# Prospective Elementary Mathematics Teacher Content Knowledge: An Introduction

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Portland State University PDXScholar Mathematics and Statistics Faculty Publications and Presentations Fariborz Maseeh Department of Mathematics and Statistics 2014 Prospective Elementary Mathematics Teacher Content Knowledge: An Introduction Christine Browning Western Michigan University Eva Thanheiser Portland State University, Alden Edson Western Michigan University Patrick M. Kimani Glendale Community College Dana Olanoff Widener University See next page for additional authors Let us know how access to this document benefits you. Follow this and additional works at: Part of the Elementary Education and Teaching Commons, and the Science and Mathematics Education Commons Citation Details Christine Browning, Eva Thanheiser, Alden J. Edson, Patrick M. Kimani, Dana Olahoff, Jennifer M. Tobias, Ian Whitcare. Prospective Elementary Mathematics Teacher Content Knowledge: An Introduction. The Mathematics Enthusiast, (2), This Article is brought to you for free and open access. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications and Presentations by an authorized administrator of PDXScholar. For more information, please contact

3 TME, vol. 11, no. 2, p. 203 Prospective Elementary Mathematics Teacher Content Knowledge: An Introduction Christine Browning Western Michigan University Eva Thanheiser Portland State University Alden J. Edson Western Michigan University Patrick M. Kimani Glendale Community College Dana Olanoff Widener University Jennifer M. Tobias Illinois State University Ian Whitacre Florida State University The first and fundamental requisite for every teacher is that he have thorough command of the subject matter which he teaches; that he have mastered it so well that he speaks with his own authority; only so can he hope to lead the pupil to the corresponding feeling of independent mastery. J. W. A. Young, 1920 This Special Issue on the mathematical content knowledge of prospective elementary teachers (PTs) provides summaries of the extant peer- reviewed research literature from 1978 to 2012 on PTs content knowledge across several mathematical topics, specifically whole number and operations, fractions, decimals, geometry and measurement, and algebra. Each topic- specific summary of the literature is presented in a self- contained paper, written by a subgroup of a larger Working Group that has The Mathematics Enthusiast, ISSN , Vol. 11, no. 2, pp The Author(s) & Dept. of Mathematical Sciences-The University of Montana

4 Browning et al., p. 204 collaborated across several years, resulting in this Special Issue sharing the final work. The authors hope this summative look at prospective teacher content knowledge will be of interest to the mathematics education community and will be a useful resource when considering future research as well as designing mathematics content courses for prospective elementary teachers. The following sections in this issue provide background information on our overarching framework for the mathematical content knowledge of prospective elementary teachers as well as our rationale for conducting the summary of research. We briefly describe the intent and history of the Working Group that conducted the summaries, followed by the methods utilized in the summary process. Finally, we provide a description of what follows in each subsequent paper and close with our intentions of how this Special Issue might be used by our readers. Background The mathematical preparation of K 8 students is a challenge both in the United States and internationally. Studies from many countries report students coming away from their elementary education having memorized facts and procedures with varying degrees of success but not developing robust mathematical conceptions or flexibility in their reasoning (e.g., Reys et al., 1999; Stigler & Hiebert, 1999). Students who struggle in school mathematics have limited career options. Even those who perform well in mathematics courses are unlikely to enjoy mathematics or take an interest in science, technology, engineering, and mathematics careers if they experience the subject as dry and procedurally focused. Mathematics instruction can emphasize conceptual understanding and the engagement in mathematical practices. In order to positively influence the

5 TME, vol. 11, no. 2, p. 205 direction of mathematics teaching and learning, our elementary teachers must be adequately prepared. Research over the last few decades has shown that the work of teaching mathematics requires a different knowledge base than the mathematical knowledge required for other professions (Ball, Hill, & Bass, 2005; Ball, Thames, & Phelps, 2008; Conference Board of the Mathematical Sciences [CBMS], 2012). Ball and her colleagues identify this knowledge as Mathematical Knowledge for Teaching (MKT), which they define as the mathematical knowledge needed to perform the recurrent tasks of teaching mathematics to students (Ball et al., 2008, p. 399). They developed a framework for MKT consisting of two domains, subject matter knowledge and pedagogical content knowledge (see Figure 1). Figure 1. Mathematical knowledge for teaching framework (Ball et al., 2008, p. 403).

6 Browning et al., p. 206 For these summary papers, we chose to focus primarily on what could be considered prospective teachers subject matter knowledge, or content knowledge. Included in mathematics content knowledge are Common Content Knowledge (CCK), which is described as the mathematical knowledge that everyone should know; Specialized Content Knowledge (SCK), described as the mathematical knowledge that is special to the work of teaching; and Horizon Content Knowledge, which is an understanding of how mathematical topics fit together and make up a curriculum. In addition to these three types of knowledge, we also include Knowledge of Content and Students (KCS), which involves understanding students thinking and difficulties with mathematics. While Ball and her colleagues include KCS in pedagogical content knowledge, we argue that understanding children s thinking can help in the development of PTs specialized content knowledge, and thus should be included in a summary of PTs content knowledge. Unfortunately, PTs often do not come to teacher education with adequate subject matter knowledge (e.g., Ball, 1990). Even after having taken their college mathematics courses, many PTs do not reason mathematically in flexible or sophisticated ways (e.g., Yang, Reys, & Reys, 2009). These deficiencies in mathematics content knowledge are also seen in practicing elementary teachers in the United States and many other countries (e.g., Ma, 1999; Reys et al., 1999). Thus, there is evidence that PTs mathematics content knowledge does not necessarily improve on its own once they become teachers. Rather, the responsibility falls on mathematics teacher educators to help PTs develop a strong base in their content knowledge during their college years. Researchers, teacher educators, and organizations have noted the need to improve PTs mathematics content knowledge and have called for efforts to that effect (Ball, 1990;

7 TME, vol. 11, no. 2, p. 207 CBMS, 2001, 2012; Mathematics Teacher Preparation Content Workshop Program Steering Committee, 2001). Research is necessary to support these efforts. In particular, research concerning PTs mathematical thinking in specific content areas would inform instruction. As the authors of The Mathematical Education of Teachers observe, The key to turning even poorly prepared prospective elementary teachers into mathematical thinkers is to work from what they do know (CMBS, 2001, p. 17). Mathematics teacher educators need to understand the conceptions with which PTs enter their classrooms in order to design instruction that builds on those conceptions (Brown, Bransford, & Cocking, 1999). By reviewing mathematics education research concerning PTs content knowledge, we seek to establish what is known up to this point in time. Summarizing what is known also enables us to identify areas for further research. We envision these summary papers to be a starting point for future directions in research on PTs content knowledge and the development of that knowledge, as well as providing information useful in the design and development of mathematics courses for PTs. Brief History and Intent of the Working Group The current set of authors has participated in one or more Working Groups at the Psychology of Mathematics Education North American (PME- NA) Chapter, the National Council of Teachers of Mathematics (NCTM), or the Association of Mathematics Teacher Educators (AMTE) meetings on a regular basis since 2007 and has presented at several of those meetings (e.g., NCTM 2007, PME- NA 2009, AMTE 2009, AMTE 2010), as well as at the International Congress on Mathematical Education (ICME 2012) and PME At PME- NA 2007, the Working Group agreed on the need for the construction of a research base for the study of prospective teacher content knowledge. This included a need to summarize

8 Browning et al., p. 208 existing (completed and current) peer- reviewed research and to develop a research agenda. At PME- NA 2009, the Working Group grew in membership and allowed for the work of summarizing the existing literature to be divided into five content areas: whole number and operations, fractions, decimals, geometry and measurement, and algebra. The group was divided into five subgroups with each focusing on one of the five content areas. The ultimate goal was to have each group produce a summary paper of the existing literature on prospective teacher content knowledge. A secondary goal was to establish continued collaborations and to grow professionally through developing our pedagogy, especially as it relates to the teaching of content courses for prospective elementary teachers. To achieve the first goal we designed the following research questions: 1. What research has been conducted on elementary prospective teachers content knowledge? 2. What have we learned about prospective teachers content knowledge? At AMTE 2010, the group met to refine the guidelines for creating these individual paper summaries, and at PME- NA 2010, a rough draft of the combined summaries was refined. Over the years, we continued this collaboration and worked together to refine the methodology and the parameters of the overarching study. Methods As the Working Group met over the years, the scope of the research review was extended. Initially in 2007, the Working Group focused on a current perspective to provide an in- depth description of what is known about prospective elementary teachers content knowledge from a review of peer- reviewed research articles of the last decade. Initially this time period was , which was eventually extended to include 2010,

9 TME, vol. 11, no. 2, p. 209 as the work of the group continued over several years. At a subsequent PME- NA meeting, the Working Group decided to include a historical look to describe what was known in the specific content area prior to Lastly, the Working Group made the decision to update the current perspective to include recent peer- reviewed articles in 2011 and provide a view of the horizon to present future directions that built upon prior sections by examining peer- reviewed articles in 2012 and conference proceedings published in While the actual review process did not follow a chronological timeline, in this Special Issue, each content group presents its summaries of the literature on mathematics content knowledge of prospective teachers in the following three time periods: A Historical Look, A Current Perspective, and A View of the Horizon. Thus, the description of the common methods will follow this chronological timeline as well. A Historical Look For the time period prior to 1998, each content group conducted an ERIC search using general search terms such as preservice, prospective, elementary, teacher, education, and content knowledge, as well as specific content search terms such as number, whole number, addition, subtraction, geometry, and algebra. Combinations of search terms were entered into the ERIC database. Since all countries do not use the same grade- level classification system as the U.S., we decided to look at findings from studies of PTs preparing to teach children aged 3 14 to account for cases with combined middle and elementary certifications. The title and abstract of each research article resulting from the two searches were read to determine whether the article focused on elementary PTs mathematics content knowledge. If the title and abstract did not suffice to make a determination of fit, reviewers read the whole article.

10 Browning et al., p. 210 A Current Perspective For the time period between 1998 and 2011, we conducted an ERIC search using the same keywords described in the historical perspective above. Likewise, a determination of fit for each article was made using the same process as described above. A list of 23 journals from which articles were found for summarizing was compiled. Subsequently, it was brought to our attention that one article published between 1998 and 2011 in one of the 23 journals was not included in the results from our ERIC search. We then found that ERIC does not contain all years for all journals included in the database. Thus, each journal was then carefully reviewed for additional articles focusing on PTs content knowledge between 1998 and 2011 to ensure all articles focusing on PTs content knowledge in those identified journals were found. This review produced more articles that were published in a year not included in ERIC database or were not indexed with any of the previously listed search terms. A View of the Horizon For the final time period, , we conducted an ERIC search for the year 2012 using the same process described earlier. In addition, we reviewed the conference proceedings from both the International Group and the North American Chapter of the Psychology of Mathematics Education (PME and PME- NA) for the years 2011 and For this review, each content group carefully searched for keywords in the titles and abstracts of all papers found in the proceedings. If the title and abstract did not suffice to make a determination of fit, reviewers read the whole paper.

11 TME, vol. 11, no. 2, p. 211 Inclusion/Exclusion Criteria The Working Group established exclusion criteria across all content groups and excluded articles that had (a) a general description of content knowledge that lacked specific attention to three primary content areas (thus, our claims are restricted to these three content areas): numbers and operations (including whole numbers, fractions, decimals, and operations), geometry and measurement, and algebra; (b) a sole focus on perceptions about mathematics not connected to content knowledge needed for teaching (we make no claims about PTs beliefs in this Special Issue); (c) a focus on describing classroom practice or activities with a lack of attention to research design methods; and (d) a primary focus on high school PTs, mathematics majors, or inservice elementary teachers (our claims are restricted to prospective elementary teachers). For each content group, at least two researchers met to discuss the inclusion/exclusion of articles in their related content area. All disagreements about inclusion/exclusion into the database were resolved through discussion. Database The database included peer- reviewed research articles focusing on the mathematical content knowledge of elementary PTs in any of the content areas described earlier. The studies in our database listed the reference, content area, research questions, study type, research design, lens or approach used, selection criteria, description of participants, conditions of and procedures for data collection, data analysis, findings, and conclusions and implications.

12 Browning et al., p. 212 Description of What Follows The results of the Working Group are summarized in each of the subsequent papers in this Special Issue. Each paper focuses on a different content topic (whole numbers and operations, fractions, decimals, geometry and measurement, and algebra) and is organized into the following categories: historical, current, and horizon. The papers are presented as literature reviews in terms of what is known regarding PTs content knowledge. A Focus on Whole- Number Concepts and Operations The research illustrates that prospective teachers knowledge of this topic is largely dependent upon standard algorithms for solving a given type of task. In addition, they struggle with justifying why the algorithms work. The authors note that more research is needed regarding the types of conceptions PTs have when entering teacher education programs, as well as a need to document how their understanding develops. A Focus on Fractions Research shows that PTs are often able to solve fraction problems algorithmically but not justify the algorithm or represent the situation with a correct model, such as a word problem or diagram. In addition, PTs understanding of fractions, in general, tends to be limited, in that most think of fractions only in terms of a part- whole interpretation. More research is needed to better understand PTs conceptions of fractions and ways to improve their understanding, as well as to document how their understanding develops. A Focus on Decimals Though PTs may be able to successfully solve computational problems with decimals, they tend to lack a conceptual understanding of the structure of decimals. Historical and current research illustrates that PTs difficulties stem from their

13 TME, vol. 11, no. 2, p. 213 understanding of place value, incorrectly transferring whole number algorithms to decimals, and with their understanding of the density of decimals. Future research is needed to systematically examine how PTs understanding of decimals develops. A Focus on Geometry and Measurement Prior research has documented that PTs understanding of geometry and measurement is limited largely to memorized procedures. Though the research literature does not address every topic within geometry and measurement, the research that has been done suggests that (a) the van Hiele levels, (b) dynamic geometry software, and (c) methods fostering PTs understanding of concept images and definitions can be useful in improving PTs conceptual understanding of these topics. Future research is needed to address ways to develop PTs understanding of geometry and measurement as well as to address the topic gaps that still exist in the literature. A Focus on Algebra The summary of the algebra research shows that PTs can readily use symbolic representations with variables, expressions, and equations; however, they have difficulties with interpreting and connecting various representations to each other or to a problem situation. In addition, PTs computational methods are often inflexible, inefficient, or incorrect. Recent research suggests developing PTs understanding by focusing instruction around justifying multiple representations and solution methods. However, more research is warranted regarding how this understanding develops, as most has focused on either PTs incoming conceptions or analysis of pre/posttests. A common theme throughout the papers is that PTs understanding of each topic is limited to using algorithms, and difficulties lie in their ability to justify why the algorithms

14 Browning et al., p. 214 work. In addition, previous research has focused chiefly on describing PTs conceptions of these content topics, whereas recent research is moving toward documenting how PTs mathematical understandings develop. More research is needed as each summary paper in this Special Issue illustrates that the research base regarding PTs understanding is limited. Final Thoughts The intent of this Special Issue is to summarize and share what research suggests regarding the mathematical content knowledge of prospective elementary teachers. Have they met the expectations described in Young s 1920 statement of mastering the subject so well that prospective teachers can lead their students to the corresponding feeling of independent mastery? Given the summary information provided, we believe this Special Issue could be a resource in: graduate course work and seminars, prompting ideas for future research directions and topics; the design and development of current and future research on the mathematical content knowledge of PTs, providing a summary of the extant research literature through 2012 in the selected mathematical topics; the design of mathematics content courses for PTs, providing information about PTs common misconceptions, as well as strategies and tools that may help PTs work through the misconceptions and develop a better understanding of the mathematics;

15 TME, vol. 11, no. 2, p. 215 the design of content- specific professional development for elementary teachers, where information on misconceptions and strategies and tools for learning may still provide useful information for improving the teachers content knowledge. Thus, we hope this Special Issue will be a useful reference for future research as well as strategies for practice related to the development of the mathematical content knowledge of prospective elementary teachers. References Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The elementary school journal, Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 14-17, 20-22, Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59 (5), Brown, A. L., Bransford, J., & Cocking, R. R. (1999). How people learn: brain, mind, experience, and school. National Academy Press. Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Washington, DC: American Mathematical Society. Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. Washington, DC: American Mathematical Society. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates. Mathematics Teacher Preparation Content Workshop Program Steering Committee. (2001). Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. National Academies Press. Reys, R., Reys, B., McIntosh, M., Emanuelsson, G., Johansson, B., & Yang, D. C. (1999). Assessing number sense of students in Australia, Sweden, Taiwan, and the United States. School Science and Mathematics, 99, Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York: The Free Press.

16 Browning et al., p. 216 Yang, D. C., Reys, R. E., & Reys, B. J. (2009). Number sense strategies used by pre- service teachers in Taiwan. International Journal of Science and Mathematics Education, 7, Young, J. W. A. (1920). The teaching of mathematics in the elementary and the secondary school. New York: Longmans, Green and Company.

### PROSPECTIVE MIDDLE SCHOOL TEACHERS KNOWLEDGE IN MATHEMATICS AND PEDAGOGY FOR TEACHING - THE CASE OF FRACTION DIVISION

PROSPECTIVE MIDDLE SCHOOL TEACHERS KNOWLEDGE IN MATHEMATICS AND PEDAGOGY FOR TEACHING - THE CASE OF FRACTION DIVISION Yeping Li and Dennie Smith Texas A&M University, U.S.A. In this paper, we investigated

### EXPLORING LESSON STUDY IN TEACHER PREPARATION

EXPLORING LESSON STUDY IN TEACHER PREPARATION Maria L. Fernández Florida State University Prospective secondary mathematics teachers continue to lack images of reformed teaching. This investigation studied

### THE IMPACT OF DEVELOPING TEACHER CONCEPTUAL KNOWLEDGE ON STUDENTS KNOWLEDGE OF DIVISION

THE IMPACT OF DEVELOPING TEACHER CONCEPTUAL KNOWLEDGE ON STUDENTS KNOWLEDGE OF DIVISION Janeen Lamb and George Booker Griffith University, Brisbane, Australia This study investigated children s knowledge

### Supporting the Implementation of NGSS through Research: Curriculum Materials

Supporting the Implementation of NGSS through Research: Curriculum Materials Janet Carlson, BSCS/Stanford University Elizabeth A. Davis, University of Michigan Cory Buxton, University of Georgia Curriculum

### A STATISTICS COURSE FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS. Gary Kader and Mike Perry Appalachian State University USA

A STATISTICS COURSE FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS Gary Kader and Mike Perry Appalachian State University USA This paper will describe a content-pedagogy course designed to prepare elementary

### K-12 Mathematics Framework

DRAFT IN PROCESS San Diego City Schools Institute for Learning MATHEMATICS DEPARTMENT K-12 Mathematics Framework Introduction Mathematics Content Mathematics Processes Note: The content in this document

### TRU Math Conversation Guide

Release Version Alpha TRU Math Conversation Guide Module A: Contextual Algebraic Tasks This TRU Math Conversation Guide, Module A: Contextual Algebraic Tasks is a product of The Algebra Teaching Study

### The South Africa Symposium of Singapore Maths Strategies 2016 PRE-PRIMARY SCHOOL PRESENTER MS PEGGY ZEE

The South Africa Symposium of Singapore Maths Strategies 2016 THEME: GO BEYOND THE BASICS USING SINGAPORE MATHS STRATEGIES DATES: 20-22 JULY 2016, VENUE: EDUPLEX PRIMARY SCHOOL, SOUTH AFRICA PRESENTERS

### Masters of Education Degree with a specialization in Elementary Mathematics

Masters of Education Degree with a specialization in Elementary Mathematics Program Proposal School of Professional and Continuing Studies Northeastern University February 2008 Revised 2/18/2008 Revised

### Key Components of Literacy Instruction

Key Components of Literacy Instruction The longer I write and read, the more I learn; writing and reading are lifelong apprenticeships --Donald M. Murray W e, in MPS, believe that effort creates ability.

### Introduce diagnostic teaching

Introduce diagnostic teaching Alan Bell and the Toolkit team Summary Many students who appear to understand a topic at the end of the teaching unit, do not retain that even for a few months. Better long-term

### Revisioning Graduate Teacher Education in North Carolina Master of Arts in Elementary Education Appalachian State University

Revisioning Graduate Teacher Education in North Carolina Master of Arts in Elementary Education Appalachian State University A. A description of how the proposed program has been revisioned to reflect

### Elementary pre-service mathematics teachers and technology: are they ready?

Elementary pre-service mathematics teachers and technology: are they ready? Abstract Matthew Boggan Mississippi State University Sallie L. Harper Mississippi State University Elizabeth Bifuh-Ambe University

### BEYOND TELL ME WHAT YOU NOTICED : USING VIDEO TO IMPROVE INSTRUCTION

NCSM 2015 Annual Meeting Boston, MA April 15, 2015 BEYOND TELL ME WHAT YOU NOTICED : USING VIDEO TO IMPROVE INSTRUCTION Deborah Loewenberg Ball, Nicole Garcia, and Meghan Shaughnessy COMMON WAYS OF USING

### A Self and Peer Assessment Intervention in Mathematics Content Courses for Pre-Service Elementary School Teachers Xin Ma, Richard Millman, Matt Wells

A Self and Peer Assessment Intervention in Mathematics Content Courses for Pre-Service Elementary School Teachers Xin Ma, Richard Millman, Matt Wells University of Kentucky Introduction We explored in

### Common Core State Standards General Brief

Common Core State Standards General Brief The Common Core State Standards Initiative is a state-led effort to establish consistent and clear education standards for English-language arts and mathematics

### Mathematics Curriculum Evaluation Framework

MARCY STEIN and DIANE KINDER, University of Washington, Tacoma; SHERRY MILCHICK, Intermediate Unit #14, Pennsylvania State Department of Education Mathematics Curriculum Evaluation Framework Abstract:

FACTS ABOUT ABOUT ALABAMA COLLEGE AND CAREER STANDARDS ALABAMA DEPARTMENT OF EDUCATION Thomas R. Bice, State Superintendent of Education THE FACTS ABOUT ALABAMA COLLEGE AND CAREER STANDARDS The Standards

### Instructional Strategies for Teaching Algebra in Elementary School:

Instructional Strategies for Teaching Algebra in Elementary School: I ncorporating algebra into the elementary grades has become a focus for teachers, principals, and administrators around the country.

### The Comprehensive Mathematics Instruction (CMI) Framework: A new lens for examining teaching and learning in the mathematics classroom

The Comprehensive Mathematics Instruction (CMI) Framework: A new lens for examining teaching and learning in the mathematics classroom Scott Hendrickson, Department of Mathematics Education, Brigham Young

### PROMPTING GROWTH FOR PROSPECTIVE TEACHERS USING COGNITIVE DISSONANCE

PROMPTING GROWTH FOR PROSPECTIVE TEACHERS USING COGNITIVE DISSONANCE Jo Clay Olson Meghan Colasanti Karmin Trujillo University of Colorado at Aurora Public School Aurora Public School Denver & Health Sciences

### Ministry of Education. The Ontario Curriculum Exemplars Grade 6. Mathematics. Samples of Student Work: A Resource for Teachers

Ministry of Education The Ontario Curriculum Exemplars Grade 6 Mathematics Samples of Student Work: A Resource for Teachers 2002 Contents Introduction......................................................

### Mathematical Knowledge for Teaching Teachers: The Case of Multiplication and Division of Fractions

Syracuse University SURFACE Mathematics - Dissertations Mathematics 2011 Mathematical Knowledge for Teaching Teachers: The Case of Multiplication and Division of Fractions Dana E. Olanoff Syracuse University

### There has been a lot of attention given to the three shifts inherent in the common core math. While all three are important, coherence is the result

1 There has been a lot of attention given to the three shifts inherent in the common core math. While all three are important, coherence is the result of the progressions that are contained within the

### Differentiating Mathematics Instruction So EVERYONE Learns

WHITE PAPER Differentiating Mathematics Instruction So EVERYONE Learns by Jennifer Taylor-Cox, Ph.D. Educational Consultant, Author and Speaker Differentiated mathematics instruction is powerful way to

### Metacognition and Mathematical Problem Solving: Helping Students to Ask The Right Questions

Metacognition and Mathematical Problem Solving: Helping Students to Ask The Right Questions Shirley Gartmann and Melissa Freiberg The acquisition of problem solving, reasoning and critical thinking skills

### Calculus Students Understandings of Division and Rate Cameron Byerley, Neil Hatfield, Patrick W. Thompson Arizona State University

Byerley, C., Hatfield, N., Thompson, P. W. (2012). Calculus students understandings of division and rate. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.) Proceedings of the 15 th Annual Conference

### PRESERVICE SECONDARY MATHEMATICS TEACHERS COMPARATIVE ANALYSES OF TURKISH AND AMERICAN HIGH SCHOOL GEOMETRY TEXTBOOK

Ekim 2006 Cilt:14 No:2 Kastamonu Eğitim Dergisi 509-516 PRESERVICE SECONDARY MATHEMATICS TEACHERS COMPARATIVE ANALYSES OF TURKISH AND AMERICAN HIGH SCHOOL GEOMETRY TEXTBOOK Hasan ÜNAL Yıldız Technical

### Study Guide. Developing Literate Mathematicians: A Guide for Integrating Language and Literacy Instruction into Secondary Mathematics

Study Guide Developing Literate Mathematicians: A Guide for Integrating Language and Literacy Instruction into Secondary Mathematics Wendy Ward Hoffer The purpose of the book, Developing Literate Mathematicians,

### The Problem-Solving Cycle: Professional Development for Middle School Mathematics Teachers

The Problem-Solving Cycle: Professional Development for Middle School Mathematics Teachers The Facilitator s Guide Draft. Please do not cite. This material is based upon work supported by the National

### Principles to Guide the Design and Implementation of Doctoral Programs in Mathematics Education

Principles to Guide the Design and Implementation of Doctoral Programs in Mathematics Education A Task Force Report for the Association of Mathematics Teacher Educators Forward This report, in some ways,

### Editorial: Continuing the Dialogue on Technology and Mathematics Teacher Education

Thompson, D., & Kersaint, G. (2002). Editorial: Continuing the Dialogue on Technology and Mathematics Teacher Education. Contemporary Issues in Technology and Teacher Education [Online serial], 2(2), 136-143.

### Preparing Teachers to Teach Mathematics With Technology

Preparing Teachers to Teach Mathematics With Technology Margaret L. Niess Oregon State University, USA niessm@onid.oregonstate.edu Abstract: The challenge for mathematics teacher educators is to identify

### Study Guide for the Mathematics: Proofs, Models, and Problems, Part I, Test

Study Guide for the Mathematics: Proofs, Models, and Problems, Part I, Test A PUBLICATION OF ETS Table of Contents Study Guide for the Mathematics: Proofs, Models, and Problems, Part I, Test TABLE OF CONTENTS

### The Number Line: An Overview

The Number Line: An Overview Learning to Think Mathematically with the Number Line One of the most overlooked tools of the elementary and middle school classroom is the number line. Typically displayed

### Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4 6 Texts Used in Singapore

The Mathematics Educator 2004, Vol. 14, No. 1, 42 46 Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4 6 Texts Used in Singapore Sybilla Beckmann Out of the

### Van de Walle (2007) describes divid ing one fraction by another in this way: Invert the divisor and multiply

6 Rich and Engaging Mathematical Tasks: Grades 5 9 Measurement and Fair-Sharing Models for Dividing Fractions Jeff Gregg and Diana Underwood Gregg Van de Walle (007) describes divid ing one fraction by

### INTRODUCTION BENEFITS FOR TEACHERS AND THEIR STUDENTS COURSE CURRICULUM

INTRODUCTION The Intel Math course is a scaled-up adaptation of the Vermont Mathematics Initiative (VMI), a content-intensive professional development program developed by Dr. Kenneth Gross, Professor

### Fractions: A Problem-Solving Approach

Fractions: A Problem-Solving Approach DeAnn Huinker, University of Wisconsin-Milwaukee (Submitted for publication in the Wisconsin Teacher of Mathematics, March 2010.) "What's a fraction?" Students, unfortunately,

### THE REFLECTIVE STUDENT LEARNING PORTFOLIO. Department of Administration, Rehabilitation & Postsecondary Education. A Requirement for the:

THE REFLECTIVE STUDENT LEARNING PORTFOLIO Department of Administration, Rehabilitation & Postsecondary Education A Requirement for the: Master of Arts Education with a Concentration in Educational Leadership

### Streatham Wells Primary School. Mathematics Policy

Streatham Wells Primary School Mathematics Policy Mission Statement We believe that children will be most successful at Mathematics when they enjoy what they are doing, when they are motivated to achieve

### New York Student Learning Objective Studio in Art (Grade 8)

New York Student Learning Objective Studio in Art (Grade 8) May 2013 Table of Contents Introduction...3 What Is an SLO?... 3 What Is an Annotated SLO?... 3 How to Use This Document... 3 New York Contextual

### EXAMPLE FIELD EXPERIENCE PLANNING TEMPLATE CCSU MAT Program

EXAMPLE FIELD EXPERIENCE PLANNING TEMPLATE CCSU MAT Program Secondary Education (Math, History/Social Studies, Science, World Languages) and Special Education (K-12) Goal: The MAT program focuses on preparing

### Integrating Technology in Teaching and Teacher Education: Implications for Policy and Curriculum Reform Dr. Charalambos Vrasidas Dr. Marina S.

Vrasidas & McIsaac 1 Integrating Technology in Teaching and Teacher Education: Implications for Policy and Curriculum Reform Dr. Charalambos Vrasidas Dr. Marina S. McIsaac About the Authors Dr. Charalambos

### Inquiry-Based Primary Mathematics Curriculum Development. Friday 17 October 2014 Session 1: 10:15-11:15

Inquiry-Based Primary Mathematics Curriculum Development Friday 17 October 2014 Session 1: 10:15-11:15 Vienna International School Vienna, Austria Dr. Christine Orkisz Lang Primary School Maths Coach Grade

### THE ROLE OF INSTRUCTIONAL ASSESSMENT AND TEACHER ASSESSMENT PRACTICES

THE ROLE OF INSTRUCTIONAL ASSESSMENT AND TEACHER ASSESSMENT PRACTICES Janet K. Pilcher University of West Florida Abstract Graue s (1993) description of instructional assessment endorses the social constructivist

### To answer the secondary question, if hands-on activities would increase student interest and comprehension, several hands-on activities were used:

1 The Effect of an Overarching Topic on Student Performance in Algebra II Christine Consoletti and David Collins J.E.B. Stuart High School Fairfax County (VA) Public Schools June 2003 Question Does an

### Teaching Reading and Writing to Struggling Middle School and High School Students: The Case for Reciprocal Teaching

Teaching Reading and Writing to Struggling Middle School and High School Students: The Case for Reciprocal Teaching WAYNE H. SLATER and FRANKLIN R. HORSTMAN Wayne H. Slater is an associate professor in

### PRIMARY TEACHER TRAINEES MATHEMATICAL SUBJECT KNOWLEDGE: THE EFFECTIVENESS OF PEER TUTORING

PRIMARY TEACHER TRAINEES MATHEMATICAL SUBJECT KNOWLEDGE: THE EFFECTIVENESS OF PEER TUTORING Patti Barber and Caroline Heal University of London, Institute of Education As a matter of course, primary teacher

CORE PROGRESS FOR MATH LEARN MORE About Core Progress Math The National Math Panel recommends that mathematics courses provide a focused, coherent progression of mathematics learning, with an emphasis

### Math Counts: Issues That Matter

Math Counts: Issues That Matter A P R O F E S S I O N A L S E R I E S, V O L U M E 6 MASTERING THE BASIC FACTS nderstanding and learning basic U facts to automaticity is a primary cornerstone in mathematics

### The lesson study model of professional. Improving Teaching through

Improving Teaching through Lesson Study The lesson study model of professional development that originated in Japan is becoming increasingly popular in the United States (Lesson Study Research Group 2009;

### Understanding Multidigit Whole Numbers: The Role of Knowledge Components, Connections, and Context in Understanding Regrouping 3+-Digit Numbers

Portland State University PDXScholar Mathematics and Statistics Faculty Publications and Presentations Fariborz Maseeh Department of Mathematics and Statistics 2012 Understanding Multidigit Whole Numbers:

### The Role of E-Learning in Science Education

Introduction NSTA Position Statement The Role of E-Learning in Science Education The National Science Teachers Association (NSTA) supports and encourages the use of e-learning experiences for prek 16 science

### The Effective Mathematics Classroom

What does the research say about teaching and learning mathematics? Structure teaching of mathematical concepts and skills around problems to be solved (Checkly, 1997; Wood & Sellars, 1996; Wood & Sellars,

1 NCTM CAEP Standards (2012) Elementary Mathematics Specialist (Advanced Preparation) Standard 1: Content Knowledge Effective elementary mathematics specialists demonstrate and apply knowledge of major

### The effects of highly rated science curriculum on goal orientation engagement and learning

APA 04 Proposal The effects of highly rated science curriculum on goal orientation engagement and learning Purpose The paper to be presented describes a study of goal orientation (Pintrich, 2000), engagement

### Rachel Sorensen ED 419. Professor Ragland. September 21, 2003. Effective Teaching in High School Mathematics

Effective Math 1 RACHEL SORENSEN: I wrote this paper for my Senior Studies course in Secondary Teaching Methods. Some of the questions that guided my research included "What practices and instructional

### Providing Curriculum Alternatives to Motivate Gifted Students

Providing Curriculum Alternatives to Motivate Gifted Students ERIC EC Digest #E524 Authors: Susan Winebrenner and Sandra Berger 1994 How to get the best performance from every student is a challenging

### Teacher Education Portfolio Guidelines and Rubric

Teacher Education Portfolio Guidelines and Rubric The Teacher Education Portfolio is developed throughout a s program. Most Teacher Education courses have a capstone project designed to meet the InTASC

### MATHEMATICAL CREATIVITY THROUGH TEACHERS PERCEPTIONS

MATHEMATICAL CREATIVITY THROUGH TEACHERS PERCEPTIONS Maria Kattou, Katerina Kontoyianni, & Constantinos Christou University of Cyprus This study examines elementary school teachers conceptions of creativity,

### Purpose of the Study Theoretical Framework and Prior Study

DEVELOPING PROSPECTIVE TEACHERS KNOWLEDGE OF ELEMENTARY MATHEMATICS: A CASE OF FRACTION DIVISION Jane-Jane Lo Raven McCrory Sarah Young Western Michigan University Michigan State University Michigan State

### PRESERVICE ELEMENTARY TEACHERS UNDERSTANDING OF GREATEST COMMON FACTOR STORY PROBLEMS. Kristin Noblet University of Northern Colorado

PRESERVICE ELEMENTARY TEACHERS UNDERSTANDING OF GREATEST COMMON FACTOR STORY PROBLEMS Kristin Noblet University of Northern Colorado Little is known about preservice elementary teachers mathematical knowledge

### Maths Progressions Number and algebra

This document was created by Clevedon School staff using the NZC, Maths Standards and Numeracy Framework, with support from Cognition Education consultants. It is indicative of the maths knowledge and

### Foundations of the Montessori Method (3 credits)

MO 634 Foundations of the Montessori Method This course offers an overview of human development through adulthood, with an in-depth focus on childhood development from birth to age six. Specific topics

### The Impact of Teaching Algebra with a Focus on Procedural Understanding

The Impact of Teaching Algebra with a Focus on Procedural Understanding Andy Belter Faculty Sponsors: Jon Hasenbank and Jennifer Kosiak, Department of Mathematics ABSTRACT Present research indicates that

### Mathematics Content Courses for Elementary Teachers

Mathematics Content Courses for Elementary Teachers Sybilla Beckmann Department of Mathematics University of Georgia Massachusetts, March 2008 Sybilla Beckmann (University of Georgia) Mathematics for Elementary

### Concrete Materials and Teaching for Mathematical Understanding. Patrick W. Thompson. Center for Research in Mathematics and Science Education

Concrete Materials and Teaching for Mathematical Understanding Patrick W. Thompson Center for Research in Mathematics and Science Education San Diego State University Running Head: Concrete Materials P.

### For many years, there have been efforts to promote

Common Core State Standards: A Tool for Improving Education We believe that this initiative is a critical step in state efforts to provide every student with a comprehensive, content rich education. These

### Teaching to the Common Core mathematics standards. Why?

Comments? Like PDK at www. facebook.com/pdkintl Why the Common Core changes math instruction It s not the New Math exactly, but the Common Core calls for sharp changes in how math is taught and ultimately

### Equity for all Students Economic Development

The New Illinois Learning Standards Incorporating the Next Generation Science Standards (NILS-Science) Illinois Vision of K-12 Science Education All Illinois K-12 students will be better prepared for entrance

### Standard 9: Content for Special Education

Standard 9: Content for Special Education The program ensures that teacher candidates content preparation aligns with increasingly rigorous state standards in the grades they are certified to teach. Why

### Helping a Young Child Connect Fact Family Addition and Subtraction using Tools Terri L. Kurz, Arizona State University H.

Helping a Young Child Connect Fact Family Addition and Subtraction using Tools Terri L. Kurz, Arizona State University terri.kurz@asu.edu H. Bahadir Yanik, Anadolu University hbyanik@anadolu.edu.tr Jorge

### Mathematics Education Master of Education Degree Program PROFESSIONAL PORTFOLIO RUBRIC Candidate: ID#: Date: Evaluator:

Mathematics Education Master of Education Degree Program PROFESSIONAL PORTFOLIO RUBRIC Candidate: ID#: Date: Evaluator: Recommendations: Evaluator Signature: Grade: PASS FAIL Developing Acceptable Optimum

### Mathematics Education Program in Malaysian Universities: Curriculum Emphasis and Preparedness of Students to Become Teachers Abstract Introduction

Mathematics Education Program in Malaysian Universities: Curriculum Emphasis and Preparedness of Students to Become Teachers Aida Suraya Md.Yunus PhD, Ramlah Hamzah PhD, Habsah Ismail PhD Faculty of Educational

### Teaching Approaches Using Graphing Calculator in the Classroom for the Hearing-Impaired Student

Teaching Approaches Using Graphing Calculator in the Classroom for the Hearing-Impaired Student Akira MORIMOTO Tsukuba College of Technology for the Deaf, JAPAN Yoshinori NAKAMURA Miyagi School for the

### Tom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.

Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find

### To date, more than 45 states and the District of Columbia have adopted the CCSS.

Parents Guide to New Assessments IN Illinois In 2010, Illinois began implementing the Common Core State Standards (CCSS). The CCSS provide a consistent, clear understanding of what students are expected

### UNDERSTANDING FRACTIONS: WHAT HAPPENS BETWEEN KINDERGARTEN AND THE ARMY?

Understanding Fractions: What Happens Between Kindergarten And The Army? UNDERSTANDING FRACTIONS: WHAT HAPPENS BETWEEN KINDERGARTEN AND THE ARMY? Karim Noura Bayside Secondary College In this paper I will

### SCAFFOLDING SPECIAL NEEDS STUDENTS LEARNING OF FRACTION EQUIVALENCE USING VIRTUAL MANIPULATIVES

SCAFFOLDING SPECIAL NEEDS STUDENTS LEARNING OF FRACTION EQUIVALENCE USING VIRTUAL MANIPULATIVES Jennifer M. Suh George Mason University Patricia S. Moyer-Packenham George Mason University This collaborative

### SECTION 1. Why Coaching?

SECTION 1 Why Coaching? CHAPTER 1 What Is Literacy Coaching? What exactly is a literacy coach? How are literacy coaches different from reading specialists and mentors? What is the value of literacy coaching?

### Becoming an Engineer in Public Universities. Pathways for Women and Minorities. Edited by Kathryn M. Borman, Will Tyson, and Rhoda H. Halperin.

http://genderandset.open.ac.uk Becoming an Engineer in Public Universities. Pathways for Women and Minorities. Edited by Kathryn M. Borman, Will Tyson, and Rhoda H. Halperin. Reviewed by Jörg Müller Internet

### Greatest Common Factors and Least Common Multiples with Venn Diagrams

Greatest Common Factors and Least Common Multiples with Venn Diagrams Stephanie Kolitsch and Louis Kolitsch The University of Tennessee at Martin Martin, TN 38238 Abstract: In this article the authors

### Teaching Students with Learning Disabilities: Constructivism or Behaviorism?

Volume 8, Number 10 April, 2005 ISSN 1099-839X Teaching Students with Learning Disabilities: Constructivism or Behaviorism? Marcee M. Steele University of North Carolina Wilmington There is much controversy

### MULTIPLICATION OF FRACTIONS 1. Preservice Elementary Teachers Understanding of Fraction Multiplication

MULTIPLICATION OF FRACTIONS 1 Running head: MULTIPLICATION OF FRACTIONS Preservice Elementary Teachers Understanding of Fraction Multiplication Presented at the 2014 NCTM Research Conference, New Orleans,

### PA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis*

Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed

### Finnish education modelpedagogical. Marianne Matilainen

Finnish education modelpedagogical approach Marianne Matilainen 1. Finnish education model Education is considered to be one of the fundamental rights of all citizens, regardless of age, domicile, financial

### INTRODUCING LANGUAGE TEACHER COGNITION

INTRODUCING LANGUAGE TEACHER COGNITION Simon Borg, Centre for Language Education Research, School of Education, University of Leeds The Origins of Teacher Cognition Research Teacher cognition research

### THE EFFECT OF MATHMAGIC ON THE ALGEBRAIC KNOWLEDGE AND SKILLS OF LOW-PERFORMING HIGH SCHOOL STUDENTS

THE EFFECT OF MATHMAGIC ON THE ALGEBRAIC KNOWLEDGE AND SKILLS OF LOW-PERFORMING HIGH SCHOOL STUDENTS Hari P. Koirala Eastern Connecticut State University Algebra is considered one of the most important

### Principles of Data-Driven Instruction

Education in our times must try to find whatever there is in students that might yearn for completion, and to reconstruct the learning that would enable them autonomously to seek that completion. Allan

### WHO ARE THEY? STUDENT TEACHERS' BELIEFS

WHO ARE THEY? STUDENT TEACHERS' BELIEFS Reidar Mosvold, Janne Fauskanger, Raymond Bjuland & Arne Jakobsen University of Stavanger Student teachers beliefs about teaching are arguably important to their

### Learning and Teaching

B E S T PRACTICES NEA RESEARCH BRIEF Learning and Teaching July 2006 This brief outlines nine leading research-based concepts that have served as a foundation for education reform. It compares existing

### ICT in pre-service teacher education in Portugal: trends and needs emerging from a survey

Interactive Educational Multimedia, Number 11 (October 2005), pp. 153-167 http://www.ub.es/multimedia/iem ICT in pre-service teacher education in Portugal: trends and needs emerging from a survey João

### Jessica F. Ellis Curriculum Vitae

Ellis: Curriculum Vitae 1 Jessica F. Ellis Curriculum Vitae Colorado State University Department of Mathematics Weber 121 Fort Collins, CO 80523-1874 Assistant Professor Mathematics Education ellis@math.colostate.edu

### Department of Applied Arts and Sciences. University of Montana- Missoula College of Technology

1 01/19/2010 Department of Applied Arts and Sciences University of Montana- Missoula College of Technology Unit Standards and Procedures for Faculty Evaluation and Advancement Instructions These unit standards

### INTEGRATING TECHNOLOGY INTO THE MATHEMATICS CLASSROOM: THE ROLE OF TEACHER PREPARATION PROGRAMS

The Mathematics Educator 2005, Vol. 15, No. 1, 18 24 INTEGRATING TECHNOLOGY INTO THE MATHEMATICS CLASSROOM: THE ROLE OF TEACHER PREPARATION PROGRAMS Regina M. Mistretta Today s technology standards (International

### Study Guide for the Elementary Education: Content Knowledge Test

Study Guide for the Elementary Education: Content Knowledge Test A PUBLICATION OF ETS Copyright 2011 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRE, and LISTENING. LEARNING.