Working Principles & Metrics

for (Collaborations in) the

Mathematical Education of Teachers

Why “Principles and Metrics”?

We live in a moment with many tensions. If we are to make any progress on the mathematical education of teachers, we must be able to hold civil discourse and take considered and principled action on complex issues. Our hope for JimFest is for all presenters and moderators to model nuance, courage, open-mindedness, and compassion in speech and action.  

Two major goals of JimFest are to:

• Articulate principles and metrics that may guide mathematics departments in designing mathematical work for teachers so that students from historically under-served communities will flourish.
• Find common ground among emerging movements in PK16 mathematics and their implications for the mathematical education of teachers, so that students from historically under-served communities will flourish.

With this in mind, we are asking all participants to reflect on the principles and metrics that may guide their work. Principles are practical, ethical, or even aesthetic stances that we turn to when we need to find a way to keep going, or to find a new path. When we act to improve a system, we are well-served by ways to measure what needs to be changed and what we aim to do. Metrics help us keep track of progress. The focal questions for each session are meant to be hard questions. It is our experience that by asking hard questions and genuinely listening across perspectives, and by finding common ground between perspectives, that we can identify underlying principles that guide us to better ideas and actions. Throughout the conference, we will also hear how speakers have used metrics to identify and characterize progress. Together, principles and metrics are used in any significant, complex endeavor, such as the mathematical education of teachers.

Synthesis

Prior to the conference, we asked presenters and moderators to share principles and metrics from their own experiences. Some themes that stand out from these pre-conference contributions, combined with the conference presentations and interactions, are the following.

• Mathematics can be joyful, beautiful, and powerful. Students and teachers should have opportunities to conjecture, hypothesize, reason, and make sense of new ideas in their mathematical work.
• Mathematics is more than what has been and will be written down. It is also about the people who create and do mathematics, in classrooms and in their professions. Students, including teachers we may teach, all deserve to have their competence recognized. It is for and up to instructors to be intensely curious about how students learn and what is in their minds, to learn to “assign competence”^[1], and to ensure that all students have their strengths seen and built on. There are emerging metrics for measuring participation, and by whom, in classrooms.
• Teaching mathematics requires a special kind of mathematical knowledge. Teachers should have opportunities to learn and harness such knowledge in their preparation. Learning and developing one’s own mathematical knowledge for teaching can be challenging, fulfilling, and joyful. Paula Jakopovic talked about feeling like a “mathematician of early childhood mathematics”.
• Culture operates at various levels. Within the classroom, there is “a density of discretionary spaces”^[2]. In these discretionary spaces, we find critical moments where messages are communicated and received about mathematical competence and mathematical identity^[3].
• Leadership in education is broad! It includes administration, government, as well as others we might already think of. Leaders in mathematics education should seek to engage beyond the spheres you might normally engage or influence, including scientists, engineers, and university administration
• Sustained changes need partnerships across different stakeholders. These partnerships take work to build, and do not happen by accident. Even if meeting of future partners happens by accident, the sustained relationships that define a partnership take time, openness, and commitment.
• We are not always going to agree, when it comes to something as big and complex as the mathematical education of teachers. It is important to make genuine efforts to understand, and to listen generously. There are always opportunities for questions that lead to common ground, even when it seems difficult to find.  
• Seek to understand rather than to win.
• There is always more to learn. Carrying out the mathematical education of teachers requires expertise in mathematics and in teaching, knowledge and experience of teaching students and teaching teachers, and more. It is unlikely that any one person can be the leader in all things. We should strive to improve our work through learning from others, even and perhaps especially those with whom we find some disagreements. The product that comes from collaboration is better than what we might be able to make individually.
• Let’s get to work!

At the conference, some of the following themes and ideas were raised.

Conceptual framing for work to improve mathematics education

• Definition (false dichotomy). A logical fallacy in which a spectrum of possible options is misrepresented as an either-or choice between two mutually exclusive things. (DLB)
• False dichotomies impede our ability to think clearly about teaching. One way to confront false dichotomies is to focus on the work of teaching mathematics, especially understanding the density of the discretionary spaces within mathematics teaching. (DLB) 
• Within the work of teaching, there are critical moments where messages about mathematical competence and identity are communicated; and so we as teacher educators need to develop communicative awareness to use these moments for benefit rather than harm (CEW II); exclusion can happen physically, socially, and intellectually (UN).
• Be able to see what is “mathematical” and what is “competent” (CEW II)
• To improve teaching, we need teachers – practicing teachers – in the room.
• To understand the demands of teaching and of teachers, we need to hear from and work with those who visit classrooms and who have been or are teachers. (PJ, DSch)

Spaces for boundary crossing

Boundary crossing happens when individuals cooperate, and those individuals come into the work with different commitments, backgrounds, or expectations for what may be most important. Cooperation can happen despite these differences, and can result in products that are more robust and innovative than those created by groups without as many differences. For boundary crossings to be successful, there are often “boundary objects” such as shared documentation or shared experiences that facilitate cooperation. At the conferences, various presenters shared examples of spaces for boundary crossing.

• Co-teaching, co-research, and reading groups involving people with different backgrounds and positions is a space for learning (JC); we can take small steps like reading groups, too, on the way to transform our teaching (CEW II). W. Gary Martin gives examples of Action Research Clusters of the Mathematics Teacher Education-Partnership that intentionally included mathematicians, mathematics educators, and practicing teachers in teams.
• Committee work is a space for boundary crossing; Gail Burrill mentions that when forming committees for NCTM, she always had at least one mathematician and at least one teacher on each committee; Matt Larson and Joan Ferrini-Mundy discussed processes for ensuring reviews from a wide variety of perspectives. On the last day of the conference, participants suggested that as a specific example, we could locally convene mathematicians, statisticians, mathematics educators, graduate students, and teachers to discuss forthcoming drafts of MET III and SET II when reviews are invited, and then submit a collective review potentially in addition to individual reviews.
• Immersion experiences in mathematics can be spaces for learning in community. (GS)
• Convergence has been defined by the NSF as “Collaboration around compelling, vexing problems that address social needs”, and that “innovation happens at seams, gaps, and boundaries.” We need to embrace this now. What we’re doing in universities isn’t working. (DSp)
• Teachers and teaching are boundary crossers of mathematics and teaching. Denise Spangler asked us to think about course sequences in teacher education and how they set up teachers to value mathematics and its intersection with pedagogy. For instance: What if there were a 1 credit methods course that preceded the content courses?

Openness and communication

• You want to communicate not in your language, but in a language that everyone speaks. (GB) As participants pointed out on the last day of the conference, for this principle to be followed, we need boundary crossers who ‘speak’ multiple languages (e.g., mathematicians with K-12 experience; or people with experiences in mathematics and education more broadly).
• Making connections beyond the spheres you might normally engage or influence, including sciences and engineering, and including university leadership (JFM)
• You need to invite your most ardent critics in, and genuinely listen. (ML) Participants discussed on the last day of the conference that for documents such as the forthcoming MET III or SET II, critical friends must come from different communities, including mathematicians, teachers, mathematics educators, and more. Critical friends must not only represent the communities the authors come from, but also – crucially – the intended audience. If the message doesn’t land with the audience, then we have lost an opportunity and also wasted time.
• “When you have a hard decision to make, you need to let people know that you will listen to their opinions, and that you will really listen, but that in the end, not all opinions can prevail.” (JFM, quoting Jim Lewis)
• Be direct, and honest.
• We need norms that more intentionally try to illuminate the hidden curriculum and invisible rules that allow more people to participate in whatever the process/task/activity/teaching is. This applies to instructors and students: what structures do we put in place to help people know how to interact in particular spaces and with particular people?

Cultural change

We need to be growing a generation of students and teachers who do not disidentify with mathematics. There are 50 million children in this country. The problem we are tackling is one of enormous scale.

• We need to incorporate attention to equity, not just in teaching (equitable teaching practices, etc.) but also in how we work with each other.
• Consider: Whose voices are represented? Who else should be “at the table”? Does your team reflect a focus on access and equity? (WGM)
• Institutional culture around teaching can change; it takes time, commitment, and coherence.
• ‘It takes a department to raise a student.’ A single individual might make a difference in one student’s life in one term; a united department can make a difference in culture over several years and sustained effort. (BF)
• Culture operates at various levels. Departmental change is brave, and necessary. Ben Ford illustrated the possibility of departmental change toward ‘servingness’ through the TIPS project. One level is the classroom, and within the work of teaching. As participants pointed out on the last day, departments are often a productive ‘unit’ for change, but departments are not isolated from culture. Department culture spans classroom culture to community culture, with the need to re-habituate norms across all these levels.
• Sustainable change efforts need to be anchored to structures (such as a Math Improvement Task Force, course coordination, placement policies) not only people. People change. Effective culture change plans for sustainability from the outset.
• Allan Donsig gave the example of Nebraska’s cultural change in the mathematics department to value teaching as a core mission. The department was able to do this by having teaching be part of the interview process, instilling a routine of applying for college and university level teaching awards, valuing pedagogical knowledge in faculty and graduate students, and valuing mathematics education faculty as full members of the faculty (who taught upper level mathematics courses, graduate level courses in their research area, etc.)
• Change agents need support, too. (WGM)
• “Whatever people love, hate, or not, keep walking and think of how you can impact others.” (TW, on lessons learned while forming the HSI solicitation at the NSF)

Mathematical flourishing, in teaching and learning

• We can think of mathematical flourishing as encompassing both mathematical understanding and mathematical identity. (DLB)
• “No one comes to class on the first day with the goal of messing with their teachers; they come to mathematics with their lived experience. This is true of students and teachers whom we teach. If we can re-orient our view to the classroom with this in mind, we can begin to understand our students, their identities, and their views of the discipline.” (DSp)
• It’s never too late to find joy in mathematics (PJ, DSch). Instructors can make a difference. And, teachers need us in their corner to advocate for their autonomy and flourishing. (PJ)
• Teaching requires knowing mathematics in the way that we want students to know mathematics, and more. (MS)
• We have to think about how to support both our novice teachers and our practicing teachers, and understand how they have experienced mathematics. We need to come in with compassion and empathy, along with the mathematics and pedagogy.

Designing structures that support the learning of teachers

• Teachers benefit when we embed mathematics and pedagogy within each of content and methods courses (JE)
• By the time teachers come to preparation or development programs they have learned mathematics in their own PK12 education, from their home, community, play; from mentor teachers, children in field placements; from their own planning. We need to learn how to harness these experiences. (DSp)
• Mathematical knowledge for teaching can be motivated by examples of student thinking and episodes of teaching. This does not necessarily mean that mathematics instructors need to do a deep dive into the work of teaching, but more that examples and episodes can illustrate problems of teaching practice where teachers’ personal mathematical knowledge can be harnessed.
• Mathematical coherence can be a principle of content sequencing. Sybilla Beckmann gave an example of a sequencing from fractions and measure to linear equations that she uses in teaching elementary teachers.
• We need to be explicit about the links we intend. Denise Spangler gave what might be akin to a parable of education, where when elementary teachers see base 10 blocks, they have an ‘a ha’ moment about base 10 arithmetic as they layer conceptual knowledge onto their procedural knowledge. They are excited about this insight, and so they use base 10 blocks with their students. But the students don’t know the procedures yet, and so they play with blocks and then still struggle with a worksheet. And so the teachers lose faith in the base 10 blocks, and then in teacher education writ large, and then the ivory tower. There are links from what happens in the classroom in a teacher education course to how it might transform into teaching children. Without being explicit about these links, and designing ways to make those links part of the experience of teacher education, we risk continuing to reproduce this parable in our educational system.
• Respect teachers for the professionals that they are. What it means to support teachers is to trust teachers as a whole to teach in the way that they think works, to set up professional development that is not just a ‘sit and get’ but something where they can work together to think about mathematics and teaching.
• It is possible for teachers to think of themselves as mathematicians. Daniel Schaben gave the example of how he did not used to think of himself that way, but after taking Math in the Middle courses, and working through challenging experiences, that he now does.

Encountering technology

• Leaders need to attend to mathematics, how it is taught, and insidious differentials in how technology is learned and used.
• AI can currently answer practically any standardized assessment question. It can ‘solve’ many homework problems. This can be thought of as a threat; but we can find opportunities to use AI to help teach mathematical thinking. Denise Spangler gave the hypothetical example of a teacher asking an AI to give them examples of children’s thinking while planning. It is harder and harder to get into schools now, for many reasons. AI might be able to give us the back end of simulations of working with students, including special education students. Gail Burrill gave examples of showing students where AI falls short, and also how students could improve on the AI answers. Or, as Lee Zia quipped, “If you could be replaced by a CD/computer/AI, then maybe you should be.”

Lessons learned and new perspectives taken

On Saturday, we discussed lessons learned and new perspectives taken. These include:

• We teach students mathematics rather than teaching mathematics to students.
• “Can we get to together to talk about ___”

• We already have permission to invite people to talk
• We don’t already have a proposed solution that we aim for buy-in; we can work together on identifying the problem first, and then work together on the solution

• Overcoming resistance involves changing narratives of “good work” and “punishment”

• One of the resistances that comes up is - “if we want to increase the number of math majors, then we need to teach the 1st and 2nd year courses better, but then few people want to figure out how to do this, let alone to the actual teaching”
• But it’s sometimes seen as a punishment to teach the “lower” classes - in high schools and in university.

• Moving local winds is worthwhile
• Tension between stability and growth - “glacial melt”

• We need partnerships in place before we take on big ideas something, and so we need to create this climate, and possibly re-connect them where they once existed
• “We carve people into ‘who gets it’ and ‘who does not’ and we forget to nurture new relationships” - glacial melt - the melt happens over time, and we don’t see it happen. Generations move on. “We assume things keep going”.

• Open questions

• How do educators address issues of equity and identity when state laws work against our goals?
• How do we guide students on course choices, keeping in mind that mathematical performance may be best assessed via multiple measures?  
• How do we conceive mathematical requirements for university in a way that sets up students for success in equitable ways? What are processes for changing and evaluating mathematical requirements for university?

Reflections shared by presenters and moderators

The following are reflections on “principles or metrics that they have found useful in their collaborations across stakeholders in the mathematical education of teachers”. Prior to the conference, we asked those willing to share a reflection to identify three such principles and metrics for participants to consider. We appreciate the generosity of those who shared their reflections below so that we can learn from them and use them as a lens for our own work.

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As a mathematics educator in a math department, I teach both mathematics and mathematics education courses for preservice secondary mathematics teachers. In mathematics education, good mathematics teaching has been characterized in a number of ways. The NCTM Compendium for Research in Mathematics Education includes several perspectives on high-quality math teaching. Here, I will focus on three principles that I find useful in guiding mathematics experiences for teachers.

The first principle parallels a well-known piece by Dorothy Law Nolte: Children Learn What They Live. I will share just a few lines to jog our memories:

“If children live with criticism, they learn to condemn… If children live with encouragement, they learn confidence.”

Extending this idea to the practice of teaching, we quickly arrive at the apprenticeship of observation, a term coined by Dan Lortie in Schoolteacher: A Sociological Study to describe how student teachers carry with them into their teacher training programs and their eventual classrooms the many thousands of hours of experience they had of being a student. In both cases, the key principle is that we learn the things that we have experienced. This principle forces us to ask, “What will their mathematical experiences tell them about what mathematics is?”

This leads us to a second principle: Mathematics is more than just a body of facts, theorems, and even proofs. It is a shared (or at least a shareable) way of thinking and of making sense of the world. For many years at the University of Delaware, we have had the Institute for Transforming Undergraduate Education. One piece of the work of this institute has been to shape the secondary mathematics education program so that preservice teachers get to experience both the kinds of important mathematics we hope they could teach their future students and the kind of high-quality instruction we hope they will use to teach their future students. I will provide three such examples in my talk.

These three examples point to a third principle: The equitable education of teachers requires attention to both individual and social processes. If, for example, we take equitable teaching to include providing more opportunities for students to learn, we must ask whether students actually have access to those opportunities. What barriers, either social or specific to the individual student, may be keeping a student from seeing those opportunities as available to them? What kinds of mathematical activity, such as posing their own mathematical problems, may be accessible to every student so that they can be positioned as a successful, active doer of mathematics? What image of mathematics is represented in the instructional choices? That image of mathematics—that answer to the question, “What is mathematics?”—will ultimately be what they teach to their students.

---Jinfa Cai

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There are three principles or metrics that I have found helpful in my time.

First: Using practice-based teacher education pedagogies. Those include but are not limited to doing rehearsals, using student work, using video recordings of classroom instruction, and simulations.

Second: The practice of assigning/acknowledging competence^[4]. This practice allows me as an instructor to attend both to mathematical content and equity. Specifically, to practice well, I have to be aware of status issues in my class, I have to use mathematical content flexibly and I have to have strategies and techniques to assign competence to students in the moment.

Third: Using the Equity QUantified in Participation (EQUIP) tool to see patterns in my instructional practice. Specifically, using this tool allows one to see which students they are calling across social demographic markers (i.e. race, gender, language, etc.) the nature of the interactions (i.e., what type of question did the teacher ask and the teacher's response evaluative), and students responses (i.e. did they ask a question, explained their thinking, etc.). This tool is powerful because it provides data about one's practice, which could then be used as a powerful tool to intentionally disrupt inequities that occur in classrooms.

---Charles E. Wilkes II

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1. Acknowledge and capitalize on the expertise of those with diverse backgrounds.

2. Recognize that a collaborative product is most likely to be a better product.

3. Acknowledge the limitation of my perspective and work to reconcile it with other perspectives.

---Gail Burrill

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1. Be patient and make genuine efforts to understand positions and perspectives of others.

2. Ask questions and watch for opportunities to synthesize and articulate common ground.

3. Draw on educational research to help ground assumptions and claims.

---Joan Ferrini-Mundy

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I'll share the principles we've used in our work on the META Math project:

1. Habit of respect: In teaching, it is the instructor's responsibility to take an asset-based view of students;

2. Active Engagement: Prospective teachers should have opportunities to make, explore, and validate conjectures and make sense of new ideas in their mathematical work;

3. Recognition of mathematics as a human activity: The practice of teaching involves interacting human beings about mathematics content

---Elizabeth A. Burroughs

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1. Mathematics is for everyone.

2. All students deserve access to inclusive mathematics learning environments where they can participate meaningfully.

3. Mathematics learning environments should be co-constructed with learners, teachers, and their communities.

---Ursula Nguyen

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1. An emphasis on mathematical habits of mind that build on low threshold, high ceiling mathematical investigations and that support independence and creativity in facing unfamiliar mathematical challenges.

2. A belief that mathematics is a deeply human activity best experienced within a richly interacting and mutually supportive community of learners including high school students, undergraduate and graduate students, and experienced mathematical researchers.

3. A commitment to meeting students in their own school environments through mathematics activities led by mathematically experienced teachers in their schools.

---Glenn Stevens

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1. You should strive to help your students see that the mathematics they are learning has meaning.

2. You should try to figure out what is actually going on in the heads of your students, even if that scares you.

3. Procedural and conceptual knowledge are deeply intertwined. When you talk about procedures you should use conceptual language, when you talk about concepts you should use procedural examples.

---William McCallum

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Here are one each for recruiting potential teachers into 4-year programs, the programs themselves, and in-service education.

1. Four-year institutions should actively increase their recruiting of mathematics students in 2-year colleges into their teacher preparation programs. The point here is that this is another way to increase the diversity of mathematics teachers across several axes.  

2. Students in mathematics teacher preparation programs should have access to courses of study in the mathematical and statistical sciences that reflect the mathematical and statistical sciences in the 21st century (and thus prepare them to teach the standards/curricula that are emerging).

3. Teachers deserve substantial, content-rich professional development that includes training in technology, and which reflects the mathematical and statistical sciences as we see them in the 21st century. (This is a principle but can also be turned into a metric).

---Charles Steinhorn

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1. Mathematics, including school mathematics, is beautiful, deep, and richly interconnected at every level. Thinking deeply about school mathematics can be a joyful, challenging, and worthwhile intellectual experience.

2. Mathematics education is a collective, shared responsibility. Sharing the responsibility requires openness, compassion, learning with and from each other, trying ideas, and being willing to change our thinking.

3. Consider teaching mathematics courses for teachers that are coherent, logically structured, and derived from a few core ideas in elementary mathematics, and that also draw on and develop teachers’ own reasoning and sensemaking.

---Sybilla Beckmann

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1. Seek to understand before you seek to be understood.

2. Assume good will and good intentions on the part of others.

3. Articulate a piece of common ground as a starting place, and return to it often.

---Denise Spangler

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1. I have found it helpful for partners to read articles and books together that provide everyone with an opportunity to understand the major issues surrounding mathematics education.

2. Within the partnerships that I have been, I have found it useful to leverage each other's strengths.

3. Within partnerships, I think distributed leadership is important.

---Marilyn E. Strutchens

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1. All students are mathematicians.

2. Always try each question.

3. Respect yourself, others, and this classroom.

---Jill Edgren

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First, we as teacher educators and leaders need to ensure that all stakeholders have a strongly aligned vision for and common language around high quality mathematics instruction. Organizations such as AMTE, NCTM, and NCSM provide guidelines, research, and resources that can help to support initial and ongoing conversations among stakeholders to determine the Why, the What, and the How when it comes to effectively serving all students and families in our communities. We must engage in challenging but necessary conversations so that we can identify and address inequities in our system so that we can effectively support those students who have been historically marginalized and made to feel inferior or invisible in mathematics spaces.

Second, it is vital that all stakeholders are critically conscious of the assets, biases, potential stereotypes, and assumptions that we all bring to the conversation around equitable mathematics instruction. When leaders and educators do not have similar backgrounds or experiences as the students we are called to serve, it is imperative that we get to know our communities and families at a personal, humanistic level. How do we seek out the assets our students and families bring into educational spaces and uplift them, rather than assume that we as educational professionals are needed to “fix” what is broken with mathematics education?

Finally, framing tools, resources, and professional development around a vision of “ambitious mathematics instruction” where all children are given access to and expected to engage with rigorous, high quality, contextualized, and meaningful mathematics, is a key to making meaningful change in the field. As we build the vision and the community, we can then collaboratively enter into the deep work of teaching mathematics with a truly centralized focus on equity.

---Paula Jakopovic

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The three main principles I am focusing on with my teaching:

 1. The first principle I practice is to grade equitably - My assessment practices are patterned from "Grading for Equity" by Joe Feldman

 2. Make mathematics relevant and "trick free". For example, do they understand the connections between distance, pythagorean theorem, absolute value, and standard deviation. Can they explain what they mean and when to appropriately apply algorithms when they say ""skip dot flip"", ""butterfly"", ""cross multiply"" or other such nonsense. Should we spend a week trying to learn that algorithm built from the days without technology for efficient pencil and paper minimizing calculation, when we can now visualize it in seconds?

3. The third principle I want to follow is implementation of authentic problem solving and communication. For example - I would love to switch my classroom structure to 100% Philip Exeter Academy. https://exeter.edu/mathproblems. Or to what our very own Shleby Aaberg does to radically transform the mathematics learning environment at Scottsbluff High School. Make math courses problem based rather than traditional set and get content based. This third one is the one I am spending most of my time on now with my own Math competition team. I would love more collaboration and development in this area.

---Daniel Schaben

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“If you give people the resources they need and expect them to succeed, then they will.”

---Allan Donsig, quoting Jim Lewis

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First, mathematicians (and mathematics departments) must own teacher preparation as a central part of their mission, rather than viewing it as a necessary evil. Without effective mathematics teachers, K-12 students will continue to enter the university underprepared to engage in high-quality mathematics.

Second, mathematics teacher educators (and education departments or colleges) must seek out and value the contributions of mathematicians as central to effective teacher preparation. As Jim Lewis, one of the founders of MTEP, emphasized to me many times over the years, accusation and blame are not good foundations for collaboration.

Third, we need to put in the work to make this happen!

---W. Gary Martin

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