Chapter 8 Mathematics Teacher Education and edTPA:

Complex Assessing

Dianne S. McCarthy
SUNY Buffalo State, USA
Barbara A. Burns
Canisius College, USA

ABSTRACT

The development of the educative teacher performance assessment (edTPA) might be considered as beginning over a century ago as mathematics, mathematics teacher education, and the teaching profession strove to improve student learning. Professional teaching organizations such as the National Council of Teachers of Mathematics, the National Board of Professional Teaching Standards, industry, and government agencies have been seeking ways to improve teaching, to differentiate among teacher candidates to predict who will be successful teachers and who will not, and to raise the level of student achievement of all students. Along with these goals is the aspiration of recognizing teaching as a profession. To achieve this, complex assessment is necessary. Assessment of teachers, students and teacher preparation programs is necessary. edTPA could lead the way.

INTRODUCTION

The advent of performance assessment for teacher educators is a multifaceted endeavor. This endeavor has been at least a decade in development and encompasses research on the historical development of teaching in the content area, on the desire to see education become a true and recognized profession, and on the making of what would best be considered quality teaching and learning.

This chapter looks at a specific performance assessment, the edTPA, and its special relationship with the content area of mathematics. Mathematics is complex. It can elicit visceral responses from people who feel they are not “good at math” or can challenge people to solve or conjecture responses to problems. It causes both confusion and creativity with the effects of both success and failure.

Mathematics education as a field of study is also complex. The challenges in mathematics teaching and learning have caused developments in P-12 schools, colleges, teacher preparation programs, and teacher certification. Throughout the history of mathematics education, it can be seen that there have been attempts to improve mathematics teaching and learning for children, for college students and for teachers.

These improvements come from an emphasis on learning mathematics with conceptual understanding, procedural fluency and mathematics reasoning and problem solving. The performance expectations of edTPA are in keeping with expectations of the common core state standards, the standards for teachers expected in National Board Certification, and society’s desire to have students who are college and career ready in mathematics.

The challenges that have faced mathematics teachers and teacher educators have resulted in many changes to content, pedagogy, and assessment over time. The most recent answer to the challenge is the edTPA. Although edTPA faces criticism, it may be the best development in solving the puzzle of mathematics education and in creating quality teaching and learning experiences.

HISTORY OF MATHEMATICS EDUCATION AND MATHEMATICS TEACHER PREPARATION

As the professional fields of mathematics and education made their debuts in the 1890s, courses in the teaching of mathematics began to evolve. Where no institutions had previously included these courses, the advent led to some interesting developments at the first five schools to embrace these fields of mathematics and education.

The University of Michigan, in 1892, led the way in their attempt to train high school mathematics teachers by establishing “teachers seminaries” – one in algebra and one in geometry. This was soon followed by Ypsilanti’s Michigan State Normal School which developed a program devoted to the strong academic and professional preparation of teachers. While this program included components similar to the “teachers seminaries” of the University of Michigan, it also included historical developments of the fields of algebra and geometry. In addition, this program included a required course entitled Professional Training in Arithmetic.

The University of Chicago and Teachers College in New York were the next on the scene with attention being paid to mathematics and education. While Chicago worked primarily on the formation of its mathematics department, they did pay special attention to the pedagogy of mathematics. While this focus began with a lecture series, it wasn’t long until a position was established for the assistant professor of mathematical pedagogy. At around the same time, Teachers College’s mathematics department had a threefold mission: “to prepare students to teach mathematics in elementary and secondary schools, to provide the introductory-level courses in algebra and geometry required of all students for admission to junior-year status, and to supervise mathematics instruction in the affiliated Horace Mann School” (Donoghue, 2013, p. 164).

Syracuse University rounded out this initial list of five schools to embrace mathematics and education. The mathematics department provided students an opportunity to learn the pedagogy of mathematics for those students who intended to teach mathematics.

Other institutions followed suit and began to offer specialized training in the preparation of mathematics teachers so, by the end of the first decade of the twentieth century, there were at least twenty-five institutions involved.

The introduction of these courses and programs allowed for the discussion and study of issues related to teaching school mathematics. This led to the formation of associations of groups of mathematics teachers and members began to publish articles on the teaching of mathematics.

The beginning of the century saw such leaders as Smith, Moore, and Young speak out about the improvement of the preparation of mathematics teachers. Smith compiled many of his lectures and produced the first textbook for mathematics educators, Teaching of Elementary Mathematics (Smith, 1900). His aim was to provide access to materials to those who wished to work with these mathematics educators: access he didn’t have when he first began.

Three principal pedagogical threads – the rule method, the inductive method, and the analytic method – were prevalent as main themes in textbooks throughout the 19^th century. The rule method was seen in North America from early colonial times until about the 1820s. This method, presented in mathematics textbooks during this time, focused on presenting definitions, rules and tables and the pedagogical focus was purely deductive. This method was quite different from the other two methods which complemented each other.

The inductive method allowed children to abandon rote and mechanical approaches to problem solving and to discover, for themselves, the basic principles involved. The analytic method was a companion strategy to the inductive method and was used to show a student how to think through a problem using logic and reasoning rather than focusing on rules.

Throughout the 19^th century, authors of arithmetic texts placed a strong emphasis on the mathematical content being practical and current. Teachers were largely supportive of methodologies based on conceptual understanding and real-world applications and the use of manipulatives was also widely accepted.

While these pedagogical threads were being used throughout the 19^th century, it is interesting to recall as discussed previously that the field of mathematics education traces its origins to the 1890s. It was 125 years ago that two distinct fields, mathematics and education, emerged as professional areas. Donoghue (2003) notes, “In 1890, no American institutions of any type offered a course in the teaching of mathematics in the high school” (p. 160). Toward the middle of the decade, five institutions were on board with such programs. This emergence of programs brought with it “the study and discussion of issues related to school mathematics” (p. 165). Members of educational groups began to write and to publish articles on the teaching of mathematics.

In the early years of the twentieth century, mathematics educators focused on two main areas: the teacher and the curriculum. Graduate programs were established in mathematics education. At Teachers College in New York, in 1900, four “qualities” – “general culture, professional knowledge of education, special knowledge of the teaching discipline, and technical skill in the classroom” – (Donoghue, 2003 p. 174) were the aims for the development of a teacher. It was during this period, “from 1890 – 1920, that the major components of the profession of mathematics education in the United States were established” (p. 187).

It was also during these years that mathematics was viewed as a prime vehicle for mental training and it was thought by some and advocated by the Committee of Ten that “every student regardless of probable destination deserved to have his or her reasoning power strengthened by the kind of mental exercise that intensive study of mathematics provided” (Kliebard & Franklin, 2003 p. 402). Critics to the report filed by the Committee of Ten had a differing view of gender differentiation. They believed that certain subjects, such as mathematics, were too strenuous for girls. In addition, they advocated that students who wouldn’t progress past elementary education had no need for mathematics beyond arithmetic.

These doubts led to a great decline in the number of high school students in mathematics classes. General mathematics classes were being formed to help students prepare for day-to-day responsibilities and to provide only the mathematics needed for direct application in jobs. The purpose of schooling was changing to a preparation model. This purpose held and, in 1940, Mathematics in General Education summed up the major reforms that had been occurring for the past half century. It continued to focus on mathematics as needed for practical application.

In the 1930s, with the hint of war in the air, mathematics educators worked to reverse the doubts in their subject. Mathematics educators began to discuss why students should study mathematics well beyond arithmetic. The Mathematical Association of America along with the American Mathematical Society formed a joint War Preparedness Committee. A subcommittee’s major report, Mathematics in the Defense Program, “focused attention on the anticipated great demands from both the military and industry for high school graduates with substantive mathematical preparation” (Garrett & Davis, 2003, p. 498). The utility of mathematics was erasing former doubts about the need for the higher order thinking of mathematics. More people had a necessity to know more about mathematics and to be able to apply it in applications to the war effort.

Before the end of World War II, President Franklin D. Roosevelt, asked that something be done to continue the advancements in mathematics and science that had occurred because of the war. In a letter to Vannevar Bush, director of the Office of Scientific Research, he asked him to delineate and evaluate the lessons learned from the war time experience that could be applied to peace time growth and development. Such advancements provided growth and development for the economy as well as jobs and security for the country (Ferrini-Mundy & Graham, 2003). It was clear at the close of World War II that there had been advancements in mathematics, science and medicine and a growing need for scientists and engineers. It was also clear from the experience of training the military during World War II to work on and with such technology as radar and radio and fields such as navigation and map making that the mathematics taught in high school was inadequate (Commission on Post War Plans, 1944). This began a movement to improve mathematics teaching and teacher preparation. Part of this movement was sidetracked as the number of school aged children grew following World War II and the need for teachers increased. As the need for teachers increased, certification requirements were lowered or eliminated. This trend reversed itself due to a teacher surplus in the late 1960s and early 1970s when certification requirements were raised.

The launch of the Sputnik satellite in 1957 produced a similar if not more urgent call for more mathematicians, scientists and engineers along with a desire to improve mathematics learning for all children. This movement did not just capture educators, but included mathematicians, scientists, the business community and politicians. It brought mathematics education into a national conversation. Professional papers (the report of the Commission on Post War Plans, 1944 and a second one in 1945, NCTM’s Secondary Curriculum Committee published recommendations in 1959, and a report of the Commission on Mathematics of the College Entrance Examination Board published in 1959), recommendations by professional organizations (Mathematical Association of America (MAA), National Council of Teachers of Mathematics (NCTM), American Association for the Advancement of Science (AAAS)), and extensive teacher training (funded by individual companies such as General Electric and Dupont, by the government through the newly established National Science Foundation, and by professional teaching organizations) were prevalent (Ferrini-Mundy & Graham, 2003).

Recommendations were varied, but some common themes emerged. Teacher training in mathematics must be more extensive and include more content so that teachers would be competent in mathematics before teaching it to others. Three years of high school mathematics was needed for elementary teaching. Through the MAA and its Committee on Undergraduate Programs in Mathematics, and the subcommittee or Panel on Teaching Training emerged recommendations that placed teaching in five levels. The five levels were established with level I being elementary teachers (K-6), Level II Junior High (7-10), Level III High School (grades 9-12), Level IV (grades 12-14) and Level V being college level instruction. It was expected that elementary teachers complete at least two courses in mathematics and high school teachers complete the equivalent of a major in mathematics. Until this point in time elementary teachers in particular had very limited experience with mathematics. Building content knowledge was considered very important. The Panel also listed some assumptions that should be considered when implementing the recommendations. These include a) that specific implementation depend on local conditions, b) that the recommendations are not tied to any particular school curriculum, c) that good mathematics is a sequential experience, and d) that the learning of mathematics begins at the concrete level before moving to the theoretical or abstract (MAA, 1961).

These recommendations were endorsed by professional organizations and were included in publications. Mathematics teacher education reform in the form of reports, conferences and funding had begun as a response to World War II and the launching of Sputnik. These themes emerged: strong content knowledge for teachers, more course work in mathematics at the college level for teacher candidates, and an emphasis on strong content for all students through the New Math curriculum. What was not evident in these reforms was the development of an examination that would illustrate that a teacher had the mathematical content understanding to teach mathematics and was competent in implementing the curriculum.

In the mid-1960s-1970s there was a push for competency based teacher education (CBTE). In the CBTE movement teacher candidates are expected to learn and demonstrate certain competencies. These competencies often included pedagogical expectations. There were limitations to the CBTE movement and such issues kept it from being widely accepted. Some of the issues were: Are the competencies that are deemed necessary really necessary? Is there a relationship between observable competencies and student achievement? What about conditions of the classroom such as class size? (Ferrini-Mundy & Graham, 2003)

The 1980s brought in the standards movement as the public perception was that the teachers were not the problem in the nation’s poor mathematics performance as described in A Nation at Risk, the curriculum was. In 1989 the National Council of Teachers of Mathematics published Curriculum and Evaluation Standards for School Mathematics in an effort to improve both content and pedagogy. There was less money available in the 1980s for preservice and inservice teacher education, but efforts continued. In the 1980s the mathematics education research community emerged and the use of this research became evident.

The 1990’s brought the results of the Third International Mathematics and Science Study which showed the children in the United States were not performing at a high level. This study seemed to illustrate the teachers were also part of the problem.

The National Board for Professional Teaching Standards (NBPTS) which began in 1987 required that National Board certified teachers provide evidence of both mathematical knowledge and pedagogical competence. The National Board has facilitated a conversation about the need for teachers to understand teaching, learning and mathematics and to expect teachers to provide evidence of this. Research studies based on educators as they engaged in the process of earning National Board Certification showed many positive results including improvement in quality and coherence of classroom assessment strategies and modifications of teaching based on the assessments. There was a shift from assessing to determine grades to assessing so that students learn. Teachers learned to evaluate students using evidence rather than anecdotes. Teaching practices based on the assessment data were more coherent. National Board candidates were rated more favorably by students. They were shown to emphasize questions that require explanations and encouraged class discussions. Students were given more time for self-assessment and reflection on their learning. Teachers who engaged in the board process used formative assessment before beginning a unit and included more hands on activities and discussions in the unit. (Sato, Wei, Darling-Hammond, 2008). Unfortunately, practicing teachers who engage in and earn board certification are a small population of the teaching profession. Considering how to include the positive aspects of earning board certification to earning initial teacher certification would strengthen teacher preparation programs.

There is also a changing student population during the years described above, post World War II to today. Rossi Becker and Perl (2003) describe the challenges facing different groups of students in achieving in mathematics. Issues of equity include access to course work, curriculum, and methods of instruction. While there has been an emphasis on mathematics for all, there has also been a desire to create a mathematically elite who would pursue careers in mathematics, science and engineering. Which courses and which methods for which students: male, female, college preparatory students, vocational students, honors students, and students of all races have been topics of discussion. Preparing students who really understand mathematics has been a concern as evidenced by programs developed such as one created by The University of Illinois Committee on School Mathematics that used manipulatives and unambiguous language to help students understand.

The issues of content knowledge and pedagogy have been raised again and again in the literature from the 1950s through today. Mathematics teaching and learning for all children regardless of innate ability has been a topic of conversation since World War II. Demonstrating competencies and providing evidence that a teacher can teach mathematics and students can learn with understanding is pervasive in today’s climate of accountability. The educative teacher performance assessment or edTPA has emerged today as a method for assessing teacher candidate’s content knowledge and pedagogical skills along with evidence of student learning. As a performance assessment it is a unique exam that demands that teacher candidates show competence in pedagogy along with content knowledge as evidenced by their work with children.

TEACHING AS A PROFESSION: EDUCATORS USE OF edTPA

The effort to promote teaching as a profession has been ongoing for more than half a century. In order to be considered a profession, there are certain features or characteristics that are required. According to Darling-Hammond and Hyler, (2013), three features are required: 1) members must be committed to the welfare of those they serve: they must put the best interests of their clients first, 2) the members of the profession must share a common body of knowledge that will advance their clients’ best interests and 3) the members of the profession must agree on, define, transmit and enforce standards of professional practice. The profession must monitor the performance of its members rather than be managed by people on the outside. “Professions make a compact with the public that allows them to manage their own work in exchange for holding themselves accountable for mastering the knowledge and skills that allow them to practice safely and effectively” (Darling-Hammond & Hyler, 2013, p. 12).

In the attempt to have teaching achieve status as a profession, several areas must be considered. Among these are the perception of teachers and the training and licensing of teachers. Public perceptions of teachers are not at a level of professional status. Teachers are often times viewed as part-time workers – summers and holidays off and shortened workdays. Teachers are charged with enormous responsibilities but their value is often understated. Teachers are primarily women and the pay for teaching has traditionally not been competitive with other professions especially those seen as more masculine in nature. An attempt is being made to address this perception of teachers and teaching.

Teaching has come a long way since the days when students of all ages were in one room with one school teacher in the front of the room. Students’ primary responsibility was to write down what the teacher said to prepare for the exam based on the teacher’s lectures. They were to be on their best behavior and they were to learn with minimal student input. The model of the teacher teaches and the student shows what was learned was prevalent. Today’s classroom usually looks a bit different. Students are more involved with their learning and oftentimes are engaged with small groups of students working to solve a particular problem. Technology can be a learning tool in the classroom. Today’s teachers are charged with knowing their students and their students’ needs. The perception of teachers is beginning to turn around.

The other area where the status of a profession can be affected is in the area of licensure and certification; attention has been turned to other professions where rigorous licensing and certification tests are critical requirement. Efforts are underway to ponder why and how to use such rigorous licensing exams. One of the major factors prohibiting teaching from being recognized as a profession is the lack of a nationwide licensing exam. Currently, each state has its own licensing criteria.

Nationwide licensing exams would allow candidates to demonstrate they can meet and uphold the high standards of the profession. These assessments not only allow candidates to enter the profession, they provide guidelines for the curriculum in professional schools. (Darling-Hammond & Hyler, 2013). While paper and pencil tests for licensing have been used, they have not been authored by members of the field and they typically don’t capture the knowledge and skills needed by a teacher. Teaching is complex enough that performance assessments are necessary.

edTPA as a Licensing Exam

The Stanford Center for Assessment, Learning and Equity (SCALE), taking the stance that individuals entering the teaching profession must be prepared to meet the academic needs of all students, developed the educative Teacher Performance Assessment (edTPA) to measure teacher candidates’ readiness to teach. The edTPA is the first nationally available, educator-designed performance assessment for teachers entering the profession. (SCALE, 2014). The edTPA carries the dual goals to improve assessment of teacher candidates and ultimately reform and distinguish teaching as a profession. The transfer of professional certification from a series of standardized exams to a performance assessment, which includes authentic tasks, allows the complex assessment of P-12 student learning, teacher candidate performance, and teacher preparation program review. “By evaluating teaching authentically, these performance assessments represent the complexity of teaching and offer standards that can define an expert profession.” (Darling-Hammond & Hyler, 2013, p.13) The effort to move teaching to a profession must include teacher candidates’ successful demonstration of an edTPA at a high level. A licensure exam such as the performance assessment must be embraced by professional educators and teacher preparation programs because it assesses the complex situation of teaching and learning. When all support the high standards expected then teaching will indeed be monitoring its own, will not require monitoring from outside, and will thus be a profession.

The edTPA was developed by teachers, unlike other licensure exams, which have not. Most of the currently used licensure exams for perspective educators have been created by groups outside the profession, make use of multiple choice or short answer formats, and emphasize content and pedagogy, but not the complexity of the teaching and learning process. The edTPA has been developed in twenty-seven different fields based on licensure areas. Because of the vast use of edTPA across the United States, (currently, 600 educator preparation programs in 40 states, and the District of Columbia, are participating in edTPA in some form or another and “as of early 2015, 12 states have either adopted statewide policies requiring a performance assessment for aspiring teachers or are actively considering such a step” (AACTE, 2015)), data on validity and reliability are available (www.edtpa.aacte.org). The opportunity to begin to consider the performance of a large number of teacher candidates offers the profession the opportunity to consider the rigor of preparation programs in addition to the performance of specific candidates.

The edTPA can be used in teacher preparation courses, as part of student teaching or, in the case of some states like New York State, as a certification exam. What makes the edTPA different from other assessments are the layers of evaluation that comprise the assessment.

The edTPA is designed to align with the authentic teaching practice of the teacher candidate. First, the tasks are integrated (that is, the learning goals, the instruction, and the student assessment are linked together) as they would be in the authentic work of a teacher. Second, each edTPA task requires the candidates to collect and submit direct evidence from the actual work of teaching such as student work samples or video recordings of the candidates engaged in instruction and interacting with students around the content learning goals. Third, the tasks represent not only the behaviors of the teacher, but also include the impact of the instruction on student learning as demonstrated through an analysis of student learning. And fourth, the instructional tasks are considered within the context of the subject matter content and learning goals. Given that the structure of teacher licensing in each state uses subject matter discipline or content specific categories, teacher candidates are seeking a license in a particular content field. Thus, this criterion aligns with the authentic work of teaching within a specific content-area. edTPA puts into practice those skills necessary to be a successful teacher and thus can be considered an authentic assessment as well as both a summative assessment and a formative assessment.

Background of edTPA

Authentic assessments, assessments used to directly examine performance on intellectual tasks, should assist in distinguishing candidates who should be successful teachers outweighing the way that traditional assessment, paper and pencil one answer questions, has previously been used.

An expanding number of teacher education programs are using authentic assessments of teaching as one set of tools to help novice teachers create, in a principled fashion, bridges from generalizations about practice to apparently idiosyncratic, contextualized instances of learning. Under the title of authentic assessment, we include opportunities for developing and examining teachers’ thinking and actions in situations that are experience based and problem oriented and that includes or simulates actual acts of teaching. Such acts of teaching include plans for and reflections on teaching and learning, as well as activities featuring direct interaction with students. (Darling-Hammond and Snyder, 2000, p. 524)

edTPA is a shift from earning teacher certification simply from completing a teacher preparation program to actually demonstrating proficiency (Darling-Hammond, 2012). The development of the edTPA is based on the Interstate Teacher Assessment and Support Consortium as well as the National Board for Professional Teaching Standards.

Educational research and peer reviewed publications were also considered when developing the edTPA. Black and Wiliam, in their article, Inside the Black Box: Raising Standards through Classroom Assessment, discuss their analysis of education research illustrating that formative assessment is the vehicle for improving student learning. Formative assessment is used to inform; it includes monitoring student progress during instruction and activities and allows for feedback and opportunity to improve. It also allows the teacher to determine if his/her teaching has been successful. In order to be effective, assessment must impact instruction. Their meta-analysis supports the premise that effective teachers assess students and use assessment data to drive instruction (Black & Wiliam, 1998).

The edTPA requires candidates in 27 fields to develop and teach a series of lessons called a learning segment. Three tasks are completed on the edTPA: planning, instructing and assessing. The planning task focuses on the intended teaching. Candidates are asked to consider what their students know, what they want their students to know, and what means (instructional strategies, learning tasks, and assessments) will need to be designed to support student learning. Candidates are required to think about the teaching (and learning) process. Based on their knowledge of their students, candidates are challenged to choose appropriate curriculum, pedagogical practices and formative assessments that will work best to improve student learning.

The instruction task focuses on the execution of a candidate’s teaching including their content specific instructional strategies as well as their skills in questioning and discourse. Attention is directed to the learning environment established, the deepening of content understanding developed by students’ responses, and connections to students’ prior academic learning. In addition, candidates use evidence from their instruction to reflect on their teaching practices and offer suggestions for change in an effort to more effectively meet a variety of student learning needs.

The assessing task focuses on the impact of a candidate’s teaching on student learning. Candidates gather evidence on what students have learned, provide meaningful feedback and use these measures to plan their next steps in instruction (SCALE, 2013a). They analyze evidence of student learning and provide a summary across their whole class, they provide feedback on strengths and weaknesses, they explain how they will support students to use feedback to deepen understanding and they use all of this information to determine their next steps.

In each of these tasks: planning, instructing, and assessing, candidates are scored using a series of 5 rubrics per task except in World Languages where there are fewer. The rubrics are based on a 5 point score, 1 – 5, which rates candidates’ work along a continuum from not ready to teach, depicted by a teacher focused, whole class, fragmented or indiscriminate presentation of work, scored as a 1, to a highly accomplished beginner teacher with evidence of student focused, individual or flexible groups, integrated, intentional and well executed presentation of work, scored as a 5. The scores from each of the rubrics are tallied and a final score is compared to a cut score established by the state to determine if candidates pass or fail the edTPA. It is important to note that the edTPA is scored by members of the profession after rigorous training to become a qualified scorer. Assessment is a common thread throughout the three tasks. Plans for assessment are proposed in task 1, evidence of assessment is seen in the instruction presented in task 2 and task 3 is fully dedicated to assessment and follow-up.

While each of the edTPAs encompass the three learning tasks of planning, instructing, and assessing, each does so with guiding questions that are unique to the field of study and which have been formulated based on research in that particular field as shown in the edTPA handbooks. For example, guiding questions in the secondary level assessments have the history candidates respond based on the use of “facts, concepts, and inquiry, interpretations or analyses to build arguments or conclusions”, the English candidates respond “using textual references to construct meaning from, interpret, or respond to complex test”, the World Language candidates respond “using development of proficiency in the target language”, and mathematics candidates respond using “conceptual understanding, procedural fluency and mathematical reasoning and/or problem solving”.

edTPA AND MATHEMATICS

Three of the edTPAs are focused specifically on mathematics: Elementary Mathematics, Middle Childhood Mathematics, and Secondary Mathematics. A fourth edTPA, Elementary Education, is also available. The Elementary Education edTPA dedicates three tasks to Literacy but includes an additional task, Task 4: Assessment of Students’ Mathematics Learning.

Task 4 of the Elementary Education edTPA focuses on assessment of student learning and the development of a re-engagement lesson based on data collected from a lesson segment in mathematics. There are three additional rubrics in the Elementary Education edTPA which are similarly scored on a scale of 1 – 5. These additional rubrics are focused on Task 4: Assessment of Students’ Mathematics Learning and use the following guiding questions to assess candidates in the area of Mathematics: 1) how does the candidate analyze whole class evidence to identify patterns of student learning? 2) how does the candidate use student work to analyze mathematical errors, confusions, and partial understandings? and 3) how does the candidate examine the re-engagement lesson to further student learning? (SCALE 2013a).

Task 4 provides an opportunity for teacher candidates to analyze whole class evidence of student learning, identify patterns of learning and areas of need related to conceptual understanding, procedural fluency and reasoning/problem solving, and to create, implement and examine a re-engagement lesson that is specific to the needs of a segment of the population of students.

Included in all the edTPAs is a focus on both quantitative and qualitative analysis in an effort to determine patterns of learning as well as the candidate’s impact on student learning. Candidates incorporate item analysis where they review patterns of scores and answers to the assessment items. Teacher candidates choose an assessment (usually formative). They conduct an item analysis and set up a chart or table to help represent their information. They then look for patterns of learning among their students. Focus students are chosen to highlight the results of the item analysis. In the Elementary Mathematics, Middle School Mathematics and Secondary Mathematics edTPAs, candidates provide an explanation of how the feedback provided to the students will address their individual strengths and needs and how the teacher candidate will use the information to inform their future instruction for the whole class and for the 3 focus students. In the Elementary Education Task 4, in addition to the analysis, a re-engagement lesson is then planned and executed and an evaluation of the re-engagement lesson is written.

Mathematical Knowledge in the edTPA

In the Mathematical Education of Teachers report (2001) by the Conference Board of the Mathematical Sciences, recommendations were set forth detailing the mathematics teachers should know as well as how they should come to know that mathematics.

The mathematical knowledge needed for teaching is quite different from that required by college students pursuing other mathematics-related professions. Prospective teachers need a solid understanding of mathematics so that they can teach it as a coherent, reasoned activity and communicate its elegance and power. (Conference Board of the Mathematical Science, 2001, p. IX)

With this in mind, in each edTPA where a mathematics component is evident, three mathematical abilities (conceptual understanding, procedural knowledge, and problem solving/ reasoning) are featured.

Conceptual understanding refers to “an integrated and functional grasp of mathematical ideas” (National Research Council, 2001, p. 118). Students who know more than isolated facts and methods, who understand why a mathematical idea is important, and who are able to represent mathematical situations in different ways are said to have conceptual understanding. Procedural knowledge refers to “knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently” (National Research Council, 2001, p. 121). Problem solving/reasoning refers to applying mathematics to real world situations. It encompasses routine, commonplace, and non-routine problems as well as the strategies, routine and non-routine, used to solve them.

The mathematical abilities are interwoven and no strand is considered more important than another.

As a child gains conceptual understanding, computational procedures are remembered better and used more flexibly to solve new problems. In turn, as a procedure becomes more automatic, the child is enabled to think about other aspects of a problem and to tackle new kinds of problems, which leads to new understanding. (National Research Council, 2001, p.134)

These mathematical abilities have been supported by research and theory in cognitive science reflecting the finding that “learning with understanding is more powerful than simply memorizing because the organization improves retention, promotes fluency, and facilitates learning related material” (National Research Council, 2001, p.118). In addition, “having a deep understanding requires that learners connect pieces of knowledge, and that connection in turn is a key factor in whether they can use what they know productively in solving problems” (National Research Council, 2001, p.118).

The edTPA requires mathematics candidates to make explicit use of these mathematical abilities in their planning, instructing and assessing. Not only do these candidates need to understand the nuances and connectivity of these mathematical abilities for themselves, they must know how they can use them in their classrooms and provide evidence that they are, in fact, using these mathematical abilities. This integration of conceptual understanding, procedural fluency, and problem solving/reasoning, the focus of the guiding questions in all the mathematics components of the edTPAs, makes the mathematics edTPAs different from the other edTPAs offered and opens the doorway to mathematics teacher educators to confront the inexperience many prospective mathematics candidates have in this area.

As stated in the Mathematical Education of Teachers report, “for many prospective teachers, learning mathematics has meant only learning its procedures and, they may, in fact, have been rewarded with high grades in mathematics for their fluency in using procedures” (Conference Board of the Mathematical Science, 2001, p.11). An opportunity is presented in the edTPA to have mathematics candidates abandon “their well-established beliefs about mathematics and expectations for mathematics instruction” (Conference Board of the Mathematical Science, 2001, p.10) and make planning, instructing and assessing decisions based on the integrated mathematical abilities. Candidates also look for evidence of these strands of mathematical abilities in their students’ work in an effort to analyze their impact on their students’ learning of these mathematical abilities.

This complex assessment of their teaching and student learning has a positive impact. According to Darling-Hammond (2010), “The requirement that beginning teachers evaluate student learning daily to adjust their plans and to evaluate student learning growth changes their understanding of teaching and their practice” (p.14). Unfortunately, this is something that many experienced teachers never learned to do.

The SCALE handbook Making Good Choices reminds teacher candidates that they are expected to analyze students’ thinking and learning—not just whether they know a set of facts or vocabulary terms. The document also states “keep in mind that you learn less about what your students are thinking and learning from multiple-choice questions or single-word response question than from open-ended questions, writing samples, performance tasks, projects, problem sets, lab reports or other more complex assessments” (SCALE 2014, 25). When candidates think about not only what to assess, but how to assess to gain insight into thinking, learning for students should improve. Improving candidates’ abilities to teach-assess and teach again with changes in instruction should serve to change the way students learn and understand mathematics.

Preparation for edTPA

Preparation for edTPA is an integral part of teacher education because of the authenticity of the tasks candidates complete. edTPA is an inquiry process where candidates’ practice is examined on a level different from faculty members’ idiosyncratic course assessments (Whittaker & Nelson, 2013). As teacher education programs strive to support candidate performance on edTPA they are thus improving their programs.

Candidate performance on edTPA will be used in accreditation reports for institutions of higher education and for comparison among teacher preparation programs. Curriculum mapping, selection of course materials, and field experiences that build from observation, to co-teaching, to the independent teaching illustrated in edTPA are all important undertakings for teacher preparation programs. Institutions of higher education are working to develop strategies that may assist their teacher preparation programs. These strategies include the development of parallel tasks, development of academic language, and development of relationships with partner schools around the edTPA.

Development of tasks which are parallel to the official edTPA tasks but which can be completed in methods courses with opportunities for questions and feedback (not allowed on an official edTPA assessment) is important as these parallel tasks provide elements of practice in the various tasks needed for completion of edTPA. These tasks can be developed to provide a critical aspect of teacher education: assessing student learning. It is critical that teacher education moves away from just the consideration of candidate performance in the implementation of a lesson, and examines, and helps the candidate to examine, the impact on student learning from the implementation of that lesson. It is also essential that mathematics teacher candidates know and understand the mathematics they teach, choose assessments that measure understanding, and interpret student work. “Obtaining evidence about understanding and reasoning requires the use of tasks and methods designed for that purpose” (National Council of Teachers of Mathematics, 2014, p. 92).

Development of academic language is a second strategy that can be helpful in successful preparation of candidates. This includes not only appropriate uses of the language of the content area but also the language of the profession of education. Academic language can be integrated in the parallel tasks used but it should also be developed throughout the candidates’ experiences in their teacher preparation programs.

The development of relationships with partner schools is, perhaps, the most challenging strategy to implement as there are many facets to consider. One way to begin the conversation with mentor teachers in P-12 schools is to help them to see how edTPA aligns with the annual professional performance reviews they complete. Another way is to help mentor teachers understand the guidelines for support they are able to provide to the teacher candidates.

Surveys of teacher candidates, teacher educators, and mentor teachers can provide an institution with necessary feedback for improving performance. Mentor teachers from partner schools, according to surveys, were particularly helpful with developing candidates’ abilities to identify central foci, create assessments, and analyze student work. Continued review of the data from the edTPA and surveys completed by candidates and mentor teachers should improve teacher preparation programs. Studies have found that the review of data from performance assessments assists institutions in data driven decision making, in developing a shared language of practice in education, in motivation to make specific programmatic changes, and in developing the profession. This shared and concrete knowledge of practice is crucial to change (Peck, Singer-Gabella, Sloan, & Lin, 2014).

The educative nature of edTPA pushes candidates beyond their comfort zone. One of the greatest advancements in teacher candidates’ learning stems from the realization that assessment of student knowledge can be found in various forms. Teacher candidates look to support materials to draw reasonable conclusions about students’ struggles and how to improve their teaching to have the best impact on student learning (McCarthy, et al. 2014). Currently, however, exemplars of the different components of the edTPA are not readily available to candidates. Institutes of higher education are encouraged to use candidates’ submitted portfolios in “local evaluation” of the edTPA. Faculty may choose exemplars when they compare their local evaluation scores to the “official” scores of the edTPAs. However, as the edTPA is so new, not many faculty have been comfortable with the selection of these and are calling for SCALE to release some official exemplars.

It is thought that the edTPA may also help teacher education programs and school districts determine which candidates will be most successful in improving student performance on standardized tests. edTPA is based on the Performance Assessment for California Teachers (PACT). The PACT has been analyzed for its predictive validity. That is how candidates score on the PACT relates to later teaching success and student learning as evidenced by scores on the California achievement tests. Because the PACT has been around for a longer period of time than edTPA, predictive data is available (PACT, 2015). Darling-Hammond, Newton and Wei (2013) found the relationship between candidate PACT scores and student learning gains was substantial. “Students taught by a teacher at the top of the scale (44) scored, on average, 20 percentage points higher than those taught by a teacher receiving the lowest passing score (24), controlling for their prior year scores and demographic characteristics” (Darling-Hammond, Newton, & Wei, 2013 p. 185). Noting this predictive nature of PACT is valid and that edTPA is based on PACT, it is likely that edTPA will be a valid predictor of candidate success in the teaching profession as determined by student test scores and student achievement.

The edTPA is also based on the requirements of The National Board for Professional Teaching Standards. This process, like the PACT, serves to evaluate and predict teacher quality. Cavalluzzo (2004) found “robust evidence” that National Board Certification is an effective indicator of teacher quality. The study used data from a large urban school district and focused on ninth and tenth grade mathematics. She looked at the association between a National Board Certified teacher and student gains in mathematics. She concludes that National Board Certification is effective in identifying quality teachers and can be effective in distinguishing among applicants. When one considers this study it can be concluded that edTPA may be effective in distinguishing between those who should enter the teaching profession and those who should not or not enter yet.

COMPLEX ASSESSING

Assessing is the act of evaluating, measuring and documenting and, in education, the data attained from assessment is used to measure student learning and to inform teacher practice. Teachers assess students frequently, both formally and informally in an effort to use the data to form judgments about content and pedagogy as they assess. Teachers also self-assess; they evaluate, measure, document and reflect on their own performance as teachers. They consider the content they teach, the instructional strategies they use, and the methods they choose for assessment. Assessment in the field of education is multi-layered and complex. Preparing candidates for this complexity adds another layer to their preparation as educators.

Assessing also impacts teacher preparation programs, P-12 education programs, the local community and the society at large. Assessment can become unwieldy, but it is necessary. What and how we assess in education has changed over time. What we expect teachers and students to know and be able to do has also changed over time. Based on assessments, progress is made in better preparing students for college and career as well as preparing teachers to lead their students to learning. All of these layers make for complex assessment.

Teacher preparation and assessment of student learning in mathematics has been studied, discussed and analyzed a great deal over the years. Mathematics is considered of principle importance to our economy, our national security, and our development as a society, is important in the sciences and in the arts, and requires logical thinking and problem solving which are essential skills for survival. The curriculum and pedagogy used to teach mathematics are often probed, textbooks and programs are reviewed and the preparation of mathematics teachers is reviewed and analyzed. This happens not just within the profession of education, but from outside as well. Many voices can be heard in the evaluation of mathematics content, pedagogy, and teacher preparation as yet another layer of complex assessing.

edTPA’s Contributions

The edTPA is a prime example of complex assessing. It serves the roles of assessing student learning in mathematics, assessing teacher candidates’ abilities to plan, instruct and assess in mathematics, and assessing teacher preparation programs. Much of the edTPA focuses on the candidate’s attention to student learning. In planning, the candidate selects worthwhile learning goals, instructional strategies, and appropriate assessments. In instructing, the candidate uses formative assessment to gauge student understanding and in assessing, the candidate determines the impact of his/her teaching on the whole class as well as on individual learners. The candidate determines the next steps based upon these results. This attention to student learning illustrates the connection among the three tasks focused around assessment. In the edTPA field test, candidates performed most highly on the planning task, followed by the instructing task and then the assessing task. “This conforms to other studies that have found that learning to evaluate and respond to students’ learning is one of the more challenging elements of teaching” (SCALE, 2014, p.2).

Assessing student learning and analyzing student errors are important practices of mathematics educators. Through the edTPA teacher candidates must demonstrate that they know, understand and exhibit the ability to plan mathematics lessons that optimize student learning with a focus on conceptual understandings, problem-solving and/or procedural fluency. They must show that they can instruct students in mathematics and that they can assess the success of that instruction. In some edTPAs they must show how they use the assessment to inform the next lesson. Deborah Ball (2011) has identified high leverage practices in mathematics education; practices that have the most potential for student learning. Included in her list are these that are demonstrated on the edTPA: choosing and using mathematical tasks that entail complex mathematical work, and choosing appropriate examples are in the planning stage, teaching and using academic language are shown in the instructing stage, and diagnosing common patterns of student thinking, and assessing students’ mathematical proficiency are in the assessment stage. While Ball makes the case that these practices can lead to mathematics learning for all, edTPA aims to assess by having a candidate demonstrate these practices. Demonstrating these practices should illustrate that a candidate will be an effective teacher. Preparing for edTPA would include practice in those behaviors that have been identified as effective for mathematics teaching and learning.

According to Chung (2008) candidates seem to have learned from completing the edTPA. For example, they learned about their students and their students’ learning needs. They planned and implemented a sequence of connected lessons, assessed student learning and modified their instruction based on the assessments. Teacher candidates thought about teaching in new ways and were able to enact some of these ideas in their practice. National Board Certification is a process similar to edTPA and has also been shown to facilitate teacher learning. Lustick and Sykes (2006) studied science teachers engaged in the National Board Certification Process. They found that teachers gained in knowledge and understanding of science instruction especially in the areas of scientific inquiry and assessment. They conclude that the process of pursuing board certification is professional development. Engaging in an edTPA in the area of mathematics assists teacher candidates in their understanding of instruction and assessment. It helps them build the teaching strategies and practices to be highly effective teachers. It helps them to reflect on their practice and on student learning.

The implementation of edTPA also serves as program assessment for teacher preparation programs. Through close review of candidates’ performance, faculty in higher education have the opportunity to revise courses and assignments for more relevance and rigor. Peck et.al. (2014) found that the edTPA provides more specific data and teacher educators can therefore make more specific revisions to courses. The edTPA leads to data driven decision-making in teacher education.

The edTPA contributes to learning, understanding, and collaborating among teacher candidates, higher education faculty, supervisors, and mentor teachers. Performance assessment requires teacher candidates to illustrate theories and concepts they are learning in courses as they perform in a clinical setting. “It may be the first and only time in a program that candidates and their instructors can see whether they indeed understand and can apply what they are supposed to be learning” (Darling-Hammond, 2010, p. 18-19) The use of the edTPA contributes to the development of the profession through shared language of practice (Peck, Singer-Gabella, Sloan, & Lin, 2014). As we assess the performance of teacher candidates through the edTPA we also assess the programs in which the candidates were instructed and review the candidates’ assessment of student learning in mathematics. This multilayered complex assessment should improve the teaching and learning of mathematics.

edTPA’s Concerns

The use of edTPA is not without challenges, however. Authentic assessment requires time and money to implement and sustain. Conducting research that illustrates that candidates taught in programs that support authentic teacher performance assessments like edTPA are better prepared than candidates who have not is challenging because of the number of variables that must be considered (Darling-Hammond & Snyder, 2000).

Performance assessment implementation as a top-down state mandate in states such as New York and California has caused many dissenting opinions on these assessments. While most still agree there should be a performance assessment, the politics have caused delays in its consequential operation and has challenged buy-in from institutions of higher education. Implementation was meant to occur in a slow, thoughtful, purposeful manner. California has at least ten years in the development and roll out of edTPA as an assessment. New York had a rapid roll out with only one year of field testing before edTPA became a high stakes certification exam. Faculty in higher education had little time to reflect on the data and prepare their candidates accordingly creating more opportunities for tension and push-back.

There is concern about the evaluation of a candidate’s edTPA by someone who does not know the candidate, the school community, or the population of the class. There are issues with inter-rater reliability. There is no way to monitor that the edTPA has in fact been completed by the candidates themselves. This filters a mixed message to teacher candidates and adds to their level of stress about the assessment. Other challenges include the cost of the assessment and dissatisfaction with a publishing company’s monopoly on assessment. As a certification exam edTPA can cost around $300 to complete. There is also concern about failures of the exam and how retakes may be completed and scored. Currently candidates can retake any one of the tasks or retake the entire edTPA (all retakes require additional fees).

Many candidates are expressing concern with technical issues such as videotaping, formatting, scanning, and uploading their portfolios. Greenblatt and O’Hara (2015) expressed concern that the cost of renting digital equipment is an additional financial burden for candidates who attend institutions that lack the funding to help candidates with technological needs such as videotaping, video compression and uploading. Inequities may also be found in the resources available at institution of higher education. Some schools are able to provide extensive supports (workshops, seminars, individual consultations with edTPA coordinators) in the preparation of teacher candidates while others lack the necessary resources and provide minimal supports (online modules) that provide only general information (Au, 2013, Greenblatt & O’Hara, 2015).

There is concern about the lack of time candidates have during their student teaching experience to keep up with the amount of writing that is part of the assessment. In addition, teaching candidates are finding district-mandated programs, cooperating teacher expectations and testing schedules difficult to maneuver successfully while working on this assessment. The edTPA, according to Greenblatt and O’Hara (2015), has impacted student teaching in many negative ways including changing the focus from preparation for the first year of teaching to passing this test. Decisions about placement of candidates for student teaching now include thoughts on how the test will be implemented in the placement. Many schools and teachers are not well informed about edTPA. Greenblatt and O’Hara (2015) discussed the challenges of finding placements for student teachers in districts with mandated curriculums because mentor teachers may not be willing, or allowed, to deviate from what is expected.

Another challenge that is seen with performance assessment stems from the weak research that is presently available on performance assessment.

One of the weaknesses of previous research on TPAs is that the impact of the assessment cannot be easily disentangled from the multiple sources of teacher learning in preservice programs, such as coursework, field and practicum experiences, mentorship, and supervision. Furthermore, there is little evidence that preservice teachers actually enact what they report learning in their teaching practice as a consequence of completing a TPA because of the lack of observational data corroborating the impact of such assessments on teacher practice. (Chung, 2008, p.9)

The edTPA is high stakes test for teacher candidates. The reading, writing and digital literacy expectations, the cost in money and the loss of learning in student teaching are all burdens of implementing edTPA. This complex assessment is not without criticisms, yet it offers the best we currently have to truly evaluate teacher candidates.

CONCLUSION

Performance assessments are the next step in the journey to improve teaching and learning especially in mathematics and to professionalize teaching. edTPA with its authentic tasks attempts to assess the complexities of the teaching and learning process. Candidates must illustrate their understandings of the complexities of teaching and learning with an emphasis on effective instruction based on assessments and learning for all students. Mathematical content knowledge is also assessed as candidates demonstrate teaching in the three interwoven areas of mathematics: conceptual understanding, procedural fluency, mathematical reasoning/problem solving. Teacher preparation programs can use the data of edTPA to assess their own programs as well. Assessment of student learning in candidate prepared lessons, assessment of teacher candidates’ competency through performance in a teaching segment and assessment of teacher preparation programs are all layers of complex assessment necessary for preparing effective mathematics teachers who are part of the teaching profession. The edTPA serves as a vehicle for meeting the multiple challenges of teacher preparation, teaching and learning mathematics.

This research was previously published in the Handbook of Research on Professional Development for Quality Teaching and Learning edited by Teresa Petty, Amy Good, and S. Michael Putman, pages 182-201, copyright year 2016 by Information Science Reference (an imprint of IGI Global).

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