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This study is designed to determine overall differences in quality that middle school mathematics teachers experience when teaching outside of their field (math). By combining subject content knowledge and pedagogical content knowledge scores for teachers both in and out of their field a more direct understanding of the difference in quality was revealed. Â
The topics of quality teaching, subject content knowledge, and pedagogical content knowledge have been reasonably well researched within the field of mathematics education. A review of the literature shows the deep connections between quality teachers and the primary knowledge domains of both subject and pedagogical content knowledge. Commonalities among the literature include the notion of specific teacher knowledge domains, such as subject content knowledge and pedagogical content knowledge. More simplistically, these knowledge domains represent both the teachers' encyclopedic understanding of the subject matter itself as well as the more intuitive perception of presenting such ideas in a clear, concise, and accessible manner to the majority of the students. The literature to be discussed can be applied to the current research as well as in support of the proposed methods and analysis.
Teachers in high poverty, high minority schools are more likely to be less experienced, less educated, teaching on emergency permits or waivers, and teaching subjects for which they are not qualified (Carroll, Reichardt & Guarino, 2000; Darling-Hammond, 2002; Goe, 2002; Hanushek, Kain, O'Brien, & Rivkin, 2005; Ingersoll, 2002; Lankford, Loeb, & Wyckoff, 2002; Marvel, Lyter, Peltola, Strizek, & Morton, 2007; Peske & Haycock, 2006; Scafidi, Sjoquist, & Stinebrickner, 2007; Useem & Farly, 2004). Studies on teacher attrition and mobility rates are generally in agreement that teachers are more likely to leave districts that have higher concentrations of poor, minority, and/or low performing students, lower salaries and less favorable working conditions (Hanushek, Kain, O'Brien, & Rivkin, 2005; Ingersoll, 2002; Lankford et al., 2002; Luallen, 2006; Scafidi et al., 2007). As a result, high-poverty schools are more likely to have teaching vacancies in many subjects, with mathematics being the most common (Strizek, Pittsonberger, Riordan, Lyter, & Orlofsky, 2006), and much more likely to staff classrooms with out-of-field, inexperienced and less-prepared teachers. (Ingersoll, 1999; Mayer, Mullens, & Moore, 2002).
The US Department of Education reported in 2000 that by 2010 there would large vacancies in the teaching job market: elementary - 34,422, secondary (including middle school) - 146,041, and special education (all levels) - 13,564. US Census Bureau (2012) has indicated that there will be in fact deficit of nearly half million teachers by 2018. Teacher attrition rates and retention problems (Ingersoll, 2002) compounded the issue of out-of-field teachers, increasing the strain of meeting the requirements of placing a highly qualified teacher in every classroom (NCLB, 2001). In President Bush's 2006 State of the Union Address he pledged to create an additional 30,000 new mathematics and science teachers to correct for these shortages (Bush, 2006). Interestingly, in 2009 Ingersoll concluded that the teacher shortages were no longer the leading cause of the lack of high quality teachers but rather it was due to "widespread school staffing" and management problems (p. 35). Â
Although teacher shortages may no longer be debated the issue of highly qualified teacher shortages remains (National Academy of Sciences, 2007; U.S. Department of Education, 2009). Sanders (2004) concluded that 57% of middle school students were taught by a teacher without even a minor in a related field. Furthermore, 48% of middle school students in physical science also were taught by a teacher lacking even a minor in a related field. More recently, a study by Schools and Trust (2008) found that teacher mis-assignments totaled 27% of the core courses in the nation's high-poverty schools. Mis-assignment is the assigning of a certified teacher to teach in a content area that he or she does not have an endorsement or major, and thus has insufficient content mastery. Alternatively, these teachers may be considered partially out-of field. Finally out-of-field assignments are today still quite common. In each of the six years of data collection, anywhere from 57% to 74% of math teachers, 16% to 31% of social studies teachers, and 38% to 48% of science teachers lacked a major in the field they were teaching. Out-of-field assignments were most prevalent in the first one or two years of respondents' careers (Donaldson & Johnson, 2010).
A consistent finding in the research literature is that teachers are important for student learning and that there is great variation in effectiveness across teachers (Aaronson, Barrow & Sander, 2003; Â Kane, Rockoff & Staiger, 2006; Hanushek, Kain, O'Brien, & Rivkin, 2005; Rockoff, 2004; Sanders & Rivers, 1996).
Quality teachers are defined as those that are effective, knowledgeable, skilled, in-field, and certified (Au, 2004). The need for students to be taught by quality teachers is very apparent, so it is important to look at what the effect might be on students who are taught by out-of-field teachers, a status that may negatively affect the quality of learning.
The National Middle School Association (NMSA) defines highly qualified as "teachers who demonstrate proficiency in pedagogical knowledge, skills, dispositions, classroom management and overall effective teaching practices as well as content knowledge" (Thornton, 2004, p. 6).
Kane, Rockoff and Staiger (2006) estimate that the difference in effectiveness between the top and bottom quartile of teachers results in a 0.33 standard deviation difference in student gains over the course of a school year.
In education, student performances have always been the bottom line. However, it seems that it is not just the students who are losing academic ground. "The knowledge gap between the US and Chinese teachers parallel the learning gap between US and Chinese students" (French, 2003). Ma's research was comparative in nature and set out to discover if the elementary and middle school teachers in two countries, the US and China, had unique bodies of pedagogical content knowledge. She found that although the training of the teachers in the United States is more rigorous than those taught in China, the Chinese teachers continue to out-perform the US teachers in both pedagogical content knowledge as well as subject knowledge (Ma, 1999).
Ma further noted that, "â€¦ the United States [has been unsupportive of] the development of elementary teachers' mathematical knowledge and its organization for teaching" (Ma, 1999, p. xxiv). Her study suggests that US math instructors are six times more likely, while conducting a lesson, to teach the procedural understanding only, than are Chinese teachers. While Chinese teachers design more conceptually directed lessons, the US focuses almost entirely on procedural lessons. This assertion helps solidify similar findings by Stigler and Hiebert, the authors of The Teaching Gap (1999).
The data from Stigler and Hiebert (1999), Conference Board of the Mathematical Sciences (2001), Ma (1999) and the RAND Mathematics Study Panel (2003) state quite clearly that many teachers struggle with concepts in mathematics. Stigler, Hiebert, and Ma interviewed teachers and at some point asked the teachers to correct some students work. Teachers were given some problems to correct and in one case a student divided both sides of an equation by zero. Many K-8 teachers missed this entirely. Other teachers who caught the mistake were unable to explain why dividing by zero was never appropriate (Conference Board of the Mathematical Sciences, 2001; Ma, 1999; RAND Mathematics Study Panel, 2003; Stigler & Hiebert, 2000).
Researchers, however, are not just pointing their fingers at the teachers, but also at the educational programs from which they were certified (McDonough, 2003). A report entitled, To Touch the Future, by the American Council on Education (McDonough, 2003), places the responsibility of generating quality teachers squarely on the shoulders of university presidents. Their first and foremost finding was that, "The success of the student depends mostly on the quality of the teacher" (McDonough, 2003, p. 5). Interestingly, while searching for patterns between quality teachers and the certification programs from which the teachers emerged, ACE concluded that there was little to no pattern found (McDonough, 2003).
In search of a solution to this ever present problem of low teacher quality, the RAND Mathematics Study Panel (2003) set forth a goal for the education community as to the design and direction of developing strong teacher certification programs.
In contrast to this, RAND (2003) also mentions that professional development programs in the past have been overall, ineffective. Increasing someone's subject matter knowledge through professional development is one thing; teaching someone how to teach that material, in a way that truly affects their teaching style and is accessible by the majority of students, is perhaps something entirely different.
A great number of researchers have shown that the quality of the teacher in the classroom is perhaps the most significant factor predicting student achievement (Ferguson 1998; Goldhaber, 1999, 2002; Hanushek, 1999; Â Hattie, 2008). Hanushek (1992) went so far as to show that a student can achieve a learning gain of 1.5 grade levels higher if they were given a high-quality teacher as opposed to a student who achieved a gain of only 0.5 grade level given a low-quality teacher. Hanushek showed that, "the quality of a teacher can make the difference of a full year's learning growth" (1999, p. 8).
Subject knowledge and pedagogical knowledge are two very different concepts. Knowing one's subject well does not necessarily guarantee his or her ability to aptly explain, to a group of students, the true nature of the subject in ways that are comprehendible by the masses. In 1985, Lee Shulman and colleagues introduced the phrase "pedagogical content knowledge" to the teaching community (as cited in RAND, 2003 p.54). Since then research has tried to establish which teachers have this knowledge, how many are in need of this knowledge, and what are the best ways to get it to them (McDonough, 2003; French, 2003; RAND, 2003). Passion, zeal, enthusiasm, whatever one calls it, is the essence of quality teaching that researchers and certification institutions intend to transfer from the haves to the have-nots, in a clear and measurable fashion. Desperate as they are, to begin manufacturing high quality mathematics teachers, some admit quite forwardly, that they have no answers. As the RAND Mathematics Study Panel points out (RAND, 2003, p. 20):
We know little about what teachers need to understand specifically within these areas. We do not know much about how teachers need to be able to get inside mathematical ideas to make them accessible to students. And we do not know what they need to know of the mathematics that lies ahead of them.
Subject Content Knowledge
Although there is considerable agreement among researchers that subject content knowledge contributes strongly to teacher quality, disagreement remains about the specific knowledge that best characterizes this quality. A study by Greenberg, Rhodes, Ye, and Stancavage (2004) supports the notion that subject matter competency and certification are key components of teacher quality. Hill (2005), however, claims that the level and content of the subject-matter necessary is unspecified.
In 2008, Ozden studied pre-service teachers and the impact that a valued-added approach to lesson writing coupled with a content-knowledge exam had on effective teaching. The results showed a strong positive correlation (Ozden, 2008, p. 639). Although value-added approaches are not in dispute, Hill and Ball (2009) found that degrees attained and courses taken have contributed to student achievement. More direct was Ahtee and Johnston (2006) who clearly showed that a lack in subject knowledge can lead to teaching difficulties. Many researchers agree that content knowledge is important to teach, but there is disagreement as to what extent (Cavanaugh, 2009).
John Dewey (1904, p. 33) stated that subject matter often represents "the knowledge of how-to-do." This was intended to represent what he felt to be a balance between teaching abstract reasoning (formal subject matter) and practical everyday knowledge (informal subject matter) (Dewey, 1904). Since Dewey, there has been a large focus on the subject content knowledge of the teacher in an ever challenging attempt to understand the nature of a quality teacher and student achievement (Andrews, Blackmon, & Mackey, 1980; Haney, Madaus, & Kreitzer, 1987; Kilpatrick, Swafford, & Findell, 2001; Mooney, Fletcher, & Jones, 2003; Rowland, Martyn, Barber, & Heal, 2000; Schalock, 1979; Soar, Medley, & Coker, 1983).
Research regarding the weakness of US teachers' mathematical fluency has both educators and policymakers highly concerned over the impact that this may have on the achievement and global competitiveness of future generations (Ball, Lubienski, & Mewborn, 2001; Ma, 1999). Concern about teachers' subject matter knowledge is also evident in the action taken by the US government through the No Child Left Behind (NCLB) Act. The NCLB Act has recommended that teachers should pass tests in subject matter they teach or complete a college major in the subject they teach in order to demonstrate their subject matter competency. Teachers' subject matter competency has become a central requirement of what makes a "highly qualified" teacher.
Research specific to the measurement of the mathematical knowledge needed to be a highly effective teacher is admittedly scarce. The Final Report of the National Mathematics Advisory Panel (2008), however, has stated that test items specifically designed to measure the mathematical content knowledge may be the best indicator of high subject content knowledge.
Teaching mathematics effectively so as to increase student achievement has always proven to be a multifaceted task. Concentrating on teacher content knowledge typifies the traditional research perspective (An, 2000, 2004; Cooney & Shealy, 1997; Fennema & Franke, 1992; Ma, 1999). Ma goes so far as to state that, "mathematics teachers have a profound mathematical content knowledge" (p. 5). This specialized math knowledge was also recognized by Hill, Rowan, and Ball (2005).
Subject content knowledge is a dramatically large component of what makes a "highly qualified" teacher. Subject content knowledge has been directly (and positively) linked to student achievement in mathematics education (Ahtee and Johnston, 2006; Hill and Ball, 2009; Mandeville & Liu, 1997; Ozden, 2008; Rowan, Chiang, & Miller, 1997).
Goldhaber and Brewer (1999) used data from the National Educational Longitudinal Study (NELS) of 1998 and found that among 24,000 eighth grader students of teachers with either a bachelor's or master's degree in mathematics, the students outperformed those with teachers that lacked this subject content "expertise in math" (Goldhaber & Brewer, 1999, p. 32).
Wenglinsky (2000) employed a meta-analysis on 39,140 prospective teachers who took the Praxis II examinations for teacher licensure and found that prospective teachers whose teacher preparation programs deemphasized educational theory and spent more energy on subject matter performed better than those prospective teachers whom emphasized educational theory more than subject matter.
Mathematics teachers, in particular, must have a clear and fundamental knowledge of multiple issues in the field of mathematics: concepts of functions, number theory, the fundamental theorems of algebra and calculus as well as other geometric topics such as perimeter, area, and volume. Mathematics teachers with a reasonable level of subject content knowledge understand the importance of a particular theorem, postulate, or proposition and should be capable of articulating its logic and purpose.
A reasonable level of subject content knowledge implies that a mathematics teacher is capable of understanding and articulating the multitude of mathematical representations (i.e. tables, graphs, charts, formulae, charts) as well as the relationship between them. Concepts, such as, dimensional analysis (or unit analysis) are deeply understood as a connective tissue between pure abstract reasoning in mathematical computations and real world measurements.
Pedagogical Content Knowledge
According to Rodgers and Raider-Roth (2006), "Many teachers are knowledgeable of their subject matter without necessarily being able to decompress it in a way that makes it accessible to their students" (p. 280). Hill, Blunk, Charalambous, Lewis, Phelps, Sleep, and Ball (2008) performed a qualitative videotape study designed to flesh out the relationship between teachers' Mathematical Knowledge for Teaching (MKT) and the Mathematical Quality of Instruction (MQI). This study essentially unveiled the subtle relationship between subject content knowledge and pedagogical content knowledge in teaching math. A strong correlation was found between Mathematical Knowledge for Teaching and Mathematical Quality of Instruction. Hill stated that, "a powerful relationship between what a teacher knows, how she knows it, and what she can do in the context of instruction" (p. 496). Hill, Ball, and Schilling (2008) also discovered that Knowledge of the Content and Student (KCS) was also directly associated with Mathematical Knowledge for Teaching.
Subject matter knowledge and pedagogy were often considered two distinct knowledge domains up until 1986. Lee Schulman (1986), while addressing this unique dichotomy within the field of teaching, introduced the notion of pedagogical content knowledge. Pedagogical content knowledge is in some sense the connective tissue that links pedagogy to content specific knowledge. Furthermore, it represents a teacher's ability to address curricular topics with the most effective pedagogical approach so as to maximize the teacher-learner experience.
Pedagogical content knowledge "represents a class of knowledge that is central to teachers' work and that would not typically be held by non-teaching subject matter experts or by teachers who know little of that subject" (Marks 1990, p. 9).
Once Shulman (1986) introduced the concept of pedagogical content knowledge various educational theorists attempted to describe its relative importance or association with other knowledge domains. Several of these theorists treat pedagogical content knowledge as if it was an integral part of the overall teaching knowledge structure, but its hierarchical status was left unstated (Bamett & Hodson, 2001). Turner-Bisset considered pedagogical content knowledge as the dominant or central component of a teacher's knowledge structure (Turner-Bisset, 2001). Enfield (2000) thought that pedagogical content knowledge is one of many necessary but considerably smaller components. Veal proposed that pedagogical content knowledge was only obtained by way of considerable foundation of subject content knowledge (Veal & MaKinster, 1999). There are also studies showing that pedagogical content knowledge may in fact be more important than subject content knowledge itself. Belden, Russonello, and Stewart, by means of surveying professors who provide graduate level training for teachers, found that they (the professors) considered acquiring pedagogical content knowledge to be more critical for emerging teachers than their command of the subject matter (Belden, Russonello, & Stewart, 1998).
Practically speaking, pedagogical content knowledge consists of the most useful forms of representations and ideas, such as illustrations, examples and explanations as well as the ways of representing and formulating the concept of function to make it comprehensible to others. Furthermore, pedagogical content knowledge also involves knowing and recognizing what makes the learning of specific topics, such as geometric area, easy or difficult and the conceptions and preconceptions that students bring with them to the learning of those most frequently taught topics leading up to and including the concept of area.
The one issue that has impeded the study of pedagogical content knowledge among teachers is that it can be difficult to measure. Studies surrounding subject content knowledge have benefited from knowing that a simple survey or questionnaire would be enough to accurately measure that particular knowledge domain. Ideally, multiple-choice tests for the study of pedagogical content knowledge would allow researchers to implement large scale studies on so that educators and policy makers could view pedagogical content knowledge from an entirely new and more statistically founded direction. Attempts were made.
Komrey and Renfrow (1991) used multiple-choice tests to measure content-specific pedagogical knowledge in mathematics. They began by creating a working definition of content-specific pedagogical knowledge and then developing items (questions) designed to assess this knowledge. Categories among these items included student error-analysis, communication between teacher and learner, organization of instructional material and the lesson, and learner characteristics and an understanding of developmental norms.
One of the next serious attempts to create a paper and pencil assessment for the measurement of pedagogical content knowledge was by the Teaching and Learning to Teach (TELT) study conducted at Michigan State University in 1993 (Kennedy, 1993). The researchers attempted to develop a survey that measured teachers' subject content knowledge as well astheir knowledge of effective teaching practices. In 2001, Rowan, Schilling, Ball, and Miller produced a research report entitled, Measuring Teachers' Pedagogical Content Knowledge in Surveys: An Exploratory Study. Their study used a modified Likert scale to measure either a teachers' content knowledge or knowledge of students' thinking (pedagogical content knowledge). Rowan, Schilling, Ball, and Miller concluded that, "particular facets of teachers' pedagogical content knowledge can be measured reliably" (Rowan, Schilling, Ball, & Miller, 2001, p. 18).
Subject content knowledge is easily quantifiable and therefore a well measured attribute of teachers (Boardman, Davis & Sanday, 1977; Ferguson 1991; Hanushek 1972; Hanushek, 1986; Harbison & Hanushek, 1992; Hauk, Jackson, & Noblet, 2010; Murnane, Singer, Willett, Kemple, & Olsen, 1991; Rowan, Chiang & Miller, 1997; Strauss & Sawyer, 1986; Saderholm, Â Ronau, Brown, & Collins, 2010; Tatto, Neilsen, Cummings, Kularatna & Dharmadasa, 1993; Wayne & Youngs, 2003). Although researchers have struggled with the measurability of pedagogical content knowledge they have found clear and acceptable techniques (Ball & Bass, 2000, 2003; Brown, McGatha, & Karp, 2006; Rowan, Schilling, Ball, & Miller, 2001; Saderholm, Ronau, Brown, & Collins, 2010). The blending of both subject content knowledge and pedagogical content knowledge represents the essence of quality teaching that researchers and professional developers intend to transfer from the haves to the have-nots, in a clear and measurable fashion. It has been recognized that a teacher can have high degree of subject content knowledge and little to no pedagogical content knowledge - the proverbial brilliant professor who is unable to explain the nature of the subject in terms palatable to his/her students (Ball, 1990; Ma, 1999; Ball, Hill & Schilling, 2004). Furthermore, it has been observed by Ball (1990), Shulman (1986), Wilson, Shulman and Richert (1987), that pedagogical content knowledge is somewhat dependent upon ones subject content knowledge; that is to say, one cannot teach a topic in a clever manner, that which, they themselves are unfamiliar.
Even (2000), from the Weizmann Institute of Science in Rehovot, Israel first decided to study the relationship between subject content knowledge and pedagogical content knowledge of secondary mathematics teachers in 2000. Even studied secondary teachers' fluency surrounding the concept of function. The findings showed that teachers with poor subject matter knowledge of functions were more likely to use insufficient terminology and limited imaging while attempting to explain the concept of functions to students.
Haciomeroglu (2006) performed a study with similar goals to the Even's research in that it was an investigation of subject content knowledge and pedagogical content knowledge of secondary mathematics teachers surrounding the concept of function. Haciomeroglu, however, used the Wilson, Shulman and Richert (1987) task-based model approach to gain insight into prospective secondary mathematics teacher's knowledge of the concept of functions. Haciomeroglu (2006) found that the participants that had both a weak subject matter knowledge and pedagogical content knowledge inhibited the teacher's ability to account student misconceptions and other sources of incorrect solutions.
Ball is a strong advocate for research involving the nature of teaching mathematics. In 2001, Ball, Lubienski, and Mewborn from the University of Michigan set out to develop a survey-based measure of pedagogical content knowledge in both reading and mathematics. The study targeted elementary level teachers for which the (mathematics) survey items were primarily composed of various number concepts (division by zero, place-values, etc.). The results to their study showed that there are "particular facets of teachers' pedagogical content knowledge can be measured reliably" (Rowan, Schilling, Ball, & Miller, 2001, p. 12).
Saderholm, Ronau, Brown, and Collins (2010) designed the Diagnostic Teacher Assessment in Mathematics and Science (DTAMS) that was used to measure both content knowledge and pedagogical content knowledge among middle school teachers. The population was 1,600 middle school mathematics teachers from 17 states. They began by establishing its validity and reliability by reviewing national standards for subject content and gathering questions from writing and reviewing teams composed of experts in the field of teaching and mathematics. Furthermore, teams composed of mathematicians, mathematics educators, and middle school teachers were used to develop prototype and parallel assessments. National reviewers then assessed the appropriateness of each of the items. Each reviewer was given four sets of assessments and was asked to identify the mathematical content of the item based specific list of topics, identify the knowledge type (I - Memorized Knowledge, II - Conceptual Understanding, III - Problem-Solving / Reasoning, and IV - Pedagogical Content Knowledge), and finally to indicate if the items represent important mathematics content for middle school teachers (High, Medium, or Low levels of importance). Each assessment was reviewed by at least six reviewers with at least one mathematician, one mathematics educator, and one middle school teacher.
It should be noted that experience in teaching has been shown to have a dramatic impact on student achievement especially within the first four-five years (Hanushek, 2005). There has been strong and consistent evidence that first year teachers produce student achievement gains that are significantly lower than otherwise similar teachers with ten to fifteen years of experience. After the first four years of experience gains in student achievement begin to level off in most cases (Kane, Rockoff, & Staiger, 2006; Rivkin, Hanushek & Kain, 2005; Rockoff, 2004).
Both subject content knowledge and pedagogical content knowledge have played major roles in past research in developing a better idea of what teacher quality specifically represents. The degree to which subject content knowledge is more or less important than pedagogical content knowledge as it relates to teacher quality depends primarily upon which study, researcher or expert is referenced. The paucity in the literature occurs when one attempts to determine the direct difference in teacher quality from a teaching within ones trained and chosen field and when one is outside those boundaries.
Chapter two is a review of much of the literature that supports the nature of the argument of this study. Teacher quality, subject content knowledge, pedagogical content knowledge, the effects of out-of-field teachers and research similar to this study were explored. Subject content knowledge, pedagogical content knowledge, and the proper certification and placement of teachers were shown to be highly influential in acquiring higher student achievement and therefore higher quality teaching. The research also indicates a substantial need to further understand the actual measured changes among in-field and out-of-field teachers, especially in the mathematics and science education community. Furthermore, the use of survey tools to establish these relationships and that its further use and development is a strongly desired contribution to the educational research community. The following chapter illustrates the methodology and establishes the procedural overview of this study.