Amy Ellis

• University of Wisconsin–Madison

This paper introduces a quadratic growth learning trajectory, a series of transitions in students’ ways of thinking (WoT) and ways of understanding (WoU) quadratic growth in response to instructional supports emphasizing change in linked quantities. We studied middle grade (ages 12–13) students’ conceptions during a small-scale teaching experiment...

Mathematical play has a fairly short history, with strong roots further back in time (e.g., Papert, Montessori), and understanding the role of mathematical play from early childhood to adulthood is, as yet, unmapped. This working group will provide a community space to explore and discuss mathematical play broadly, ranging from early childhood to u...

The overwhelming majority of efforts to cultivate early mathematical thinking rely primarily on counting and associated natural number concepts. Unfortunately, natural numbers and discretized thinking do not align well with a large swath of the mathematical concepts we wish for children to learn. This misalignment presents an important impediment t...

Understanding function is a critical aspect of algebraic reasoning, and building up functional relationships is an activity increasingly encouraged at the elementary and middle school levels. This study identifies how one group of middle-school students leveraged their rate of change thinking to inform the development and understanding of correspon...

A persistent challenge in supporting students' proof activity is fostering the transition from less formal, empirical arguments to formal deductive arguments. A number of researchers have begun to investigate students' thinking with examples, addressing how example use can support conjecture understanding, exploration, and proof. We extend this lin...

Examples can be a powerful tool for students to learn to prove, particularly if used purposefully and strategically, but there is a pressing need to better understand the nature of productive example use. Therefore, we examined the characteristics of the successful and unsuccessful cases of proving in the context of a number theory task across the...

This study characterizes s of quadratic function from a variation perspective in the context of a quantitatively rich instructional setting. We studied middle grade (ages 12-13) ceptions during a small-scale teaching experiment aimed at fostering an understanding of quadratic function as a growth situation with a constant difference in rate of chan...

This paper explores the nature of students' quantitative reasoning and conceptions of functions supporting their ability to symbolize quadratic function rules, and the meanings students make of these rules. We analyzed middle school students' problem solving activity during a small group teaching experiment (n=6) emphasizing quadratic growth throug...

This article presents an Exponential Growth Learning Trajectory (EGLT), a trajectory identifying and characterizing middle grade students’ initial and developing understanding of exponential growth as a result of an instructional emphasis on covariation. The EGLT explicates students’ thinking and learning over time in relation to a set of tasks and...

Examples can play a critical role in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, there is more to learn about the precise ways in which mathematicians leverage examples as they formulate proofs. In this paper, we present results from surveys and interviews with ma...

This paper explores the role examples play as mathematicians formulate conjectures. Although previous research has examined example-related activity during the act of proving, less is known about how examples arise during the formulation of conjectures. We interviewed thirteen mathematicians as they explored tasks requiring the development of conje...

Supporting students' mathematical reasoning is an important goal of mathematics instruction, but can be challenging for many teachers .We report the results of a study aimed at better understanding and identifying the ways in which teachers support student reasoning when provided with conceptually rich tasks. This study resulted in the Teacher Move...

Modeling the motion of a speeding car or the growth of a Jactus plant, teachers can use six practical tips to help students develop quantitative reasoning.

This paper addresses the role of learning progressions in informing many international standards documents, discussing the affordances and limitations of building standards adn curricula from a learning progression model. An alternate model, the hypothetical learning trajectory, is introduced and contrasted with learning progressions. Using the exa...

This paper explores the role examples play in mathematicians’ conjecturing activity. While previous research has examined example-related activity during the act of proving, little is known about how examples arise during the formulation of conjectures. Thirteen mathematicians were interviewed as they explored tasks that required the development of...

Examples play a critical role in the exploration and proving of conjectures. Although proof has been studied extensively, the precise ways in which examples might facilitate successful proofs are not well documented or understood. Working within a larger set of studies that argue for the value of examples in proof-related activity, in this paper we...

This paper presents the results of two studies aimed at identifying the ways in which successful provers (students and mathematicians) engage with examples when exploring and proving conjectures. We offer a framework detailing the participants' actions guiding a) their example choice and b) their example use as they attempt to prove conjectures. Th...

Examples play a critical role in mathematical practice, particularly in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, the role that examples play in the process of exploring and proving conjectures has not received the same attention. In this paper, results are pres...

In this Research Commentary, 3 JRME authors describe the process of publishing their research in JRME. AU 3 authors published parts of their dissertation in JRME and are sharing their stories to help (new) researchers in mathematics education better understand the process and to offer (experienced) researchers in mathematics education a tool that c...

Although students' difficulties in developing and understanding proofs in mathematics is well documented, less is known about how students' example use may support their proof practices, particularly at the middle school level. Research on example use suggests that strategic thinking with examples could play an important role in exploring conjectur...

Examples play a critical role in mathematical practice, particularly in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, the role that examples play in the process of exploring and proving conjectures has not received the same attention. In this paper, we present a fra...

Generalization is a critical component of mathematical activity and has garnered increased attention in school mathematics at all levels. This study documents the multiple interrelated processes that support productive generalizing in classroom settings. By studying the situated actions of 6 middle school students and their teacher—researcher worki...

Twenty middle-school students participated in semi-structured interviews in which they were asked to assess the validity of two mathematical conjectures. In addition to being free to develop a valid proof as a justification, students were also asked to generate numeric examples to test the conjecture. Students demonstrated strategic reasoning in th...

Understanding function is a critical aspect of algebraic reasoning, and building functional relationships is an activity encouraged in the younger grades to foster students’ relational thinking. One way to foster functional thinking is to leverage the power of students’ capabilities to reason with quantities and their relationships. This paper expl...

Research in the field of mathematics indicates that many students struggle with justification and proof. However, in non-mathematical contexts, students are relatively strong at inferential reasoning. Our research presents two parallel lines of investigation—one focused on mathematical domains, the other focused on non-mathematical domains—in order...

This article presents secondary students’ generalizations about the connections between algebraic and graphical representations of quadratic functions, focusing specifically on the roles of the parameters a, b, and c in the general form of a quadratic function, y=ax2+bx+c. Students’ generalizations about these connections led to a surprising findin...

Standardized tests in the U.S. indicate that girls now score just as well as boys in math.

This article presents a cohesive, empirically grounded categorization system differentiating the types of generalizations students constructed when reasoning mathematically. The generalization taxonomy developed out of an empirical study conducted during a 3-week teaching experiment and a series of individual interviews. Qualitative analysis of dat...

This paper reports the mathematical generalizations of two groups of algebra students, one which focused primarily on quantitative relationships, and one which focused primarily on number patterns disconnected from quantities. Results indicate that instruction encouraging a focus on number patterns supported generalizations about patterns, procedur...

Research investigating algebra students' abilities to generalize and justify suggests that they experience difficulty in creating and using appropriate generalizations and proofs. Although the field has documented students' errors, less is known about what students do understand to be general and convincing. This study examines the ways in which se...

We address the telling/not-telling dilemma in mathematics education. Telling is instructionally important, but has been downplayed because of (a) perceived inconsistencies between telling and constructivism, (b) increased awareness of the negative consequences of relying too heavily on telling, and (c) a focus on "non-telling" actions as pedagogica...

This article sets forth a way of connecting the classroom instructional environment with individual students' generalizations. To do so, we advance the notion of focusing phenomena, that is, regularities in the ways in which teachers, students, artifacts, and curricular materials act together to direct attention toward certain mathematical properti...

We use the notion offocusing phenomena to help explain how a teacher’s actions were connected to her students’ interpretations of a linear equation. This study was conducted in a high-school classroom that regularly emphasised dependency relationships in real-world situations. Seven interviews revealed a majority view ofy = b + mx as astorage conta...

When technology is implemented in classrooms, students often form ideas that are unexpected and unwanted by the teachers and the designers of the technology. This article advances the notion of the focusing effect of technology as a way of systematically accounting for the role of technology in such situations. A focusing effect refers to the direc...