Mathematics Teacher Educator Podcast

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.

Episode 25: Creating a Third Space for Learning to Design Technology-Based Mathematics Tasks

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.

Episode 22: Interventions, Tools, and Equity-Oriented Resources in the MTE Journal

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.

Episode 18: Design Principles for Examining Student Practices in a Technology-Mediated Environment

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.

Episode 15: Noticing Aloud: Uncovering Mathematics Teacher Noticing in the Moment

May 28th, 2020  |  Season 4  |  23 mins 46 secs

Understanding mathematics teacher noticing has been the focus of a growing body of research, in which student work and classroom videos are often used as artifacts for surfacing teachers’ cognitive processes. However, what teachers notice through reflecting on artifacts of teaching may not be parallel to what they notice in the complex and demanding environment of the classroom. This article used a new technique, side-by-side coaching, to uncover teacher noticing in the moment of instruction. There were 21 instances of noticing aloud during side by side coaching which were analyzed and classified, yielding 6 types of teacher noticing aloud, including instances in which teachers expressed confidence, struggle, and wonder. Implications for coaching and future research on teacher noticing are discussed.

Episode 14: Supporting Teachers to Use Formative Assessment for Adaptive Decision Making

May 18th, 2020  |  Season 4  |  32 mins 11 secs

professional development; research on professional development; formative assessment

Formative assessment helps teachers make effective instructional decisions to support students to learn mathematics. Yet, many teachers struggle to effectively use formative assessment to support student learning. Therefore, teacher educators must find ways to support teachers to use formative assessment to inform instruction. This case study documents shifts in teachers’ views and reported use of formative assessment that took place as they engaged in professional development (PD). The PD design considered the formative assessment cycle (Otero, 2006; Popham, 2008) and embedded it within a pedagogical framework (Lamberg, 2013, in press) that took into account the process of mathematics planning and teaching while supporting teachers to learn math content. Teachers restructured their definition of student understanding, which influenced how they interpreted student work and made instructional decisions. Teachers’ pre-PD instructional decisions focused on looking for right and wrong answers to determine mastery and focused on pacing decisions. Their post-PD decisions focused on student thinking and adapting teaching to support student thinking and learning. Implications for PD to support teachers to use formative assessment and research are discussed.

Episode 13: Visions of the Possible: Using Drawings to Elicit and Support Visions of Teaching Mathematics

April 21st, 2020  |  Season 4  |  26 mins 34 secs

Mathematics Teacher Educators (MTEs) help preservice teachers in transitioning from students to teachers of mathematics. They support PSTs in shifting what they notice and envision to align with the collective vision encoded in the AMTE and NCTM standards. This study analyzes drawings and descriptions completed at the beginning and end of a one-year teacher education program—snapshots depicting optimized visions of teaching and learning mathematics. This study analyzed drawings-and-descriptions by cohort and by participants. The findings suggest that the task can be used as formative assessment to inform supports for specific PSTs such as choosing a cooperating teacher or coursework that challenges problematic beliefs. It can also be used as summative assessment to inform revision of coursework for the next cohort.

Episode 12: Complex and Contradictory Conversations: Prospective Teachers Interrogating Dominant Narratives Within Mathematics Education Discourse

April 4th, 2020  |  Season 3  |  26 mins 26 secs

In this conceptual piece, I explore complex and contradictory conversations during an idea mapping task in which prospective elementary teachers interrogated dominant discourses within mathematics education, such as “mathematics is everywhere” and “being a math person.” I argue that this exercise of engaging with contradictions provided prospective teachers with opportunities to tease out nuances for reconstructing ideas that generate new perspectives for teaching and learning mathematics. Sharing my experience with the idea mapping task as a case study, I offer an alternative role for mathematics teacher educators to consider—as facilitators who create spaces for prospective teachers to interrogate complex and contradictory conversations within mathematics education.

Episode 11: Using Coordinated Measurement with Future Teachers to Connect Multiplication, Division, and Proportional Relationships

March 26th, 2020  |  Season 3  |  29 mins 43 secs

We report results from a mathematics content course intended to help future teachers form a coherent perspective on topics related to multiplication, including whole-number multiplication and division, fraction arithmetic, proportional relationships, and linear functions. We used one meaning of multiplication, based in measurement and expressed as an equation, to support future teachers’ understanding of these topics. We also used 2 types of length- based math drawings—double number lines and strip diagrams—as media with which to represent relationships among quantities and solve problems. To illustrate the promise of this approach, we share data in which future secondary mathematics teachers generated and explained without direct instruction sound methods for dividing by fractions and solving proportional relationships. The results are noteworthy, because these and other topics related to multiplication pose perennial challenges for many teachers.

Episode 10: Engaging Preservice Secondary Mathematics Teachers in Authentic Mathematical Modeling: Deriving Ampere’s Law

March 21st, 2020  |  Season 3  |  19 mins 56 secs

Incorporating modeling activities into classroom instruction requires flexibility with pedagogical content knowledge and the ability to understand and interpret students’ thinking, skills that teachers often develop through experience. One way to support preservice mathematics teachers’ (PSMTs) proficiency with mathematical modeling is by incorporating modeling tasks into mathematics pedagogy courses, allowing PSMTs to engage with mathematical modeling as students and as future teachers. Eight PSMTs participated in a model-eliciting activity (MEA) in which they were asked to develop a model that describes the strength of the magnetic field generated by a solenoid. By engaging in mathematical modeling as students, these PSMTs became aware of their own proficiency with and understanding of mathematical modeling. By engaging in mathematical modeling as future teachers, these PSMTs were able to articulate the importance of incorporating MEAs into their own instruction.