annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education *(PME-NA 2016), Tucson, AZ..

Bishop, J. P., Lamb, L. L., Philipp, R. A., Schappelle, B. P., & Whitacre, I. (2016). Uncovering structure in students' integer reasoning. *Mathematics Teaching in the Middle School*.

Whitacre, I., Bouhjar, K., Bishop, J. P., Philipp, R. A., Schappelle, B., & Lamb, L. L. (2016). Regular numbers and mathematical worlds, *For the Learning of Mathematics*, 36(2), 20-25.

Rather than describing the challenges of integer learning in terms of a transition from positive to negative numbers, we have arrived at a different perspective: We view students as inhabiting distinct mathematical worlds consisting of particular types of numbers (as construed by the students). These worlds distinguish and illuminate students' varied responses. Proficient students and adults may also inhabit multiple mathematical worlds from moment to moment. In our framework of mathematical worlds, we focus on the contrast between regular numbers and signed numbers in analyzing cases of student thinking. Implications of these ideas for educators and researchers are presented.

Bishop, J. P., Lamb, L.L., Philipp, R. A., Whitacre, I., Schappelle, B. P (2014). Using order to reason about negative numbers: The case of Violet, *Educational Studies in Mathematics, 86*(1), 39-59. (Published Online First. Available for purchase online at http://link.springer.com/article/10.1007/s10649-013-9519-x)

In this paper we present a case study that illustrates how a 2nd-grade child, Violet, used an ordinal view of number to reason about positive and negative integers and arithmetic involving integers. Violet’s ordinal view of number facilitated her ability to reason about and correctly solve some integer-related problems and constrained her solutions to others. We demonstrate how Violet’s thinking evolved over time while she extended the properties of whole numbers and addition and subtraction to the integers. Using this case study as a basis, we propose a series of developmental milestones that build toward one's understanding integers and integer arithmetic in an order-based way. We believe that understanding Violet’s order-based reasoning can help us listen to other children.Watch videos of Violet's reasoning while solving equations.

Bishop, J. P., Lamb, L.L., Philipp, R. A., Whitacre, I., Schappelle, B. P., & Lewis, M. (2014). Obstacles and affordances for integer reasoning: An analysis of children’s thinking and the history of mathematics. *Journal for Research in Mathematics Education,* *45*(1), 19-61.

We identify and document 3 cognitive obstacles, 3 cognitive affordances, and 1 type of integer understanding that can function as either an obstacle or affordance for learners while they extend their numeric domains from whole numbers to include negative integers. In particular, we highlight 2 key subsets of integer reasoning: understandings or knowledge that may, initially, interfere with one’s learning integers (which we call cognitive obstacles) and understandings or knowledge that may afford progress in understanding and operating with integers (which we call cognitive affordances). We analyzed historical-mathematical writings related to integers as well as clinical interviews with 6–10-year-old children to identify critical, persistent cognitive obstacles and powerful ways of thinking that may help learners to overcome obstacles.

Whitacre, I., Bishop, J. P., Philipp, R. A., Lamb, L. L., Bagley, S., & Schappelle, B. P. (2014). 'Negative of my money, positive of her money': Secondary students' reasoning about integers in relation to a money context. *International Journal of Mathematical Education in Science and Technology*, 1-16. doi: http://dx.doi.org/10.1080/0020739X.2014.

We interviewed 40 students each in Grades 7 and 11 to investigate their integer-related reasoning. In one task, students were asked to write and interpret equations related to a story problem about borrowing money from a friend. All the students solved the story problem correctly. However, they reasoned about the problem in different ways. Many students represented the situation numerically without invoking negative numbers, whereas others wrote equations involving negatives. When asked to interpret equations involving negative numbers in relation to the story, students did so in 2 ways. Their responses reflect distinct perspectives concerning the relationship between arithmetic equations and borrowing/owing. These findings raise questions about the role of contexts in integer instruction.

Whitacre, I., Bishop, J.P., Philipp, R.A., Lamb, L.L., & Schappelle, B. (2014). Dollars and sense: Students’ integer perspectives*. Mathematics Teaching in the Middle School 20*(2), 84-88.

Students use a variety of equations and interpretations when reasoning about a money context, and we encourage teachers to recognize and value students’ different ways of reasoning.

Lamb, L., Bishop, J., Philipp, R., Whitacre, I., Stephan, M., Bofferding, L., Lewis, J., Brickwedde, J., Bagley, S., & Schappelle, B. (2013). Building on the emerging knowledge base for teaching and learning in relation to integers. *Proceedings of the 35th annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education *(PME-NA 2013), Chicago, IL, 1358-1366.

Bishop, J. P., Lamb, L. L, Philipp, R. A., & Schappelle, B.P. (2013). Opportunities for algebraic reasoning in the context of integers. In Steffe, L.P., Moore, K.C., & Hatfield, L.L (eds.)

Students’ understanding of integers has major implications for their success in algebra. However, relative to other content areas, research about issues related to the teaching and learning of integers is limited. The members of this new working group seek to explore what is known in the field in relation to integers’ teaching and learning and to identify next steps for continuing to build the research base on this topic. During the first two sessions, participants will draw from facilitators’ sharing of their work in order to identify existing research in the field and collectively categorize research strands. During the third session, the group will discuss directions for future research, seek to build consensus about what constitutes a robust understanding of integers, and plan next steps for continued engagement.

*Epistemic Algebriac Students: Emerging models of students’ algebraic knowing: Papers from an invitational conference*, Wisdome Monograph, Vol. 4, University of Wyoming, Laramie, WY, 303-316.

Our work studying students’ ways of reasoning about integers has helped us to identify areas of intersection for integer reasoning and algebraic reasoning. We have found that some students are able to leverage fundamental principles of algebraic reasoning in their problem-solving strategies for integer tasks. In this chapter we discuss one area in which we saw evidence of algebraic reasoning embedded in children’s integer strategies—strategies for solving integer tasks in a category we have named logical necessity. This way of reasoning leveraged fundamental principles and underlying mathematical structures and emerged when students solved integer equations for unknowns (e.g., -3 + c = 6). Logical necessity emerged for students despite the fact that the problems we posed were extensions of arithmetic and primarily focused on numerical processes (what Sfard & Linchevski [1994] described as “algebra of a fixed value”). After sharing examples of logical necessity, we contrast this way of reasoning with the use of nonequivalent transformations that some students invoked in solving integer equations and discuss why students may struggle to understand the idea of invariant transformations.

Lamb, L L., Bishop, J. P., Philipp, R. A., Schappelle, B. P., Whitacre, I., & Lewis, M. (2012). Developing symbol sense for the minus sign. *Mathematics Teaching in the Middle School 18*(1), 5–9. Available online at http://www.jstor.org/stable/10.5951/mathteacmiddscho.18.1.0005

Research on how students make sense of and use the minus sign indicates that students struggle to understand the multiple meanings of this symbol. Teachers can support students in developing a robust understanding of each interpretation

Whitacre, I., Bishop, J. P., Lamb, L. L. C., Philipp, R. A., Schappelle, B. P., & Lewis, M. (2012). What sense do children make of negative dollars? In L. R. Van Zoest, J. -J. Lo, & J. L. Kratsky (Eds.), *Proceedings of the 34th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education* (pp. 958–964). Kalamazoo, MI: Western Michigan University. Available online at http://pmena.org/2012/proceedings.html

We interviewed 40 students in Grade 7 to investigate their integer reasoning. In one task, children were asked to write and interpret equations related to a story problem about borrowing money from a friend. Their responses reflect different perspectives concerning the relationship between this real-world situation and various numerical representations. We identify distinct ways in which integers were used and interpreted. All of the students solved the story problem correctly. Few thought about the story as involving negative numbers. When asked to interpret an equation involving negative numbers in relation to the story, about half related it to the story in an unconventional fashion, which contrasts with typical textbook approaches. These findings raise questions about the role of money and other contexts in integer instruction.

Lamb, L. L., Bishop, J. P., Philipp, R. A., Schappelle, B. P., Whitacre, I., & Lewis, M. (2012). *High school students’ conceptions of the minus sign.* * Mathematics Teaching,* (Issue 227), 40–44.

The minus sign elicits three useful meanings for students: subtraction (5 – 3), part of the symbol for a negative number (-4), and the opposite of (--4, the opposite of negative 4). High School students reason productively about the first 2 meanings but rarely appear to invoke the 3rd meaning.

Whitacre, I., Bishop, J. P., Lamb, L. C., Philipp, R. A., Schappelle, B. P., & Lewis, M. L. (2012). Happy and sad thoughts: An exploration of children's integer reasoning.* Journal of Mathematical Behavior, 31*, 356–365.

We explore the use of problems in context to investigate children’s mathematical reasoning. We report on 3 ways of reasoning about 2 related contextualized tasks involving directed magnitudes and relate these responses to reasoning about integers. We describe and analyze the responses of 3 children (in Grades 1, 3, and 5) who exemplify the ways of reasoning that we present. We view these ways of reasoning in terms of increasing levels of sophistication, potentially belonging to a single learning trajectory. Thus, we see the roots of more sophisticated integer reasoning in children’s early intuitions about directed magnitudes.

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.

In the third century, Diophantus, the "Father of Algebra" no less, described equations of the form x + 20 = 4 as "absurd." The absurdity stemmed from the fact that the result of four is obviously less than the addend of twenty. And more than 1300 years later, Pascal argued that subtracting four from zero leaves zero because of the impossibility of taking something from nothing. Surely, then, ideas this challenging are too complex for first graders--or are they? Recent research shows that children as young as six years of age can, in fact, reason about negative numbers and even perform basic calculations using them (Behrend and Mohs 2006; Wilcox 2008). The authors' goal was to build on this research to further explore young children's ideas about negative numbers. This article describes the background of their study, provides an overview of the tasks they used, discusses children's responses to these tasks, and identifies two ways that students in their study reasoned about and approached problems involving negative numbers. (Contains 1 table and 2 figures.)

Whitacre, I., Bishop, J. P., Lamb, L. L. C., Philipp, R. A., Schappelle, B. P., & Lewis, M. (2011). Integers: History, textbook approaches, and children’s productive mathematical intuitions. *Proceedings of the 33rd annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education* (pp. 913–920). Reno, NV.

We juxtapose the history of integers with current pedagogical approaches. We also present findings concerning children’s reasoning about negative numbers prior to instruction. We see alignment between children’s intuitions and the avenues that afforded progress for mathematicians, whereas the textbook approaches tend to run counter to lessons learned from history.

Bishop, J. P., Lamb, L., Philipp, R., Schappelle, B., & Whitacre, I. (2010). A developing framework for children’s reasoning about integers. In P. Brosnan, D. Erchick, & L. Flevares (Eds.)* Proceedings of the 32nd annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education *(Vol. VI, pp. 695–702). Columbus, OH: The Ohio State University.* (*Retrievable from http://pmena.org/2010/)