Standard P.3. Opportunities to Learn to Teach Mathematics
Standard P.3. Opportunities to Learn to Teach Mathematics 

An effective mathematics teacher preparation program provides candidates with multiple opportunities to learn to teach through mathematicsspecific methods courses (or equivalent professional learning experiences) in which mathematics, practices for teaching mathematics, knowledge of students as learners, and the social contexts of mathematics teaching and learning are integrated. 
P.3.1. Address Deep and Meaningful Mathematics Content Knowledge P.3.2. Provide Foundations of Knowledge About Students as Mathematics Learners P.3.3. Address the Social Contexts of Teaching and Learning 
In this section, we provide two elaborations describing how programs preparing teachers of mathematics at the high school level can support their candidates’ opportunities to learn to effectively teach mathematics, Ethics and Values for Teaching Mathematics at the High School Level and Mathematics Methods Experiences for Teachers of High School Mathematics.
HS.8. Ethics and Values for Teaching Mathematics at the High School Level
Effective programs preparing teachers of mathematics at the high school level provide multiple opportunities for candidates to develop political clarity on the profession and their advocacy roles in teaching. [Elaboration of P.3.3]
The important roles that teachers of high school mathematics play in teaching young adults during their final years of compulsory education and in potentially providing the bridge to further education, employment, and citizenship heighten the need for effective programs preparing teachers of mathematics at the high school level to provide opportunities for candidates to develop political clarity on the profession and their advocacy roles in teaching. Candidates must understand the possible influences of the curriculum and school policies on student learning and students’ identities and be willing to advocate for their students when their students’ best interests are being violated. Having political clarity requires that teachers ensure that the very students who have not been served well by the school system become the focus and the means by which we judge the efficacy of what we are doing in mathematics teaching, rather than considering the students as a single group.
In effective programs, mathematics methods experiences address the social, historical, political, and institutional contexts that affect mathematics teaching and learning and provide practicebased experiences to develop core practices and pedagogical content knowledge that honors mathematics, students’ mathematical thinking, and cultural/communitybased funds of knowledge and experiences. Instructors of mathematics methods experiences are aware of issues particular to high school students. For example, although the practice of sorting students into or away from highlevel mathematics courses begins before high school, these tracks are often set in stone by high school, thereby precluding many students from accessing the most rigorous mathematics courses. Furthermore, although deficit language pervades all levels of schooling, deficit language is ingrained in the fabric of many mathematics departments. Moreover, many teachers may think that by high school age, those students who are good at mathematics have already been defined and little can be changed in students’ dispositions toward the discipline or their motivations to learn. For example, teachers commonly gather in the lunchroom or mathematics office to complain about today’s students’ lack of skills (relative to students from the past) or to speak about groups of students as if they are a homogenous group, ignoring their individual differences and the unique strengths and experiences they bring to classrooms. Mathematics methods experiences in effective programs offer opportunities for candidates to learn to respond to and professionally challenge colleagues who may hold deficit perspectives on students.
Rather than simply learning about the latest reform initiatives in mathematics education, prospective teachers can benefit from opportunities to see that professionals do not always blindly follow recommendations of their district or state officials. Instead, candidates are exposed to both the need and specific strategies for creative insubordination, the bending of rules to adhere to higher ethical standards (Gutiérrez, 2013b, 2016) when working with colleagues. One way in which those preparing teachers of high school mathematics can accomplish this work is through offering opportunities for prospective teachers to analyze the context of mathematics teaching and learning and prepare for the political realities of the profession. That is, just as prospective teachers need time to rehearse for teaching through such activities as peer teaching or extensive lesson planning, they need opportunities to prepare for the kinds of controversial discussions and negotiations they will face as new teachers in a context of highstakes education and in a country that has consistently failed to support Black, Latinx, American Indian, and other historically marginalized communities. If a candidate believes that a school policy or curricular initiative is unfair, he or she will need to practice challenging that policy, including knowing how to do such things as effectively press for explanation, turn a rational issue into a moral one, or seek allies. They must understand that a number of factors will help them be more strategic and effective. According to Gutiérrez (2016),
Choosing an appropriate strategy requires we first recognize the kind of issue at stake (i.e., What power dynamics are operating? How does this issue relate to student learning and social justice?) and then consider the speaker(s), our relationship with them, and the context in which we find ourselves. (p. 54)
Given the growing knowledge base of both prospective and practicing teachers using creative insubordination, teacher education programs can draw on cases of effective teachers employing this practice.
Vignette 7.6 provides an example of the kinds of dilemmas and reflective exercises that secondary mathematics teacher education programs may provide as a means for beginning teachers to consider the ethical decisions they may face in their future teaching.
Vignette 7.6. Helping Candidates Consider Ethical Decisions They May Face
Marcia is attending her mathematics department meeting. She likes her colleagues and generally agrees with their approaches to teaching. However, today, she learns that the district has decided to change their textbooks to a series that is text heavy and requires significantly more reading for students to understand the examples and begin the homework. She recognizes the need for occasional changes in texts to keep up with current reforms, but she is worried that the emergent multilingual students in her school may now require more of her attention because they may be less likely to use the text as a resource or they may be less likely to persevere in textheavy word problems in homework sets. She asks herself, “Will students become confused with the extraneous words or unfamiliar contexts?” Many of the students she knows who speak more than one language have been transitioned out of multilingual education classes and may not appear to need special attention. She would like to raise this issue at her meeting, but she is in only her 2^{nd} year of teaching and wonders if she should just stay quiet on policy issues until she has earned tenure. She hopes someone else in her department will speak up on behalf of the emergent multilingual students in their school. After some time in the meeting, however, she realizes that her colleagues do not recognize the potential hurdle these new texts may pose for students. Those preparing teachers of high school mathematics could ask teacher candidates to reflect on how they might handle the situation if they were in Marcia’s shoes.
 Suppose Marcia decides to speak up. How might she frame her concerns so that she does not appear to be simply complaining and has no solutions? What kind of language could she use in this public meeting that will promote dialogue and action rather than selfdefense or resistance by her colleagues? What particular strategies for creative insubordination might she employ? What kinds of examples or information might elicit understanding and empathy from her colleagues or administrators rather than deficit perspectives toward students? How might she speak to individuals who do not share her expertise or commitment to emergent multilinguals? If, as a department, they decide to accept the district’s choice, could Marcia convince the department to structure ways to support students with the new texts? What might she suggest? Can the old texts be retained as resources for students? If Marcia is unsatisfied with the outcomes of the meeting, can she engage her emergent multilingual students in ways that would not require knowledge by or consensus with other teachers?
 Suppose Marcia decided not to speak up at the meeting but wanted to follow up later on. How might she decide where to direct her energies? Would raising the issue with the department chair be the right place to start? Or, should she begin with a sympathetic colleague who has more teaching experience or more years in the school? What role could data or research articles play in her attempts to convince her colleagues about a particular action? Could the students and parents play roles in helping her advocate for emergent multilingual students and others who may struggle with the new textbook? Might she find allies (e.g., nonmathematics teachers in the building) with more seniority? What collective effort might they make?
In such reflective exercises, prospective teachers might develop essays to provide professional advice to Marcia, produce a script, or role play to rehearse how Marcia might raise these issues and enlist the help of others in her mathematics department to advocate for the emergent multilinguals in the school. This practice of rehearsing through the activity of “In My Shoes” is being used in mathematics teacher education programs in various universities across the United States (see, e.g., Gutiérrez & Gregson, 2013; Gutiérrez, Gerardo, & Vargas, in press).
Finally, effective programs launch their students on trajectories of continued professional growth, providing candidates opportunities for building “their emerging leadership by being capable and effective advocates for excellence in mathematics teaching and learning and by holding themselves and their colleagues responsible for the mathematical success of all students” (MTEPartnership, 2014, p. 5).
HS.9. Mathematics Methods Experiences for Teachers of Mathematics at the High School Level
Effective programs preparing teachers of mathematics at the high school level provide candidates multiple opportunities to learn to teach mathematics effectively through the equivalent of three mathematicsspecific methods courses. [Elaboration of P.3.4]
In effective programs preparing teachers of mathematics at the high school level, methods courses or equivalent experiences address deep and meaningful mathematics comprised of the interrelationship among concepts, procedures, reasoning and justification, problem solving, and development of productive dispositions (Kilpatrick, Swafford, & Findell, 2001). Effective programs include experiences where candidates revisit high school mathematical content from a rich and meaningmaking perspective. Even having content courses focused on mathematics content relevant to teaching mathematics, as recommended in HS.7, will not, however, overcome the influence of a candidate’s own K–12 learning experience if it was focused primarily on procedures. Furthermore, given the variety of paths taken by prospective teacher prior to credential programs, some students in mathematics methods courses will not have completed three especially designed mathematics courses for high school. Thus, mathematics methods courses must be focused not only on the methods of teaching mathematics but also on the high school mathematics for which those teaching methods are intended. In programs that offer fewer than three mathematics methods courses, experiences of integrating rich and deep mathematics with issues of teaching and learning must be included in other program components. In programs certifying mathematics teachers at both middle and high school, at least half of the content must be focused explicitly on highschoollevelmathematics topics.
In effective programs, teachers must also be prepared to hold strong foundations of knowledge about students as mathematics learners, including knowing about researchbased progressions within welldefined content domains, understanding students’ ways of engaging in mathematical practices, helping students share their mathematical thinking, and leveraging the diversity of students’ thinking to advance instruction. Consequently, experiences for prospective high school teachers to investigate high school students’ mathematical thinking are also incorporated into mathematics methods courses and early field experiences.
Mathematics methods experiences in effective programs also address the social, historical, political, and institutional contexts that affect mathematics teaching and learning and provide practicebased experiences to develop core practices and pedagogical content knowledge that honors mathematics, students’ mathematical thinking, and cultural/communitybased funds of knowledge and experience.
As important as it is for prospective teachers to grapple with issues of mathematics, students’ thinking, and equity, the experience is more valuable for candidates when these issues arise simultaneously. How might mathematics methods instructors address issues of equity alongside issues of mathematical content and students’ thinking? Consider an example by Civil (2016) of a prospective teacher who, after reading an article about algorithms taught in Latin American countries that differed from those taught in U.S. schools, said, “This is nice, but they need to learn to do things the U.S. way” (Civil, 2016, p. 220.) This case raises a myriad of issues, including valorization of knowledge, whereby the knowledge held by immigrant children is not valued as much as the knowledge that is taught in U.S. schools; the role of convention and principle in algorithms; and the great pedagogical value in understanding and building upon students’ mathematical thinking. All these issues must be given space to arise and be discussed in mathematics methods classes.