American presenters described teacher preparation in the United States and offered some perspectives on the challenges of ensuring access to and equity in mathematics education in a very diverse nation.
John Staley set the stage for the discussion by noting that the structure for teacher education in the United States is complex and reflects a variety of options and influences. States’ policies for teacher certification requirements vary significantly, he explained, and many offer alternative certification pathways that make it easier for individuals to switch to teaching from another career, for example. Prospective teachers may earn degrees at 2- or 4-year institutions or through online programs, whose curricula and requirements vary. Requirements for secondary mathematics teachers, however, include not only mathematics content knowledge but also general pedagogical knowledge, mathematics pedagogical knowledge, and knowledge of mathematics curricula. Note that there are also some states that have similar requirements for elementary teachers.
Jennifer Marshall of Farmington Area Public Schools used a program at the University of Minnesota to illustrate one pathway for prospective secondary (middle and high school) mathematics teachers. It is a 5-year
program. Prospective teachers first earn a 4-year undergraduate degree in the content area they would like to teach. Future mathematics teachers begin with a grounding in college-level mathematics, and after a fifth year of study they earn a teaching certificate.
During that fifth year, the students take two types of academic courses. Foundational education courses that are required for licensure for all teachers cover such topics as child psychology, learning and assessment, technology, English learners, reading in a content area, and special needs. Prospective secondary mathematics teachers also take courses in mathematics pedagogy, which cover the teaching of arithmetic, algebra, and geometry. The prospective teachers also complete a 10-week teaching practicum in the fifth year, in which they gradually take on teaching responsibilities under the supervision of a collaborating teacher.
At the end of the year, the prospective teachers take a performance assessment, the EdTPA.1 This assessment is required for all prospective teachers in Minnesota, a policy designed to bring more consistency to the expectations for new teachers.2 As part of the assessment, students are videotaped giving lessons through which they can demonstrate their mastery of mathematics concepts and of research-based strategies for teaching them. They are also asked to write reflections on their teaching practice.
Prospective teachers may complete a sixth year of study if they wish to earn a master’s degree in education. The master’s requirement is four additional courses, one of which is a research project. However, Marshall noted, the university encourages students to complete the master’s degree after gaining several years of teaching experience. Districts are often reluctant to hire teachers who have just completed a master’s degree because they are eligible for higher salaries than other teachers, even though they do not yet have much experience.
Marshall closed with observations about how her own experience at the University of Minnesota had prepared her to make curricular and instructional decisions using current research-based best practices in mathematics teaching, but also continually seek out current research and finding ways
1 The EdTPA is an assessment system for prospective teachers developed by the American Association of Colleges for Teacher Education and Stanford University; see http://edtpa.aacte.org/about-edtpa [accessed October 1, 2016].
2 Minnesota law requires the Board of Teaching to implement a performance assessment as part of the teacher preparation experience. In collaboration with the Minnesota Association of Colleges for Teacher Education, the Board of Teaching selected EdTPA as that performance assessment.
to implement it in lessons to improve student learning outcomes. The program offered “a nice blend of math content, pedagogy, and practical experience.” She did pursue the master’s degree, and liked having the opportunity to stay connected with the university and gain further training as she also gained teaching experience in the classroom. During these experiences, she was able to practice classroom management techniques while planning and implementing mathematics instruction. The classroom experiences also included extensive reflective elements, such as the EdTPA, that encouraged her to continually refine her teaching skills. Collaborating with colleagues was one challenge she encountered once she began working in a school. She found that there were significant differences among her colleagues in teaching strategies and beliefs about the best ways to teach math—she attributed those differences to inconsistencies across the institutions that prepare prospective teachers in Minnesota.
Janine Remillard of the University of Pennsylvania followed up by highlighting circumstances that create challenges for teacher preparation in the United States. There are approximately 500,000 teachers of mathematics for students in grades 7 through 12, she noted (teachers of younger students are usually responsible for multiple subjects). These teachers work in a system that is educating more than 50 million students (15 million of whom are in grades 9 through 12),3 so scale is a challenge in thinking about any aspect of U.S. schools.
Another challenge is the high rate of turnover among public school teachers, especially among math and science teachers: 33,000 math and science teachers left the field after the 2008 school year (10,000 of those were retiring). Lack of autonomy in the classroom, weak professional development, problems with student discipline, low compensation, job insecurity due to frequent budget crises, and the appeal of other career options are some of the primary reasons teachers leave their job. In math and science, the number of teachers leaving the system each year is about equal to the number of teachers completing teacher preparation, so “we don’t have a lot of wiggle room filling these positions with new teachers.” The new teachers are not always readily available in the schools where they are most needed, particularly those that serve disadvantaged student populations. In many
of those schools, “we’re losing math and science teachers more quickly than we’re able to prepare them.”
A third challenge is that the way in which teachers are prepared varies significantly both from state to state and within states. Historically, it has been colleges and universities that were primarily responsible for providing teacher preparation. More recently, alternative programs that are not part of a college or university have been certified by states to prepare teachers. As part of their authority over public education, states determine the requirements for teacher certification or licensure and also the quality requirements for the institutions that prepare teachers.4 These institutions may offer bachelor’s or master’s degrees or only a teaching certificate. A state may identify any of these qualifications as the minimum requirement for licensure. Teacher preparation programs vary in the time required to earn the same degree (from 1 to 5 years), in the required prerequisites and courses, and in the nature and extent of classroom field experience.
Remillard added that assessing teachers’ qualifications is very difficult. The approaches most commonly used include program recommendations, written state tests, performance assessments, and the use of student test scores to assess teacher performance, but each presents challenges.
Another critical issue, Remillard concluded, is that the teacher workforce does not “match the ethnic diversity of our student body.” The K–12 student population is becoming more diverse very quickly. The teacher workforce is also becoming more diverse, but at a much slower rate. Among public school teachers in 2011, for example, 16.5 percent were members of racial and ethnic population subgroups, while 41 percent of students in grades K through 12 were members of these groups, and 10 percent were English language learners (Ingersoll and May, 2011).
Several of the presenters focused on the challenge of preparing mathematics teachers to work effectively with an increasingly diverse student population.
4 Public school teachers earn certificates or licenses to teach in particular states, which may or may not be transferrable to other states. Private schools in the United States, which do not receive public funding, may set their own requirements for teachers.
Challenges for Mathematics Teachers
Alejandra Sorto of Texas State University oriented the discussion of diversity by showing a short video that illustrated some of the difficulties related to cultural differences that prospective teachers are likely to find when they reach the classroom. The video showed the story of a young Hispanic man whose own mathematics education was affected by negative assumptions about how well he was likely to do as a mathematics student, but who nevertheless became a mathematics teacher.
Sorto asked the group to discuss the challenges the video highlighted. More than one participant commented that it was an effective reminder that many teachers are not well enough prepared to work with students from other cultures. Teachers may be both well informed and well intentioned yet still make unconscious judgments about students’ mathematical ability, one commented. Another noted that teachers need more tools to recognize mathematical ability so they can effectively address the areas where students struggle. One person commented that the problem is not so much that individual teachers may not be well enough prepared, but that many children get “signals from the whole environment that they are not effectively able to learn.” Changing the classroom culture will be more valuable than just addressing the students individually, one person observed.
Sorto identified three challenges that are central to providing equal access to mathematics for all students:
- Preparing teachers to provide access to mathematics for students, such as the one in the video, who have grown up in a culture that has not been supportive of their learning
- Preparing teachers whose own primary and secondary education limited their own access to mathematics
- Recruiting teachers from diverse communities to become mathematics teachers
Sorto noted that there are promising efforts to address these challenges. “There’s an interesting phenomenon in Texas,” because the state is very large and diverse. Many districts in the southern portion, which borders Mexico, perform better at educating English language learners than districts in other parts of the state do. She and her colleagues have conducted research to identify the factors that contribute to the mathematics learning of the English language learners in the south Texas schools that perform so well.
One practice they have identified is the use of two languages in the instruction. This approach encourages students to “think in Spanish” so they can retrieve useful prior knowledge that is relevant to the material they are studying, such as word cognates. Some students use a mixture of the two languages as they progress, and teachers should be prepared to work with them in both languages. Using two languages also allows the students who are developing proficiency in English to participate in Spanish, while those who are more proficient in English can help Spanish speakers interpret what is said. Sorto used another video to demonstrate how teachers use this strategy—it showed a teacher organizing work groups in which fluent bilingual children assist other students who are still developing their English skills, and also showed the teacher giving assistance in two languages to students working individually and in groups.
Developing the Mindsets of Prospective Elementary Teachers
Amber Candela of the University of Missouri–St. Louis described a program at her institution that prepares prospective teachers in elementary education with a specialty in either special education or educating English language learners. The prospective teachers in the program, she noted, are primarily white females from middle-class backgrounds who are being prepared to teach in diverse classrooms. The university has received a grant that is supporting the work of an interdisciplinary team, which includes specialists in mathematics education, literacy, special education, English learning, and clinical teacher supervision, to explore ways to strengthen the university’s pedagogical methods classes and better prepare students to address students’ strengths and needs.
Candela offered a quote expressing the problem to be addressed: “If we wrongly assume that a competent individual cannot learn and understand, and restrict opportunities as a result, we’ve done a great disservice” (Hussman, 2015, p. 5). With that idea in mind, she tries to emphasize that prospective teachers should “presume competence,” and recognize that their job is to help all students use the mathematical knowledge they have and to find ways to increase every student’s access to mathematics. The program’s goal is to “eliminate the deficit perspective” and prepare prospective teachers to think in terms of challenging all students to learn more, rather than supporting some of them in overcoming deficits. The program aims to prepare prospective teachers to identify where students are in their learning, be inclusive, and find ways to engage all students in group discussions and activities.
Taking this approach can influence many decisions teachers make, and Candela used the design of mathematics tasks for students to illustrate. Mathematics tasks that are designed to be open-ended can give all students cognitive challenges while still allowing students at different stages in their understanding to identify a place to start in solving them. If the teacher has given thought to the different ways a task can be solved and the tools that might be helpful, he or she will be much better able to help any students who struggle with it. Opportunities to discuss both correct and incorrect solutions and use discourse to examine the reasons why some solutions are not correct provide important learning experiences.
Candela said that giving prospective teachers opportunities to reflect on a task they have used in practice teaching helps them build their understanding of the strategies the students used.
Learning to Lead Mathematics Discussions
Meghan Shaughnessy of the University of Michigan described another teaching practice that can help teachers reach all students in a diverse classroom.5 Leading a whole-class discussion is a complex task for which teachers need to be prepared. Figure 5-1 is a model for thinking about the components of the planning process, a series of steps through which the teacher makes a plan for accomplishing his or her goals for a class discussion.
Selecting the mathematics problem the class will tackle is a critical part of this preparation, and Shaughnessy used a kindergarten mathematics problem that can be approached in several ways to demonstrate how teachers can do this (see Figure 5-2). The figure includes the prompt, “Use Xs and Os to make 7 in different ways,” and also suggests a variety of ways that students might respond. Some students might think the task is asking them to use Xs and Os in different configurations that total seven, as shown in the thought bubble on the lower left. Some might think the task is asking them to use the Xs and Os to make a figure that looks like the numeral 7, as shown on the lower right. Other children may not know what “different ways” actually means and just arrange a set of Xs and Os in different ways, and some may just feel puzzled and wonder what they are expected to do.
5 Shaughnessy provided several citations for her remarks: Boerst, T., L. Sleep, D. Ball, et al., 2011; Chapin, S., C. O’Connor, and N. Anderson, 2013; Kazemi, E., and A. Hintz, 2014; and Smith, M. and M. Stein, 2011.
Some novice teachers may assume that students will understand the task and can begin work. Others may “over-scaffold the task,” she went on, reducing the cognitive demand by providing so much guidance that there is little thinking left for students to do. For example, a novice teacher who worried that students would feel lost might provide them with a graphic that indicates that they should put an X or an O in each of seven spaces. Both of these approaches would be problematic for many students. Some children may not know how to start without some guidance about what the task requires, but students who are given too much guidance may not engage in any real mathematical work.
To help novice teachers avoid those two extremes, Shaughnessy and her colleagues have developed an approach for “rehearsing” the way a task is set up. They begin by examining the task with their peers to make sure they understand what students are expected to learn from it, and plan an approach that provides an appropriate degree of guidance. The prospective teachers rehearse as a group, offering one another feedback. One prospective teacher takes on the role of the teacher, and the others act as the students. The teacher educator offers coaching during the rehearsal, highlighting important elements, and guides the prospective teachers in a discussion of what they learned from the rehearsal.
Teacher Recruitment and Targeted Preparation
Janine Remillard turned the conversation to ideas for recruiting and preparing mathematics teachers who themselves are from diverse communities and may have had limited access to mathematics learning. In the United States, there has been increased attention to the importance of identifying students who may not see themselves as skilled in mathematics or as potential teachers and encouraging them to recognize their potential. Efforts to address the way students relate to mathematics, however, can discourage many students.
Teacher education in the United States has generally been situated within the traditional higher education system. Unfortunately, some prospective teachers can face challenges gaining access to traditional institutions or teacher preparation programs. For some, cultural differences may have impeded their access to the mathematics instruction they received at the elementary and secondary level. Difficulties with mathematics courses at the university level mean that such students are not well prepared; they may lack the academic credentials required for admission to teacher preparation programs and score relatively poorly on required tests. Students who encounter such barriers are less likely than others to view themselves as candidates for a 4-year degree or a teaching career. They also may lack resources to support a college education or may live in an area with few academic opportunities.
Remillard described some promising avenues for such prospective teachers. She noted that, although assessments play an important role in ensuring that candidates becoming teachers have the requisite skills, some may exclude candidates unnecessarily. Alternatives, such as performance assessments to supplement written tests, may allow additional promising
teacher candidates to gain access to programs. Community colleges (which offer 2-year degrees and generally accept all students with a high school degree or equivalent) offer important “access points,” allowing students to develop their mathematics skills and earn credentials needed for admission to 4-year programs. Targeted recruitment of candidates from community college backgrounds is another strategy to increase access to becoming a teacher.
Another approach is to expand teacher education opportunities at institutions that educate large numbers of students from diverse populations. Remillard noted that such institutions have often been overlooked as sites for teacher preparation—she identified as “minority-serving” institutions those whose student populations are 20 percent minority or greater. Nontraditional teacher programs,6 which may be independent of postsecondary institutions or work in collaboration with them, are also promising avenues for reaching new students.
Participants had time to reflect in groups about equity issues and then to share some of their observations. Exploring the experiences of students who struggle is really helpful for prospective teachers, one participant noted. Prospective teachers can gain insights from concrete examples into the ways students may misunderstand, especially if they discuss with their peers possible ways to respond to the specific struggles. This participant emphasized that, though prospective teachers need training to correct mistaken assumptions about particular groups, it is important that they also realize that every student struggles at times. Another participant agreed, adding that asking students to elaborate on statements that seem incorrect or puzzling can also be very helpful. Another noted that prospective mathematics teachers have usually been successful mathematics students themselves, and may have trouble understanding the thinking of students who think differently from them. Interviewing someone who has not liked studying mathematics is another source of insight, the participant observed. Asking prospective teachers to
6 Alternatives to traditional teacher education programs housed within universities include programs for midcareer professionals who already have a bachelor’s or higher degree in another field, internships and other experiences organized in cooperation with university faculty, and other options; see http://www.nctq.org/teacherPrep/review2014/alternativeCertification/ [accessed November 1, 2016] for more information about nontraditional teacher preparation.
work through a problem or task that is truly a challenge for them is another way to expand their awareness of the process of solving the problem as well as the emotions and thinking they bring to the task.
Hyman Bass of the University of Michigan concluded with some thoughts about the role of professional mathematicians in K–12 mathematics education. He was overwhelmed by the profound differences between the U.S. and Finnish systems. A key difference is that in Finland educational decisions are made by professional educators, whereas in the United States “everyone is entitled to an authoritative opinion.” For example, in the United States many decisions about curriculum are made by school boards, which are elected or appointed bodies often made up of parents and community members. Professional mathematicians are regarded as responsible for shaping the “content” of mathematics instruction, through their role in the development of curriculum materials (standards, textbooks, and assessments) that need to be “mathematically correct, rigorous, comprehensive, and ambitious.” Their role in other sorts of decision making is less clearly defined, though nevertheless important.
Three major reforms in mathematics education in the United States illustrate the important role of mathematicians. A move to what was called “new math” in U.S. schools during the 1960s was prompted by national security concerns, he noted. The Soviet Union’s launch of the Sputnik 1 satellite spurred concern about the technical workforce in the United States. This opened up an opportunity to change curriculum in the schools, at a time when many mathematical problems of long standing were being solved. Mathematicians hoped to build some of these ideas into the curriculum, an endeavor that Bass described as idealistic but “naïve, pedagogically.” The approach did not work well, though some vestiges of it remain.
The new math reform was a vision directed mainly by mathematicians, not educators. It was mathematically aspirational, but pedagogically naïve, introducing highly abstract approaches even in the early grades. A backlash to this led to a return to a basic facts mode of instruction; but, with the growth of technology and related higher demands of the workplace, this basic-level education began to erode U.S. economic competitiveness. The business community felt this, having to invest heavily in new skills training for high school graduates. This crisis was dramatized in the U.S. Department of Education’s report, “A Nation at Risk.” Though this educational
challenge was national in level, and systemic, the United States does not have a tradition or the infrastructure to address educational reform at the national level; indeed there is no national agency with the standing to do so. In this “authority vacuum,” the National Council of Teachers of Mathematics (NCTM), the world’s largest mathematics education organization, took the bold (and courageous) step, in collaboration with other professional organizations, to develop and promulgate (in 1989), a set of standards for mathematics education in the United States that, though not “official,” represented a truly national vision. NCTM standards were widely adopted. They stimulated the development of “standards-based curriculum materials,” and launched the era of “standards-based reform” in science and in many other disciplines.
Mathematicians, who had not as a group been significantly involved in developing the NCTM standards, became alarmed at some of what they saw in the new standards. Their concerns led to what became known as the “math wars”—debates between advocates of traditional mathematics instruction and a reform approach. In this climate, the individual states developed their own mathematics standards, a situation that European colleagues viewed as “insanity.” Today, the Common Core State Standards in mathematics reflect much more consensus (see http://www.corestandards.org/about-the-standards/).
Mathematicians are not interested only in mathematics curriculum and materials, Bass continued, but also in the mathematics knowledge and skills of teachers. Because teachers need a deep and broad knowledge of the mathematics their students are expected to learn, mathematicians believe they generally need more advanced mathematics courses to keep up.
However, mathematicians have typically had little training in general or mathematics pedagogy. They have little understanding of “the mathematical thinking in another person’s head” or training in how understanding of the learning process can be integrated into instruction. A mathematician is likely to view preparing a coherent and well-conceived presentation of mathematical ideas as a teacher’s primary goal, and to believe that “if the students fail to learn, the fault lies with them, not with the teacher.”
Bass also noted that teachers and the teaching profession receive much less respect in the United States than they do in Finland. “There’s a lot of blaming” in the United States. The relationships among college and university mathematics departments and schools of education are a significant problem in the United States. Mathematicians frequently lament that many school mathematics teachers do not have strong mathematical knowledge,
yet these teachers generally have studied mathematics in the university departments in which mathematicians teach. However, “the boundary between pedagogy and content is becoming less sharp,” which means that improved coordination between mathematics departments and schools of education is increasingly important. Some institutions are beginning to appreciate this, in his view. This type of cross-boundary coordination is not typically rewarded in university mathematics departments, in terms of status or judgments about promotion or tenure.
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