Preparation of School Teachers:

The Journal

**Patterson, Blain** *Real Analysis Mathematical Knowledge for Teaching: An Investigation* (February 2021)

**Abstract.**The goal of this research was to investigate the relationship between real analysis content and high school mathematics teaching so that we can ultimately better prepare our teachers to teach high school mathematics. Specifically, I investigated the following research questions. (1) What connections between real analysis and high school mathematics content do teachers make when solving tasks? (2) What real analysis content is potentially used by mathematics teachers during the instructional process?

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**Bowen, Brian** *First Generation Common Core Era Teachers� Understanding of Fractions* (February 2021)

**Abstract.**Implementation of the Common Core Mathematical Standards and Practices (CCSS) has emphasized the importance of instruction that develops both procedural knowledge and conceptual understanding. Novice teachers have experienced the implementation of the CCSS during their PreK-12 experience, as well as within their teacher preparation program. It may be that the renewed emphasis on productive mathematical dispositions and conceptual understanding might be realized in the content and pedagogical knowledge of these novice teachers. This study intends to contribute to the research on teacher knowledge by focusing on the knowledge of fractions held by first generation Common Core Era novice teachers.

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**Obara, Samuel; Welsh, Stewart** *Is e^p > p^e? Pre-Service Teachers Making Sense of Numbers* (October 2020)

**Abstract.**This paper is about making conclusions from the attempts of two calculus II pre-service teachers who were presented with the following problem: How can it be shown that e^p>p^e without the use of a calculator? Jane employed knowledge of calculus to solve the problem but she evinced gaps in her mathematical background when using operations of logarithmic and exponential functions. On the other hand, John used two approaches to solve the problem: one using calculus I techniques and the other, infinite series ideas. Interestingly his second approach enabled him to amend his first approach which resulted in a correct solution. Jane demonstrated gaps in mathematical understanding that were not witnessed in John�s work. The problem makes an excellent exercise in a first-semester course in calculus.

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**Numberger-Haag, Julie; Singh, Rashmi; Wernet, Jamie L.** *An Atypical Approach To Improve Typical Issues With Pre-Service Teachers' Geometric Shape Knowledge* (July 2020)

**Abstract.**Although geometric shape content should be mastered by early high school, many pre-service teachers (PSTs), particularly those in elementary and special education, have limited yet deeply engrained shape concepts. This poses challenges for how to teach shape concepts in content courses for PSTs. Thus, this study tested an approach atypical for a content course. PSTs� positive perceptions of this 40-minute activity and the specific ways their content knowledge did and did not change were reported. The results could influence instruction of mathematics content courses for PSTs in powerful ways with minimal time as well as open research opportunities.

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**Beach, Janessa; Alvarez, James A. Mendoza; Jorgensen, Theresa** *Mathematics Teacher Preparation Standards in a Research-based Mathematics Course for Prospective Secondary Mathematics Teachers: Exploring Student Experiences and Curriculum Writers� Perspectives* (April 2020)

**Abstract.**Recommendations for preparing teachers of mathematics have been published by multiple professional organizations in the United States; however, there is limited research on whether student course perceptions affirm the instructor-identified alignment of mathematics content courses for prospective secondary mathematics teachers (PSMTs) to these recommendations. We investigate the alignment of research-based tasks and explorations used in a content course for PSMTs to the Association of Mathematics Teacher Educators Standards for the Preparation of Teachers of Mathematics and recommendations in the Mathematical Education of Teachers II. Analyzing end-of-course interviews with 11 students enrolled in this course, we employ thematic analysis to identify themes in student perceptions that reflect central ideas in these documents. Four aspects of the course arose that likely support these themes and related standards and recommendations. This study generates other questions about the extent to which mathematics content courses focusing on research-based knowledge and practices for PSMTs should naturally elicit solid links to these standards.

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**Patterson, Blain** *Connections Between Real Analysis and Secondary Mathematics* (April 2020)

**Abstract.**In this study, we examine the connections between real analysis concepts and secondary mathematics content. This is accomplished by conducting a textbook analysis of a commonly used real analysis textbook. Concepts from the real analysis textbook were mapped to the Common Core State Standards-Mathematics (2010) to determine the relationship between real analysis and high school mathematics. The results of this textbook analysis suggests that several connections between real analysis and high school mathematics do exist with a particular emphasis on functions. Implications, limitations, and future research are also discussed.

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**Kalinec-Craig, Crystal A.; Prasad, Priya; Mira, Raquel Vallines; Walls, Carey** * Unpacking the Algorithm: Rethinking Elementary Pre-Service Teacher's Strategies for Solving Multi-digit Addition Problems* (February 2019)

**Abstract.** This paper details an exploratory study of 14 elementary prospective teachers (PTs) solving purposefully crafted, two-digit addition problems. The numbers in each problem were chosen to elicit diverse solution strategies. We coded 95 responses based on the ways the PTs completed the calculation (for example, by referring to the number as individual digits and/or by using strategies that attended to the value of the number). The findings suggest that although an overwhelming number of strategies explicitly referred to the numbers as distinct digits while adding, there was some evidence that PTs utilized more diverse strategies or sub-strategies (e.g., counting up or down strategies, derived facts of doubles or base ten knowledge). Our study seeks to unpack for mathematics teacher educators the typical narrative of PTs reliance on algorithmic strategies while highlighting their use of more nuanced strategies as well.

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**Max, Brooke; Amstutz, Molly** * The Intersection of MET II Content Domains and Mathematical Knowledge for Teaching in Mathematics Content for Elementary Teachers Courses* (February 2019)

**Abstract.** Mathematics content for elementary teachers (MCfET) courses generally provide prospective elementary teachers (PTs) with opportunities to engage with mathematical content and consider it from a teacher's perspective, yet little is known about the intended curriculum of such courses. Findings from an analysis of MCfET course activities using the Mathematical Knowledge for Teaching framework indicated that activities developing the Mathematical Education of Teachers II recommended content domains of Geometry and Measurement & Data provided fewer opportunities for PTs to develop specialized content knowledge (SCK) than arithmetic- and algebraic-based activities. This led to recommendations of research on developing SCK in MCfET courses.

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**Monandi, Okemwa Ogega** * Pre-Service Elementary Teachers’ Knowledge of Number Properties and Patterns in the Context of early Algebra* (August 2018)

**Abstract.** The purpose of this study was twofold: 1) investigate preservice teachers’ knowledge of the associative property for addition and multiplication and the distributive; 2) investigate preservice teachers’ knowledge of explicit and recursive representation of numerical pattern. In the Common-Core State Standard for Mathematics (CCSSM), fifth-grade students are expected to recognize the usefulness of number properties in simplifying and interpreting numerical expressions. In the CCSSM, fifth-grade students are also expected to learn to analyze patterns and relationships and to generate a numerical pattern given a rule. So, this study was conducted to examine preservice elementary teachers’ knowledge of the content they are eventually expected to teach.

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**Busi, Rich; Jacobbe, Tim** * The Impact of Analyzing Student Work on Preservice Teachers� Content Knowledge and Beliefs about Effective Mathematics Teaching* (August 2018)

**Abstract.** This study investigated the impact of analyzing student work on preservice teachers� knowledge and beliefs about mathematics teaching. Forty-two prospective teachers participated. Data were collected using established instruments from the Integrating Mathematics and Pedagogy (IMAP) project and from the Learning Mathematics for Teaching (LMT) project. Results are shared along with the protocol used to guide the treatment. Findings show that preservice teachers who learn through the analysis of student work experienced a significant impact on their beliefs regarding how mathematics should be taught. Findings also show student work analyses did not diminish content knowledge development. Implications for undergraduate mathematics education courses are presented.

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**Jung, Hyunyi; Strikwerda, Sarah L.; Park, Joyce (JuHye); Newton, Jill** *Preservice Teachers� Perspectives on Their Opportunities to Learn about Algebra* (August 2018)

**Abstract.** This study investigates secondary mathematics preservice teachers� (M-PSTs�) opportunities to learn about algebra as reported by five teacher preparation programs. Ten focus group interview transcripts were analyzed using systematic grounded theory. When M-PSTs mentioned specific algebra tasks during the focus group interviews, the instructors� interview data and course materials were used to support the M-PSTs� assertions. M-PSTs presented diverse learning opportunities that aligned with current policy recommendations, but also reported their preferences to learn about algebra in particular ways. Listening to voices of future teachers has the potential to improve teacher preparation programs, ultimately impacting secondary students� opportunities to learn algebra.

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**Murray, Eileen; Baldinger, Erin; Broderick, Shawn; Wasserman, Nicholas; White, Diana ** *Connecting Advanced and Secondary Mathematics* (August 2017)

**Abstract.** There is an ongoing debate among scholars in understanding what mathematical knowledge secondary teachers should have in order to provide effective instruction. We explore connections between advanced and secondary mathematics as an entry point into this debate. In many cases, advanced mathematics is considered relevant for secondary teachers simply because the content is inherently related. In this paper, we instead argue that there are connections between advanced mathematics and secondary mathematics that directly influence teaching. These are not discussions of the mathematical connections, per se, but rather discussions of specific ways in which knowing mathematical connections might influence secondary teachers� pedagogical choices in the classroom. With a focus on abstract algebra, we exemplify three categories of connections between advanced and secondary mathematics that influence teaching practice. Through exploring these examples, we aim to identify and situate more advanced mathematics knowledge within a practice-based lens of teacher knowledge.

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**Alvey, Christina; Hudson, Rick A.; Newton, Jill; Males, Lorraine M. ** *Secondary Pre-Service Teachers� Algebraic Reasoning About Linear Equation Solving* (September 2016)

**Abstract.** This study analyzes the responses of 12 secondary pre-service teachers on two tasks focused on reasoning when solving linear equations. By documenting the choices PSTs made while engaging in these tasks, we gain insight into how new teachers work mathematically, reason algebraically, communicate their thinking, and make pedagogical decisions. We will share qualitative results from our examination of teacher knowledge through pre-service teachers� explanations, models, language, and conjectures about student thinking.

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**Hinton, Vanessa; Burton, Megan; Flores, Margaret; Curtis, Rebecca ** *An Investigation into Pre-service Special Education Teachers� Mathematical Skills,
Self-Efficacy, and Teaching Methodology* (March 2015)

**Abstract.** The purpose of this mixed method study was to investigate future special education teachers� preparation for effectively teaching mathematics. During the last semester of their program, pre-service special education teachers completed elementary level mathematics computation and problem solving assessments, a mathematics efficacy beliefs survey, and an open-ended question about their teaching methods. The assessments were analyzed using qualitative and quantitative methods. Analysis revealed that participants who described their instruction as solely procedural in nature rated their teaching outcome expectancies lower and achieved lower computation scores than other participants. The results and their implications will be discussed.

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**Chesler, Joshua ** *Reading the News: The Statistical Preparation of Pre-Service Secondary Mathematics Teachers* (March 2015)

**Abstract.** Undergraduate mathematics programs must prepare teachers for the challenges of teaching statistical thinking as advocated in standards documents and statistics education literature. This study investigates the statistical thinking of pre-service secondary mathematics teachers at the end of their undergraduate educations. Although all had completed a required upper-division two-course sequence in probability and statistics, most were challenged by two tasks which required a critical analysis of the use of statistics in newspaper articles. Some patterns emerged in the incorrect answers, including a tendency to focus on potential sampling issues which were not relevant to the tasks. The results have implications for, and reaffirm concerns about, the undergraduate statistics preparation of secondary teachers in the United States.

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**Jukić, Ljerka; Brückler, Franka Miriam ** *What Do Croatian Pre-Service Teachers Remember from Their Calculus Course?* (June 2014)

**Abstract.** This paper reports a study on retention of core concepts in differential and integral calculus by examining the knowledge of two pre-service mathematics students. The study is conducted using a mixed method approach and the obtained data were analyzed using theory of three worlds of mathematics. The results showed that having good understanding of the concepts in the conceptual-embodied world can reflect on the long-term retention of mathematical knowledge in the proceptual-symbolic and formal-axiomatic world as well.

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**Stavrou, Stravros Georgios** *Common Errors and Misconceptions in Mathematical Proving by Education Undergraduates* (March 2014)

**Abstract.** Ninety-seven education students majoring or minoring in mathematics had their math homework examined in a Number Theory or Abstract Algebra course. Each student�s homework was observed for the purpose of identifying common errors and misconceptions when writing mathematical proofs. The results showed that students collectively made four recurring errors: assuming the conclusion in order to prove the conclusion, proving general statements using specific examples, not proving both conditions in a biconditional statement, and misusing definitions. In the same courses taken subsequently by 91 new students, we informed them about these common errors prior to assigning their homework to see how the students� proving processes would differ. The results showed that more exercises were left blank with comments such as �I�m not sure how to start the proof�, and that many students provided unnecessary examples to supplement their valid proofs.

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**Lee, Ji-Eun** *Developing New Views on Taken-for-Granted Assumptions: The Case of Division of Fractions* (December 2013)

**Abstract.** This article outlines a discussion had by a group of prospective teachers regarding questions that remained after their previous experiences as students. The discussion was focused on the rules associated with the division of decimals. Prospective teachers� initial discussion showed a strong tendency to handle the rules as taken-for-granted facts. As discussion progressed, however, they showed more accountability for their learning by connecting previously known concepts/explanations to justify the rules in the target question.

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**Bonner, Emily P.; Ruiz, Elsa C.; Travis, Betty** *Investigating Content Knowledge of Traditionally vs. Alternatively Prepared Pre-service Secondary Mathematics Teachers* (February 2013)

**Abstract.** Teacher content knowledge has been repeatedly linked to student achievement. Alternative Mathematics Teacher Education Programs are popular and prevalent, but do they prepare teachers with the content knowledge needed to teach secondary mathematics? This study reports on a quantitative analysis comparing scores between traditionally and alternatively prepared teachers on a secondary mathematics state licensure test. Results show that neither group had a passing mean, and traditionally prepared teachers typically score higher on the state licensure test, though only significantly higher in particular domains and total score.

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**Welder, Rachael M.; Simonsen, Linda M.** *Elementary teachers� mathematical knowledge for teaching prerequisite algebra concepts* (January 2011)

**Abstract.** The current study investigated the effects of an undergraduate mathematics content course for pre-service elementary teachers. The participants� content knowledge was quantitatively measured using an instrument comprised of items from the Mathematical Knowledge for Teaching Measures (Hill, Schilling, & Ball, 2004). Using a one-group pretest-posttest design, matched pairs t-tests showed significant gains (p = .000) in both common and specialized content knowledge and in two areas of prerequisite algebra concepts (numbers and equations/functions). Results provide evidence of pre-service teachers developing mathematical understanding beyond common content knowledge within collegiate course settings.

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**Matthews, Michael; Rech, Janice; Grandgenett, Neal** *The Impact of Content Courses on Pre-service Elementary Teachers� Mathematical Content Knowledge* (December 2010)

**Abstract.** In response to research documenting the mathematical deficiencies of pre-service elementary teachers, many teacher preparation programs are requiring mathematical content courses specifically focusing on the mathematics taught at the elementary level. This study considers what impact two such courses (one course focusing on Arithmetic, and the other course focusing on Geometry and Measurement) had on the mathematical content knowledge and attitude towards mathematics by comparing a group of pre-service elementary teachers who took these courses to a group of pre-service elementary teachers who took only a more general mathematics course (such as College Algebra). Results indicated that those teachers who took the specialized content courses had significantly higher mathematical content knowledge compared to those pre-service elementary teachers who took more general mathematics courses, but not significantly better attitudes towards mathematics.

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**Thanheiser, Eva; Browning, Christine A.; Moss, Meg; Watanabe, Tad; Garza-Kling, Gina ** *Developing Mathematical Content Knowledge for Teaching Elementary School Mathematics* (December 2010)

**Abstract.** In this paper the authors present three design principles they use to develop preservice teachers' mathematical content knowledge for teaching in their mathematics content and/or methods courses: (1) building on currently held conceptions, (2) modeling teaching for understanding, (3) focusing on connections between content knowledge and other types of knowledge. The authors share results of individual research projects and teaching approaches focusing on helping preservice elementary teachers develop such knowledge. Specific examples from different content areas (whole number, fractions, angle, and area) are discussed.

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**Cunningham, Robert F.; Roberts, Allison** *Reducing the Mismatch of Geometry Concept Definitions and Concept Images Held by Pre-Service Teachers* (September 2010)

**Abstract.** Twenty-three female elementary pre-service teachers were assessed on their ability to answer questions involving geometry concepts. Despite being given the definitions of the altitude of a triangle and the diagonal of a polygon on the pretest, limited understanding of these concepts was evident. A treatment using both graphic organizer and concept attainment teaching strategies improved performance on the posttest but not to the level required for teachers to supplement textbooks that lack adequate definitions or non-prototypical examples. Careful attention to challenging geometry concepts is needed so that pre-service teachers will be able to provide their students with improved conceptual understanding.

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**Flores, Margaret M.; Patterson, DaShaunda; Shippen, Margaret E.; Hinton, Vanessa; Franlin, Toni M.** *Special Education and General Education Teachers� Knowledge and Perceived Teaching Competence in Mathematics* (September 2010)

**Abstract.** The purpose of this study was to conduct a survey of special education and general education teachers� mathematical skill and perception of competence. The participants (n=206) were current (n-32) or future (n=174) elementary (K-6) and middle level (4-8) general and special education teachers enrolled in two major state universities in the Southern United States. The participants completed surveys of K-6 mathematics content and reported their perceived level of teaching competence. Findings indicate no differences in mathematics skills by certification area, but significant differences exist by grade level and perceived level of competence in teaching mathematics. The results have implications for teacher preparation.

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**Gleason, Jim** *Reliability of the Content Knowledge for Teaching-Mathematics Instrument for Pre-service Teachers* (January 2010)

**Abstract.** The Content Knowledge for Teaching Mathematics instrument was developed by the Study for Instructional Improvement and Learning Mathematics for Teaching projects at the University of Michigan to measure elementary school and middle school in-service teachers� mathematical knowledge for teaching to assist in the evaluation of professional development programs for mathematics teachers. This instrument is currently in widespread use among colleges and universities for the purpose of evaluating mathematics education programs for prospective elementary and middle school teachers. Since this is an �off-label use of this instrument, this article establishes the reliability of the instrument among this new population of pre-service teachers.

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**Wilburne, Jane M.; Long, Michael** *Secondary Pre-Service Teachers� Content knowledge for State Assessments: Implications for Mathematics Education Programs * (January 2010)

**Abstract.** Seventy secondary mathematics preservice teachers from two universities were assessed on their content knowledge, vocabulary knowledge, and their perceived confidence in teaching the content addressed on the eleventh grade state assessment. The results indicate the preservice teachers had significant content weakness in data analysis, algebra, and pre-calculus concepts. Correlation analyses revealed strong associations between their content knowledge and vocabulary knowledge, and between their content knowledge and their perceived comfort with teaching the content. The study supports the need for mathematics education programs to include a strong emphasis on making connections between the college-level content courses, secondary methods courses, and secondary mathematics content across the curriculum.

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**Varghese, Thomas** *Secondary-level Student Teachers� Conceptions of Mathematical Proof * (June 2009)

**Abstract.** Recent reforms in mathematics education have led to an increased emphasis on proof
and reasoning in mathematics curricula. The National Council of Teachers of
Mathematics highlights the important role that teachers� knowledge and beliefs play in
shaping students� understanding of mathematics, their confidence in and outlook on
mathematics education, and their ability to use math to solve fundamental problems. It is
crucial that teachers, especially the uninitiated, understand on a deep level the
mathematical concepts that they are expected to teach to adolescents. Thus, it becomes
critical for teacher educators to assess the understanding and abilities of student
teachers in constructing mathematical proof. The analysis in this study is based on three
factors: 1) meaning of proof, 2) ideas about teaching methods on proof, and 3) ideas
about the use-fullness of proof in a mathematics classroom. An analysis of the data
collected from this study indicates that current student teachers� conceptions of
mathematical proof are limited. The uneasiness expressed by student teachers about
mathematical proof may suggest an examination of students� experiences with the
mathematical proof in both secondary and post secondary classrooms.

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**Burton, Megan; Daane, C.J.; Giesen, Judy** *Infusing Mathematics Content into a Methods Course: Impacting Content Knowledge for Teaching * (May 2008)

**Abstract.** This study compared content knowledge for teaching mathematics differences between elementary pre-service teachers in a traditional versus an experimental mathematics methods course. The experimental course replaced 20 minutes of traditional methods, each class, with an intervention of elementary mathematics content. The difference between groups was measured by the Content Knowledge for Teaching Mathematics Measure. Using a mixed ANOVA design, analysis revealed that experimental mathematics methods course participants demonstrated greater growth in mathematics knowledge for teaching. This study suggests that even limited amount of mathematical content embedded in a methods course may make a difference in mathematical content knowledge for teaching.

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**Halat, Erdogan** *Pre-Service Elementary School and Secondary Mathematics Teachers� Van Hiele Levels and Gender Differences * (May 2008)

**Abstract.** The aim of this study was to find and compare the pre-service elementary school and secondary mathematic teachers� reasoning stages in geometry. There were a total of 281 pre-service teachers, 125 elementary school teachers and 156 secondary mathematics teachers, involved in the study. The researcher employed a multiple-choice geometry test. This test was developed to find out one�s geometric reasoning levels. After the collection of the data, the independent samples t-test with alpha = .05 was used to analyze the data. The study found that there was no statistically significant difference in regard to the reasoning stages between the pre-service elementary school and secondary mathematics teachers, and that although there was a difference with reference to van Hiele levels between male and female pre-service secondary mathematics teachers favoring males, there was no sex-related difference found between male and female pre-service elementary school teachers.

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**Latterell, Carmen M.** *A Snapshot of Ten Pre-Service Secondary Mathematics Teachers * (May 2008)

**Abstract.** Reform in secondary mathematics education has created changes regarding how to educate teachers. It is time to check the result of such changes. This study provides a snapshot look at the commonalities of ten preservice teachers in regards to their mathematics knowledge and their attitudes about mathematics and mathematics education. Four themes emerged. The preservice secondary mathematics teacher enjoys and has knowledge of secondary mathematics; does not enjoy nor have a deep understanding of undergraduate mathematics; is drawn to teaching and not overly drawn to mathematics; and has a medium-level of commitment to the National Council of Teachers of Mathematics.

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**Benken, Babette M.; Brown, Nancy** *Integrating Teacher Candidates� Conceptions of Mathematics, Teaching, and Learning: A Cross-University Collaboration* (January 2008)

**Abstract.** This study reports findings from an elementary teacher education initiative advanced between a department of mathematics and a school of education in a large, state-supported university. The design incorporated the interconnectedness of teacher candidates� conceptions related to mathematics, teaching, and learning and sought to explore how conceptions related to views of practice. Results suggest that an integrated program can foster an increased willingness to teach mathematics, greater recognition that mathematics is taught in lower elementary grades, a switch in focus of lessons to an understanding of content, and an integration of new content understandings in lesson design.

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**Turnuklu, Elif B.; Yesildere, Sibel ** *The Pedagogical Content Knowledge in Mathematics: Pre-Service Primary Mathematics Teachers� Perspectives in Turkey* (October 2007)

**Abstract.** The purpose of this research was to determine the pre-service primary mathematics
teachers� competency of pedagogical content knowledge in mathematics. The data
were collected by means of four open ended problems from the participation of 45
primary mathematics teacher candidates. Teacher candidates� responses were
analyzed based on pre-determined criteria. According to findings it was found that
having a deep understanding of mathematical knowledge was necessary but not
sufficient to teach mathematics. This finding pointed out the connection between
knowledge of mathematics and knowledge of mathematics teaching. It is suggested
that primary mathematics teacher candidates should be educated both from
�mathematics knowledge� and �pedagogical content knowledge� aspects.

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**Matthews, Michael E.; Seaman, Walter I.** *The Effects of Different Undergraduate Mathematics Courses on the Content Knowledge and Attitude towards Mathematics of Preservice Elementary Teachers* (July 2007)

**Abstract.** Preservice elementary teachers have been shown to generally possess poor mathematical knowledge (e.g. Goulding, Rowland, & Barber, 2002) and also strong negative attitudes toward mathematics (e.g. MacNab & Payne, 2004). Recently, national organizations have proposed interventions to address these issues (Conference Board of the Mathematical Sciences, 2001). This paper analyzes the impact of a content course intervention. When compared to a control group, the experimental group had a significantly more positive attitude toward mathematics. When previous achievement was partially controlled for, the experimental group scored significantly higher than the control group on a measure of content knowledge.

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**van der Sandt, Suriza** *Pre-Service Geometry Education In South Africa: A Typical Case?* (May 2007)

**Abstract.** A two year study investigating the state of pre-service teachers� (PTs�) (n=254), teachers� (n=18) and students� knowledge (n=103) of Grade 7 geometry (using the van Hiele theory (1986) and acquisition scales of Gutierrez, Jaime & Fortuny (1991)) is reported. Results indicate that both teachers and PTs fail to reach the expected level of geometric thinking and degree of acquisition. A relationship between student learning and the teachers� pre-service education and years of teaching experience seems to exist. Some pre-service preparation programs had no significant impact by either maintaining or positively impacting on the already attained thought levels. Results may hold implications for pre- and in-service training. Features of an improved program are suggested.

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**Pickreign, Jamar** *Rectangles and Rhombi: How Well Do Preservice Teachers Know Them?* (February 2007)

**Abstract.** This descriptive study investigated preservice teachers' understanding of
the properties and relationships among parallelograms. Forty preservice teachers in a pre-methods mathematics course for elementary education provided written descriptions of the terms rectangle and
rhombus. These responses were sorted into categories according to similarities
in description. Results show that only nine respondents articulated adequate
descriptions of rectangle and only 1 respondent articulated an adequate
description of rhombus. The findings are consistent with the body of literature
showing teachers of young children are in need of further development of their
conceptions of mathematics ideas - particularly in geometry.

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**Lee, Ji-Eun; Kim, Kyoung-Tae** *Elementary School Teacher Candidates' Perceptions of Good Problems* (September 2005)

**Abstract.** This study describes a classroom action research activity regarding a group of elementary school teacher candidates� perceptions of good mathematics problems. A questionnaire containing 20 problems was given, and the candidates were asked to rate the quality of each problem on a 5-point scale. The results revealed that the majority of the teacher candidates considered typical routine problems good and showed strong resistance to some non-routine problems that have atypical characteristics. Although the need of new perspective towards the nature of problem solving was identified throughout the reflection process, the teacher candidates expected difficulties in utilizing atypical problems in their future classrooms due to the lack of systemic support or individual teacher�s confidence. Implications of the results and the need for new directions are discussed.

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**Pomerantsev, Ludmila; Korosteleva, Olga** *Do Prospective Elementary and Middle School Teachers Understand the Structure of Algebraic Expressions?* (December 2003)

**Abstract.** A large number of students' mistakes in algebra are due to their inability to see the structure of a mathematical expression. This study analyzes and compares the typical mistakes made by prospective elementary and middle school teachers as these students progress through the courses at California State University at Long Beach. The study shows that the students have difficulties recognizing structures of algebraic expressions not only at the introductory level but also later as the students take calculus and senior level courses.

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**Meel, David E.** *Prospective Teachers' Understandings: Function and Composite Function* (October 1999)

**Abstract.** The current education reform efforts have placed greater emphasis on conceptual understanding and have focused attention on teacher preparation especially on the adequacy of teachers' mathematical knowledge of the material they will be engaged in teaching. This paper discussed the responses of 29 prospective elementary and special education mathematics specialists to questions focused on conceptualization of the function concept as well as facility with composite functions. The results point to the conclusion that many of the prospective teachers held historical definitions tied to formulaic rules and this negatively affected their ability to solve composite function problems.

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**Bryan, Tommy J.** *The Conceptual Knowledge Of Preservice Secondary Mathematics Teachers: How Well Do They Know The Subject Matter They Will Teach? * (October 1999)

**Abstract.** The success of the current reform movement in mathematics education depends on teachers' conceptual knowledge of the school mathematics subject matter. The prospective secondary mathematics teacher typically completes a major or minor in mathematics in order to gain certification. Through the process of interviews in order to explore and describe the knowledge of the school mathematics subject matter held by nine preservice secondary mathematics teachers at varying stages in the completion of their college-level content requirements, it was found that the subject matter knowledge of these prospective teachers was generally lacking in conceptual depth.

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