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Partner disciplines vary by institution but usually include the physical sciences, the life sciences, computer science, engineering, economics, business, education, and often several social sciences. It is especially important that departments offer appropriate programs of study for students preparing to teach elementary and middle school mathematics. Recommendation B.4 is specifically for these prospective teachers.

*Ensure that mathematical sciences faculty cooperate actively with faculty in partner disciplines to strengthen courses that primarily serve the needs of those disciplines;**Determine which computational techniques should be included in courses for students in the partner disciplines;**Develop new courses to support student understanding of recent developments in the disciplines;**Determine the appropriate uses of technology in courses for students in the partner disciplines;**Develop applications for mathematics classes and undergraduate research projects to help students transfer to their own disciplines the skills learned in mathematics courses;**Explore the creation of joint and interdisciplinary majors.*

**Strengthening Mathematics Courses to Support Future STEM Study**

The MAA Online webpage The Curriculum Foundations Project contains summaries of the contributions of faculty from partner disciplines to the Curriculum Foundation workshops that have appeared in *Focus* as well as the full text of all the Curriculum Foundation workshop reports. The book form of the reports, Curriculum Foundations Project: Voices of the Partner Disciplines is available on line as well as in printed form. The reports are intended to serve to initiate discussions between mathematics faculty and faculty from other disciplines at colleges and universities.

Project Kaleidoscope (PKAL) is an informal national alliance working to build strong leaning environments for undergraduate students in mathematics, engineering, and the various fields of science. PKAL hosts interdisciplinary conferences, publishes reports, and supports consultants to work with individual departments. For example, PKAL undertook the study of interdisciplinary initiatives in quantitative aspects of undergraduate biology and identified a number of exemplary programs: Capital University, College of Wooster, Davidson College, Duquesne University (component 1, component 2), Harvey Mudd College, James Madison University, Macalester College, Montana State University, Pomona College, Seattle Central Community College, University of Detroit Mercy, University of Mary Washington (contact Suzanne Sumner), University of Redlands, University of Richmond (contact Kathy Hoke), University of Wisconsin-La Crosse, and Wheaton College. These are discussed in the article ’Quantitative Initiatives in College Biologyâ? in *Math & Bio 2010: Linking Undergraduate Disciplines*, ed. Lynn Arthur Steen, MAA, 2005. The appendix to this volume contains an extensive list of related websites, including undergraduate programs and courses and teaching resources.

The movement to strengthen calculus courses has been significantly affected by interaction between mathematics faculty and faculty in partner disciplines. A website hosted by the University of Tennessee is especially rich in information and links to a variety of sites with suggestions for calculus instruction.

The Long Island Consortium for Interconnected Learning in Quantitative Disciplines (LICIL) is a comprehensive, interdisciplinary project, sponsored by the National Science Foundation, aimed at improving how faculty teach and how students learn in mathematically based subjects. A primary goal of the project is to promote a greater degree of realistic applications in mathematics offerings and a greater degree of mathematical sophistication in the offerings of client disciplines. Ten institutions participate in the consortium. At Farmingdale State University of New York, the project led to a complete overhaul of courses in developmental mathematics, precalculus, calculus, and differential equations. An article by S. Gordon and J. Winn about the Farmingdale experience is scheduled to appear in an MAA Notes volume on quantitative reasoning, edited by Rick Gillman, Valparaiso University.

Tevian Dray (a mathematician) and Corinne Manogue (a physicist) worked together to develop materials to use in a vector calculus course to try to align the study in the course with the way the concepts are used in engineering. The Vector Calculus Bridge Project is developing supplemental materials, especially small group activities, that emphasize the geometry of highly symmetric situations. Some have been developed for use with an otherwise traditional vector calculus course, and others are designed for a new, upper-division physics course on related material. The activities are intended to bridge the gap between the types of problems and methods of solution students will encounter in their chosen specialization and traditional vector calculus and its applications.

The Foundation Coalition, funded by the National Science Foundation, was established as an agent of systemic renewal for the engineering educational community. Its First-Year Engineering Curricula and Sophomore Engineering Curricula were developed with cooperation among faculty from mathematics, science, and engineering at the participating institutions. Links from the foundation’s web pages contain descriptions and evaluations of the programs at the participating institutions: Arizona State University, University of Massachusetts Dartmouth, University of Wisconsin Madison, Rose-Hulman Institute of Technology, and Texas A&M University.

Physicists Mel S. Sabella and E. F. Redish in Student Understanding of Topics in Linear Algebra present an analysis from a physicists point of view of five research papers that present objectives and recommendations for linear algebra courses. One paper, by Tsi-Wei Wang, discusses the philosophy, assignments, and student feedback for a linear algebra course designed for graduate students in chemical engineering at the University of Tennessee. Two other papers, by David Carlson, presents results of the work of the Linear Algebra Curriculum Study Group, which was significantly affected by input from partner discipline faculty.

In his talk, ’On the Centrality of Linear Algebra in the Curriculum,â? given at the Joint Mathematics Meetings in 1997, Carl Cowen, Purdue University, cited the effect of interaction with faculty in disciplines that use linear algebra on his understanding about how the course should be presented. Cowen’s talk included suggestions for teaching a linear algebra course that would meet the needs of students in a variety of programs of study, and he discussed some ways he believes computers can be used effectively in the classroom.

Between 1998 and 2001 a joint committee of ACM and IEEE-CS developed a set of curriculum guidelines for programs in computer science, referred to as CC2001. The following topics, labeled ’discrete structures,â? are included as part of ’core knowledgeâ? (with suggested number of instructional hours as noted): Functions, relations, and sets (6), Basic logic (10), Proof techniques (12), Basics of counting (5), Graphs and trees (4), Discrete probability (6). The website contains suggested outlines of discrete structures courses: a one-semester course, which is strongly recommended for first-year students, and a two-semester course, which is recommended whenever possible as an alternative to the one-semester version. The guidelines can be used as a basis for discussions between mathematics and computer science faculty about introductory discrete mathematics courses for computer science students.

As part of two MAA Professional Enhancement Programs (PREP) workshops in 2003 and 2004 and workshops at the annual meetings of the ACM Special Interest Group for Computer Science Education (SIGCSE), Bill Marion of Valparaiso University and Peter Henderson of Butler University organized an ongoing, online repository of examples and applications that can be used in discrete mathematics courses.

Although Kate Cowles's course, Statistical Methods and Computing, at the University of Iowa started out as a beginning course for undergraduate statistics majors, it was open to other students. When word got out that it was a hands-on data-analysis class using the software SAS, some other programs ’ industrial engineering and certain biology concentrations ’ started requiring it of their students.

A description of Allan Broughton’s Methods of Image Processing course at Rose-Hulman Institute of Technology is included in Part 1, Section 4. Broughton also developed a presentation about the course.

David Bressoud (chair of CUPM) has written a series of ’Launchings,â? which describe aspects of the CUPM Curriculum Guide 2004.. The following deal with interdisciplinary cooperation: Computational Science in the Mathematics Curriculum and Math & Bio 2010.

The term learning community refers to a set of two or more courses taken in distinct disciplines by a common set of students. Normally, faculty teaching the courses meet to arrange coordination, the extent of which may vary from one institution to another. The Fund for the Improvement of Post-Secondary Education (FIPSE) supported projects to build learning communities at 18 institutions: 6 each of community colleges, regional universities and research universities. The Learning Community Commons contains information about summer institutes and regional networks and gatherings as well as a collection of monographs and much other resource material. The document Assessment in and of Collaborative Learning contains a set of assessment tools for college teachers and staff involved in collaborative learning and in learning communities. The Learning Communities National Resource Center** **has links to a variety of helpful resources.

Examples of joint and interdisciplinary majors are contained in Part 2, Section C.5.

*Creating, solving, and interpreting basic mathematical models;**Making sound arguments based on mathematical reasoning and/or careful analysis of data;**Effectively communicating the substance and meaning of mathematical problems and solutions.*

**Research on Teaching and Learning**

In recent years a new area of study has emerged: research on the teaching and learning of mathematics at the college level. A significant amount of this research has concerned courses taken primarily by students in partner disciplines: calculus, differential equations, linear algebra, statistics, and courses for prospective elementary and middle school teachers, but research has also been conducted on students’ learning of proof and on proof-emphasizing courses. The AMS online bookstore contains tables of contents and some prefaces for the volumes in the CBMS series Issues in Mathematics Education; John and Annie Selden edit the MAA Research Sampler, an online collection of articles on research in undergraduate mathematics education; and the Research in Undergraduate Mathematics Education Community makes information about articles written by its members on the organization website.

A report, Computer Algebra Systems in Calculus Reform, written by Lisa Denise Murphy while a graduate student at the University of Illinois at Urbana-Champaign, includes some history, evaluation of several projects, and a literature review. The book *Changing Calculus* by Susan Ganter, Clemson University, is a report on evaluation efforts and national impact of calculus reform efforts from 1988-1998.

**Improving Students’ Abilities to Think about and Do Mathematics**

In 1995 the mathematics department at Rutgers University lengthened the weekly ’recitationâ? sections of Calculus I and II for the Mathematical and Physical Sciences from 55 to 80 minutes. Instructors use more than half of the longer period for workshop problems. Graphing calculators are used both in recitation and in lecture ’ but students are expected to master them on their own. Undergraduate peer mentors are provided for all recitations to help with group work and routine grading in order to free TA time for reading and grading the workshop write-ups. (The department won two internal university grants ’ one to train and pay the peer mentors; one to buy calculators to lend to the TA's and the peer mentors and calculator projectors to lend to faculty.) In its first year, these changes involved about 15 faculty members, 15 teaching assistants, 30 undergraduate mentors, and 600 enrolled students. A common final exam for Calculus and Intensive Calculus showed that time spent on workshop problems did indeed elicit better performance.

Although some students objected to the extra work, the Engineering School responded that good technical communication in speaking and writing is important for professional success and supported the change. Faculty member Amy Cohen reported that the strongest students seem to have benefited the most; mediocre students less so; the rate of D's, F's, and W's was not immediately reduced. She added that the peer mentors may have been the biggest winners. They enjoy the work, the interactions with TA's and faculty, and the fact that academic success gave them paying jobs. They receive training in general ’people skillsâ? from the Learning Resource Center, and in ’discipline specificâ? skills from mathematics faculty. The department now hires about 50 peer mentors each term. Not all of them are mathematics majors. Indeed most of them had intended to be science or engineering majors. But a number of them have completed a second major in mathematics in part because of their close engagement with the department’s courses and faculty.

Some sample workshop problems are at the mathematics department’s Old Workshops and Exams webpage. Additional ones are on Stephen Greenfield’s homepage (click on 151 and 152). Cohen cautioned that some problems are chosen to remedy an ineffective lecture or to reinforce a topic students handled poorly on text-book homework. Problems appear to benefit students most when each instructor writes them to fit the particular needs of his or her own class, its text, and its syllabus. She also cautioned that workshop problems are not a replacement for weekly sets of exercises and routine problems that support the current material. Students who skipped routine homework frequently had to spend too much time on basic technique to attend properly to the deeper material in good workshop problems. Often students do not do homework if it is not graded.

A different kind of mathematical thinking is emphasized in the Introduction to the Design and Analysis of Experiments course at Mount Holyoke College. This course annually attracts 20-25 students, mostly biological science majors and some statistics majors. The prerequisite is a semester of college-level work in mathematics or statistics. The course uses the text *An Introduction to the Design and Analysis of Experiments* by George Cobb. The goals of the course are to teach students (i) to choose sound and suitable design structures; (ii) to recognize the structure of any balanced design built from crossing and nesting; (iii) to explore real data sets using a variety of graphs and numerical methods; (iv) to assess how well the standard assumptions of analysis of variance (ANOVA) fit a data set, and if the fit is poor, to choose a suitable remedy such as transforming to a new scale; (v) to decompose any balanced data set into ’overlaysâ? (components corresponding to factors in the design) and to find the parallel decompositions of the sums of squares and degrees of freedom; (vi) to construct the interval estimates and F-tests of formal inference; and (vii) to interpret numerical patterns and formal inferences in relation to the relevant applied context.

Project CALC, developed by David Smith and Lang Moore, Duke University, is a curriculum, funded by the National Science Foundation, which produced materials for a three-semester calculus course that emphasizes real-world problems, hands-on activities, discovery learning, writing and revision of writing, teamwork, intelligent use of available tools, and high expectations of students. Available materials include laboratory manuals using Derive, Maple, Mathematica, Mathcad, and HP48G/GX that accompany the basic texts: *Calculus: Modeling and Application* and *Calculus: Modeling and Application Multivariable Chapters*. A research study comparing Project CALC students and students in a traditional calculus course indicated that the Project CALC students outperformed the traditional students overall.

At the University of Rochester, students can choose from among several calculus sequences including the honors calculus sequence, part of the university-wide Quest program. The aim of the honors sequence is to introduce students to how research mathematicians think and their techniques for solving problems. It designed for all talented students who want mathematical insight to be part of their intellectual toolbox. The four-semester sequence awards 20 academic credits (instead of the usual 16) and includes the material for a fifth course ’ linear algebra. For non-mathematics majors this accelerated sequence is often the only way they can fit linear algebra into their programs. A second Quest calculus sequence emphasizes problem solving, de-emphasizes theory, and omits the extra linear algebra material. The lecture sections are small, and some are oriented toward students with a special interest in a related field, such as physics or biology.

See also the description of inquiry-guided learning courses at North Carolina State University in Part 2, Section C.1.

**Writing in Introductory and Service Courses**

In responding to recommendations about linear algebra in the core curriculum, Steve Friedberg wrote in *Confronting the Core Curriculum* (p. 72) ’In every major mathematics course at Illinois State University, students are given Guidelines for Writing Mathematics ’ a list of rules for good mathematical writing. In the sophomore linear algebra course students are given weekly written assignments. For example, they may be asked to prove or disprove that a given subset of **R*** ^{n}* is a subspace, that a subset of a linearly independent set is linearly independent, or that the trace of

Six week-long writing assignments have been a main learning and assessment tool in the sophomore-level linear algebra course taught by Douglas Kurtz at New Mexico State University. Approximately every other week students are assigned a group project by means of which they are to learn part of the core material of the course. The assignments begin by asking students to learn certain essential material by reading the textbook. Groups of three or four students work to explain key points from the reading, respond to a set of questions about it, and solve an applied problem based on it. For example, after a reading on null spaces and column spaces, one assignment asks ’Suppose that *Nul* A={0}. If A**x**=b is consistent, how many solutions does it have? Explain your answer and give an example to demonstrate this situation. Be sure to explain why your example demonstrates the situation.â? This problem comprises about 20% of that assignment. In the weeks these assignments are due, class periods are used as labs with students working in their groups. This allows Kurtz to watch students working together, answer questions, and insure that every group has had time to meet. The six assignments count for 60% of the course grade, and material from them appears on both the midterm and the final examinations. A group’s grade is based on the extent to which the problems are solved and the work is explained in a clear and well-organized manner. The use of this type of writing assignment was originally developed for calculus classes; a modified version has been used in mathematics classes for elementary education majors.

See Part 1, Section 2 and Part 2, Section C.1 for additional items about developing mathematical thinking and communication.

*Some courses in statistics and discrete mathematics should be offered without a calculus prerequisite;**Three-dimensional topics should be included in first-year courses;**Prerequisites other than calculus should be considered for intermediate and advanced non-calculus-based mathematics courses.*

Various colleges and universities are drawing students in to mathematics majors and minors through entry paths other than the traditional three semesters of calculus. For instance, at Macalester College discrete mathematics has no prerequisite and can be used as a prerequisite for both linear algebra and number theory. At __Oberlin__ and __Lake Forest__ Colleges, one semester of calculus is the prerequisite for discrete mathematics and discrete mathematics can be used as the prerequisite to linear algebra. In addition, at Lake Forest College, the discrete mathematics course, entitled ’Introduction to Abstract and Discrete Mathematics,â? emphasizes techniques of proof and can serve as the prerequisite for number theory, modern algebra, real analysis, complex analysis, probability, combinatorics, and cryptography. At the University of Portland, discrete mathematics also has one semester of calculus as the prerequisite. After taking it students may continue directly to number theory or real analysis or topology. Students at Juniata College are allowed to take linear algebra with only a high school mathematics background, and students may take abstract algebra after completing linear algebra and a mathematical foundations course. At DePaul University the two-quarter discrete mathematics sequence, which does not have calculus as a prerequisite, can in turn be used as the prerequisite for three further courses: applied linear algebra, combinatorics, and number theory. In addition, the combination of discrete mathematics and either number theory or linear algebra is one of the ways to satisfy the prerequisite for abstract algebra.

The report FIPSE Courses at Mount Holyoke College discusses how seven advanced courses were reworked in order to reduce the prerequisites to at most two semesters of college-level mathematics. The goal was to attract students who were not mathematics majors, but who would enjoy and be able to use some of the ideas encountered in traditional junior-senior level courses. The article *Do We Need Prerequisites *by D. O’Shea and H. Pollatsek offers additional discussion on the topic of reducing prerequisites. The Mount Holyoke course Discrete Markov Chain Monte Carlo examines contemporary topics and applications with a prerequisite of only linear algebra. More information on this course is given in Part 2, Section C.3.

**Including 3-Dimensional Topics in the First Year.**

The Colorado School of Mines has a three-course calculus sequence that includes multivariate topics in the second course. In supplementary Academic Excellence Workshops, held one evening a week, participants collaboratively solve worksheets that cover current course material.

The three-course calculus sequence at the University of Redlands also includes multivariable topics in the second course. A discussion of the issues that led to the decision to revamp the sequence in this manner is given in Report for West Point Core Curriculum Conference in Mathematics by Janet Beery, Portia Cornell, Allen Killpatrick, and Alexander Koonce.

Grinnell College offers a two-course calculus sequence with the following topics in the second course: functions of more than one variable: partial and total derivatives, multiple integrals, vector-valued functions, parametrized curves, and applications to differential equations.

An alternative to a major revision of the order of topics in the calculus sequence is to revise the sequence enough to allow students to enter Calculus III (multivariable topics) immediately after Calculus I (differentiation and basic integration). According to faculty member Harriet Pollatsek at Mount Holyoke College it is possible for any solid student who requests moving from Calculus I to Calculus III to do so, and this way of sequencing the courses is not uncommon, especially for economics majors.

*A solid knowledge’at a level above the highest grade certified’of the following mathematical topics: number and operations, algebra and functions, geometry and measurement, data analysis and statistics and probability;**Mathematical thinking and communication skills, including knowledge of a broad range of explanations and examples, good logical and quantitative reasoning skills, and facility in separating and reconnecting the component parts of concepts and methods;**An understanding of and extensive experience with the uses of mathematics in a variety of areas;**The knowledge, confidence, and motivation to pursue career-long professional mathematical growth.*

Guidance to Colleges and Universities

To guide colleges and universities in the mathematical preparation of future teachers, the professional mathematical societies have issued recommendations in the publications *A Call for Change* (MAA, 1991), *Professional Standards for Teaching Mathematics* (NCTM, 1992), and the *Mathematical Education of Teachers (MET*) (CBMS, 2001). *MET* contains detailed recommendations for prospective teachers in grades K-12 focusing on two general themes: (1) the intellectual substance of school mathematics; and (2) the special nature of the mathematical knowledge needed for teaching. The report from the CRAFTY Curriculum Foundations Workshop for Teacher Preparation: K-12 Mathematics recommends the following additional references for those involved in planning teacher preparation programs: *Knowing and Teaching Elementary Mathematics* (Ma, 1999), *The Teaching Gap* (Stigler & Hiebert, 1999), and the National Research Council’s review of research on learning and teaching mathematics in Grades K-8 in *Adding It Up: Helping Children Learn Mathematics *(National Academy of Science, 2001).

Faculty at Virginia Commonwealth University are leading a 10-college effort to strengthen programs and prepare many more mathematics teachers at the middle-school level through the Virginia Collaborative for Excellence in the Preparation of Teachers. Faculty are also involved in a number of other collaborative efforts, including the Virginia Mathematics and Science Coalition and the publication of the Journal of Mathematics and Science: Collaborative Explorations.

The Michigan Mathematics Teacher Educators are a new group, formed in 2003, to promote high quality math education at all levels by drawing together educators to share information and discuss issues involved in teacher preparation programs and K-12 curriculum development projects. The group sponsors annual day-long conferences and encourages on-going collaboration among participants.

**Programs for Elementary Teachers**

In recent years a number of states have increased the mathematics requirements for prospective elementary school teachers. For example, in Virginia future teachers seeking a license from the state are required to have completed at least 12 credits of college-level mathematics. While universities may submit programs for approval with less mathematics, most require at least this minimum. For example, at Longwood University in Virginia all students earn a K-6 license through the Liberal Studies Major, in which they are required to complete a general education mathematics course followed by nine credits of mathematics at the upper level. These courses cover numeration systems, including ancient systems, base systems, and the real number system; geometry, including coordinate and transformational geometry; probability and statistics; and functions. Problem-solving skills are stressed throughout, and manipulatives and technology are emphasized.

At the University of Northern Colorado the program for students planning to become elementary school teachers leads to a B.A. in Interdisciplinary Studies with a Liberal Arts (Elementary Education) Emphasis with one of nineteen areas of concentration, one of which is mathematics. The mathematics concentration includes two courses (6 credits) in the required general education program, one course (4 credits) in the core program, an additional five courses, and a directed study (17 credits).

**Programs for Middle School Teachers**

At Western Kentucky University students seeking middle school certification in the single field of mathematics must complete at least 30 semester hours of work in mathematics, including either trigonometry or college algebra and trigonometry, two semesters of calculus, two semesters of mathematics for elementary teachers, statistics, introduction to linear algebra, geometry, problem solving for elementary and middle school teachers, history of mathematics, and teaching middle school mathematics. If a student begins his or her college mathematics courses at a level higher than the early courses, the department may grant permission to substitute higher level mathematics courses. Students seeking middle school certification for teaching both mathematics and a second field must complete at least 25 semester hours in mathematics.

At James Madison University the middle school mathematics certification program is built around the mathematics/science concentration in the Interdisciplinary Liberal Studies (IDLS) academic program. The 21 required mathematics credits include 9 credits of lower-division mathematics courses, designed according to *MET* guidelines for the mathematical preparation of elementary teachers. The remaining 12 credits are based on the guidelines for additional mathematics preparation for teachers in grades 6-8 and were developed with the support of a grant from the National Science Foundation. They include upper-division courses in algebra, geometry, probability and statistics, and discrete and continuous calculus. Taking an additional course leads to a state-approved algebra I add-on endorsement. The Essential Mathematics for Middle School Teachers website describes the program and contains links to a number of additional resources for programs for middle school mathematics specialists. Students obtaining certification in certain other areas may obtain state endorsement to teach Algebra I by taking 24 credits, including Elementary Functions, Fundamentals of Mathematics I and II, Mathematical Problem Solving, Principles of Algebra, Principles of Geometry, Principles of Analysis, and Principles of Probability and Statistics.

East Carolina University's (ECU) middle school mathematics teacher preparation program was among four honorees in the first year of the Department of Education’s National Awards Program for Effective Teacher Preparation. East Carolina's program includes a year-long internship, ongoing guidance from clinical faculty, and required courses that focus specifically on middle school preparation and that blend theory with practice. Candidates prepare portfolios during their internships illustrating how effective they are at teaching to the state standards. The courses in ECU's 27-hour mathematics concentration are algebraic concepts and relationships, data analysis and probability, Euclidean geometry, discrete mathematics, pre-calculus concepts and relationships, and a two-semester sequence in calculus for the life sciences. In addition, an applied mathematics via modeling course and a teaching of middle grades mathematics course provide a ’look backâ? at the mathematics in the concentration through the vehicle of mathematical modeling and a ’look aheadâ? at continued development of the pedagogical content knowledge of middle-grades mathematics.

At SUNY-Fredonia, prospective middle school mathematics teachers participate in a certification program for grades 5-9 called the Middle Childhood Education Mathematics Specialist Program (click on ’Curriculumâ?, ’Programs in Mathematics,â? and then ’Mathematics: Middle Childhood Educationâ?). Students pursuing this degree take two semesters of calculus, discrete mathematics, linear algebra, geometry, history of mathematics, statistics, the three-semester sequence of mathematics courses for the elementary teachers (of which the third semester is only for those concentrating in mathematics), Senior Seminar, Reading and Writing Mathematics (a course intended both for these students and for the secondary mathematics education students), a mathematics middle school teaching methods course, a laboratory science course, a general course on teaching in the middle school, several other education courses, and student teaching.

Portland State University offers a minor in middle school mathematics. The minor program consists of eleven courses: Foundations of Elementary Mathematics I, II, III, Computing in Mathematics for Middle School Teachers, Experimental Probability for Middle School Teachers, Problem Solving for Middle School Teachers, Geometry for Middle School Teachers, Arithmetic and Algebraic Structures for Middle School Teachers, Historical Topics in Mathematics for Middle School Teachers, Concepts of Calculus for Middle School Teachers, and one approved elective course.

Ira Papick, University of Missouri, worked with several other mathematicians and mathematics educators to develop four foundational mathematics courses, with accompanying support material, for middle school teachers in the Connecting Middle School and College Mathematics (CM)^{2} project. The courses are intended to provide middle grade mathematics teachers with a strong mathematical foundation and to connect the mathematics they are learning with the mathematics they will be teaching. Courses focus on algebraic and geometric structures, data analysis and probability, and the mathematics of change, and employ standards-based middle-grade mathematics curricular materials as a springboard to explore and learn mathematics in more depth. For instance, in the algebraic structures course, students are introduced to fundamental notions in number theory (e.g., greatest common divisor, least common multiple, the Euclidean algorithm) through problems taken from NSF-funded middle school curricula.

The University of Northern Colorado offers a B.A. in Mathematics with Emphasis in Middle School Teaching. Students in this program take 43 credits of mathematics courses, 40 hours of general education courses and have 42 hours of teaching experience.

At Virginia Commonwealth University, students preparing to teach mathematics at the middle school level may choose interdisciplinary science as their academic major. The major includes 30 hours of mathematics: one semester of calculus, linear algebra, mathematical structures, mathematical modeling, statistics, and contemporary applications of mathematics, applied abstract algebra and geometry. Students also complete approximately 20 hours of science; methods courses in teaching mathematics and science and two interdisciplinary mathematics and science courses designed for prospective middle school teachers. This program was developed as part of Virginia Collaborative for Excellence in the Preparation of Teachers and the follow-up project Preparation of Middle School Mathematics and Science Teachers in Virginia.

An extensive set of notes by Carl Lee and Jakayla Robbins were written for use in the geometry for middle school teachers course at the University of Kentucky.

Slides from talks, handouts, and URLs from the 2007 MAA session ’Content Courses for the Mathematical Education of Middle School Teachers,â? featuring 18 presentations and organized by Laurie Burton, Maria G. Fung and Klay Kruczek, are available through the given link.

The Indiana University Indiana Mathematics Initiative Partnership is designed to establish linkages between Indiana University's pre-service program and nine urban school districts in Indiana to enhance the ability of the districts to both attract and retain qualified mathematics teachers.

**Research to Practice ’ Elementary and Middle School Teachers**

In 2001, Suzanne Wilson, Robert Floden, and Joan Ferrini-Mundy published *Teacher Preparation Research: Current Knowledge, Gaps, and Recommendations* for the U.S. Department of Education and the Office for the Educational Research and Improvement. The report was a product of the Center for the Study of Teaching and Policy in collaboration with Michigan State University. The purpose of this report was ’to summarize what rigorous, peer-reviewed research does and can tell us about key issues in teacher preparation.â? This executive team examined over 300 published research reports on teacher preparation and found 57 that met its criteria for inclusion. Five major questions were addressed:

* What kinds of subject matter, and how much of it, do prospective teachers need?

* What kinds of pedagogical preparation, and how much of it, do prospective teachers need?

* What kinds, timing, and amount of clinical training (’student teachingâ?) best equip prospective teachers for classroom practice?

* What policies and strategies have been used successfully by states, universities, school districts, and other organizations to improve and sustain the quality of pre-service teacher education?

* What are the components and characteristics of high quality alternative certification programs?

The report suggested that the research base on teacher preparation is not sufficiently well developed and that continued research in the area is needed to inform future practice.

The conference Critical Issues in Education: Teaching Teachers Mathematics was held at the Mathematical Sciences Research Institute (MSRI) May 30-June 1, 2007. Previous MSRI workshops have addressed issues related to Assessing Students’ Mathematics Learning, The Mathematical Knowledge Needed for Teaching, and Raising the Floor: Progress and Setbacks in the Struggle for Quality Mathematics Education for All. Building on the issues investigated in these previous workshops, materials from which are also available on the MSRI website, this workshop focused concretely on courses, programs and materials that aim to increase teachers’ mathematical knowledge for teaching. The following questions guided the workshop design:

* How can courses and programs for teachers be designed and structured so as to increase teachers’ mathematical knowledge for teaching? What are the features, strengths, and drawbacks of different approaches?

* What materials exist for use in teaching teachers mathematics? What are their features? What is known about their usability and effectiveness?

* Who teaches teachers mathematics and how is their work supported? What can we learn about the factors that enhance the teaching of mathematics for teachers?

The program contains abstracts for most presentations, and the main website for the conference has a link to a zip file with all the collected presentation materials. Talks from the workshop are available on streaming video through another linked website. (Scroll down the list of videos for Spring 2007 to the ones from the conference.)

The Center for the Study of Mathematics Curriculum (CSMC) is a consortium consisting of Michigan State University, University of Missouri-Columbia, Western Michigan University, University of Chicago, Horizon Research, Inc., Columbia MO Public Schools, Grand Ledge Public Schools, and Kalamazoo MI Public Schools, funded by the National Science Foundation. ’The CSMC serves the K-12 educational community by focusing scholarly inquiry and professional development around issues of mathematics curriculum. Major areas of work include understanding the influence and potential of mathematics curriculum materials, enabling teacher learning through curriculum material investigation and implementation, and building capacity for developing, implementing, and studying the impact of mathematics curriculum materials.â?

The Integrating Mathematics and Pedagogy (IMAP) project at San Diego State University is a federally funded Interagency Education Research Initiative (IERI) designed to integrate information about children's thinking about mathematics into mathematics content courses for college students who intend to become elementary school teachers. The IMAP researchers are studying how students' participation in a Mathematics Early Field Experience (MEFE) affects their beliefs about mathematics, its teaching and learning, and their mathematics content learning. The primary research questions are:

* Might integrating content with pedagogy change initial beliefs that are in conflict with the present consensus on how mathematics is learned and how it should be taught?

* How do beliefs affect what is learned in content courses?

* Do these effects depend on whether the integration of content and pedagogy occurs early in the prospective teachers' program or late in the program?

* How do variables such as age and experience with children affect the findings?

* Is the use of software programs developed for children and aimed at developing conceptual understanding, then used in both content courses and in early field experiences, effective in helping prospective teachers focus on conceptual learning and understanding?

Because of the size of San Diego State University’s teacher preparation program, opportunities exist to investigate these questions experimentally with large groups of students.

Reconceptualizing Mathematics: Courseware for Elementary and Middle Grade Teachers is a research-based curriculum development project from San Diego State University that focused on the development of courseware and instructional materials for mathematics courses for prospective elementary and middle school teachers, and for the professional development of practicing teachers. Additional links: an Online Video of a talk by Judith Sowder about the project at the Mathematical Sciences Research Institute; a syllabus for a course that uses materials developed by the project from Andrea Levy at Seattle Central Community College; an article The Mathematical Education and Development of Teachers by Judith T. Sowder, San Diego State University, which examines the question of teacher education from a broad perspective.

Sybilla Beckmann, University of Georgia, developed the materials that led to her textbook *Mathematics for Elementary Teachers* over a period of years of experimentation in her courses and one year of teaching mathematics to a class of sixth graders. Her website contains advice on teaching the course. Some sample syllabi for courses using the book are from Perla Myers, University of San Diego, course 1 and course 2; Art Duval, University of Texas at El Paso; Ben Ellison, University of Wisconsin.

Thomas H. Parker and Scott J. Baldridge developed two textbooks to be used in conjunction with elementary school texts from Singapore. Elementary Mathematics for Teachers and Elementary Geometry for Teachers. The Singapore books are used as a source of problems and to repeatedly illustrate several themes, including: (a) How the nature of a mathematics topic suggests an order for developing it in the classroom. (b) How topics are developed through "teaching sequences" which begin with easy problems and incrementally progress until the topic is mastered. (c) How the mathematics builds on itself through the grades. A syllabus for a course that uses the book is from Loren Spice at the University of Michigan.

**Programs for Mathematicians Teaching Future Teachers**

The Mathematical Association of America (MAA) conducts a National Science Foundation-funded, multi-dimensional program Preparing Mathematicians to Educate Teachers (PMET) to help meet the need for better preparation of the nation's mathematics teachers by mathematics faculty. The PMET program has four major components:

· *Faculty Development* - Workshops and mini-courses help mathematicians to be better prepared to provide high-quality mathematical education to teachers.

· *Information and Resources* - PMET is providing the mathematics community with information about the mathematical education of teachers by multiple means, including talks, articles, and websites with course resources.

· *Regional Networks* - PMET is building an infrastructure of regional networks to help initiate, support and coordinate efforts at individual institutions to improve the mathematical education of teachers. Initially, PMET is concentrating activities in five states-- California, Nebraska, New York, North Carolina, and Ohio --in order to build model networks.

· *Mini-grants* - PMET supports efforts by mathematicians at individual institutions to improve their teacher education programs and to develop new instructional materials. A listing of mini-grant awards and progress reports is on the PMET website.

The goal of the Appalachian Collaborative Center for Learning, Assessment, and Instruction in Mathematics (ACCLAIM) is to build a mathematics infrastructure in the Appalachian regions of Kentucky, Ohio, Tennessee, and West Virginia, in order to provide a model and resources for other isolated, rural, poverty-stricken areas across the country. ACCLAIM works to (1) build mathematics capacity and expertise through advanced degree programs in mathematics, job-embedded professional development for middle and high school mathematics, and research that connects mathematics and rural education, and (2) improve the quality of mathematics teacher education programs and mathematics teaching at the middle and high school levels in these Appalachian regions through the development of collaborative networks and innovative delivery systems. ACCLAIM receives funding from the National Science Foundation.

The Institute for Mathematics and Science Education (IMSE) at the University of Illinois at Chicago (UIC) was established to promote UIC efforts to improve pre-college and undergraduate education in the areas of mathematics and science. Funding is primarily from the National Science Foundation. The project involves collaboration of research mathematicians and scientists with education researchers and teachers in four areas: curriculum development, professional development and outreach efforts, faculty involvement in education improvement initiatives, and research. Parts of the project join faculty at UIC with faculty at five Chicago-area universities and six community colleges. Some of the other activities include a Secondary Math Improvement Project and Family Math Leadership Workshops.

Additional collaborative efforts in Virginia, Virginia Collaborative for Excellence in the Preparation of Teachers and Preparation of Middle School Mathematics and Science Teachers in Virginia are in the ’Preparing Middle School Teachersâ? section above.

Videos are increasingly being used in teacher preparation classes. The following are some of the sites offering resources:

* The Mathematics Teacher Education Resource Place: Video Resources

* __Northwest Regional Educational Laboratory: Publications and Video__

* __Eisenhower National Clearinghouse: Exploring Teacher Practice __

* __Teaching Mathematics in Seven Countries: TIMSS Video Study__

* __Mathematics Teaching and Learning to Teach Project__

* __Computing Technology for Math Excellence: Professional Development__

* __Heinemann__

* __The Math Forum @ Drexel__

Video clips can be very powerful, but several authors suggest that clips cannot stand on their own (e.g., Arcavi & Schoenfeld, April 2003; Ball & Findell, 2001; Lampert & Ball, 1998; Seago, 2000). The following serve as resources to assist mathematics departments in the implementation of video for the education of future teachers:

* Arcavi, A. & Schoenfeld, A. H. (April 2003). Using the unfamiliar to problematize the familiar. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL.

* Dede, C. et al. (2005). An Overview of Current Findings From Empirical Research on Online Teacher Professional Development. Harvard Graduate School of Education.

* Ball, D. & Findell, B. (2001). Video as a delivery mechanism. In Gail Burrill (Ed.), *Knowing and learning mathematics for teaching *(pp. 98’103). Washington, DC: National Academy Press.

* Lampert, M. & Ball, D. L. (1998). *Mathematics, teaching, and multimedia: Investigations of real practice.* New York: Teachers College Press.

* Seago, Nanette M. (2000). Using video of classroom practice as a tool to study and improve teaching. In *Mathematics education in the middle grades: Teaching to meet the needs of middle grades learners and to maintain high expectations *(pp. 63’74)*. *Proceedings of a National Convocation and Action Conference. Mathematical Sciences Education Board, National Research Council. Washington, DC: National Academy Press.

* Nemirovsky, R. and Balvis, A. (2004). Facilitating Grounded Online Interactions in Video-Case-Based Teacher Professional Development. *J. of Sci. Ed. & Tech.,* (13) 1.

Additional information and resources for high school mathematics teacher preparation are contained in Part 2, Sections C (for general mathematics major resources) and D.1 (for specific resources for majors preparing to become high school teachers). Information about the use of technology for visualization and exploration is contained in __Part 1, Section 5.__