A perpetual challenge of mathematics teacher education is bridging the gap between ideals, theory, and practices learned in teacher education courses and realities encountered in student teaching and fieldwork. Too often school systems do not support the kind of equitable, ambitious, and complex instruction featured in teacher education courses (Litke, 2015; 2020). Even more, pre-service teachers often do not see the methods and values from their teacher education reflected in their mentor teachers’ classrooms (Campbell & Dunleavy, 2016; Feiman-Nemser & Buchmann, 1985). The “apprenticeship of observation” (Lortie, 1975) is a strong way that teachers learn to teach, and so it is critical that the gap between university learning and classroom teaching be bridged.
We have found ways to begin to bridge this gap in our work with pre-service teachers. Specifically, Dr. Amick has had success supporting pre-service teachers learning to be critical consumers of curriculum and to lesson plan with evidence-based instructional practices, even when they go into schools using more traditional curricula. There are five key suggestions we can share for mathematics teacher educators (MTEs) to use in methods classes.
1. Select rich tasks
First and foremost, it is important for MTEs to select rich tasks. Pre-service teachers need exposure to rich tasks even if they will be working in districts using more traditional materials. The fact that the same task can be fruitful across many grade bands is shocking for many pre-service teachers. Encountering tasks embedded in contexts more complex than a simple word problem can be similarly surprising. If pre-teachers do not understand the value of these tasks, they can easily diminish them into more familiar “naked” mathematical problems. Lotan (2003) defined the characteristics of tasks that support teamwork in mathematics class, and she coined these tasks as group-worthy. Group-worthy tasks:
- Are open-ended and require complex problem-solving,
- Provide students with multiple entry points and multiple opportunities to show competence,
- Deal with content that is mathematically important,
- Require positive interdependence and individual accountability, and
- Include clear evaluation criteria for what teams produce (p. 72).
Such tasks are cognitively demanding and support productive struggle.
Pre-service teachers do not just need exposure to these tasks; they need to learn how to teach through them. Dr. Amick uses CPM’s 6-12 curricular materials to fill the need for rich tasks in her secondary mathematics teacher education courses (this often requires omitting the scaffolding questions present below the main problem). We will use examples from CPM materials since both authors are familiar with it and since it is openly available to any MTE who requests it (https://cpm.org/university). Elementary-focused MTEs can find similar materials available from the Math Learning Center at https://www.mathlearningcenter.org/about/giving-back.
As an important note, group-worthy tasks are important for all K-12 students, including those who have a history of struggling in mathematics. Tracking (in general) is not an equitable practice, yet K-12 students who have struggled in mathematics are often segregated to remedial classes that do back-to-basics work. However, usually, these K-12 students can and should learn through cognitively demanding tasks during teamwork (Jasien & Hayes, 2022). For an example of a group-worthy task, see this sample lesson annotated with design rationale from CPM’s Inspirations & Ideas supplemental 8th-grade course that aims to bring rich mathematics and equitable instruction to all students (https://cpm.org/iandi).
2. Model teaching these tasks
After MTEs compile a bank of rich tasks to draw from, it is important for MTEs to model teaching these tasks and pre-service teachers experience doing cognitively demanding mathematics. Again, many pre-service teachers have not seen equitable, ambitious, complex instruction. Modeling teaching these tasks provides a window into a new way to teach. As you model teaching the task, you can engage in the 5 practices for orchestrating discourse (Smith & Stein, 2018) and illustrate best teaching practices. Modeling of best practices should always be accompanied by reflection on how there is no one-size-fits all instructional recipe since teaching is inherently relational and situated. Indeed, some argue that fetishizing methods by making specific teaching moves and routines more central to teacher education than foundational principles and a commitment to justice is highly inequitable (Phillip et al., 2019). We suggest that best practices be both modeled (and rehearsed, see our fourth point) and analyzed for how, when, and why these instructional moves support particular learners.
Beyond the pedagogical benefits of modeling teaching with rich tasks, pre-service teachers gain content knowledge when they engage as learners in solving rich mathematical tasks. Debriefing after these tasks can help them grow in their mathematical knowledge for teaching (Hill et al., 2005), including:
- How and when they might introduce and explain terms and concepts to students,
- How they might interpret particular statements and solutions that were made during their teamwork and whole-class mathematical discussion,
- Judging and correcting textbook treatments of particular topics, and
- Using mathematical representations accurately in the classroom (p. 373).
In addition to any narration you do while modeling, this debrief can also highlight for pre-service teachers how standards of mathematical practice are engaged in lessons.
3. Provide meaningful tools for lesson planning
Pre-service (and in-service!) teachers need more than a lesson planning template to plan for instruction. They need instructional strategies for supporting collaborative learning, pocket questions, hands-on activities, launch and closure activities, eTools, formative assessment practices, differentiation strategies, and more. CPM’s Teacher Edition materials contain resources for these needs.
For example, CPM has a library of “study team-teaching strategies” (STTS) that give pre- and in-service teachers ideas for how to help 6-12 students learn from each other and keep struggle productive during teamwork. An example is the “I Spy” strategy, where one team member gets up to listen in on another team’s work and then reports what they learned to their team. This strategy allows students to maintain mathematical authority in the classroom. Another example is the “Elevator Talk” in which students have about 30 seconds to explain a problem, topic, or concept to their shoulder partners. The teacher circulates and selects some students to share out. Teachers can use this strategy to initiate a conversation in which students engage in the mathematical practice of constructing viable arguments and critiquing the reasoning of others. Rather than reinventing the wheel, having resources to pull from can help pre-service teachers plan engaging lessons that feature teamwork even when their base curriculum is designed in a way that affords primarily direct instruction.
4. Facilitate rehearsals of instructional strategies that support teamwork
After pre-service teachers plan a lesson using the resources mentioned above, they can rehearse part of the lesson, trading off acting as the teacher and as students. In Dr. Amick’s mathematics methods class, pre-service teachers are partnered up and assigned a rich mathematical task that they must co-teach to their peers. They are required to demonstrate one type of co-teaching strategy (e.g., team teaching, parallel teaching, or station teaching), asked to model one STTS, and facilitate the task through the framework of the 5 practices for orchestrating discourse (Smith & Stein, 2018).
Rehearsals can be generative for pre-service teachers and support them to “actually be able to do [socially and intellectually ambitious] teaching when they get into classrooms” (Lambert et al., 2013, p. 240). Rehearsals with debriefs can be structured in a way that supports pre-service teachers to hold themselves and their colleagues accountable to equitable, ambitious, complex instruction (Horn & Campbell, 2015). In fact, even pre-service teachers who are not acting as the “rehearser” in rehearsals can benefit from engaging in thought experiments after the debrief concludes (Munson et al., 2021).
5. Implement a curricular analysis project
In addition to the strategies above, we suggest MTEs implement a curricular analysis task where pre-service teachers analyze two curricula simultaneously. When Dr. Amick does this task, she uses the local school district’s curriculum (more traditional) and CPM curricula. She provides her students with a rubric to help guide them towards noticing particular features of the curriculum.
This curricular analysis task is important for pre-service teachers. Without opportunities to learn to critically consume curricula in terms of what to attend to and how to interpret and respond to it (see Males et al., 2015), pre-service teachers’ curricular noticing can be “haphazard or indiscriminate and possibly restricted by graphical design and layout” (Dietiker et al., 2018, p. 525). Amador et al. (2015) found that pre-service teachers need opportunities to evaluate explicit and implicit properties of multiple curricula with different design principles and philosophies. Specifically, Amador and colleagues noted that pre-service teachers benefit from using curricular analysis tools (such as rubrics) that support pre-service teachers to notice content, practices, equity, assessment, and technology in curricula. Depending on your pre-service teachers’ prior knowledge, you may want to have them develop their own rubric as part of the curricular analysis task. After doing a curricular analysis project, pre-service teachers may also benefit from designing group-worthy tasks together (Lotan, 2003). These tasks directly support the development of mathematical knowledge for teaching by helping students be critical consumers of curricula.
How do you bridge the gap?
We would love to hear ways that you support pre-service teachers to bridge what they learn in their teacher education courses with the realities they experience in their student-teaching and fieldwork. We have linked a two question survey. If we get enough responses then we will submit what we learn from the survey to AMTE’s Connections. You can find the survey here https://forms.gle/4XMYbHydYbZVfVED6. The form is set so that respondents can see how others have responded.
References
Amador, J., Earnest, D., Males, L., & Dietiker, L. (2015). Dimensions of Curricular Noticing. Paper presented at a National Council of Teachers of Mathematics conference. https://nctm.confex.com/nctm/2015RP/webprogram/Manuscript/Session33909/Dimensions%20of%20Curricular%20Noticing_NCTM_2015.Paper.pdf
Campbell, S. S., & Dunleavy, T. K. (2016). Connecting university course work and practitioner knowledge through mediated field experiences. Teacher Education Quarterly, 43(3), 49-70. https://files.eric.ed.gov/fulltext/EJ1110331.pdf
Dietiker, L., Males, L. M., Amador, J. M., & Earnest, D. (2018). Research Commentary: Curricular Noticing: A Framework to Describe Teachers' Interactions With Curriculum Materials. Journal for Research in Mathematics Education, 49(5), 521-532.
Feiman-Nemser, S., & Buchmann, M. (1985). Pitfalls of experience in teacher preparation. Teachers College Record, 87, 53–65.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406.
Horn, I. S., & Campbell, S. S. (2015). Developing pedagogical judgment in novice teachers: Mediated field experience as a pedagogy for teacher education. Pedagogies: An International Journal, 10(2), 149-176.
Lampert, M., Franke, M., Kazemi, E., Ghousseini, H., Turrou, A.C., Beasley, H., Cunard, A., & Crowe, K. (2013). Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching. Journal of Teacher Education, 64, 1-18.
Litke, E. G. (2015). The state of the gate: A description of instructional practice in algebra in five urban districts (Doctoral dissertation).
Litke, E. (2020). The nature and quality of algebra instruction: Using a content-focused observation tool as a lens for understanding and improving instructional practice. Cognition and Instruction, 38(1), 57-86.
Lortie, D. (1975). Schoolteacher. University of Chicago Press.
Lotan, R. A. (2003). Group-worthy tasks. Educational Leadership, 60(6), 72-75.
Males, L. M., Earnest, D., Dietiker, L., & Amador, J. M. (2015). Examining K-12 Prospective Teachers' Curricular Noticing. North American Chapter of the International Group for the Psychology of Mathematics Education. https://files.eric.ed.gov/fulltext/ED584212.pdf
Munson, J., Baldinger, E. E., & Larison, S. (2021). What if…? Exploring thought experiments and non-rehearsing teachers’ development of adaptive expertise in rehearsal debriefs. Teaching and Teacher Education, 97, 103222.
Philip, T., Souto-Manning, M., Anderson, L., Horn, I., J. Carter Andrews, D., Stillman, J., & Varghese, M. (2019). Making Justice Peripheral by Constructing Practice as “Core”: How the Increasing Prominence of Core Practices Challenges Teacher Education. Journal of Teacher Education, 70(3), 251–264. https://doi.org/10.1177/0022487118798324
Stein, M. K., & Smith, M. (2018). Five practices for orchestrating productive mathematics discussions. National Council of Teachers of Mathematics.