Episode 31: Learning to Launch Cognitively Demanding Tasks: A Practice-Based Unit for Secondary Methods
June 1st, 2021 | Season 7 | 23 mins 1 sec
cognitively demanding tasks; noticing; pedagogies of practice; practice-based teacher education; secondary methods course; task launch
Cognitively demanding tasks provide important opportunities for students to develop an understanding of mathematics; however, they are challenging to launch and implement. The authors designed a secondary methods unit on launching tasks. Participants in the study were enrolled in five different methods courses. Using a noticing framework, findings suggest that by engaging in the unit, preservice teachers developed a greater understanding of the four aspects of an effective task launch. When viewing video examples, preservice teachers were able to talk about the four aspects of a task launch with increased specificity. Additionally, they began to identify ways of developing common language without reducing cognitive demand. We discuss implications of this work and offer suggestions for future teacher education research.
May 25th, 2021 | Season 7 | 30 mins 11 secs
professional development; pk–12 teachers; argumentation
To support teachers in implementing ambitious reform efforts, professional developers and teacher educators need to know more about teachers’ thinking about argumentation. Specifically, there is a need to understand more about teachers’ views and evaluations of students’ mathematical arguments as they play out in practice. In this article, we share a tool developed to elicit teachers’ pre- and post evaluations of students’ mathematical arguments on a problem-solving task. We discuss the design of the tool and provide evidence of its utility. Our findings indicate that the tool can be used to (a) identify changes in teachers’ evaluations of student mathematical arguments over time and (b) inform the design of professional learning experiences
May 18th, 2021 | Season 7 | 28 mins 32 secs
professional development; grades 3–5
Research has shown that the ways in which teachers engage in professional development activities vary widely. Farmer et al. (2003) identified three levels of teacher appropriation within professional development, with their inquiry stance indicative of teachers engaging in self-sustaining practices. In our project, we modified the demonstration lesson format so that teachers took an active role in changing an observed lesson and then viewing the impact of those changes as a second lesson was taught. We share evidence that this modified structure provided opportunities for teachers to engage in an inquiry stance on teaching and discuss implications for professional development providers in structuring activities to foster an inquiry stance.
Episode 28: Supporting Preservice Secondary Mathematics Teachers’ Professional Judgment Around Digital Technology Use
May 11th, 2021 | Season 7 | 30 mins 12 secs
mathematics methods course; grades 6–12; use appropriate tools strategically; technology
The pervasiveness of digital technology creates an imperative for mathematics teacher educators to prepare preservice teachers (PSTs) to select technology to support students’ mathematical development. We report on research conducted on an assignment created for and implemented in secondary mathematics methods courses
requiring PSTs to select and evaluate digital mathematics tools. We found that PSTs primarily focused on pedagogical fidelity (ease of use), did not consider mathematical fidelity (accuracy), and at times superficially attended to cognitive fidelity (how well the tool reflects students’ mathematical thinking processes) operationalized as the CCSS for Mathematical Practice and Five Strands of Mathematical Proficiency. We discuss implications for implementing the assignment and suggestions for addressing PSTs’ challenges with identifying the mathematical practices and five strands.
Episode 27: Editorial February 2021: Considering Connections Across Research Questions, Data, Methods, and Claims
May 4th, 2021 | Season 7 | 18 mins 23 secs
A key component of a Mathematics Teacher Educator (MTE) journal article is a description of the innovation or tool that was used with teachers and a report of the details of the research on that innovation/tool. In our September editorial we highlighted the innovation. In this editorial, we will focus on the importance of aligning research questions, data, and claims with existing research and theories to present a strong and coherent argument about the contribution the innovation/tool makes to mathematics teacher education.
Episode 24: Representing Student Voice in an Approximation of Practice: Using Planted Errors in Coached Rehearsals to Support Teacher Candidate Learning
January 4th, 2021 | Season 6 | 29 mins 55 secs
approximations of practice; authenticity; coached rehearsal; responding to errors; whole-class discussion
Approximations of practice provide opportunities for teacher candidates (TCs) to engage in the work of teaching in situations of reduced complexity. A problem of practice for teacher educators relates to how to represent student voice in approximations to engage TCs with interactive practices in meaningful ways. In this article, we share an analysis of our use of “planted errors” in coached rehearsals with secondary mathematics TCs focused on the practice of responding to errors in whole-class discussion. We highlight how different iterations of the planted errors affect the authenticity of how student voice was represented in the rehearsals and the resulting opportunities for TC learning. We offer design considerations for coached rehearsals and other approximations of practice.
Episode 26: Fostering Middle School Teachers’ Mathematical Knowledge for Teaching via Analysis of Tasks and Student Work
January 4th, 2021 | Season 6 | 36 mins 59 secs
proportional reasoning; mathematical knowledge for teaching; middle school teachers; student work
Mathematical knowledge for teaching is a complex web of knowledge domains. In this article, we share findings from an 18-month professional development project that aimed to improve middle school mathematics teachers’ mathematical knowledge for teaching (MKT) of proportional reasoning by focusing on the critical analysis of mathematical tasks and student work. Although multiple studies have shown that professional development can contribute to teachers’ MKT globally, little is known about how this knowledge grows and how specific domains of MKT can be targeted through professional development. Findings in this study show how professional development positively influenced participants’ knowledge of content and teaching and knowledge of content and students, two domains of MKT, through teachers’ twinned analyses of tasks and student work in proportional reasoning.
Episode 23: Exploring Real Numbers as Rational Number Sequences With Prospective Mathematics Teachers
January 4th, 2021 | Season 6 | 35 mins 22 secs
prospective mathematics teachers; quantitative reasoning; real numbers
The understandings prospective mathematics teachers develop by focusing on quantities and quantitative relationships within real numbers have the potential for enhancing their future students’ understanding of real numbers. In this article, we propose an instructional sequence that addresses quantitative relationships for the construction of real numbers as rational number sequences. We found that the instructional sequence enhanced prospective teachers’ understanding of real numbers by considering them as quantities and explaining them by using rational number sequences. In particular, results showed that prospective teachers reasoned about fractions and decimal representations of rational numbers using long division, the division algorithm, and diagrams. This further prompted their reasoning with decimal representations of rational and irrational numbers as rational number sequences, which leads to authentic construction of real numbers. Enacting the instructional sequence provides lenses for mathematics teacher educators to notice and eliminate difficulties of their students while developing relationships among multiple representations of real numbers.
January 4th, 2021 | Season 6 | 35 mins 28 secs
preservice teachers; technology; school based partners
In this article, we examine the ways in which the creation of a third space can bridge the divide between coursework and practice for preservice secondary mathematics teachers (PSTs) taking a technology, pedagogy, and content course. A university-based instructor partnered with two high school teachers to create a space in which PSTs draw upon and use both academic and practitioner knowledge while creating technology-based tasks for high school students to use. Our results revealed increased focus on pedagogical decisions in areas such as technology-task design and questioning techniques. The data also indicate that the success of this collaboration was connected to fair distribution of work, feeling valued, and personal benefit and challenges centered on maintaining rejection of hierarchy.
December 13th, 2020 | Season 6 | 24 mins 9 secs
The Mathematics Teacher Educator journal is co-sponsored by the National Council of Teachers of Mathematics and the Association of Mathematics Teacher Educators. In June, both organizations released statements that call for mathematics teachers and mathematics teacher educators (MTEs) to “engage in anti-racist and trauma-informed education in our daily practices as processes of learning and adjustments” (NCTM, 2020) and to “actively work to be anti-racist in our acts of teaching, research, and service” (AMTE, 2020). This editorial highlights equity-related interventions and tools that can be implemented by MTEs. We reiterate statements made by NCTM and AMTE, describe key features of interventions and tools, and share equity-related resources published in the journal for MTEs to use with teachers.
Episode 21: Undergraduate Research in Mathematics Education: Using Qualitative Data About Children’s Learning to Make Decisions About Teaching
September 18th, 2020 | Season 5 | 30 mins 45 secs
undergraduate research; design-based research; clinical interviews; formative assessment; classroom data analysis
Undergraduate research is increasingly prevalent in many fields of study, but it is not yet widespread in mathematics education. We argue that expanding undergraduate research opportunities in mathematics education would be beneficial to the field. Such opportunities can be impactful as either extracurricular or course-embedded experiences. To help readers envision directions for undergraduate research experiences in mathematics education with prospective teachers, we describe a model built on a design-based research paradigm. The model engages pairs of prospective teachers in working with faculty mentors to design instructional sequences and test the extent to which they support children’s learning. Undergraduates learn about the nature of systematic mathematics education research and how careful analyses of classroom data can guide practice. Mentors gain opportunities to pursue their personal research interests while guiding undergraduate pairs. We explain how implementing the core cycle of the model, whether on a small or large scale, can help teachers make instructional decisions that are based on rich, qualitative classroom data.
Episode 20: Learning to Launch Complex Tasks: How Instructional Visions Influence the Exploration of the Practice
September 11th, 2020 | Season 5 | 25 mins 53 secs
instructional vision; practice-based teacher education; teacher learning cycle; launch; complex tasks
This study investigates how the exploration phase of the teacher learning cycle provides 11 novice mathematics teachers with the opportunity to learn about the high-leverage practice of launching a complex task. Findings suggest that the exploration phase of the teacher learning cycle provides novice teachers with opportunities to reflect on how
to launch a complex task within the context of their own instructional practice. Because of this opportunity to deeply consider the pedagogical resource and reflect on it, novice teachers’ instructional visions were a filter through which they interpreted key instructional strategies offered up during the exploration phase of the teacher learning cycle. Further, the authors discuss three key takeaways for teacher educators who are attempting to implement the teacher learning cycle into their teacher education coursework
Episode 19: Do You See What I See? Formative Assessment of Preservice Teachers’ Noticing of Students’ Mathematical Thinking
September 4th, 2020 | Season 5 | 37 mins 27 secs
formative assessment; professional noticing; approximations of practice
Developing expertise in professional noticing of students’ mathematical thinking takes time
and meaningful learning experiences. We used the LessonSketch platform to create a learning
experience for secondary preservice teachers (PSTs) involving an approximation of teaching
practice to formatively assess PSTs’ noticing skills of students’ mathematical thinking.
Our study showed that approximations of teaching practice embedded within platforms
like LessonSketch can enable mathematics teacher educators (MTEs) to carry out effective
formative assessment of PSTs’ professional noticing of students’ mathematical thinking
that is meaningful for both PSTs and MTEs. The experience itself as well as its design features
and framework used with the assessment can be applied in the work of MTEs who develop teachers’ professional noticing skills of students’ mathematical thinking.
August 28th, 2020 | Season 5 | 36 mins 53 secs
technology; function; preservice secondary mathematics teachers
In this article, we present a set of design principles to guide the development of
instructional materials aimed to support preservice secondary mathematics teachers
(PSMTs) examining student practices in technology-mediated environments. To
develop design principles, we drew on the literature related to technological
pedagogical content knowledge (TPACK; Niess, 2005), video cases as learning objects
(Sherin & van Es, 2005), and professional noticing (Jacobs, et al., 2010). After presenting
the design principles, we share a task created using these design principles. Finally, we
share PSMTs’ reflections about changes in their own understanding after examining
students’ practices. Their responses provide insights into the usefulness of the
design principles for deepening PSMTs’ mathematical knowledge and knowledge
of students’ understanding, thinking, and learning with technology.
Episode 17: Editorial: Analyzing Eight Years of Mathematics Teacher Educator Articles: Where We Were, Where We Are, and Where We Are Going
June 29th, 2020 | Season 5 | 34 mins 57 secs
In this editorial, an analysis of articles published in the Mathematics Teacher Educator journal (MTE) from 2012 to 2020, which describes the knowledge base for mathematics teacher educators addressed by MTE authors, is presented. This analysis builds on similar work conducted four years ago (Bieda, 2016). These more recent findings demonstrate that articles focusing on teacher knowledge; mathematical content; student thinking and reasoning;
and models of teacher preparation or in-service professional development (PD) have been the most frequently published in MTE. In contrast, a limited number of articles have focused on discourse; diversity, equity, and language; technology; and methods of research. This examination allows us to assess as a community where we were, where we are, and where we might go in the future.
Episode 16: Diverge then Converge: A Strategy for Deepening Understanding through Analyzing and Reconciling Contrasting Patterns of Reasoning
June 18th, 2020 | Season 4 | 31 mins
classroom discourse; enacting mathematical practices; pre-service content courses
One of the challenges of teaching content courses for prospective elementary teachers (PTs) is engaging PTs in deepening their conceptual understanding of mathematics they feel they already know (Thanheiser, Philipp, Fasteen, Strand, & Mills, 2013). We introduce the Diverge then Converge strategy for orchestrating mathematical discussions that we claim (1) engenders sustained engagement with a central conceptual issue and (2) supports a deeper understanding of the issue by engaging PTs in considering both correct and incorrect reasoning. We describe a recent implementation of the strategy and present an analysis of students’ written responses that are coordinated with the phases of the discussion. We close by considering conditions under which the strategy appears particularly relevant, factors that appear to influence its effectiveness, and questions for future research.