Introduction
Two years ago, inspired by the movement toward standards-based learning in K-12 schools, I embarked on a journey to reinvent my assessment practices for the mathematics methods courses I teach at a mid-size public university. I began to move away from the traditional, points-based structure common in higher education toward a standards-based approach. The emphasis on reflection and growth embedded in standards-based learning, in which teachers work with students to determine proficiency levels and set goals for improvement (Heflebower et al., 2019), appealed to my desire to frame teacher education as a learning process rather than an evaluative process. It also aligned with the growth mindset emphasized in Jo Boaler’s (2016) Mathematical Mindsets, the textbook I utilize in my undergraduate K-8 methods course, which has the largest enrollment of the various methods courses I teach.
In this essay, I share insights I have gleaned from the process of retooling my K-8 mathematics methods course and evaluating my students’ work with respect to AMTE’s 2017 Standards for Preparing Teachers of Mathematics. My deep dive into the Standards has enabled me to identify strengths and weaknesses in my own instruction and to make improvements accordingly to better support my pre-service teachers (PSTs)—a majority of whom plan to teach in elementary school classrooms—in achieving AMTE’s vision.
Developing Standards-Based Course Objectives
When I first became interested in standards-based learning in 2018, I was unaware of AMTE’s Standards, so I developed objectives for my mathematics methods course based on the Danielson Framework (2009), which my department had adopted for assessment purposes. Like Danielson, I utilized a 4.0 scale and wrote criteria for each of the four proficiency levels for each objective. Since the Danielson Framework is not discipline specific, I referred to NCTM’s Principles to Actions (2014) when preparing the language for my objectives. The Mathematical Tasks Framework (Stein et al., 2009), with its four levels of cognitive demand, served as a guide in developing my proficiency level descriptions, with “doing mathematics” roughly aligning with expectations for proficient performance; “procedures with connections” with nearing proficient; “procedures without connections” with basic; and “memorization” with unsatisfactory.
In the Mathematical Tasks Framework, “doing mathematics” involves applying non-algorithmic thinking to solve open-ended problems and is recognized as the pinnacle of mathematical proficiency. Hence, to attain proficiency in tasks such as lesson planning, my PSTs must demonstrate an ability to plan lessons that position students as the primary doers and thinkers in activities that encourage critical thinking and a synthesis of skills, not merely rote application. Although there is a time and place for practicing formulas (“procedures without connections”) and reciting facts (“memorization”) within mathematics classrooms, planning for these alone does not constitute rigorous instruction, as emphasized in the Standards. Stein et al. (2009) admonished, “use of these types of tasks may improve student performance on tests that consist of low-level items and may lead to greater efficiency of time and effort in solving routine aspects of problems that are embedded in more complex tasks. However, focusing exclusively on tasks of these types can lead to a limited understanding of what mathematics is and how one does it” (p. 5). Making connections between procedures and concepts and between procedures and representations (“procedures with connections”) is crucial for supporting students’ understanding of mathematics, but this too falls short of full mathematical proficiency, which requires an ability to apply mathematics to novel contexts and arrive at novel solutions (AMTE, 2017; Stein et al., 2009).
Once I had established learning objectives and criteria for each proficiency level, I organized these into a single rubric against which all assignments—both formative and summative—would be evaluated throughout the semester. I wanted my students to view their learning as a progression, gradually tackling the objectives and working their way toward proficiency. I did not want to replicate traditional grading by generating individual rubrics for individual assignments such that each assignment would be viewed in isolation rather than as contributing to overall growth.
I utilized my initial rubric for two semesters before becoming aware of AMTE’s Standards. In comparing my existing rubric to the Standards, I immediately noticed a glaring omission: equity. While my existing rubric incorporated differentiation, it did not otherwise explicitly address issues related to equity, as stressed so unambiguously in the Standards. I updated my rubric to better reflect the Standards, adding an advocacy and equity domain to my existing domains of content knowledge, objectives and assessments, knowledge of resources, instructional strategies, reflection and growth, and professionalism. A copy of the revised rubric I now utilize in my methods courses is included in the appendix.
Adjusting Course Assignments and Timelines
A principle tenet of standards-based learning is the opportunity for students to receive meaningful and timely feedback with respect to course objectives and to improve their performance accordingly (Heflebower et al., 2019). To enable students to continue to work on assignments after receiving feedback, I needed to adjust my course schedule to accommodate revision cycles. The main change I made was to move up the due dates for my three major assignments—a teaching reflection, an Indian Education for All lesson plan[1], and a unit plan—and to add a final portfolio to the course. The final portfolio would comprise revisions to the three major assignments based on feedback that students received from me during the semester. To ensure students would invest the time necessary to make meaningful revisions, I dedicated the last 2 weeks of the semester to preparing the portfolios.
Evaluating Student Performance
I assess my students’ performance against the course rubric multiple times throughout the semester. While I do score each assignment, none of the scores on early drafts are pegged to my gradebook—only the culminating portfolio and a handful of reading reflections factor into the final grade. At first I did not provide scores on assignment drafts, but this left my students in a panic throughout the semester because they never knew their grades. Reluctantly, I gave up trying to convince college students to forget about their grades and focus on their growth and instead decided to take advantage of their concern about grades to motivate them to take revisions seriously when it came time to prepare their final portfolios. Earning a low score on an early draft seemed an adequate motivation for most students—although there have been some growing pains in getting my students to understand that one point on a four-point proficiency scale is much different than one point on a 100-point assignment. At first, many of my students assume that if they only need to improve their score by a few points, then they only need to make minor tweaks to their drafts in order to obtain proficiency, which is not the case.
Lessons Learned
Interested in responding to Berk and Hiebert’s (2009) call for mathematics teacher educators to engage in deliberate, systematic study of student learning outcomes with respect to a set of learning goals shared by the broader mathematics education community (which finally took shape in the form of the Standards), I launched an in-depth study of my K-8 methods students’ assignments from the Fall 2019 cohort (N = 21). What I learned through this process ended up having more to do with my own gaps in instruction than with my students’ abilities per se.
My study’s method involved re-evaluating every initial and final submission of two lesson planning assignments—the Indian Education for All lesson plan and the unit plan—to search for differences between initial and final drafts that might indicate areas in which my students had improved (or not) with respect to the Standards as a result of the feedback and revision cycles I had incorporated into the course. As Morris and Hiebert (2017, p. 536) asserted, “Although planning a lesson is not the same as teaching a lesson, the evidence suggests that how teachers plan lessons is related to how they teach lessons.” As I evaluated my students’ lessons, I engaged in my own revision process, continually adjusting my rubric to better capture my students’ work. I also generated a codebook of examples to illuminate each objective and each proficiency level, which will be a useful reference for future students as well as other mathematics teacher educators.
Many of my PSTs made observable improvements with respect to the various domains and objectives in my rubric when revising their assignments. But the only objective for which a majority of students demonstrated proficiency was in aligning learning objectives with learning standards. For objectives related to incorporating prerequisite knowledge and skills, level of cognitive demand of objectives and assessments, differentiation, and perspectives, most of my students remained at a basic level of proficiency even after making revisions, while for other objectives most students demonstrated nearing proficient performance in their final portfolios.
The weakest area overall in my students’ work has been in equity, which had not been included in my original rubric. Once I defined my criteria for equity to be in line with AMTE’s Standards, I realized that I was not providing my students with the opportunities they needed to meet these standards. My criteria for proficiency in the perspectives category of my advocacy and equity domain (which also encompasses differentiation) is, “Incorporates multiple perspectives and contexts into lesson and unit plans, including those of non-dominant and historically marginalized groups, in a way that challenges stereotypes and predominant paradigms.” The criteria aligns with objectives C.4.2-4 of the Standards. While the Indian Education for All unit I have incorporated into my course represents a concerted effort to address this objective in my teaching, I am not doing enough to support my students in meeting these criteria for other assignments. For example, how does one challenge dominant paradigms in an ordinary unit plan? I have not been providing explicit content in my methods course that would answer this question for my students, which is an issue I must address moving forward.
Concluding Remarks: How Setting Standards Raises Expectations for PSTs and Teacher Educators Alike
When I embarked on my mission to transition to standards-based learning, I had envisioned the benefit as being improved student performance. I had not initially thought about the endeavor as a way of improving my own performance as a mathematics teacher educator, but this has in fact been the most notable outcome of my transition. By taking a deep dive into the Standards, I recognized areas where my own instruction was falling short, which prompted me to set improvement goals for my courses. Learning and growing is indeed a work in progress, and just like our students, we teacher educators can only achieve proficiency by reflecting on and adjusting our performance with respect to a set of rigorous standards.
References
Association of Mathematics Teacher Educators. (2017). Standards for preparing teachers of mathematics.
Boaler, J. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. Jossey-Bass.
Danielson, C. (2007). Enhancing Professional Practice: A Framework for Teaching, (2nd ed.). Association for Supervision & Curriculum Development.
Heflebower, T., Hoegh, J. K., Warrick, P. B., & Flygare, J. (2019). A teacher’s guide to standards-based learning. Marzano Research.
Morris, A. K., & Hiebert, J. (2017). Effects of teacher preparation courses: Do graduates use what they learned to plan mathematics lessons? American Educational Research Journal, 54(3), 524-567. https://doi.org/10.3102%2F0002831217695217
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). Implementing standards-based mathematics instruction: A casebook for professional development (2nd ed.). Teachers College Press.
[1] In my state, K-12 schools are required to incorporate lessons on the history, culture, and contemporary issues impacting Native American communities; the law mandating this requirement is referred to as Indian Education for All.
Appendix
Danielson-AMTE Rubric for Pre-Service Mathematics Teacher Educators
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Proficient for Pre-Service Expectations |
Nearing Proficient for Pre-Service Expectations |
Basic |
Unsatisfactory |
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Demonstrating Knowledge of Content (Danielson 1a; AMTE C.1.1, C.1.4-5, C.2.2, C.3.1-2)
Mathematical concepts
Significance
Prerequisite knowledge and skills
Student thinking |
Mathematical Concepts |
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Demonstrates a solid understanding of mathematical concepts and their relationship to procedures; tasks and prompts help student to link procedures and concepts. |
Demonstrates sufficient understanding of mathematical concepts and their relationships to procedures but exhibits some gaps in knowledge; tasks and prompts help students to connect procedures to concepts but PST overlooks an aspect of the procedure or concept. |
Demonstrates limited understanding of mathematical concepts and struggles to connect procedures and concepts; tasks and prompts focus on answers. |
Demonstrates significant gaps and misunderstandings in mathematical knowledge; procedures and concepts are misunderstood or inappropriately linked. |
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Significance |
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Incorporates mathematical and/or real-world scenarios that aptly illuminate mathematical concepts and naturally lend themselves to modeling. |
Incorporates mathematical and/or real-world scenarios that are relevant and lend themselves to modeling but are not fully developed. |
Incorporates mathematical and/or real-world scenarios that are tangential or contrived, or that are disconnected from the rest of the lesson. |
Incorporates mathematical and/or real-world scenarios that are inappropriate, or does not provide a purpose for the mathematical activity. |
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Prerequisite Knowledge and Skills |
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Invokes key background knowledge and skills in lesson plans, providing a solid foundation for building new knowledge and skills. |
Invokes relevant background knowledge in lesson plans but overlooks a relevant concept or skill, or incorporation of prior knowledge or skills requires further development to provide a strong foundation for building new knowledge and skills. |
Identifies relevant background knowledge but does not attempt to use it as a foundation to build new knowledge and skills; or identifies and incorporates some relevant knowledge but overlooks a fundamental concept. |
Does not identify or incorporate relevant background knowledge or skills, or incorporation of knowledge or skills is inappropriate. |
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Student Thinking |
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Anticipates an appropriate range of potential solution strategies and responses and anticipates potential points of confusion. |
Anticipates potential solution strategies and responses but overlooks a key strategy or response; may not anticipate potential points of confusion. |
Expects a single specific solution strategy and response; does not anticipate potential points of struggle. |
Does not indicate expected or anticipated solution strategies or responses; does not anticipate potential points of confusion. |
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Setting Instructional Outcomes and Assessment (Danielson 1c/e/f, 3c-d; AMTE C.2.2)
Alignment between learning objectives and learning standards
Alignments between learning objectives and assessments
Level of cognitive demand of objectives and assessments
Developmental progression and pacing |
Alignment Between Learning Objectives and Learning Standards |
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Learning objectives align with an appropriate grade-level standard in a clear, specific, and measurable way. |
Learning objectives align with an appropriate grade-level standard in a way that is either clear, specific, or measurable, but not all three; or learning objectives align with a standard that is similar to the standard indicated in the lesson in a clear, specific, and measurable way but the actual standard that the lesson addresses is a different standard than the one indicated in the lesson.
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Learning objectives appear to align with an appropriate grade-level standard but are not fully clear, specific, or measurable. |
Learning objectives do not align with an appropriate grade-level standard or are too unclear to decipher. |
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Alignment Between Learning Objectives and Assessments |
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Objectives and assessments align in terms of both content and process and are supported by the lesson’s main activities; examples of satisfactory student work and explanations that would meet the objectives are provided. |
Objectives and assessments are mostly aligned in terms of both content and process and are supported by the lesson’s main activities but may require further development and/or examples of satisfactory student work and explanations that would meet the objectives are not provided. |
Objectives and assessments align in terms of content or process but not both, or objectives and assessments align but are not supported by the main activities in the lesson. |
Objectives and assessments are unaligned and/or inappropriate, or alignment cannot be discerned due to lack of clarity. |
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Level of Cognitive Demand of Objectives and Assessments |
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Objectives and assessments reflect a high level of cognitive demand and are designed to elicit information about both conceptual and procedural understanding; opportunities are provided for student self-evaluation. |
Objectives and assessments reflect a high level of cognitive demand and are designed to elicit information about both conceptual and procedural understanding but are not fully developed and/or opportunities are not provided for student self-evaluation. |
Objectives and assessments reflect a low level of cognitive demand and primarily emphasize procedural understanding. |
Objectives and assessments emphasize memorization only. |
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Developmental Progression and Pacing |
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Learning objectives reflect appropriate attention to the development progression of a concept, supporting procedural fluency by first developing conceptual understanding. |
Learning objectives reflect a slightly rushed or prolonged developmental progression of a concept but afford some opportunity to develop conceptual understanding. |
Learning objectives proceed far too quickly or slowly through the developmental progression of a concept, rushing or overlooking concepts to focus on procedures. |
Learning objectives do not follow a logical developmental progression. |
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Demonstrating Knowledge of Resources (Danielson 1d, 3c; AMTE C.1.4/6, C.2.3)
Suitability of resources
Use of resources |
Suitability of Resources |
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Selected resources are optimally suited to the mathematical content intended to be conveyed in a lesson or unit. |
Selected resources are suited to the mathematical content intended to be conveyed in a lesson or unit but a particularly salient resource has been overlooked. |
Selected resources may be used for the mathematical content intended to be conveyed in a lesson or unit but other resources would be better suited to the content and/or selected resources are unnecessarily limited. |
Selected resources are ill suited to the mathematical content intended to be conveyed in a lesson or unit. |
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Use of Resources |
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Resources are utilized to support student exploration and sense making and to illuminate important mathematical connections; students are given choice in how resources are utilized whenever appropriate. |
Resources are utilized to support student exploration and sense making and to illuminate important mathematical connections but efforts require further development to be fully effective. |
Resources are utilized in a prescriptive or non-mathematical manner with little or no opportunity for exploration, sense making, or illuminating important mathematical connections. |
Resources are utilized incorrectly or inappropriately, or manner of use cannot be determined from the lesson. |
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Instructional Strategies (Danielson 1e, 3b-d; AMTE C.1.4, C.2.2-3)
Coherence
Nature of tasks
Nature of prompts |
Coherence |
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Lessons and units feature a well-defined, unifying theme and a logical flow, with opportunities for reflection and closure. |
Lessons and units feature a unifying theme, a mostly logical flow, and opportunities for reflection and closure, but one or more of these elements requires further development. |
Lessons and units feature a mostly logical flow but include abrupt transitions, do not include a discernible theme, and/or opportunities for reflection and closure are absent.
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Lessons and units feature an inappropriate theme and/or the flow of the lesson or unit is not logical. |
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Nature of Tasks |
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Tasks position students as the primary doers and feature a high level of cognitive demand, permitting non-algorithmic thinking and multiple solution strategies and fostering conceptual as well as procedural understanding. |
Tasks foster conceptual as well as procedural understanding but are teacher directed rather than student driven, or tasks feature the characteristics of proficiency but require further development to be fully effective. |
Tasks feature student engagement but the level of cognitive demand is low, emphasizing prescribed routines or procedures without meaningful connections to concepts. |
Tasks are entirely teacher led with no opportunity for active student engagement, and/or tasks emphasize memorization only. |
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Nature of Prompts |
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Prompts are designed to elicit information about conceptual and procedural understanding and to encourage reflection, justification, and connections. |
Prompts are designed to elicit information about conceptual and procedural understanding but do not encourage reflection, justification, and/or connections, or prompts incorporate the features of proficiency but more such prompts are needed for the lesson to be fully effective. |
Prompts emphasize answer and procedures only. |
Prompts are inappropriate or absent. |
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Advocacy and Equity (Danielson 1e-f; AMTE C.2.1, C.4.1-4)
Differentiation
Perspectives |
Differentiation |
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Lessons and units feature multiple entry points and specific supports for struggling students as well as appropriate extensions for student ready for a challenge. |
Lesson and units feature either multiple entry points and specific supports for struggling students or appropriate extensions for students ready for a challenge but not both, or lessons and units incorporate the features of proficiency but require further development to be fully effective. |
Lessons and units suggest plans for differentiation, but plans are vague or are not linked to the concept under study. |
Lessons and units do not incorporate plans for differentiation, or plans for differentiation are inappropriate or alter the level of cognitive demand for different learners. |
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Perspectives |
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Incorporates multiple perspectives and contexts into lesson and unit plans, including those of non-dominant and historically marginalized groups, in a way that challenges stereotypes and predominant paradigms. |
Incorporates multiple perspectives and contexts into lesson and unit plans, including those of non-dominant and historically marginalized groups, but these require further development to effectively challenge stereotypes and predominant paradigms. |
Lesson appears to be framed from the perspective of the dominant group without due diligence to verify whether commonly held assumptions are shared by other groups. |
Incorporates inaccurate, offensive, or disrespectful information into lessons and unit plans such that negative stereotypes are reinforced. |
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Reflection and Growth (Danielson 4a/e; AMTE C.1.3, C.2.4)
Learning from others
Reflecting on performance |
Learning from Others |
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Takes seriously all opportunities to learn from peers, instructor feedback, personal experience, and the literature. Articulates lessons learned in a detailed manner, citing specific evidence that has contributed to increased knowledge and understanding. |
Takes most opportunities to learn from peers, instructor feedback, personal experience, and the literature. Incorporates some details and supporting evidence to articulate lessons learned but some details and evidence are missing. |
Engages minimally in opportunities to learn from peers, instructor feedback, personal experience, and the literature. Articulates lessons learned in a vague manner, without specific supporting evidence or details. |
Does not engage in opportunities to learn from peers, instructor feedback, personal experience, or the literature. Does not articulate lessons learned. |
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Reflecting on Performance |
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Reflects on personal strengths and weaknesses as well as those of others in a detailed manner. Continually improves performance based on lessons learned. |
Reflects on strengths and weaknesses but reflections require more detail. Makes efforts to improve based on lessons learned but efforts require more diligence. |
Reflects minimally on strengths and weaknesses in a vague manner without detail. Makes few efforts to improve based on lessons learned.
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Does not demonstrate a willingness to be self-critical and/or blame is placed on factors outside of one’s control. Does not make efforts to improve. |
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Professionalism (Danielson 4f; AMTE C.2.5)
Engagement in communities of practice
Adherence to professional standards |
Engagement in Communities of Practice |
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Participates fully in class and field experiences except in the case of emergencies and excused absences. Contributes consistently and thoughtfully to tasks and discussions, engaging with instructor, peers, and students in a cooperative manner.
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Participates regularly in class and field experiences but occasionally misses class. Contributes to most tasks and discussions but is sometimes uninvolved or inattentive. |
Participates minimally in class and field experiences, missing class frequently. Contributes minimally to tasks and discussions. |
Does not attend class and does not contribute to tasks and discussions and/or participation is disrespectful or inappropriate. |
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Adherence to Professional Standards |
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Submits required work on time. Communicates promptly if situations arise that may prevent unintended delays and plans ahead for anticipated absences. Work is clear, complete, well organized, free of errors, and written in the student’s own words, except for occasional cited quotations. |
Submits some required documents late. Communicates about absences or delays but does not always plan ahead for anticipated absences. Work is mostly clear but is not always complete or free of errors. Although properly cited, work often relies heavily on the thoughts and wording of others. |
Submits many required documents late. Does not always communicate about absences or delays and does not plan ahead for anticipated absences. Work is not always clear, complete, or free of errors. Credit is given to the thoughts of others but work is not properly cited. |
Submits most required documents late. Does not communicate about absences or delays and does not plan ahead for anticipated absences. Work is unclear, incomplete, and contains frequent errors and/or work is plagiarized. |
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