Michelle Cirillo

• Doctor of Philosophy
• Professor (Associate) at University of Delaware

Eliciting student thinking is a core practice of ambitious mathematics teaching, yet it is one practice that is difficult for novice teachers to do consistently and proficiently. To better understand how to support pre-service teachers (PSTs) to enact this practice, our study sought to investigate how secondary PSTs engage in eliciting when they ex...

The recent push toward active learning – engaging students in the learning process – is meant to benefit students. Yet there is still much to learn about students’ perceptions of this phenomenon. We share results from an interview study of students’ perceptions of features of two active learning models institutionalized at a large doctoral-granting...

This chapter synthesizes three analyses of a 2-day secondary mathematics lesson presented through classroom transcripts. Specifically, three different researchers viewed the same data set through the lenses of mathematical justification, argumentation, or proof. Here, we looked across these three chapters and compared and contrasted the research qu...

Argumentation, justification, and proof are conceptualized in many ways in extant mathematics education literature. At times, the descriptions of these objects and processes are compatible or complementary; at other times, they are inconsistent and even contradictory. Regardless of the descriptions of these processes, however, given the importance...

We report findings from an investigation of one teacher's instruction as he guided students through the proofs of 21 theorems in a Grade 8 Honors Geometry course. We describe a routine involving four distinct phases, including Setting up the Proof and Concluding the Proof. Results from an end-of-course proof test are also presented to attest to the...

The Covid-19-induced closure of schools forced many instructors throughout the world to develop ways to deliver instruction online. Video-conferencing applications became one prominent tool for instructors to continue instruction. We report the perceptions of undergraduate mathematics students and preservice mathematics teachers who interacted in o...

Decades of research have established that solving geometry proof problems is a challenging endeavor for many students. Consequently, researchers have called for investigations that explore which aspects of proving in geometry are difficult and why this is the case. Here, results from a set of 20 interviews with students who were taught proof in sch...

Twenty students who earned A or B course-grades in the proof unit(s) of a secondary course that addressed proof in geometry were asked to work on two proof tasks while sharing their thinking aloud and using smartpens. Students were classified into two categories: those who were successful with both proofs and those who were unsuccessful with both p...

Across three years, more than 1000 students enrolled in courses that addressed proof in secondary geometry were tested at the beginning of the school year and after completing the proof unit(s). In 4 of the 20 study teachers' classrooms, students were introduced to proof through a set of 16 specially designed lesson plans that addressed particular...

(...) In this editorial, we discuss the first of the five overarching problems: defining and measuring learning opportunities precisely enough to study how to maximize the quality of the opportunities experienced by every student.

Although often asked tactfully, a frequent question posed to authors by JRME reviewers is “So what?” Through this simple and well-known question, reviewers are asking: What difference do your findings make? How do your results advance the field? “So what?” is the most basic of questions, often perceived by novice researchers as the most difficult q...

In our recent editorials (Cai et al., 2019a, 2019b), we discussed the important roles that research questions and theoretical frameworks play in conceptualizing, carrying out, and reporting mathematics education research. In this editorial, we discuss the methodological choices that arise when one has articulated research questions and constructed...

In our March editorial (Cai et al., 2019), we discussed the nature of significant research questions in mathematics education. We asserted that the choice of a suitable theoretical framework is critical to establishing the significance of a research question. In this editorial, we continue our series on high-quality research in mathematics educatio...

Decades of research suggests that students have struggled with learning proof in geometry. We identify and describe five common misconceptions and errors that inhibit students' success with proof as evidenced by multiple data sources. We then share sample tasks that can be used to diagnose and remedy these misconceptions.

Use these ideas to diagnose and address common conceptual obstacles that inhibit students’ success.

In 2002, the National Research Council (NRC) released Scientific Research in Education , a report that proposed six principles to serve as guidelines for all scientific inquiry in education. The first of these principles was to “pose significant questions that can be investigated empirically” (p. 3). The report argued that the significance of a que...

Students who earned high marks during the proof semester of a geometry course were interviewed to understand what high-achieving students actually took away from the treatment of proof in geometry. The findings suggest that students had turned proving into a rote task, whereby they expected to mark a diagram and prove two triangles congruent.

Argumentation, justification, and proof are conceptualized in many ways in extant mathematics education literature. At times, the descriptions of these objects and processes are compatible or complementary; at other times, they are inconsistent and even contradictory. The inconsistencies in definitions and usages of the terms argumentation, justifi...

Argumentation, justification, and proof are conceptualized in many ways in extant mathematics education literature. At times, the descriptions of these objects and processes are compatible or complementary; at other times, they are inconsistent and even contradictory. The inconsistencies in definitions and usages of the terms argumentation, justifi...

Argumentation, justification, and proof are conceptualized in many ways in extant mathematics education literature. At times, the descriptions of these objects and processes are compatible or complementary; at other times, they are inconsistent and even contradictory. The inconsistencies in definitions and use of the terms argumentation, justificat...

Reconsider typical discourse strategies when discussing homework and move toward a system that promotes the Standards for Mathematical Practice.

We describe our ongoing efforts to design materials for supporting secondary mathematics teachers in using a set of Teacher Discourse Moves purposefully in order to develop classroom discourse that is both productive and powerful for students' learning. We focus on secondary mathematics classroom discourse because mathematical language and meanings...

By examining findings and research methodology across studies focused on reasoning-and-proving in mathematics textbooks, this paper provides commentary on the nature of reasoning and proving and curriculum analysis in mathematics education. Large variations across the studies were noted with regard to the framings of reasoning-and-proving as well a...

Background: Homework is a key component of students' school mathematics experiences, especially at the secondary level. Past studies have shown that because a substantial portion of class time is spent going over homework assignments, homework does not remain an at-home activity. Yet, little is known about what takes place during the classroom acti...

Although popular media often provides negative images of mathematicians, we contend that mathematics classroom practices can also contribute to students' images of mathematicians. In this study, we examined eight mathematics teachers' framings of mathematicians in their classrooms. Here, we analyze classroom observations to explore some of the char...

Through the Standards documents, NCTM has called for changes related to Reasoning and Proof and Geometry. There is some evidence that these recommendations have been taken seriously by mathematics educators and textbook developers. However, if we are truly to realize the goals of the Standards, we must pose problems to our students that allow them...

In this article, we describe aspects of mathematical language that could be problematic to English-language learners, provide recommendations for teaching English-language learners, and suggest activities intended to foster language development in mathematics.

What I wish I had known about teaching proof before I taught geometry

“Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well” (The Teaching Principle, NCTM 2000, p. 16).

This article examines a collaborative study group's discussions about “revoicing,” an idea from linguistics that has been identified as an important discourse strategy in the teaching of mathematics as well as other content areas. This group, made up of eight middle grades (grades 6–10) mathematics teacher-researchers (TRs), one university professo...

In this article, the author explores the history and the mathematics used by Newton and Leibniz in their invention of calculus. The exploration of this topic is intended to show students that mathematics is a human invention. Suggestions are made to help teachers incorporate the mathematics and the history into their own lessons. (Contains 3 online...