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Learning progressions have become an important construct in educational research, in part because of their ability to inform the design of coherent standards, curricula, assessments, and instruction. In this paper, I discuss how a learning progressions approach has guided our development of an early algebra innovation for the elementary grades and provide examples of how this approach can help challenge a settled mathematics learning status quo about the kind of algebra students can learn, when they can learn it, and how all students can be successful. Empirically derived learning progressions are an important part of designing early algebra innovations that can open new curricular pathways for teaching and learning algebra, creating accessible and effective avenues of learning for all students. [For the complete proceedings, see ED630210.]

This research reports on the teacher language and gesture that contributed to shifts in thinking about the equal sign and equations observed in twenty kindergarteners who took part in an early algebra intervention. Our analysis revealed ways in which the teacher used language and gesture to support students in moving from describing and working with the equal sign operationally (i.e., as a signal to compute) to describing the symbol as indicating the equivalence of two amounts and successfully working with equations of various forms. We detail four kinds of language and two kinds of gesture specifically related to mathematical equivalence that we believe contributed to students' growth. [For the complete proceedings, see ED630210.]

We share results from a one-year early algebra classroom intervention. One kindergarten, one first-grade, and one second-grade classroom participated in the intervention and three classrooms of the same grade levels served as control sites. The intervention addressed the structure of even and odd numbers, mathematical equivalence and equations, properties of arithmetic, the representation of varying unknown quantities, and functional thinking. Students in the control classrooms received their regular mathematics instruction. We compare growth in algebraic thinking across the two groups as measured by an online assessment completed by 40 students per condition. Findings suggest that students who participated in the early algebra lessons experienced more overall growth in their understanding of early algebra concepts as compared to control students.

This research focuses on the retention of students' algebraic understandings one year following a Grades 3-5 early algebra intervention. Participants included 1455 Grade 6 students who had participated in a cluster randomized trial. Approximately half of these students received an early algebra intervention as part of their regular instruction in Grades 3-5, while the other half received only their regular mathematics instruction. Results show that, as was the case at the end of Grades 3, 4 and 5, treatment students outperformed control students at the end of Grade 6 on a measure of algebraic understanding. This was despite the fact that treatment students experienced a significant decline in performance and control students a significant increase in performance after the intervention. An item-by-item analysis performed within condition revealed the areas in which students in the two groups experienced a change in performance. [For the complete proceedings, see ED606556.]

This research explores the interplay between students' understandings of proportional and functional relationships. Approximately 90 students participated in an early algebra intervention in Grades 3- 5. Before the intervention and after each year of the intervention, we evaluated their understandings of proportional and functional relationships. Data revealed that among Grades 4 and 5 students who identified a correct function rule, a higher percentage were unsuccessful solving a proportional reasoning problem than those who were not able to identify a correct function rule. Namely, the data suggest that students' development of functional thinking may interfere with their development of understanding proportional relationships. [For complete proceedings, see ED581294.]

Through a theoretical framework emphasizing the importance of fidelity of implementation (FOI), this paper explores how 3rd and 4th grade teachers implemented an early algebra intervention, and the extent to which the FOI related to student learning. The data for this report are taken from the first two years of an experimental research project. Videotaped classroom observations, our primary measure of FOI, were coded by adding to and adapting the Mathematical Quality of Instruction (MQI) instrument, and student performance was measured by overall score (correctness) on an algebra assessment. Results revealed a significant positive relationship between teachers' implementation and their students' performance. [This paper was published in: Galindo, E., & Newton, J., (Eds.), "Proceedings of the Thirty-Ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education" (pp. 299-302). Hoosier Association of Mathematics Teacher Educators.]

In this paper, we describe our learning progressions approach to early algebra research that involves the coordination of a curricular framework, an instructional sequence, written assessments, and levels of sophistication describing the development of students' thinking. We focus in particular on what we have learning through this approach about the development of students' abilities to generalize and represent functional relationships in a grades 3-5 early algebra intervention by sharing the different levels of responses we observed in students' written work and the percent of students situated at each level across different tasks.

We detail a learning progressions approach to early algebra research and how existing work around learning progressions and trajectories in mathematics and science education has informed our development of a four-component theoretical framework consisting of: a curricular progression of learning goals across big algebraic ideas; an instructional sequence of tasks based on objectives concerning content and algebraic thinking practices; assessments; and posited levels of sophistication in children's reasoning about algebraic concepts within big ideas of early algebra. This research balances the goals of longitudinal research on supporting students' preparedness for algebra while attending to the practical goals of establishing connections among curriculum, instruction, and student learning. [This paper was published in: Bartell, T. G., Bieda, K. N., Putnam, R. T., Bradfield, K., & Dominguez, H. (Eds.) (2015). "Proceedings of the 37th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education" (p201-204). East Lansing, MI: Michigan State University.]

In this paper, we analyze individual semi-clinical interviews conducted with one kindergarten and one first-grade student. We build on prior research to offer evidence, via excerpts from these interviews, that children as young as kindergarten have a powerful, intuitive sense of generality and indeed naturally draw upon it to reason through mathematical scenarios. We identify within these children's utterances four features of generalizing for which educators might attend. [For the complete proceedings, see ED583989.]

Third- through fifth-grade students participating in a classroom teaching experiment investigating the impact of an Early Algebra Learning Progression completed pre- and post-assessment items addressing their abilities to engage in functional thinking. We found that after a sustained early algebra intervention, students grew in their abilities to shift from recursive to covariational thinking about linear functions and to represent correspondence rules in both words and variables. [For the complete proceedings, see ED584829.]

Third- through fifth-grade students participating in a classroom teaching experiment investigating the impact of an Early Algebra Learning Progression completed pre- and post-assessments documenting their abilities to represent or describe unknown quantities. We found that after a sustained early algebra intervention, students grew in their abilities to represent related unknown quantities using letters as variables. [For the complete proceedings, see ED585874.]

This conference proceedings contains three plenary session reports, 12 working group and 79 research reports, 35 short oral reports, 60 poster session reports, and two discussion group reports. Major papers (excluding "short orals" and "posters") include: (1) "Semantical Obstacles in Mathematics Understanding" (Carlos Arteaga and Manuel Santos); (2) "Mathematical Beliefs and Conceptions of College Algebra Students" (Victor V. Cifarelli and Tracy Goodson-Espy); (3) "Mathematical Intimacy: Local Affect in Powerful Problem Solvers" (Valerie A. DeBellis); (4) "A Modeling Approach to Non-Routine Problem Situations" (Helen M. Doerr); (5) "Posing Questions or Reformulation of Problems as an Activity to Perceive the Structure of Mathematical Problems" (Juan Estrada); (6) "An Analysis of Mathematics Problems in Middle School Textbooks" (Yeping Li); (7) "Recognizing Isomorphism and Building Proof: Revisiting Earlier Ideas" (Ethel M. Muter and Carolyn A. Maher); (8) "Students' Strategies for Searching the Instantaneous Flow in a Graphical Representation of Time versus Volume" (Rodolfo Oliveros); (9) "An Investigation of the Ability to Process and Comprehend Text on Student Success in Solving Algebraic Word Problems" (David K. Pugalee); (10) "Variables Influencing the Effectiveness of Small-Group Work: A Year Long Study of Collaborative Math Problem Solving in Two Fourth-Grade Classrooms" (Rose Sinicrope, Marion A. Eppler, Marsha Ironsmith); (11) "Creating the Conditions for Conceptual Change" (Judy Clark and Joan D. Lukas); (12) "Assessing Students' Attitudes toward Mathematics from a Multilevel Perspective" (George Frempong); (13) "Studying the Impact of Reformed Mathematics Curricula" (Mary C. Shafer and Norman Web); (14) "Connecting Students' Everyday Mathematics and School Mathematics" (Marta Civil, Melanie Ayers, Jose David Fonseca, and Leslie Kahn); (15) "A Vygotskian Action-Research Model for Developing and Assessing Conceptual Models and Instructional Materials Inter-Actively" (Karen C. Fuson, Bruce Sherin, and Steven T. Smith); (16) "Relating Equity to Classrooms Which Promote Understanding: Identifying Relevant Issues" (Lynn Liao Hodge); (17) "The Power of One in a Mathematics Classroom: Lisa and Divisibility by Four" (Kathy M.C. Ivey); (18) "Students' Use of Function Ideas in Everyday Activities" (Joanna O. Masingila and Helen M. Doer); (19) "Mathematics Power: Developing the Potential to Do Work" (Rodney E. McNair); (20) "Using the Metaphor of Voice to Investigate the Mathematical Experiences of African American Students" (Vivian R. Moody and Patricia S. Moyer); (21) "The Impact of a Secondary Preservice Teacher's Beliefs about Mathematics on Her Teaching Practice" (Babette M. Benken and Melvin (Skip) Wilson); (22) "Implementing Mathematics Reform: A Look at Four Veteran Mathematics Teachers" (M. Lynn Breyfogle and Laura R. Van Zoest); (23) "Reflections about Listening and Dialogue as Evidence of Teacher Change" (George W. Bright, Anita H. Bowman, and Nancy N. Vacc); (24) "Shifting Beliefs: Preservice Teachers' Reflections on Assessing Students' Mathematical Ideas" (Patricia S. Moyer and Vivian R. Moody); (25)"Understanding How Prospective Secondary Teachers Avoid Accommodating Their Existing Belief Systems" (Daniel Siebert, Joanne Lobato, and Stacy Brown); (26) "Self-Estimating Teachers' Views on Mathematics Teaching--Modifying Dionne's Approach" (Gunter Torner); (27) "Learning to Teach Mathematics Using Manipulatives: A Study of Preservice Elementary School Teachers" (Sunday A. Ajose); (28) "Prospective Elementary Teachers' Constructions of Understanding about Multiplication with Rational Numbers" (Diane S. Azim); (29) "Narrative as a Tool for Facilitating Preservice Mathematics Teacher Development" (Olive Chapman); (30) "Teacher Change during an Urban Systemic Initiative" (Thomas G. Edwards and Sally K. Roberts); (31) "Cases as Contexts for Teacher Learning" (Megan L. Franke, Elham Kazemi, Jeffrey Shih, and Stephanie Biagetti); (32) "Preservice Elementary Teachers Learning to Integrate Mathematics with Other Subjects through Interdisciplinary Instruction" (Susan L. Hillman); (33) "Using Conceptual Design Tools to Foster Learning in the Game Design Environment" (Yasmin B. Kafai, Megan L. Franke, Jeffrey C. Shih, and Cynthia C. Ching); (34) "Redefining 'The Object' of Assessment in Clinical Interviewing" (Rochelle G. Kaplan and Peter M. Appelbaum); (35) "Professional Development and Systemic Reform: Some Important Connections" (Andrea Lachance and Jere Confrey); (36) "An Analysis of Two Novice K-8 Teachers Using a Model of Teaching-In-Context" (Cheryl A. Lubinski, Albert D. Otto, Beverly A. Rich, and Patricia A. Jaberg); (37) "Preservice Teachers' Ideas about Teaching Mathematics: Contributions to Frameworks for Teacher Education Courses" (E.E. Oldham, A.E. van der Valk, H.G.B. Broekman, and S.B. Berenson); (38) "Classroom Teachers Becoming Teacher Educators: 'Just' Facilitators or Active Agents?" (Deborah Schifter and Susan Jo Russell); (39) "Using Thought-Revealing Activities to Stimulate New Instructional Models for Teachers" (Roberta Y. Schorr and Richard Lesh); (40) "The Role of Formal and Informal Support Networks as Teachers Attempt to Change Their Practice from Traditional to Innovative Reform-Oriented Teaching of Mathematics" (David Feikes); (41) "Enhancing the Pedagogy of University Faculty through Mentoring and Reflection: Preliminary Observations" (Lynn C. Hart); (42) "Developing Teachers' Ability to Identify Student Conceptions during Instruction" (Miriam G. Sherin); (43) "Characterizing a Perspective on Mathematics Learning of Teachers in Transition" (Martin A. Simon, Ron Tzur, Karen Heinz, Margaret Kinzel, and Margaret Schwan Smith); (44) "Investigating Teachers' Insights into the Mathematics of Change" (Janet S. Bowers and Helen M. Doerr); (45) "Technology, Students' Understanding of Graphs, and Classroom Interactions" (Maria L. Fernandez); (46) "Learning the Standard Addition Algorithm in a Child-Centered Classroom" (Betsy McNeal and Betty Tilley); (47) "Megan: 'Seventeen Take Away Sixteen? That's Hard!'" (Catherine A. Pearn); (48) "Thinking in Units: An Alternative to Counting as a Basis for Constructing Number" (Grayson H. Wheatley and Anne M. Reynolds). (ASK)

This conference proceedings contains three plenary session reports, 12 working group and 79 research reports, 35 short oral reports, 60 poster session reports, and two discussion group reports. The titles of all papers (excluding "short orals", "posters", and brief discussion group reports) are: (1) "On Relationships between Psychological and Sociocultural Perspectives" (Pat Thompson and Paul Cobb); (2) "On Theory and Models: The Case of Teaching-In-Context" (Alan H. Schoenfeld); (3) "Building Mathematical Structure within a Conjecture Driven Teaching Experiment on Splitting" (Jere Confrey); (4) "The Role of Advanced Mathematical Thinking in Mathematics Education Reform" (M. Kathleen Heid, Joan Ferrini-Mundy, Karen Graham, and Guershon Harel); (5) "Visions of Algebra in Diverse Instruction" (David Kirshner, Carolyn Kieran, Tom Kieren, and Analucia D. Schliemann); (6) "Representing Algebraic Relations Before Algebraic Instruction" (Analucia D. Schliemann); (7) "As It Happens: Algebra Knowing in Action: The Polynomial Engineering Project" (Tom Kieren); (8) "Theories and Experiments in Collegiate Mathematics Education Research" (Ed Dubinsky, Francisco Cordero, Joel Hillel, and Rina Zazkis); (9) "Gender and Mathematics: Integrating Research Strands" (Suzanne Damarin, Diana Erchick, Jere Confrey, Dorothy Buerk, Linda Condron, Ruth Cossey, Laurie Hart, Peter Appelbaum, and Patricia Brosnan); (10) "Geometry and Technology" (Doug McDougall, Dan Chazan, Chronis Kynigos, and Rich Lehrer); (11) "Learning to Reason Probabilistically" (Carolyn Maher, Susan Friel, Cliff Konold, and Robert Speiser); (12) "Research on Rational Number, Ratio and Proportionality" (Tom Post, Kathleen Cramer, Guershon Harel, Thomas Kieren, and Richard Lesh); (13) "Representations and Mathematics Visualization" (Fernando Hitt, James Kaput, Norma Presmeg, Luis Radford, and Manuel Santos-Trigo); (14) "Using Socio-Cultural Theories in Mathematics Education Research" (Judit Moschkovich, Karen Fuson, Yolanda de la Cruz, Joanna Masingila, and Marta Civil); (15) "Developing a Conceptual Framework for Mathematics Teaching" (Martin Simon, Deborah Ball, Rijkje Dekker, and Susan Jo Russell); (16) "Teacher Change in a Reform Calculus Curriculum: Concepts Related to the Derivative" (Thomas Fox); (17) "A Logo-Based Microworld as a Window on the Infinite" (Luis Moreno-Armella and Ana Isabel Sacristan R.); (18) "Sociomathematical Norms and Student Autonomy in Calculus II Honors Students" (Chris L. Rasmussen and Karen D. King); (19) "The Role of a Formal Definition in Nine Students' Concept Image of Derivative" (Michelle J. Zandieh); (20) "What Kind of Notation Do Children Use to Express Algebraic Thinking?" (Carol W. Bellisio and Carolyn A. Maher); (21) "Effects of Algebra Instruction on the Recognition of the Mathematical Structures of Word Problems" (Emily D.F. McFadden); (22) "Middle School Students' Algebraic Reasoning: New Skills and Understandings from a Reform Curriculum" (John P. Smith III, Elizabeth A. Phillips, and Beth Herbel-Eisenmann); (23) "Developing an Imagistic Basis for Understanding Binomial Multiplication" (Erna Yackel and Diana Underwood); (24) "The Effects of Staff Development Focused on Assessment Practices" (Daniel J. Brahier); (25) "Case Studies of Preservice Elementary Teachers' Development in Mathematics Assessment" (Anne Raymond); (26) "Assessing and Documenting Student Knowledge and Progress in Early Mathematics" (Bob Wright, Rita Stewart, Ann Stafford, and Richard Cain); (27) "The Dualistic Nature of School Mathematics Discourse" (Eric J. Knuth and Dominic Peressini); (28) "Examining Teacher and Classroom Supports for Student Invention" (Elizabeth S. Senger); (29) "Tracing Children's Construction of Fractional Equivalence" (Elena Steencken and Carolyn Maher); (30) "Graphing of Discrete Functions versus Continuous Functions: A Case Study" (Fernando Hitt and Orlando Planchart); (31) "Perception of Graphs and Equations of Functions and Their Relationships in a Technology-Enhanced College Algebra Course" (Armando M. Martinez-Cruz); (32) "Understanding Mathematical Experience" (Tracy Noble, Ricardo Nemirovsky, Tracey Wright, and Cornelia Tierney); (33) "Unspecified Things, Signs, and 'Natural Objects': Towards a Phenomenological Hermeneutic of Graphing" (Wolff-Michael Roth); (34) "Re-Thinking Covariation from a Quantitative Perspective: Simultaneous Continuous Variation" (Luis Saldanha and Patrick W. Thompson); (35) "Analyzing Children's Length Strategies with Two-Dimensional Tasks: What Counts for Length?" (Jeffrey Barrett and Douglas Clements); (36) "How Has Measurement Been Taught in Mexico?" (Mariana Saiz); (37) "Understanding Angles from the Perspective of a High School Cerebral Palsy Student" (Kenneth L. Shaw and Paul Durden); (38) "Analyzing Students' Learning in Social Context: A Student Learns to Measure" (Michelle Stepha); (39) "The Learning of Concepts of Stochastic Implied in the Binomial Distribution by Means of the Use of Different Representations and Contexts" (Jesus Colin Miranda); (40) "Comparing Data Sets: How Do Students Interpret Information Displayed Using Box Plots?" (Susan N. Friel); (41) "Students' Statistical Thinking" (Graham Jones, Carol Thornton, Cynthia Langrall, Edward Mooney, Bob Perry, and Ian Putt); (42) "Tracing the Origins and Extensions of Mathematical Ideas" (Regina Kiczek and Carolyn Maher); (43) "Justifications to 5-8 Year-old Students' Responses to Decision Problems" (M.en C. Araceli Limon Segovia); (44) "Supporting Students' Reasoning about Data" (Kay McClain and Paul Cobb); (45) "An Analysis of Students' Statistical Understandings" (Maggie McGatha, Paul Cobb, and Kay McClain); (46) "Middle School Students' Misuse of the Phrase '50-50 Chance' in Probability Instruction" (James E. Tarr). (ASK)

The study reported here uses a quasi-experimental design to compare the algebraic thinking of students after a 3-year, grades 3--5 early algebra intervention to students in more traditional (arithmetic-focused) classrooms. Data sources are students' responses to grades 3--5 written assessment items, designed by the project team to measure learning throughout the intervention, that focus on students' use of variables to represent generalizations across four situations: (1) algebraic expressions; (2) equations; (3) functional relationships; and (4) arithmetic generalizations. Results shows that intervention students performed significantly better than comparison students in both grades 3 and 4 post-tests. More interesting, however, is that intervention students were significantly more successful writing a function rule with variables than with words.