The University of Arizona has begun implementing initiatives and programs to address the critical state of mathematics education and teacher preparation in Arizona and in the country. This presentation focuses on the "Four R's": Recruitment of students, Retention of teachers, Reward for faculty, and Reform of the curriculum. In particular it examines:
The oncampus Center for Recruitment and Retention of Mathematics Teachers.  
The University program for promotion and tenure of mathematics faculty whose contributions are in education, scholarship, and outreach.  
The reform curriculum for secondary school mathematics teachers. 
The University of Arizona is establishing a Center for Recruitment and Retention of Secondary Mathematics Teachers in the Department of Mathematics. The Center is in response to a critical shortage of qualified secondary mathematics teachers. This crisis is national in scope; experienced teachers are retiring or quitting in droves; it is estimated that 35% of new mathematics teachers leave the profession within five years and 30% of University students who get a teaching degree in mathematics do not enter the classroom at all, but opt instead for the higher salaries of other professions.
We mirror the national situation here in Tucson. Many of our finest teachers are retiring after thirty years of service. They not only leave a gap in personnel but also a gap in leadership. Fewer students are preparing for mathematics teaching positions; ten years ago the University of Arizona would provide the school system with at least 15 new mathematics teachers per year; this year we have only three on track to graduate.
The Center for Recruitment and Retention will address this emergency on four different fronts.
Attract new teachers. An allout attempt will be made to identify and encourage promising high school students to strongly consider a career in mathematics teaching. Our plans range from honoring such students and their teachers at an annual banquet at the University to providing tutoring opportunities, to offering scholarship money when they enter the University.  
Keep prospective teachers interested. Once students have shown an interest in mathematics teaching we will provide opportunities for them at the University to hone their skills. We will establish contacts between future teachers and the local schools. The students can work as classroom aides during the regular school day, as their schedule permits. The students can help with tutoring in the afternoons. Any of these possibilities could offer the students university credit or tutoring wages. In addition we will provide full scholarships for deserving university students.  
Mentor student teachers. Once students in the teacher preparation program reach their last term at the University they go into the schools to student teach. It is imperative that this be a good experience. Therefore students must be carefully paired with outstanding teachers in the districts. And the progress of the student during this crucial semester must be carefully monitored.  
Help current teachers. After students leave the University and begin their careers they need support; the first few years are crucial to their development. One such activity is linking together experience teachers with new teachers. This linkage will provide support of all types: from pedagogical, educational, mathematical, to just plain morale boosting. We will encourage teachers to attend professional meetings by providing financial support. We plan on providing an electronic hookup where teachers can communicate questions of a mathematical or pedagogical nature and get immediate feedback. We will also provide tuitionfree scholarships for current teachers who wish to take more mathematics classes. 
On the recruiting front we are piloting a program employing 20 tutors from the University and Pima Community College and 15 cooperating teachers from two middle schools and six high schools. The tutors are taking a one credit hour course and receiving $10 per hour for their tutoring. In the spring semester we begin a second program placing as many as 40 high school students in middle schools and high schools. In the second year we hope to expand our programs even further, perhaps involving as many as 80 tutors and 40 coordinating teachers. This is not only an initiative to help students and teachers in the schools but it is also a recruiting tool. There are many university students who have an aptitude for mathematics and who love to help others. We want to give them the opportunity to experience what it is like to use their talents in a meaningful way.
On the retention front we are initiating a program this November where new teachers and experienced teachers can link together. The goal of this project is to link exemplary teachers with teachers new to the profession in an effort to provide meaningful professional development opportunities and positively impact the retention of both groups. New teachers will gain much needed support and exemplary teachers will benefit from the recent knowledge the new teachers bring from their recent education courses. Fifteen first and second year mathematics teachers have been identified and have already been paired with mentors. We hope that this will alleviate the exodus from the classroom by first and second year teachers.
We also have a large scholarship initiative to attract young students into teaching. This fall we have secured funding to provide 13 scholarships to juniors and seniors at the University who are considering a teaching career. In the future we hope to have as many as 25 scholarships and include high school seniors and university freshmen and sophomores as recipients.
Finally, we have secured funding so that fifteen teachers can attend the annual NCTM meeting in Las Vegas in April 2002.
Let us examine in more detail our two major initiatives: the tutoring program and the
mentor teacher/new teacher collaborative program. These recruiting and retention efforts require thorough and extensive planning. The following steps are being planned for next year for the two programs. The duties listed are for the codirectors, the teacher coordinator at the site (for tutoring) and the new teachers and the mentor teachers (for collaboration).
Month  Responsible Party 
Event 
May 
Codir 
Develop applications for middle and high schools 
May 
Codir 
Develop applications for college tutors 
May 
Codir 
Develop applications for high school tutors 
Aug 
Codir 
Advertise program in middle and high schools 
Aug 
Codir 
Advertise program to college students 
Aug 
Codir 
Develop a prequestionnaire for participating tutors 
Sept 
Codir 
Screen middle and high school applications and select participating schools 
Sept 
Codir 
Notify participating schools 
Sept 
Codir 
Advertise at selected schools for a teacher coordinator 
Sept 
Codir 
Select teacher coordinators at each participating site 
Sept 
Teach Coor 
Advertise program at sites 
Sept 
Teach Coor 
Arrange locations and times for tutoring 
Sept 
Teach Coor 
Screen high school tutor applications and select tutors 
Sept 
Codir 
Screen college tutor applications and select participants 
Oct 
Codir, 
Train Tutors 
Teach Coor 

Oct 
Codir 
Assign college students to participating sites 
Oct 
Teach Coor 
Identify students to be tutored 
Oct 
Teach Coor 
Determine schedules of tutors 
OctMay 
Teach Coor 
Monitor tutoring 
OctMay 
Tutors 
Tutor students 
NovMay 
Codir 
Monitor tutoring through school visits 
Mar 
Codir 
Develop post questionnaire for tutors 
Mar 
Codir 
Develop and distribute project evaluation forms to sites 
May 
Teach Coor 
Distribute and collect tutor questionnaires 
Jun 
Codir 
Summarize and analyze evaluations 
Month 
Responsible 
Event 
April 
Codir 
Advertise program to middle and high school teachers 
April 
Codir 
Develop applications for new and mentor teachers 
MayAug 
Codir 
Select new teachers 
MayAug 
Codir 
Select mentor teachers 
Aug 
Codir 
Mentor teacher training 
Aug 
Codir 
Meet with new teachers 
Aug 
Codir 
Develop application for teacher teams to attend local, regional and/or national 
conferences, distribute applications 

August 
Codir 
Select and notify recipients for conference attendance 
SeptApril 
Mentor Te, 
Attend conferences 
New Te 

Sept, Nov 
Codir 
All day inservices with new and mentor teachers 
Jan, Mar 
Mentor Te, 
Make site visits to new teachers 
Codir 
" 

Oct, Nov 
" 
" 
Feb, April 
" 
" 
Oct, Jan 
Mentor Te 
Mentor teachers observe new teachers 
Year long 
New Te 
New teachers observe exemplary teachers 
March 
Codir 
Develop evaluation forms of project for mentor teachers 
March 
Codir 
Develop evaluation forms for new teachers 
April 
Codir 
Distribute evaluation forms to new and mentor teachers 
May 
Codir 
Summarize and analyze data from evaluations 
This is an expensive undertaking. Setting up the Center requires finding space, hiring directors and staff, and providing for operating funds. We have an office in the mathematics building and we have hired two codirectors and they, in turn, have hired a supervisor for the mentor teachers. The tutoring initiative and the collaboration initiative cost a lot of money but we are intent on paying good salaries because our mission is to bring class, dignity and respect to the teaching profession. We have begun operations thanks to generous gifts from private individuals. These funds, along with some state funds, have given the Center the chance to carry out its plans for the first year and even begin outlining plans for the second year. We have also applied for grants to help us through the first few years. The Center has the support of the University Administration. While it is not yet incorporated into the University structure we have been given assurances that if the Center can make a tangible difference in the next few years, the University will provide necessary resources and fund it long term. We are hopeful that the work of the Center will make a clear difference in the near future so that we can continue this work.
For more details contact Fred Stevenson: frstv@math.arizona.edu or the codirectors:
Sue Adams at: adams@math.arizona.edu
Ann Modica at: modica@math.arizona.edu
In this section we describe the mechanism that the University of Arizona has in place to recognize and support faculty member engaged in mathematics and science education activities. We will limit our description to the case of department of mathematics, but the mechanism is for any faculty member in the college of science who plays a substantial role in precollege mathematics or science education (at the end of this section we include the official procedures adopted by the College of Science).
In the eighties the department experienced considerable activity in terms of outreach programs (e.g., teacher enhancement projects). Several faculty members started devoting considerable amounts of time to precollege mathematics education activities. One example is the very successful coop program through which about 6 local school teachers (middle school and high school) spend one year in our department teaching entry level courses and taking courses for their own professional development. In 1987 the department hired a nationally recognized mathematics educator to lead the teacher preparation program and outreach endeavors. A weekly mathematics instruction colloquium was created. The aim of this colloquium is mostly to discuss K12 mathematics education topics. Local teachers regularly attend these colloquia (as well as faculty members and students in the department of mathematics). In 1989, the deans of the College of Science and of the College of Education created a Universitywide Science and Mathematics Education Center (SAMEC). Any faculty member with an interest in precollege mathematics and science education can be part of SAMEC . Through monthly meetings, SAMEC members have an opportunity to talk about the different projects they are engaged in and to network with other individuals with similar interests. SAMEC membership is quite diverse and not restricted to faculty members only. It is open to individuals in the community who work in K12 mathematics/science education. In 1990, the Department of Mathematics hired an assistant professor in Mathematics Education. Her teaching was going to be primarily in the teacher preparation program and her research in traditional mathematics education topics. A mechanism to ensure that such individual (and others in the future) would be evaluated in an appropriate manner was developed. In 1991 the deans of the College of Science and of the College of Education created the Science Education Promotion and Tenure Committee (SEPTC). We include the official procedures at the end of this section. So far, in the Department of Mathematics, under the SETPC system, we have had two promotions to full professor and one promotion and tenure granting. The basic idea behind the SEPTC mechanism is that this committee is formed by individuals with an interest and expertise in mathematics/science education. In all likelihood at least one of these individuals is a faculty member from the College of Education. For example, in the case of the assistant professor hired in 1990, her research is in traditional mathematics education. A faculty member from the College of Education is more likely to be familiar with this line of research and thus be able to evaluate it appropriately.
How does the mechanism work?
A candidate for promotion or tenure submits his/her dossier to SEPTC.  
SEPTC solicits evaluations from appropriate outside and inside referees, including scholars in the appropriate content who have an interest in education, and distinguished educators who have an interest in the appropriate content.  
SEPTC evaluates all materials, prepares a report and recommendation and sends them to the department’s Promotion and Tenure Committee.  
The department Promotion and Tenure Committee prepares its own report based on the College of Science criteria and on the evaluation letters from SEPTC. SEPTC’s report is included in the documents sent to the College of Science Promotion and Tenure Committee.  
The rest of the process follows the same steps as any other case. The main difference is the initial SEPTC step. 
From the Promotion and Tenure and Continuing Status Guidelines
The University of Arizona
PROCEDURES FOR EVALUATION OF FACULTY MEMBERS WHO PLAY A SUBSTANTIAL ROLE IN PRECOLLEGE MATHEMATICS AND SCIENCE EDUCATION (Adopted by the College of Science 7/23/92)
Promotion and Tenure cases of faculty members whose appointments carry a substantial component of responsibility for education of precollege science and mathematics and teachers are also reviewed by the Science Education Promotion and Tenure Committee (SEPTC). The following procedures and the accompanying criteria are meant both to assure high quality scholarship and to assure faculty who choose to participate in such activities that they will be evaluated in an appropriate manner. No faculty member will be evaluated using these procedures without a written agreement between the faculty member and the department head. Department heads are expected to consult with their respective faculties before entering into such agreements.
Procedures: These procedures and criteria are written broadly enough so that some faculty whose primary appointment is in the College of Education may also be evaluated under them when appropriate.
A written agreement should be reached between each individual faculty member and his or her department head as to what percent of the faculty member’s time is to be spent on precollege mathematics or science education.
When the percentage agree to in item 1 is greater than zero, and the faculty member wishes to be evaluated by SEPTC, appropriate papers should be submitted to SEPTC concurrently with or previous to submission of such papers to the Departmental evaluation committee. The percentage agreed to in item 1 should also be communicated to SEPTC.
SEPTC will solicit evaluations from appropriate outside and inside referees. These must include scholars in the appropriate content areas who have interest and expertise in education, and distinguished educators who have interest and expertise in the appropriate content areas.
Before making a formal report, SEPTC will meet with the faculty member to advise him or her about SEPTC’s preliminary evaluation and to consult with the faculty member about possible further information and alternative actions.
SEPTC will evaluate all materials and send them, with SEPTC’s recommendation to the appropriate
department head and evaluation committee. SEPTC’s report will become a part of the permanent record.
Criteria: The purpose of mathematics and science education is to improve the teaching and learning of mathematics and science. Evaluation of faculty members who play a substantial role in mathematics and science education should take into account the impact they have and, are having, and are likely to have, on the teaching and learning of mathematics and science. Both the magnitude of the impact and its direction should be considered.
Written evaluations by distinguished colleagues and others, both within and without the University, will necessarily play an important role in determining the magnitude and the quality of a professor’s impact. Efforts that will be evaluated for mathematics and science education should be directed toward the systematic improvement of science and mathematics education beyond the faculty member’s classroom and advising activities. Examples of such efforts might include: scholarly works that make a contribution toward teaching and learning, innovative textbooks that substantially impact on teaching and learning, leadership in service activities, etc., but in all cases, the magnitude and quality of the impact is the essential issue.
Further evidence of achievement may be found in the initiation and development of educational programs, in the obtaining and managing grant support, in service on advisory and policy boards that have a substantial influence, and in other similar activities.
Traditional categories (research, teaching, service) may be inappropriate for evaluating science and mathematics educators because the lines between the categories are often blurred. If these categories are to be used, however, caution must be exercised to avoid assigning creative scholarly work to the service or teaching category (where it ordinarily receives less weight in the overall process) simply because it is different from traditional research.
Research or its Creative Equivalent: The University of Arizona College of Science Guidelines for Judging Stature and
Excellence in Research (Section II B 1) are appropriate for mathematics and science education, but some of the specifics may
differ from more conventional evaluations within the College of Science.
Worthy contributions could include scholarly books that make a significant contribution, textbooks that are substantially different
from, and better than, previous textbooks (if any) on a worthy subject, articles in refereed, respected journals that describe and
advocate better practice or that present research results relating to learning science and mathematics, improved methods and
instruments for evaluation, computer software, movie or television productions that enhance education, and so on.
No one person, of course, will make contributions in all of these ways, but any of these activities, and many similar ones, should be thought of as legitimate research or creative activities. The quality and impact of the work must be seen as the important issues.
Evaluation committees must consider with some care the actual origin of materials. If a textbook, for example, was designed and developed by employees of the publishing company, the “author” should receive little credit for it. If coauthored articles or books were written largely by the other authors, that fact should be considered. In situations where possibilities of this sort exist, the evaluation committee has an obligation to establish the nature and magnitude of the faculty member’s contribution.
Instruction: In addition to the University of Arizona College of Science Guidelines for Judging Stature and Excellence in Instruction (Section II B 2) special consideration will be given to the development of new and innovative courses, and to the creation of new
courseware or laboratory activities that substantially enhance existing teaching practice. Unusually strong commitment to student
advising (such as being a Faculty Fellow) should be taken into account. It is appropriate to consider the career outcomes of former
students, and to solicit their evaluations of the faculty member.
It is also important to recognize and evaluate activities that impact the quality of science and mathematics teaching in the schools.
This includes inservice training of teachers, and the development of courses or materials that substantially benefit instruction in the
schools.
Service: Reference the University of Arizona College of Science Guidelines for Judging Stature and Excellence in Service (Section
II B 3). Because a major goal of university mathematics and science education is to improve teaching and learning in the schools,
service may carry greater weight in the consideration than it does for other members of the College of Science. Such service may
include scholarly contributions to professional organizations, to government and other agencies, to the University, to the College, to
the Department, to local schools, etc. It may also include speeches and workshops at professional meetings, and similar activities.
There may appear to be some overlap between “research or its creative equivalent” and “service” as used here. Many of the opportunities to provide service on the national or international level may be indicators of a distinguished reputation, and therefore of high quality research and creativity. However, speaking, service, etc., should not be taken ipso facto evidence of research and creativity. The research and other contributions must be considered directly, and the opportunities for service taken as only one indicator of the quality of that research and creative contribution.
The University of Arizona secondary mathematics education program features two core courses: geometry and algebra/number theory. The courses have a lot in common. Both emphasize problem solving, both require students to work on projects during the semester, and both have end of the term presentations.
In the geometry class, the course ends with a “Geometry Fair”. This is a culmination of a month long exploratory project. Students can work by themselves or in groups of two or three. They tackle an openended problem, write up their results, give an oral presentation, assembly a display sciencefair style. The fair is an event complete with food, judges, prizes, certificates, and awards and it is open to all faculty and students to attend.
Here is an example of a project: Find at least one slanted cube inside a three dimensional integer lattice. A cube is slanted if none of its sides are parallel to a coordinate axis. The counterpart to this problem in two dimensions is easy and students start their work here. The smallest slanted square can be constructed by joining the lattice points (1,0), (2,1), (1,2), and (0,1). This fits nicely into the square lattice of points (x,y) where 0 £ x £ 2, 0 £ y £ 2, where x and y are integers. There are plenty of other larger examples to find and there is interesting mathematics in the twodimensional case. But solutions in threedimensions are not easy to come by and there are surprises about the nature of the lengths of the sides of such a cube. )Notice that in the square above the sides are of length Ö2.) Students who choose this project build large cubes with strings within showing the various lattice points.
In algebra/number theory, the presentation also is the culmination of a monthlong effort by groups of students. The students write up their results in a paper and deliver a 30 minute long presentation in front of their colleagues. Nowadays the project is presented in powerpoint.
Here is an example of an algebra/number theory project. Find the smallest number whose consecutive index exceeds 500. The consecutive index of a number n, is the number of different ways that a sequence of consecutive positive integers can add to n. For example 15 = 7+8 = 4+5+6 = 1+2+3+4+5. So the consecutive index of 15 is 3. A further question in this project asks for a formula for the consecutive index of a number. Often students generate data with great quantities of spreadsheets before the decide to try to figure out what is really going on with the numbers.
Both courses have traditional aspects; homework, proofs, hour tests, and final exams, to go along with the projects. Both courses have a miniproject at midterm to go along with the major presentation at the end. The presentation of the final project performs at least three purposes: it gives the students the opportunity to play with mathematical ideas, it encourages students to work together in groups, and it gives the students the opportunity to write and to teach mathematics. All of these activities are essential to the preparation of a future teacher.
The courses are taught from books written by two University Mathematics Professors. The table of contents of each book is below.
Geometry by Discovery, David Gay, University of Arizona, Wiley, 1998, 410 pgs.
Contents
1 Getting Started: Strategies for Solving Problems
The Problem of Five Planes in Space
Strategies for Solving Problems
Baby Gauss and a Formula
A Collection of Problems
A Trick Yields Another Formula
Proofs Without Words
MiniProjects
References.
2 Episodes in the Measurement of Length, Area, and Volume
The Odometer Problem
Making Sense of the Formula for the Circumference of a Circle
The Garden Plot Problem
Area Principles and Strategies
The Circular Garden Plot Problem
Volume Measurement
The Sphere Volume Problem
The Triangular Pyramid Problem
Archimedes’ Solution to the Sphere Volume Problem
Notes
References
3 Polyhedra
What is a ThreeDimensional Counterpart to a Polygon in the Plane?
Regular Polyhedra
Classification of Regular Polyhedra
Possible Regular Polyhedra
Numerical Data Associated with a Polyhedron
Euler’s Formula
An Application of Euler’s Formula
Predicting with Euler’s Formula
The Soccer Ball
Notes
References
4 Shortest Path Problems
The Speaker Wiring Problem
The Problem of the Milkmaid, the River, and the Cow
Paths of Light: Mirrors
The Cowboy Problem
Ellipses
Parabolas
The Three Cities Optical Network Problem
How to Construct the Special Point for the Three Cities Problem
Inscribed Angles in a Circle: A Pause for Review
The Three Cities Problem Concluded
Notes
References
5 Kaleidoscopes
Two Mirror Kaleidoscopes
Good Kaleidoscope Angles
Bad Kaleidoscope Angles: An Analysis
ThreeMirror Kaleidoscope I All Mirrors Perpendicular to a Fourth Plane
Good Triangular Kaleidoscope
Making a Kaleidoscope
Three Mirror Kaleidoscope II: Three Mirrors Meeting in a Point
Notes
References
6 Symmetry
Mirror Symmetry and Transformational Geometry
Mirror Symmetry as a Function
Rotational Symmetry
Kaleidoscopes
Successive Mirror Reflections in Two Parallel Lines
Transformational Symmetry
Products of Mirror Reflections
What Are We? How Did We Get Here?, Where Are We Going?
Products of Three Reflections
Symmetries of Frieze patterns and Wallpaper Designs
Symmetries in Space
Mirror Reflections in Space
Rotations in Space
Symmetries of a Regular Tetrahedron, Part I
Product of a Mirror Reflection With a Rotation
Symmetries of a Regular Tetrahedron, Part II
Symmetries of a Cube
Symmetries of a Regular Tetrahedron and a Cube, Concluded
A ThreeDimensional Kaleidoscope
Notes
References
7 What Shapes are Best?
The Garden Problem
The Storage Problem
More Garden, Storage, and Building Problems to Solve
Related Problems and the Next Step
The Post Office problem
The Cereal Box Problem
The Isoperimetric Problem
Dido’s Problem
Picking the Raisins off the Cake
ThreeDimensional Analogue to the Isoperimetric Problem
Notes
References
8 Symmetry
The Irrigation Problem
The Soup Can Packing Problem
The City Elementary School Partitioning Problem
Jensen’s Inequality
A Proof that the Hexagonal Arrangement is Best
Modeling Natural Phenomena with Arrangements of Circles
The Wine Bottle Packing problem
The Ball Bearing Shipping Problem
Sphere Packing Experiments
A Return to the Ball Bearing Problem
Caps for the Beehive Cells
Body Centered Cubic Packing
Beehives and the Isoperimetric Problem for Spacefilling Polyhedra
The Cubic Close Packing Space Grid
The Cubic Close Packing Space Grid from a Cannonball Perspective
Notes
References
9 Where to Go From Here? Project Ideas
Project Ideas
Resources
Communicating Project Results
NittyGritty on Putting Together an Effective Display
Project Evaluation
The Bottom Line
Exploring the Real Numbers, Frederick W. Stevenson, University of Arizona, Prentice Hall, 365 pgs.
Contents
1 The Natural Numbers
1.1 The Basics
1.2 The Fundamental of Arithmetic
1.3 Searching for Primes
1.4 Number Fascinations
2 The Integers
2.1 Diophantine Equations
2.2 Congruence Arithmetic
2.3 Pell and Pythagoras
2.4 Factoring Large Numbers
3 The Rational Numbers
3.1 Rational Numbers as Decimals
3.2 Decimals as Rational Numbers
3.3 Continued Fractions
3.4 Solving Equations on the Rational Plane
4 The Real Numbers
4.1 Algebraic Representations
4.2 Geometric Representations
4.3 Analytic Representations
4.4 Searching for Transcendental Numbers
5 Mathematical Projects
5.1 Rings of Factors
5.2 Sums of Consecutive Numbers
5.3 Measuring Abundance
5.4 Inside the Fibonacci Numbers
5.5 Pictures at an Iteration
5.6 Eenie Meenie Miney Mo
5.7 Factoring with the Pollard r Method
5.8 Charting the Integral Universe
5.9 Triangles on the Integral Lattice
5.10 The Gaussian Integers
5.11 Writing Fractions the Egyptian Way
5.12 Building Polygons with Dots
5.13 The Decimal Universe of Fractions, I
5.14 The Decimal Universe of Fractions, II
5.15 The Making of a Star
5.16 Making Your Own Real Numbers
5.17 Building 1 the Egyptian Way
5.18 Continued Fraction Expansions of ÖN
5.19 A Special Kind of Triangle
5.20 Polygon Numbers
5.21 Continued Fraction Expansions
For more details contact Fred Stevenson: frstv@math.arizona.edu and
dgay@math.arizona.edu.