Relationship between teacher knowledge of concepts and ...achievement in middle grades mathematics Mourat A. Tchoshanov Published online: 24 September 2010 ... mathematics coursework,

Relationship between teacher knowledge of conceptsand connections, teaching practice, and studentachievement in middle grades mathematics

Mourat A. Tchoshanov

Published online: 24 September 2010# Springer Science+Business Media B.V. 2010

Abstract The mixed method sequential nested study examines whether and how thecognitive type of teachers’ content knowledge is associated with student achievement, andcorrelated with teaching practice. In the context of this study, the cognitive type refers to thekind of teacher content knowledge and thinking processes required to accomplish a tasksuccessfully, in terms of knowledge of facts and procedures (Type 1), knowledge of con-cepts and connections (Type 2), and/or knowledge of models and generalizations (Type 3).A sample of 102 middle school mathematics teachers (grades 6–8) was assigned to thestudy from a number of school districts in an urban area in the Southwestern US. Teacherswere tested using the Teacher Content Knowledge Survey. Student level data of about 2,400middle grades students’ standardized test passing rates including percentage of studentsmeeting the state standards by objectives were collected. The type of teachers’ contentknowledge was assessed and tested for association with student achievement on the state-mandated standardized test using multivariate methods including tests for variance andindependence. The nested research consisted of three phases. Substudy-1 focused onquantitative analysis of the association between cognitive type of teacher contentknowledge and student achievement. Substudy-2 aimed at the correlation between cognitivetype of teacher content knowledge and teaching practice. Finally, substudy-3 was a casestudy on examining middle grades mathematics teachers’ knowledge and understanding offraction division.

Keywords Teacher content knowledge . Cognitive type of teacher knowledge .

Student achievement

1 Introduction

In the recent decade, a body of solid research regarding teacher knowledge was producedby international scholars (for example, see special issue of Educational Studies in

Educ Stud Math (2011) 76:141–164DOI 10.1007/s10649-010-9269-y

M. A. Tchoshanov (*)University of Texas at El Paso, EDU612, 500 W. University Avenue, El Paso, TX 79968, USAe-mail: [email protected]

Mathematics on Forms of Mathematical Knowledge, volume 38, 1999) (Tirosh, 1999).Some studies on teacher knowledge have a focus on particular countries. Studies conductedin the USA claim that American teachers lack essential knowledge for teachingmathematics (Ball, 1991; Stigler & Hiebert, 1999; Ma, 1999). It has also been wellreported that: teachers’ mathematical knowledge is important for teaching (Hill, Shilling, &Ball, 2004; Rowland, Huckstep, & Thwaites, 2005; Davis & Simmt, 2006), teacher’ssubject-matter knowledge grows through teaching (Mathematics Education Library, Vol.10, edited by Bishop, Mellin-Olsen, Van Dormolen 1991), teachers’ content knowledgeaffects student success (National Mathematics Advisory Panel, 2008a), and studentachievement parallels teacher knowledge (Tchoshanov, Lesser, & Salazar, 2008).

The US Task Group on Teachers and Teacher Education of the National MathematicsAdvisory Panel (2008b) conducted a comprehensive review of existing research on therelationship between teacher content knowledge and student achievement. The task groupidentified three different ways teacher knowledge was measured: type of teachercertification, mathematics coursework, and test of teacher content knowledge. The groupconcluded that, “Across all studies, the general results are mixed but overall do confirm theimportance of teachers’ content knowledge. However, because most studies have relied onproxies for teachers’ mathematical knowledge (such as teacher certification or coursestaken), existing research does not reveal the specific mathematical knowledge [emphasisadded] and instructional skill needed for effective teaching, especially at the elementary andmiddle school level” (National Mathematics Advisory Panel, Report of the Task Group onTeachers and Teacher Education, 2008b p. 8).

This study stemmed from a similar assumption on teacher content knowledge andstudent achievement. We believe that it does not only matter how much mathematicscoursework a teacher has, but rather, what specific type of mathematical knowledge ateacher possesses. In order to detect this knowledge, “More precise measures are needed tospecify in greater detail the relationship among... middle school teachers’ mathematicalknowledge... and students’ learning” (National Mathematics Advisory Panel, 2008b p. xxi).This study had the following main purpose—to measure the cognitive type of middlegrades teachers’ content knowledge and its association with students’ achievement in state-mandated standardized test. The study also looked at the relationship between the type ofteacher knowledge and teaching practice. Overall, it examined the following researchquestions: (a) what kind of teacher content knowledge is critical for student success? (b) towhat extent is the cognitive type of teachers’ knowledge associated with studentachievement? (c) how does the cognitive type of teacher knowledge correlate with teachingpractice and lesson quality? The study explores whether teachers’ knowledge of facts andprocedures (Type 1 knowledge) will have a different effect on students’ achievement,relative to knowledge of concepts and connections (Type 2 knowledge), or knowledge ofmodels and generalizations (Type 3 knowledge). Type 1 knowledge requires recall andapplication of basic mathematical facts, rules, and algorithms to perform routineprocedures. For example, if a teacher is able to recall a rule for fraction division or tosolve simple fraction division problem such as 1 3

4 � 12 ¼ , then we say that she has

procedural knowledge of fraction division. The notion of procedural knowledge is wellreported and discussed in works of Skemp (1978); Stein et al. (2000) and others. Type 2knowledge is quite different from type 1 knowledge in a sense that it focuses on conceptualunderstanding through increased quantity and quality of connections between mathematicalprocedures and ideas. For instance, tasks such as “Solve the following fraction divisionproblem 1 3

4 � 12 ¼ in more than one way (e.g., draw a diagram or illustrate it with

manipulatives)” or “Make up a story for the given fraction division problem” require

142 M.A. Tchoshanov

conceptual rather than only procedural knowledge. Skemp (1978) called it relationalunderstanding or conceptual knowledge. Stein et al. (2000) named such tasks as procedureswith connections. Davis (2008) used the construct of concept study to improve teachers’conceptual knowledge. Type 3 knowledge is more theoretical: it requires testingconjectures, making generalizations, proving theorems, etc. For example, more than a halfof middle grades teachers tested in the study had difficulties correctly responding to thefollowing question “Is the following statement: a

b � cd ¼ ac

bd (a, b, c, and d are positiveintegers) ever true?” Tasks such as this one require a different type of knowledge. We call itknowledge of models and generalizations. Doerfler (1991), Presmeg (1997) and otherscholars explored this type of knowledge and its forms as well as relationship betweengeneralization and different modes of representation. The cognitive types of teacherknowledge are not hierarchical in a sense that teacher should not necessarily possess theprevious type of knowledge before the next one. In our study, we had teachers with higherscores on Type 3 of content knowledge compare to Type 2 knowledge scores: the casestudy of two middle grades mathematics teachers (described in the substudy-3 later) is agood example of this kind of discrepancy in teacher content knowledge.

In order to address research questions the following objectives were fulfilled during thestudy:

▪ Design a Teacher Content Knowledge Survey (TCKS) to assess the cognitive type ofteachers’ content knowledge.

▪ Assess the type of teachers’ content knowledge and test it for correlation with studentachievement on the state-mandated standardized test.

▪ Examine association between the type of teacher content knowledge, teachingpractice, and lesson quality.

▪ Evaluate teachers’ knowledge and understanding of a specific content topic—fractiondivision.

The study was conducted at an urban area in the Southwestern USA with 70%population of Mexican origin. The teacher sample reflects the demographics of the region:76% of participating teachers were Hispanics, 16%—Whites, 3%—African-American, and5%—Other (including, but not limited to, Asians, Middle Easterns, and Pacific-Islanders).

This paper has several sections. The first section addresses main theoretical perspectiveson categories of teacher knowledge and cognitive types of teacher content knowledge. Thesecond section deals with methodology of the study and its results for each phase. The thirdsection focuses on discussion and conclusion.

2 Theoretical background

2.1 Categories of teacher knowledge

Following Shulman (1986), in the study we use a term teacher content knowledge as “theamount and organization of knowledge per se in the mind of teachers” (p. 9). It includesmathematical facts, procedures, concepts, generalizations and models as well as why theyare true and how knowledge is generated and structured. Some of the existing research onteacher knowledge has sought to identify its respective types and categories. Along withcontent knowledge or knowledge of subject matter, Shulman in his pioneering works (1986,1987) proposed several domains of teacher knowledge including but is not limited topedagogical content knowledge, curriculum knowledge, and knowledge of learners.

Relationship between teacher knowledge and student achievement 143

Research forum on “Teacher Knowledge and Teaching” that took place at the conferenceof the International Group of the Psychology of Mathematics Education (Research Forumon Teacher Knowledge and Teaching, 2009) is a compelling evidence of a strong interest ofthe research community in this topic. Sponsored by the US Department of Education, theNational Mathematics Advisory Panel (2008a, b) recommends, “teachers’ mathematicalknowledge is important for students’ achievement”. Studies conducted in Canada (Davis &Simmt, 2006), England (Rowland et al., 2005), Australia (Chinnappan & Lawson, 2005)seem to share similar vision. Davis and Simmt (2006) argue, “the issue of teachers’knowledge of mathematics has been a prominent one for several decades” (p. 294). At thesame time, “the documenting of teachers’ content knowledge for teaching has received littleattention in debates about teacher knowledge” (Chinnappan & Lawson, 2005, p. 199).Moreover, “it is widely agreed that a vital component of the complex knowledge base forteaching is the transformation of subject-matter knowledge…into a form that might enableothers to learn it” (Rowland et al., 2005, p. 121).

Studies on models and categories of teacher knowledge were intensified since the late1980s. Leinhardt and Smith (1985) define teacher subject-matter knowledge as knowledgewhich includes “concepts, algorithmic operations, the connections among differentalgorithmic procedures…” (p. 247). Researchers seem to have similar ideas on the natureof teacher content knowledge. Fennema and Loef Franke (1992) designed a model ofteacher knowledge as a development in context to include the following components:knowledge of the content of mathematics, knowledge of pedagogy, and knowledge ofstudents’ cognition. The intersection of these components is linked with teachers’ beliefs toform the context specific/situated teachers’ knowledge. Teacher content knowledgecomponent of this model “includes teacher knowledge of the concepts, procedures, andproblem solving processes…” (Fennema & Loef Franke, 1992, p. 162). (Hill, Rowan, &Ball, 2005) proposed that content knowledge could be subdivided into two categories:common content knowledge, or mathematics knowledge that is common to manydisciplines, and specialized content knowledge or knowledge specific to teaching. Theycalled the latter mathematical knowledge for teaching and defined it as “the mathematicalknowledge used to carry out the work of teaching mathematics” (p. 373). In the recent workby (Hill, Ball, & Schilling, 2008), a category of teacher knowledge of content and student isalso considered. As indicated by research literature, scholars have substantially advancedthe knowledge base and development of theory regarding teacher knowledge. Although thegroundwork on teachers’ knowledge has contributed to the field, much remains to be donein understanding the potential impact of a particular category of teacher knowledge onteacher quality. In this study, we focus on teacher content knowledge and “zoom-in” to thecognitive type of teacher knowledge that has a potential to effect student achievement.

2.2 Cognitive type of teacher content knowledge

Teaching opportunity depends on teacher content knowledge. However, an amount ofcontent knowledge a teacher might have is not enough to influence student learning andachievement positively. The type of content knowledge is important. A teacher with contentknowledge limited to mathematical procedures only has less opportunity to influencestudent success than a teacher who conceptually understands the subject.

One of the indicators of teachers’ conceptual understanding of mathematics is an abilityto engage students into meaningful discourse in the classroom, through “selectinginstructional and assessment tasks that embody learning goals” (Shepard et al., 2005).Students learn from the kind of work they do in mathematics classroom, and the tasks they

144 M.A. Tchoshanov

are asked to complete determines the kind of work they do (Doyle, 1988). Mathematicaltasks are critical to students’ learning and understanding because “tasks convey messagesabout what mathematics is and what doing mathematics entails” (NCTM, 1991, p. 24).“The tasks make all the difference” (Hiebert et al., 1997, p. 17). Tasks provide the contextin which students think about mathematics, and different tasks place different cognitivedemands on students’ learning (Doyle, 1988; Henningsen & Stein, 1997). Cognitivedemands can be defined as the kind and level of thinking required of students in order tosuccessfully engage with and solve the task (Stein et al., 2000, p. 11). Such thinkingprocesses range from memorization to the use of procedures and algorithms (with orwithout attention to connections, concepts, understanding, or meaning), to complexthinking and reasoning strategies that would be typical of “doing” mathematics, such as:conjecturing, justifying, or interpreting (Henningsen & Stein, 1997, p. 529). Resnick andZurawsky’s (2006) publication attempts to broaden the ways the notion of cognitivedemand is interpreted by scholars.

The term “cognitive demand” is used in two ways to describe learning opportunities.The first way is linked with curriculum policy and students’ course-taking options—how much math and which courses. The second way relates to how much thinking iscalled for in the classroom. Routine memorization involves low cognitive demand, nomatter how advanced the content. Understanding mathematical concepts involveshigh cognitive demand, even for basic content. Both types of cognitive demand areassociated with student performance on achievement tests, but they are not substitutesfor each other. (p. 1)

Cognitive demand is a function of teacher content knowledge (Tchoshanov et al., 2008).More specifically, it is a function of the cognitive type of teacher knowledge. In this study,we propose to use the term cognitive type of teacher content knowledge to describe bothlearning and teaching opportunities:

& how much thinking is called for in the classroom—the learning opportunity aspect;& the kind of teacher knowledge needed to sustain students’ high level thinking in the

classroom—the teaching opportunity aspect.

In the context of the teaching opportunity, the cognitive type refers to the kind of teachercontent knowledge and thinking processes required to accomplish a task successfully, in termsof knowledge of facts and procedures, knowledge of concepts and connections, and/orknowledge of models and generalizations. Given the importance of cognitive type of teacherknowledge through selection of worthwhile mathematical tasks, the next critical considerationis “What do teachers need to know to select or make up appropriate individual tasks andcoherent sequences of tasks? The simple answer is that they need to have a good grasp of theimportant mathematical ideas and they need to be familiar with their students’ thinking”(Hiebert et al., 1997, p. 34). Similarly, Grossman, Schoenfeld, and Lee (2005) posed aquestion: “What do teachers need to know about the subject they teach?” (p. 201), andprovided a fairly straightforward answer: “Teachers should possess deep knowledge of thesubject they teach” (ibid.). However, the more important question, “What kinds of knowledgeare important for teaching?” (p. 202) remains unanswered. Considering the limitation outlinedby Grossman, Schoenfeld, Lee and others, the proposed research focuses on a specific type ofteacher content knowledge that is critical to student success.

Summarizing literature review and critical considerations on limitations of previousstudies on teacher knowledge, the following theoretical framework with references to work


of scholars on categories of teacher knowledge and cognitive types of content knowledge(presented in Fig. 1) is used to design the methodology of the study.

3 Methodology

3.1 Research design and measures

In connection with the theoretical framework and research questions, mixed methodsequential nested design (Tashakkori & Teddies, 2003) was implemented in this study. Theresearch consists of three substudies presented below (Fig. 2):

& S-1: quantitative study (an initial sample of N=102 middle grades mathematicsteachers) on the association between cognitive type of teacher content knowledge andstudent achievement;

& S-2: mixed methods study (N=10, a subset of the initial sample) on the correlationbetween cognitive type of teacher content knowledge and lesson quality;

& S-3: case study (N=2, a subset of two teachers from the substudy-2) on examiningmiddle grades mathematics teachers’ knowledge and understanding of fraction division.

Research questions and design have driven the selection of measures. Measures for datacollection at every phase of the study are presented in Table 1.

Knowledge of Models and

Generalizations

(Doerfler, 1991; Presmeg, 1997)

Fig. 1 Theoretical framework for the study on cognitive types of teacher content knowledge

146 M.A. Tchoshanov

4 Substudy-1: cognitive type of teacher content knowledge and student achievement

4.1 Sample size of teachers and students

Teacher sample size consisted of 102 subjects (initially, 105 teachers were tested using theTeacher Content Knowledge Survey; data for one teacher was incomplete and two teachersas outliers were removed from further consideration and analysis); seven teachers out of102 were beginning first year teachers for whom the students’ scores on standardizedtesting were not collected, simply because the test administration takes place in the springof each academic year. Teacher sample represents 12 different middle schools from threemajor independent school districts in the region. Participating teachers’ student passingrates on standardized test were reported by campus mathematics coaches and departmentchairs. Teacher sample demographic information was self-reported by participatingteachers. Teacher sample characteristics are similar to teacher population demographicswithin the districts: about 75% of teachers are Hispanics.

Fig. 2 Mixed method sequentialnested design of the study

Table 1 Measures of teacher content knowledge, teaching practice, and student achievement

Phase Substudy Measure

S-1 Quantitative study on the association between cognitive type of teachercontent knowledge and student achievement

Teacher ContentKnowledge Survey

Student StandardizedTest

S-2 Mixed method study on the correlation between cognitive type of teachercontent knowledge and lesson quality

Classroom ObservationProtocol

Teacher Reflections

S-3 Case study on examining middle grades mathematics teachers’knowledge and understanding of fraction division

Structured Interviewswith Teachers


Student sample size was determined based on the teacher sample: the study includedabout 2,400 students of participating teachers. Student sample major demographics are asfollows: 92% of Hispanics and 84% of student body is economically disadvantaged. We arecognizant of the impact socio-economic status (SES) has on student achievement. However,since the student population was homogeneous in regards to SES, we did not consider it asa separate variable in the study.

4.2 Teacher content knowledge survey reliability and validity evidence

In this study, the Teacher Content Knowledge Survey (TCKS) instrument was designed anddeveloped to measure teachers’ content knowledge based on different cognitive types:

& Type 1: Knowledge of Facts and Procedures. This is kind of knowledge that reflectsmemorization of facts, definitions, formulas, properties, and rules, performingprocedures and computations; making observations, conducting measurements, andsolving routine problems.

& Type 2: Knowledge of Concepts and Connections. This type includes, but is not limitedto the following: understanding concepts, making connections, selecting and usingmultiple representations, transferring knowledge to a new situation, and solving non-routine problems.

& Type 3: Knowledge of Models and Generalizations. This type requires teachers’knowledge and thinking for generalization of mathematical statements, designingmathematical models, making and testing conjectures, and proving theorems.

The cognitive types helped us to describe what kind of content knowledge andunderstanding teachers possess, rather than how much knowledge they have. Theinstrument consisted of 33 multiple-choice items reflecting key standards and competenciesfor middle grades teachers’ knowledge: Number Sense, Algebra, Geometry andMeasurement, Probability and Statistics. The instrument development team includedinterdisciplinary faculty with expertise in the following domains: mathematics, mathematicseducation, statistics and statistics education, representing multiple institutions: university,community college, and local schools. Item development included the following steps: (1)selection of a test item reflecting a particular standard and competency, (2) identification ofthe cognitive type to which the item belongs, and (3) development of test items that addressthe same standard and competency for the two other cognitive types. Test items wererevised by a group of experts in mathematics, statistics and mathematics education. Theinstrument was field-tested during 2005–2006 (Tchoshanov et al., 2008). The alphacoefficient technique (Cronbach, 1951) was used to assess the reliability of the TeacherContent Knowledge Survey instrument. The value of the coefficient of.839 suggests thatthe items comprising the TCKS are internally consistent (standard error=59).

In order to establish validity evidence based on test content, a specification table(Table 2) was constructed to guide the process of test development. The table includedmajor content objectives and competencies for teachers. The objectives were closelyaligned with corresponding objectives in state-mandated standardized tests for students.

Aside from the specification table, the item analysis table (Table 3) was used to furtherensure validity evidence based on test content. The item analysis table included samples ofcompetencies and items from the Teacher Content Knowledge Survey. These competenciesand items were mapped and aligned with competencies and items from the state-standardized test for students.

148 M.A. Tchoshanov

4.3 Data collection

The study implemented data collection procedures at two different levels: (a) teacher and(b) student level. Teacher level data—measurement of teachers’ knowledge was conductedusing the TCKS instrument. Each teacher was given 90 min (2–3 min per item) to completethe survey and they were allowed to use graphic calculators during the survey. Along withteachers’ scores on the TCKS, teachers’ demographic information such as: gender andethnicity, years of teaching experiences, as well as other proxies for teacher contentknowledge (i.e., teacher certification and mathematics coursework) was also collected.Teacher level data were collected during 2007–2008 academic year.

Student level data—collection of student achievement data was conducted primarilyusing students’ passing rates on the state-standardized assessment. Student passing rate is ameasure used by state as major indicator of student achievement and school accountability.Administration of the state-standardized testing usually takes place in the spring of eachacademic year. The data were collected for the spring 2007 administration of the test.

4.4 Data analysis

In correspondence with research questions, data analysis was performed using the followingtwo major statistical techniques:

& Correlation analysis using standard ordinary least square method: the selection of thisparametric technique was determined based on the key research question of the study(relationship between teacher knowledge and student achievement), number and natureof dependent and independent variables as well as the study design and the interval typeof scale used for student achievement scores.

& Non-parametric techniques (test for variance and test for independence) were selected tomeasure the variance between independent groups of the same (not normal) distributionwith arbitrary sample sizes of each group. The selection of these tests was also based onthe ordinal (ranked) nature of data for cognitive types of teacher content knowledge.

4.5 Results

Distribution of the Teacher Content Knowledge Survey scores by cognitive types in theform of box-whisker diagrams is presented in Fig. 3.

Table 2 Content specification table for the TCKS instrument

Objectives in the teacher contentknowledge survey

Number of Type 1items



Total

Number sense and quantitativereasoning

3 3 3 9

Patterns and algebraic reasoning 3 3 3 9

Geometry and spatial reasoning 2 2 2 6

Measurement concepts 1 1 1 3

Probability and statistics 2 2 2 6

Total number of items 11 11 11 33


Figure 3 also includes values of standard errors for every type of the Teacher ContentKnowledge Survey scores: small standard error values relative to samples’ means (.2095 forType 1 (L1), .2130 for Type 2 (L2), and.2556 for Type 3 (L3)) indicate that the studysample is likely to be an accurate representative of the population. Overall correlationbetween Type 1 and Types 2 and 3 of teacher content knowledge was not significant(r=.581, p=.917 and r=.507, p=.576 correspondingly), however the correlation betweenType 2 and Type 3 was statistically significant (r=.533, p< .01).

4.6 Difference in types of teacher knowledge between teacher rating categories

The teacher sample (N=102, excluding 2 outliers) was subdivided into three teacher ratingcategories corresponding to the state accountability system based on student performanceon standardized testing:

& Unacceptable teacher performance: student passing rate is below 44%;

Table 3 Mapping the teacher content knowledge survey items with corresponding competencies for students

Student competency Teacher competency Sample of teacher contentknowledge survey item

The student is expected torepresent multiplication anddivision situations involvingfractions

Teacher understands numberoperations and computationalalgorithms, uses a variety ofrepresentations to investigate theconnections between operationsand algorithms

Type 1 item

What is the rule for fractiondivision?

A. ab � c

d ¼ acbd B.

ab � c

d ¼ abcd

C. ab � c

d ¼ cdab D.

ab � c

d ¼ adbc

Type 2 item

Which of the problems belowrepresents the operation1 34 � 1

2 ¼ ?

A. Juan has a piece of rope1 34 feet

long and cuts it in half. At whatlength should he cut the rope?

B. Maria has 1 34 liters of juice. How

many liter containers can she fill?

C. A boat in a river moves 1 34 miles

in 2 h. What is the boat’s speed?

D. Daniel divides 1 34 pounds of

coffee evenly between 2customers. How many pounds ofcoffee will each customer get?

Type 3 item

Some students mistakenly dividetwo fractions in the following way:ab � c

d ¼ acbd. If a, b, c, and d are

positive integers, then thefollowing holds

A. This equation is always true

B. This equation is true when c=d

C. This equation is never true

D. This equation is true when ad=bc

150 M.A. Tchoshanov

& Acceptable teacher performance: student passing rate is between 45% and 74%;& Recognized teacher performance: student passing rate is 75% and above.

The results of the analysis of teacher content knowledge by different rating categories arepresented in Fig. 4. As is evident, it reflects a minor variation in Types 1 (L1) and 3 (L3) ofteacher content knowledge between different rating categories. However, the variation inType 2 (L2) of teacher content knowledge is significant, as it grows from 42% for teacherswith academically unacceptable performance and 46% for teachers with academicallyacceptable performance, to 60% for teachers with recognized level of performance.

The two-way Chi-Square test for independence showed a statistically significantdifference between three subgroups of teachers with regard to Type 2 of teacher contentknowledge (χ2=6.99>5.99 [χ2 critical], df=2, p=.030<.05). Chi-Square test results showthat it is most likely that differences in teacher performance are due to differences in teacherknowledge of concepts and connections.

Another important finding of the study was the correlation between the cognitive type ofteacher content knowledge and student achievement. The relationship between students’passing rates on standardized testing and Types 1 and 3 of teacher content knowledge wasnot significant (for Type 1 Pearson’s r=.06, p=.537, for Type 3 r=.02, p=.853). In contrast,the correlation between students’ passing rate and Type 2 of teacher content knowledge wassignificant with a large effect size (r=.26, p=.009<.01, d=.89).

The study shows that there is a little evidence on the impact of Type 1 knowledgeon student achievement. However, we feel that there is not enough evidence to claimthat Type 3 of teacher content knowledge is not important for student success. Onereason for a non significant relationship between Type 3 of teacher knowledge andstudent achievement, might be due to a low number of Type 3 items in the middlegrades state-standardized test. We performed an item analysis of the state-standardized

Types N Mean Conf. (±) Std.Error Std.Dev.L1 104 7.9803 0.4155 0.2095 2.1154L2 104 5.4608 0.4225 0.2130 2.1511L3 104 5.4216 0.5070 0.2556 2.5811

L3L2L1

12

10

8

6

4

2

0

Dat

a

Boxplot of L1, L2, L3

Fig. 3 Distribution of teacher content knowledge survey scores by cognitive types


test and found that there were only 14% of test Type 3 items compare to 45% and 41%of test items of Type 1 and Type 2, respectively. Another assumption could be that theType 3 of teacher content knowledge may play more significant role in studentachievement at high school level. In the substudy-3, we will consider the case with twomiddle grades teachers who have similar Type 1 and 2 scores but contrasting score onType 3 content knowledge.

5 Substudy-2: teacher knowledge and lesson quality

5.1 Sample and measure

The second phase of the study was implemented through classroom observation of asubgroup of teachers (N=10) using validated instrument—Classroom Observation andAnalytic Protocol (Horizon Research, 2000). The protocol is designed to measure lessonquality based on the following key indicators: lesson design, lesson implementation, lessoncontent, and classroom culture. Each indicator is scored using a Likert-type scale, from1—“Not at all” to 5—“To a great extent”.

5.2 Results

Table 4 depicts a set of Pearson’s correlation coefficients for key indicators of lesson qualityand observed teachers’ type of content knowledge (*p< .05; **p< .01).

Considering the limitation of the second phase of the study (only 10 teachers wereobserved), we cannot draw strong inferences. However, the results of the classroomobservations are worth of careful consideration with regard to supporting some of thequantitative findings of the research from the first substudy. The Pearson’s values for theLesson Design (−.0485, .2806, .4502 correspondingly) showed that teachers with high Type2 and 3 scores on TCKS have a tendency to design their lessons better than those who

Fig. 4 Teachers’ average scores by cognitive types across different teacher rating categories

152 M.A. Tchoshanov

possess high scores on Type 1. Analysis of teachers’ lesson plans supported this claim.Lesson plans of teachers with high Type 2 and 3 scores reflect careful planning andorganization; good selection of resources that contributes to accomplishing the purposes ofthe instruction; thoughtful design for future instruction that takes into account whattranspired in the lesson. At the same time, teachers’ Type 1 scores on TCKS had negativecorrelation with the quality of their lesson design.

We observed the same tendency for the Lesson Implementation indicator (Pearson’s rvalues: −.1246, .6373, .1888 correspondingly). The main difference in this observation is thatteachers with high Type 2 scores on the TCKS ‘outperformed’ those with high Type 1 and 3scores. The rating for the Lesson Implementation was calculated based of the followingdescriptors: the instruction was consistent with the underlying approach of the instructionalmaterials; the instructional strategies were aligned with investigative mathematics; the teacherappeared confident in his/her ability to teach mathematics; the teacher’s classroom managementstyle/strategies enhanced the quality of the lesson; the pace of the lesson was appropriate for thedevelopmental levels/needs of the students and the purposes of the lesson; the teacher was ableto ‘read’ the students’ level of understanding and adjusted instruction accordingly; the teacher’squestioning strategies were likely to enhance the development of student conceptualunderstanding/problem solving (e.g., emphasized higher order questions, appropriately used‘wait time,’ identified prior conceptions and misconceptions); the lesson was modified asneeded based on teacher questioning or other student assessments.

Similar tendency continued on the other two Lesson Quality indicators. Moreover, thecorrelation between cognitive Type 2 of teacher content knowledge and Lesson Content andClassroom Culture became stronger (Pearson’s r values .7277 and .7384 correspondingly). Therating for the lesson mathematics content was calculated based of the following descriptors: themathematics content was significant and worthwhile; the mathematics content was appropriatefor the developmental levels of the students in this class; students were intellectually engagedwith important ideas relevant to the focus of the lesson; teacher-provided content informationwas accurate; the teacher displayed an understanding of mathematics concepts (e.g., in his/herdialogue with students); mathematics was portrayed as a dynamic body of knowledgecontinually enriched by conjecture, investigation analysis, and/or proof/justification; appropriateconnections were made to other domains of mathematics, other disciplines, and/or real-worldcontexts; the degree of “sense-making” of mathematical content within this lesson wasappropriate for the developmental levels/needs of the students and the purposes of the lesson.

The rating for the Classroom Culture was calculated based of the following descriptors:active participation of all was encouraged and valued; there was a climate of respect forstudents’ ideas, questions, and contributions; interactions reflected collegial working

Table 4 Pearson’s correlation coefficients between cognitive types of teacher content knowledge and keyindicators of lesson quality

Lesson quality key indicators Type 1 Type 2 Type 3

Lesson design −.0485** .2806** .4502**

Lesson implementation −.1246* .6373* .1888**

Lesson content .0115** .7277** .2106**

Classroom culture .0887* .7384* .0887**

Lesson quality rating −.0888** .6439** .2271**

*p<.05

**p<.01


relationships among students (e.g., students worked together, talked with each other aboutthe lesson); interactions reflected collaborative working relationships between teacher andstudents; the climate of the lesson encouraged students to generate ideas, questions, conjectures,and/or propositions; intellectual rigor, constructive criticism, and the challenging of ideas wereevident.

Overall, results of classroom observations show that teaching practices and classroomculture of teachers with high Types 2 and 3 of content knowledge are quite different comparedwith teachers with low level of corresponding types of content knowledge. It was evident viaclassroom observation that teachers who possess Types 2 and 3 of content knowledge arecapable of effectively using instructional strategies and activities relevant to students’ priorknowledge and learning styles, in order to implement high order questioning techniques thatenhance the development of student conceptual understanding. This serves to provide anappropriate level of support as needed to include classroom discourse, multiple representations,and teacher explanations. The quantitative evidence collected through classroom observationwas supported by correlation analysis of the relationship between Type 2 of teacher knowledgeand overall lesson quality. The statistics analysis showed significant results: Pearson’s r=.644,p=.045<.05, standard error=.599. These results indicate that teachers who possess concep-tual knowledge are more likely to have effective teaching practices. The higher the teacher’sconceptual knowledge, the better the quality found in the delivery of a teacher’s lesson.

The major outcomes of the substudy-2 were supported by its qualitative component—teacher reflections. Teachers were asked to reflect on the following question—“Doesteacher content knowledge affect student achievement, and if so, how?” Below are somerepresentative excerpts from teachers’ written responses:

& “Teachers’ knowledge is the main ingredient for students to become great achievers.My students won’t learn if I really don’t understand what I am teaching” (Bradley,1 8thgrade mathematics teacher).

& “A teacher with more knowledge of the content is able to better explain math conceptsto students, in depth and many different ways so thats/he reaches all students”(Michael, 8th grade mathematics teacher).

& “My personal believe [sic] has always been that teachers who have a deeperunderstanding of content knowledge can be more flexible in delivering the knowledgeas well as more capable of breaking down big concepts into smaller connected pieces toensure student learning and understanding” (Kimberly, 7th–8th grade mathematicsteacher).

6 Substudy-3: examining teachers’ knowledge and understanding of fraction division

6.1 Sample and measure

Two middle grades teachers (from the substudy-2)—let us call them Michael and Bradley—were selected for a structured interview using interactive reading technique of interviewsegments (Grbich, 2007) with emphasis on cross-case analysis method (Berkowitz, 1997)based on the contrasting scores they had on Type 3 of content knowledge.

Michael and Bradley had similar scores on Types 1 and 2: 90% and 100% on Type 1 and69% and 62% on Type 2 items of the TCKS correspondingly. However, their scores on

1 For a purpose of anonymity, teacher names were changed.

154 M.A. Tchoshanov

Type 3 TCKS items were dramatically different: Michael scored only 20% and Bradley—100%! Nonetheless, both of them performed at a recognized level with respect to studentpassing rates (82% and 84% accordingly). Most of Michael and Bradley’s othercharacteristics such as gender, ethnicity, teaching experience, degree, grade level theyteach, and level of lesson quality are also very similar (Table 5).

The structured interview consisted of the following five questions:

1. If you were to teach fraction division, what would be some important things you wantyour students to learn about this topic?

2. What is a rule for fraction division?3. Divide 1 3

4 � 12.

4. Make up a story for the fraction division problem 1 34 � 1

2. Write down your story.5. Is the following statement a

b � cd ¼ ac

bd (a, b, c, and d are positive integers) ever true?Explain your reasoning, please.

6.2 Results

The case of Michael and Bradley with regard to their scores on the TCKS and studentachievement is supporting evidence of the main outcome of the study: Type 2 of teachercontent knowledge is closely associated with student achievement compare to Types 1 and3. Bradley’s impressive score on Type 3 TCKS items does not make a big difference interms of student achievement compare to Michael. This particular result puzzled us fromthe very beginning of this study right after we have compiled the data for the substudy-1.One of our reviewers suggested to address this post hoc phenomenon and conduct furtherexploration. Following the reviewer’s suggestion, we decided to “zoom-in” and unpack thecase of teachers’ Type 3 of content knowledge in a separate substudy. Below we describeMichael and Bradley’s responses to the five interview questions, which will further revealtheir knowledge and understanding of a specific topic—fraction division. Scratch paperwith interview questions was provided to interviewees so that they could talk and also writedown their responses.

With regard to the first question on some important things they want their students tolearn about fraction division, both Michael and Bradley seemed to share similar idea on the

Table 5 Teacher characteristics of Michael and Bradley

Teacher characteristics Michael Bradley

Student passing rate on standardized test 82% 84%

Gender Male Male

Ethnicity Hispanic Hispanic

Teaching experience 7 years 11 years

Degree BIS=Bachelor of InterdisciplinaryStudies

BIS (started as ElectricalEngineering major)

Grade level 8th grade 8th grade

TCKS scores Type 1–90% Type 1–100%

Type 2–69% Type 2–62%

Type 3–20% Type 3–100%

Overall lesson quality rating 4.57 4.38


importance of students’ prior knowledge. However, they approach it differently: Michaelemphasizes fraction multiplication and possible use of fraction bars to make a transition tofraction division, whereas Bradley focuses more on the prior concept of equivalent fractionsand fraction terminology (Table 6). At the same time, they both mention multiplicativeinverse/reciprocal as a key concept for understanding the fraction division algorithm.

Along with procedural knowledge, Michael and Bradley also talked about importance ofthe meaning for fraction division (Table 7). The advantage of Bradley’s Type 3 contentknowledge is evident when he talks about the “why” algorithm works even though it is “alittle above of what the student need to know” (Table 8).

The second question asked about a rule for fraction division. Both Michael and Bradleymentioned the same algorithm “copy-change-flip/reciprocal”. Notice that Michael used theword “flip” for what Bradley refers as “some kids say it” and used the more accuratemathematical term “reciprocal”. The dramatic difference between Michael and Bradley’sType 3 knowledge is quite evident here: Bradley writes down the fraction division rule ingeneral symbolic form with some justification of why the rule works whereas Michael onlyillustrates the rule with numerical example (Table 8).

When asked to divide two given fractions 1 34 � 1

2 ¼(question 3), both Michael andBradley showed accurately how to execute the “copy-change-reciprocal” algorithm, whichwas not a surprise for us considering their high scores on Type 1 survey items (90% and100% correspondingly).

Regardless of the slightly high score Michael has on Type 2 TCKS items (69%)compared to Bradley (62%), the conceptual question on making up a story for the givenfraction division problem (question 4) was more challenging for Michael than Bradley.

In order to illustrate a scope of Michael’s intellectual struggle (it took him 17 min torespond) with this question, we decided to include major transitional points of his interviewbelow. Michael started with a pizza story and then rejected it saying, “because if you dividepizza amongst people I don’t have full whole number for people and I don’t want to go inthat direction”. Then he picked another context—cooking: “maybe something aboutcooking instead of pizza”.

Michael: So, what I would do is maybe something with cooking where I can tell themabout a story where a recipe calls for one and three fourths cups of sugar [writes it ona scratch paper, Fig. 5]… So, how much sugar is needed for six cookies [pause].Okay, that may not work [inaudible].

Table 6 Michael and Bradley’s cross-case responses on student prior knowledge for fraction division

Michael Bradley

That is challenging to teach fraction division. Uhm, Iwould start with fraction multiplication. With it,you can use the model, the multiplication model,and cut your fractions without using fraction bars.You can use…a model to represent, you know, thisfraction of another fraction and then to go tofraction division. In my experience it’s beenchallenging…to use models to represent division.Because certain fraction division—fraction by awhole number—the models can get a littleconfusing for students so I have to go withalgorithms

Oh, it is just ideas, okay. So, every time we areworking with fractions we got to…[pause]. Guess Iwill take them back and make sure they have alltheir other concepts down as far as equivalentfractions [pause], simplifying, reciprocals, mixednumbers, and improper fractions

156 M.A. Tchoshanov

After Michael realized that the “sugar context” could lead him to a multiplicationproblem instead of fraction division, he laughs at himself saying “those will be somepretty sugary cookies”. We found Michael’s attitude as positive and open-mindedduring the entire interview because he continued working on the problem constantlyself-assessing and correcting his work.

Interviewer: How will you change it into a division problem?

Michael: [Pause]…I am going to scratch the sugar…I will go to a pan, a pan ofbrownies, a one and three fourths pan of brownies. I am going to try to represent that,how many halves are in one and three fourths...

Okay, back to pizza, one and three fourths pizza left. So, one and three fourths pizzaleft…, each person [inaudible]. [Pause] I am stuck.

This was a very dramatic episode during the interview. We thought that Michael wasalmost going to give up. Luckily, he did not. He admitted that he made a mistake andwanted to fix it. His conceptual and reflective thinking was evident throughout the strugglehe had with this question. Finally, he was able “to deliver a ball”.

Michael: I have, I caught myself making a mistake so I want to fix that mistake and Iwant to try and represent that as division instead of multiplication. Okay, got pizza,brownies, sugar [pause]. I think I have to start with the halves because over here I amtrying to start with the one and three fourths and it’s not working out…[Pause] howmany half slices of pizza are in one and three fourth…I just have to write it better.Makes it sound more logical. It would represent division…If there is one and threefourth pizza left and if we tried to take half slices how many half slices are there...

Interviewer: Are you comfortable with this answer?

Michael: Uhm, yeah a little relieved, I think if I start with the half and try to fit it intothe one and three fourths where the one and three fourths is the whole then it wouldbe more relevant [inaudible].

In contrast toMichael, Bradley was quite fast and confident in his story. However, he admittedthat this type of task is difficult (“hard ones”) for both students and himself as a teacher.

Bradley: These are the hard ones. Well, it is difficult to…[inaudible], usually we havea hard time, you know, to see the numbers and put them into a word problem…Imean even as a teacher sometimes I run into that, the kids even more so…I mean it iseasy to do fraction division with numbers, it is, probably, easy even to draw it. You

Table 7 Michael and Bradley’s cross-case views on the important of the meaning for fraction division

Michael Bradley

I would show an algorithm for fraction division.Uhm, I would have to go with some vocabularylike multiplicative inverse going back to themultiplication of fractions [pause]...

…I guess we need to read what fractions really meanas part of a whole and what division really does toa number. Maybe, even take them back to howdivision affects a whole number just so we can seeif we could find patterns and, maybe, examinedoes it affect fractions in the same way

Maybe just some dialogue about what it means for afraction to be divided into another fraction...Prettymuch, how many times will this fraction fit insideof this other fraction? So, we probably could usefraction bars to see…


Tab

le8

Michael

andBradley’scross-case

responseson

thequestio

naboutarule

forfractio

ndivision

Michael

Bradley

Okay,

therule…

Ido

n’tremem

berwhatthe—

whatthetextbo

okssay-bu

titinvo

lves

themultip

licativeinverse[w

riteson

ascratchpaper]where

you…

,uhm

,takethefirst

fractio

nandyouwould

multip

lyby

theinverseof

thesecond

fractio

n[pause]andin

middleschool

wesay:

copy,change,flip.So,

weusethosethreewords

sothey

can

remem

berthealgorithm

betterby

copyingthefirstfractio

n,changing

division

into

multip

lication,

andthen

theflip

would

bethereciprocal

ofthesecond

fractio

n.So,

Ithinkthebook

says

multip

licativeinversebefore

division

offractio

ns.

WhenIteachfractio

ndivision,Iusetherule

ofmultip

lyingby

thereciprocal.Ialso

show

them

why

thatneedsto

bedo

ne[w

riteson

ascratchpaper].Itisalittle

abov

eof

whatthestudentsneed

toknow

.Basically,an

algorithm

that

Itellthem

toremem

ber

iscopy,change

andthen

reciprocal

orflip,somekids

sayit,

andthen

multip

ly.Or,

copy

-chang

e-multip

ly.

158 M.A. Tchoshanov

know, you got one and three quarters and then you cutting it, dividing it into halves[draws on a scratch paper, Fig. 6].

After Bradley illustrated the fraction division problem with the diagram, he was able tomake up a story that reflects this problem.

Bradley: So, you got one half, two halves, three and then you have half of another half, soas far as drawing themodel I am, kind of, thinking about how to set it up as a word problem.See I could, probably, come up with maybe something simple but uhm… like evenmonetary wise, how many fifty-cent units…can I get out of a dollar seventy-five. Youknow something like that but sometimes I guess, I try to take it over the top and maybe dosomething a little more tangible…[writes the story on a scratch paper, Fig. 6].

Analyzing teachers’ responses on question #4, we concluded that conceptualunderstanding is not only about getting the correct answer fast. It is more about theprocess of flexible and reflective thinking in search of the correct response.

The last question on whether “the following statement ab � c

d ¼ acbd is ever true” revealed

the advantage of Bradley’s Type 3 knowledge before Michael (Table 9). Michael tried somenumerical values to plug in the statement and concluded that it is not true. He alsomentioned that “…maybe something you can do algebraically with the variables that will

Fig. 5 Michaels’ scratch paper on the fraction division story


make it true”. However, limitations of his Type 3 content knowledge kept him at thedeclarative level not even trying to work something out algebraically. Opposite to Michael,Bradley’s response was quite different: he jumped into symbolic justification of the casewhen the statement could be true.

The case study helped us to look more closely at cognitive types of Michael andBradley’s content knowledge. We refrain from drawing any strong conclusions from thissubstudy. However, Bradley’s knowledge of models and generalizations was clearly evidentas much as Michaels’ conceptual and reflective thinking was constantly dominantthroughout the interview. Perhaps, Michaels’ knowledge of concepts and connections isenough to be associated with recognized level of his students’ performance on the state-standardized test whereas Bradley’s knowledge of models and generalizations might be “alittle above of what the student need to know”.

7 Discussion and conclusion

Overall, existing research on teacher mathematical knowledge, measured by variety of tests,and student achievement, assessed by standardized tests, shows a promising positive trend.Our study joins the pool of studies on positive correlation between teacher contentknowledge and student achievement. For instance, the study by Harbison and Hanushek(1992) found a positive effect of teacher mathematics test scores to fourth-grade tests onstudent achievement: “At fourth grade, a ten-point improvement in the mean teacher’scommand of her mathematics subject matter…would engender a five-point increase instudent achievement; this is equivalent to a 10% improvement over the mean scores offourth graders” (p. 114). Clotfelter, Ladd, and Vigdor (2007) examined the relationship ofteacher test scores to student mathematics achievement and found that higher teacher testscores are a significant predictor of higher student achievement. Hill et al. (2008) designeda test specifically assessing mathematical knowledge for teaching and found that themeasure of teacher content knowledge is a significant and positive predictor of studentsuccess in mathematics.

Our findings in the substudy-2 on teacher knowledge and lesson quality provideevidence for previous studies (Steinberg, Haymore, & Marks, 1985), which claimed that the

Fig. 6 Bradley’s scratch paperon fraction division story

160 M.A. Tchoshanov

Tab

le9

Michael

andBradley’sresponseson

thequestio

n-5whether

statem

enta b�

c d¼

ac bdisever

true

Michael

Bradley

Michael:Iguessitcouldbe

true

ifthey

areallones

[pause].But,they

aredifferent…

Bradley:Ido

n’tunderstand,arethereotherstipulations

forcanddbesidesbeing

positiv

e?Can

they

bethesame?

Interviewer:Doestheproblem

specifyit?

Interviewer:Doestheproblem

specifyit?

Michael:Theyarealldifferentintegers.Justby

lookingat

it,itcouldbe

no…

Iam

infavo

rof

saying

“No”

justby

look

ingat

itbu

ttheremay

beacase

[writesdo

wn],

maybe,aspecialcase…

Bradley:Well…

[inaud

ible]So,

ifcanddareequalthen

Igu

essitwou

ldn’treally

matterwhenIdo

mycopy

change

they

arestill

thesamevalue…

[writeson

ascratch

paper]

[Pause]maybe

something

youcando

algebraically

with

thevariablesthat

will

makeit

true.

Itcouldbe

true…ifcisequaltodin

thiscase.I

couldprob

ably

saythatIam

done

with

itandthen

Iam

probably

goingto

bethinking

aboutitallnight[laughs],tryingto

see

ifthereisanothercondition.


teachers whose mathematical knowledge appeared to be connected and conceptual werealso more conceptual in their teaching, while those without this type of knowledge weremore rule-based. Fennema and Loef Franke (1992) echo the previous claim in saying,“When a teacher has a conceptual understanding of mathematics, it influences classroominstruction in a positive way” (p. 151). The case of Michael (lesson quality rating 4.57) andBradley (4.38) revealed “a cognitive anatomy” of these two teachers’ content knowledgeand its potential relationship to classroom practice.

The research presented in this paper is a focused study specifically tailored to measurethe cognitive type of teachers’ content knowledge and its impact on students’ achievement.The study explored whether teachers’ content knowledge of facts and procedures havedifferent effects on student achievement and teaching practice, relative to knowledge ofconcepts and connections or knowledge of models and generalizations. We believe that themajor outcome of this study that contributes to the field of mathematics education researchcould be summarized in the following way: teacher content knowledge of concepts andconnections is significantly associated with student achievement and lesson quality inmiddle grades mathematics.

However, we are cognizant of the limitation of our study, regarding the narrow focus onteacher content knowledge. At the same time, we recognize that the Type 2 of teachercontent knowledge “crosses boundaries” of other types of teacher knowledge including butnot limited to pedagogical content knowledge. It is evident from teacher responses belowthat not only teacher content knowledge is important, but also ability to teach it effectively.The teacher content knowledge, isolated from other categories of teacher knowledge, maynot provide a complete picture of a relationship between teacher knowledge and studentachievement. Therefore, future studies on integrated teacher knowledge and its impact onstudent success are needed.

Results of this study suggest a trend: teacher knowledge of concepts and connectionshas a potential to be a good predictor of successful teachers who might positivelyimpact middle grades students’ mathematics achievement. This trend is also supportedby teachers’ responses during the second substudy reflections on the followingquestion—“What kind of mathematical content knowledge should a teacher possess inorder to impact student achievement?” Below are some representative excerpts fromteachers’ written responses:

& “In order to impact student achievement a teacher must have a thorough understanding[emphasis added] of mathematics being taught” (Ivan, 6th–7th grade mathematicsteacher).

& “No matter how much content knowledge a teacher possesses, she must have an abilityto deliver the knowledge to students” (Mary, 6th–7th grade mathematics teacher).

& “I think a teacher should know and understand ways to solve problems [emphasisadded] but not necessarily memorize formulas and algorithms” (Kimberly, 7th–8thgrade mathematics teacher).

The outcomes of this research not only support the National Mathematics AdvisoryPanel claim that “Teachers’ mathematical knowledge is important for students’ achieve-ment” but zoom further in the type of teacher content knowledge that is critical for studentsuccess in middle school. One of the practical implications of this study is that it suggestsplacing targeted emphasis on the development of teachers’ knowledge of concepts andconnections while providing content-focused professional development specificallydesigned to improve student achievement.

162 M.A. Tchoshanov

Acknowledgements This research was partially supported by several agencies: Texas Education Agencygrant #04/05-350 on Improving Student Achievement in Mathematics, NSF-funded Mathematics andScience Partnership grant #EHR-0227124, and the Teachers for a New Era (TNE) grant #B7458.R02Afunded by the Carnegie Corporation of New York. Some results of this study have been presented andpublished in the Proceedings of the Joint Meeting of the International Group for the Psychology ofMathematics Education (PME 32 and PME-NA XXX) in Morelia, Mexico, 2008 (Tchoshanov, 2008). Theauthor expresses his gratitude to Dr. Larry Lesser and Mr. James Salazar for collaboration in the TeacherContent Knowledge Survey item development, and deep appreciation for graduate students of his spring2008 MTED5320 class for the survey administration and data collection efforts.

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