This session focused on the preparation of middle school teachers and on the collaborative efforts of three universities to develop programs for these students.

The session started with a review of the first three explorations taken from a course guide for faculty teaching a “Concepts of Calculus for Middle School Teachers” course developed by faculty at Portland State University and other universities in collaboration with The Math Learning Center. (This course guide, and similar guides for courses designed for middle school teachers in Geometry, Experimental Probability and Statistics, Computing in Mathematics, and Problem Solving, are available from The Math Learning Center (MLC). To obtain information on acquiring copies of any of them, contact Tom Schussman at MLC, 1-800-575-8130.)

The calculus explorations were used to illustrate the difference between the calculus course for middle grades teachers compared to a business calculus or standard calculus course. In the first exploration, Rates of Change, students build open boxes from grid paper, compare and graph the heights and volumes, and use the resulting graph to develop the definition of the derivative as a measure of the average and instantaneous rate of change of volume compared to height. The second exploration enables students to develop for themselves the formula for the derivative of a product of functions. Students construct three dimensional card stock models of the product of two different functions of x and then comparing neighboring “slices” of the model to discover how the product changes as x does. The third exploration uses the open boxes  from the first exploration to investigate the average volume of the set of boxes, which leads to a discussion of the average value of a function on an interval. The fourth exploration uses cube models to lead to formulas for summing the positive integers, their squares, and their cubes.

The examples from calculus led to a general discussion of courses for middle grades teachers and how they differ from courses for mathematics majors or typical courses for non-majors.

Marj Enneking described briefly the seven mathematics courses for middle school teachers and the background and development of these courses at Portland State University:

Mth 490/590 Computing in Mathematics for Middle School Teachers
Mth 491/591 Experimental Probability and Statistics for Middle School Teachers
Mth 492/592 Problem Solving for Middle School Teachers
Mth 493-593 Geometry for Middle School Teachers
Mth 494-594 Arithmetic and algebraic Structures for Middle School Teachers
Mth 495-595 Historical topics in Mathematics for Middle School Teachers
Mth 496/596 Concepts of Calculus for Middle School Teachers

She then described the collaboration of faculty and teachers-in-residence at Portland State University, Southern Oregon University, and Western Oregon University to build and/or enhance courses for middle school teachers on all three campuses. The collaboration was fostered by close faculty and teacher connections made through the long standing Oregon  TOTOM (Teachers of Teachers of Mathematics) organization and the Oregon Council of Teachers of Mathematics. It was partially supported by the National Science Foundation project, Oregon Collaborative for Excellence in the Preparation of Teachers (OCEPT). Teachers–in-residence worked with faculty in developing course materials and mentored faculty on teaching and assessment practices. Faculty on one campus taught their course by distance to include students from a second campus in the course in all aspects of the course, including in-class group activities and discussions by students on both campuses, thus enabling faculty on the second campus to learn more about the courses. Faculty from these  three institutions, and several others in the state, met to share ideas and materials and conduct workshops for one another.

Laurie Burton described the over-all mathematics program at Western Oregon University including the mathematics courses taken by students planing to teach in any of Oregon’s four authorization levels of teacher licensure: early childhood, elementary, middle level, and secondary. Students normally take a program leading to two of these authorization levels.   Within this general framework she focused on the courses taken by prospective middle grades teachers and engaged session participants in an algebra activity , using toothpicks:


WOU TEACHER EDUCATION ACADEMIC PROGRAM DESCRIPTIONS

At Western Oregon University [WOU] the College of Education offers undergraduate education programs with four authorization levels:

Early Childhood [Age 3 - Grade 4, Elementary School]
Academic Requirements General breadth in all areas including five quarter-long mathematics classes for pre-service Elementary Education majors [10 semester hours].

Early Childhood/Elementary [Age 3 - Grade 8, Elementary School]
Academic Requirements General breadth in all areas including five quarter-long mathematics classes for pre-service Elementary Education majors [10 semester hours].
Academic Specialization Requirements Students choose two focus areas. Mathematics is a popular focus area choice. Student at this authorization level, specializing in mathematics, take three additional quarter-long mathematics classes for pre-service Elementary
Education majors [6 more semester hours].

Elementary/Middle Level [Grade 3 - Grade 10, Elementary School or Middle School]
Academic Requirements General but increased breadth in all areas including seven quarter-long mathematics classes for pre-service Elementary Education majors [14 semester hours].
Academic Specialization Requirements Students choose one focus area. Mathematics is a popular focus area choice. Student at this authorization level, specializing in mathematics, take four additional quarter-long mathematics classes for pre-service Elementary Education majors [8 more semester hours].

Middle Level/High School [Grade 5 - Grade 12, Middle or High School] .
Academic Requirements BA or BS degree in the chosen subject area.

All of these undergraduate programs include four terms of teacher education curriculum beyond the core academic work.


WOU ELEMENTARY EDUCATION PRE-SERVICE MATHEMATICS COURSES
All pre-service teachers at WOU are required to take the following five mathematics courses:

Foundations of Elementary Mathematics [Three term sequence, each course: 3 quarter hours] Intended for prospective elementary teachers. Introduction to problem solving, sets, whole numbers, number theory, fractions, decimals, percent, ratio and proportion, integers, rational and real numbers. Introduction to probability and statistics, measurement, and geometry.

Manipulatives in Mathematics [3 quarter hours] Using concrete models to teach mathematics. Learning theory from concrete to abstract. Models include Cuisenaire rods, Bean sticks, Wooden Cubes, Geoboards, Decimal Squares, Pattern Blocks and Multibase Blocks.

Elementary Problem Solving [3 quarter hours] Goals for this class are to help elementary teachers become better mathematical problem solvers, to introduce techniques and materials helpful in improving student problem solving abilities, and to suggest ways to organize the curriculum and daily instruction to achieve problem solving goals.

The following courses are either required for authorization level 3 or fulfill the mathematics specialization requirements for authorization levels 2 or 3.

College Algebra for Elementary Teachers [3 quarter hours, required for all students in Authorization level 3] Algebraic skills; solving linear and quadratic equations; inequalities; functions; graphs; systems of linear equations.

Introduction to Abstract Algebra for Elementary Teachers [3 quarter hours] An introduction to abstract mathematics as a structured mathematical system. The system of whole numbers, elementary group theory, and integers are examined. Students are expected to make conjectures and prove them true or false with a deductive proof or counter example. Some elementary logic is also examined.

Probability and Statistics for Elementary Teachers [3 quarter hours] Using basic elements of probability and statistics to solve problems involving the organization, description and interpretation of data. Concrete application will be explored.

Introduction to Geometry [3 quarter hours]  A brief examination of intuitive geometry including construction, basic Euclidean
geometry, proof and measure.

Elementary Integrated Mathematics [3 quarter hours] The study of computational skills, geometry, probability and statistics, data collection and number theory in applied problem solving. Extensive use of group activities, technology, and real-world applications will be used to gain an understanding of the underlying mathematics and an appreciation of the utility and value of mathematics. The goals of the classes are for students to achieve learning to value mathematics, becoming confident in one’s own ability,  becoming a mathematical problem solver, learning to communicate mathematically, and learning to reason mathematically.

Elementary School Mathematics [3 quarter hours] The study of mathematical topics relevant to the elementary and middle school curriculum. All topics will be studied with emphasis on problem solving and use of multiple strategies for solving the problem.

Expanded Description: College Algebra for Elementary Teachers
All Elementary/Middle School pre-service teachers at WOU take this course. Math focus Early Childhood/Elementary pre-service teachers may take this course.

Most of the classwork for this course is done in an exploratory lab setting. The current course features Visual Math [now Math Alive!] & Math in the Mind's Eye materials developed at the Math Learning Center in collaboration with Portland State University  [http://www.mlc.pdx.edu/]. Students work through an algebra packet using black and red tiles to model integers and algebra pieces to model algebraic terms. The development of algebraic thinking in this course is tremendous. Students' use of visual models greatly enhances their overall understanding of the algebraic concepts. Students regularly share solutions in class emphasizing the importance of the clear communication of as well as the understanding of mathematical ideas.

Curriculum Example: College Algebra for Elementary Teachers Session participants investigated the beginning of the Math in the Mind's Eye, Unit IX, Activity 1 lesson: Toothpick Squares: An Introduction to Formulas to experience an introduction into this method of instruction.

In the classroom setting this activity is a transition lesson between modeling integer operations with black and red tiles and modeling algebraic operations with algebra pieces. In this lesson and during the DC Summit session: Students start out by exploring toothpick patterns visually and numerically Students then describe the general pattern using words Students then assign symbols [e.g. S = the number of sections, T = the total number of toothpicks] and translate their words into symbolic expressions.

References [all available from the Math Learning Center]:
“Math In The Mind's Eye (MME) Algebra Collection”
“Algebra Pieces” [includes “Black & Red Tiles”]
The MLC also creates course packs with student pages from MME lessons if you ask for them.

Expanded Description: Abstract Algebra for Elementary Teachers
All Elementary/Middle School and math focus Early Childhood/Elementary pre-service teachers may take this course. All math focus Elementary/Middle School pre-service teachers must take this course.

Most of the class work for this course is done in an active group collaborative setting. Students explore the structure and properties of number systems [whole, integer, rational, real] to gain further depth in their understanding of the fundamental structures and algorithms they will use as an Elementary or Middle School teacher.

In addition to working with real number properties, students are introduced, at a basic level, to the structure of groups. The study of finite groups, in particular the integers modulo n, allows the students to explore the abstract, yet in this case accessible, nature of an algebraic system. The students compare and contrast the abstract structural properties of the finite groups to the additive and multiplicative group structures of integers, rational and real numbers.

Concrete examples and applications are shared with the students to help them to relate to the material. For example, students designed and created clock arithmetic art posters and worked with basic cryptology during the 2000-2001 offering of this course. The 2000-2001  WOU students wrote lesson plans and designed student activity pages, appropriate for a grade 5 - 8, covering the topics of number systems, basic clock arithmetic and basic cryptology [secret decoder rings] for their final course project.

Curriculum Example: Abstract Algebra for Elementary Teachers The following is an excerpt from a module of the course.  Students are introduced through class activity [human clocks] and exploratory group work to clock arithmetic prior to the following
problems:

Mod 4 Addition
As a group, fill out the following chart for addition mod 4 (use your four o’clock clock if it helps)

(Z4, +4)

Closure:
As a group, decide what it would mean for Z4 to be closed under the operation of adding mod 4 (+4). Is (Z4, +4) closed?

As a group, decide what it means in general to say that Zn is or is not closed under the operation of adding mod n (+n). Is (Zn, +n) closed? Explain your answer in detail here (use complete sentences):

Identity:
As a group, decide what it would mean for Z4 to have an identity element under the operation of adding mod 4 (+4). Does (Z4, +4) have an identity element?

Is the identity element that you have found unique? Explain your answer in detail here (use complete sentences):

As a group, decide what it means in general to say that Zn has or does not have an identity element under the operation of addition. What does that mean? Explain your answer in detail here (use complete sentences):

Associativity:
As a group, decide what it would mean for Z4 to be associative under the operation of addition.

Show at least one case of your group's example computations here:

Is (Z4, +4) associative?

As a group, decide what it means in general to say that Zn is or is not associative under the operation of addition. What does that mean? Explain your answer in detail here (use complete sentences):

In this same way, students explore the properties of Unique Closure, Commutativity and Additive Inverses for (Zn, +n), n = 2, 3, 4, …
Students also consider the connection between the additive identity and the pairs of additive inverses. Students then are introduced to the formal definition of a finite group [under the operation of addition]. Students now move to exploring multiplicative structures. Students are comfortable using the notation +n for addition mod n and xn for multiplication mod n.

Mod 4 Multiplication Table:
As a group, fill out the following chart. Use multiplication mod 4.
Students are provided with a mod 4 multiplication table.

Multiplicative Closure, Identity & Inverses:
As a group, change the closure, identity and inverse properties that we have been using for addition, mod 4 to closure, identity and inverse properties for multiplication, mod 4. Write out each property here.

Students then explore these properties in more detail.

As a group, make a conjecture about the possible group structure of (Z4, x4) and write it here:

Students then explore other modular structures and eventually find that (Zp*, xp), p: prime, has the multiplicative group structure they are looking for.

Expanded Description: Geometry for Middle School Teachers
All Elementary/Middle School and math focus Early Childhood/Elementary pre-service teachers may take this course. All math focus Elementary/Middle School pre-service teachers must take this course.

Most of the class work for this course is done in a group activity or computer lab setting. Students in the 2000-2001 course taught by Sandy Kralovec worked though a set of problem-solving oriented geometry materials for Middle School teachers developed by
Michael Arcidiaciano for Portland State University. The students in 2000-2001 course offering were challenged by working with geometry in a problem-solving environment. These students encountered both depth and breadth while developing skills in the geometric content of the course. The students found they needed to investigate a variety of resources to help them with the underlying and fundamental geometric concepts they needed to work with new ideas. Students used technology, in particular, Geometer's Sketchpad, to work through and present some of their work in this course.

Curriculum Example: Geometry for Middle School Teachers
Session participants were given the "Week Eight Assignment" from:  Geometry for Middle School Teachers Course Guide (available from The Math Learning Center)

Expanded Description & Curriculum Example: Probability and Statistics for Elementary Teachers
All Elementary/Middle School and math focus Early Childhood/Elementary pre-service teachers may take this course. All math focus
Elementary/Middle School pre-service teachers must take this course.

Most of the class work for this course is done via exploratory group activities. For example, to understand the chi square relationship students use a spinner to simulate the selection of one of four equal choices. This is repeated 20 times and the chi square value is calculated using a spreadsheet. After each group has done this 10 times. The class has enough data to discuss the probability of any possible outcome. To simulate other activities the class also uses graphing calculators to generate random integers and Excel spreadsheets and Fathom to explore data sets. Following these explorations the students are expected to write articles that present their findings both graphically and verbally. Some data sets are generated in class and some data sets are taken from other sources such as the Internet or the Fathom data set libraries. Students in this course are expected to have a deeper understanding of Z score and fit lines than students in the elementary foundations courses. Most of the work in probability is experimental with some follow-up on the theoretical results.

Marj shared information on the program offered at Southern Oregon University, including course syllabi from two SOU courses.

Math 481/581 Experimental Probability & Statistics Dr. John Engelhardt

TEXT: NCTM Addenda book: Dealing with Data and Chance. Mathematics Teaching in the Middle School: Focus Issue- Data & Chance. Other readings and assignments will be distributed during the term.

WHY TAKE THIS COURSE? This course is designed for those students who wish to pursue authorization to teach mathematics at the middle school level. It is one of several courses we offer focusing on middle school mathematics. Middle school level is distinct from elementary and secondary and as such deserves special emphasis. The course is designed to help students grow in their awareness and understanding of uncertainty and how to quantify it, as well as how to collect, organize, present, interpret and infer from data. The course provides an active, hands-on approach to investigations and explorations that is appropriate for the middle grades, but from an advanced point of view.

COURSE OBJECTIVES The course is intended to foster: (1) positive growth in attitude towards probability and statistics; (2) building basic concepts of probability and statistics; (3) enhanced data representation and interpretation skills; (4) statistical literacy; (5) the organization, analysis and communication of information; (6) decision making; (7) understanding of misuses of statistics; (8) recognition of misconceptions of probability; (9) development of cooperative problem solving skills; (10) computer use to enhance and conduct simulations of probability experiments; (11) a repertory of teaching ideas, models, and materials for classroom use.

COURSE CONTENT Study of probability and statistics through laboratory experiments, simulations, and applications.

ASSESSMENT There will be 9 weekly assessments due on the Wednesdays of weeks 2 through 10 of the course. Some of these will be write-ups due from the previous week, others may be presentations to be given during class, or a longer-term project which is developed over the term. Late papers are subject to a 10% penalty per day, to a max of 50%. There will also be a final exam which will be an opportunity to pull together the various dimensions of probability and statistics we explore over the term.

EVALUATION I intend to use a 5-point scale to grade weekly assignment problems. The attached scoring template gives you an idea of what constitutes each of the values 0 - 5. I anticipate each weekly assignment will be worth 20 points, with a letter grade corresponding to the scale on the scoring template. I anticipate a project and final exam each worth 40 points. At this level, I anticipate Aaverage work@ to be AB@ work. If you distinguish yourself as not average, one way or the other, the grade should show that.

ATTENDANCE You are expected to be present for each class. Meeting only once a week means each gathering is 10% of our total time! Attendance is a necessary but not sufficient condition for class participation and course growth. If a student is absent from two class meetings (20% of the course), there will be a letter grade drop in the course grade. If a student is absent from three class meetings, the student will be dropped from the course. If there is an accident, contact the Dean of Students who will notify all your instructors.

STUDY GROUPS I encourage you to work together, but each person must submit a writeup in his/her own words.

CALCULATORS You will need a calculator for this course, preferably one which has statistical and probability functions. You can find these on TI-30 calculators, but you will probably be happier with a graphing calculator (TI-83 my preference) in the long run.  These allow you to enter multiple data sets and perform multiple data analysis tests or computations on them.

PREREQUISITES The course has a prerequisite of Math 212, Math 243, or Math 251. This means that students should have been exposed to some of the ideas of probability and statistics (Math 212 or 243) or be in a position to quickly catch on to the underlying foundation of what we will undertake (Math 251). It is designed to allow students from various mathematical backgrounds to come to this course with a reasonable expectation of success.

EXPECTATIONS Sometimes students come into these courses with false expectations. This course is not a methods course, where we spend time discussing sequence of topics and how to teach them. This course is an active investigation and exploration class where we uncover mathematics that is appropriate for middle school students, and provide the necessary background for teachers to understand, develop and extend appropriate activities. As such, we will need to consider the mathematical background to our investigations.



Math 481/581 Geometry Dr. John Engelhardt

COURSE RESOURCES:
Essentials Geometry I by Research & Education Association.
Mathematics Teaching in the Middle School, March-April 1998. Focus issue on Geometry. (A more recent focus issue exists but it was not possible to get it in time for this course.) Other readings and assignments will be distributed at appropriate class sessions.

WHY TAKE THIS COURSE?
This course is designed for those students who wish to pursue authorization to teach mathematics at the middle school level. It is one of several we offer focusing on middle school mathematics. Middle school level is distinct from elementary and secondary and as such deserves special emphasis. The course is designed to help students grow in their awareness and understanding of geometry of one, two and three dimensions. It provides a medium to investigate mathematics that is appropriate for the middle grades, but from an advanced point of view appropriate for future teachers. This approach will complement the college geometry taken by the typical math major (Math 411 at SOU) as well as the   exploratory geometry of the elementary prepared student (Math 213 at SOU).

COURSE OBJECTIVES
The course is intended to foster:
positive growth in attitude towards geometry
ability to form generalizations by formulating and testing conjectures
ability to communicate one's thinking by carefully writing up problem investigations
ability to communicate one's thinking by verbally sharing problem approaches in class
awareness of learning models related to development of geometric concepts
a repertory of teaching ideas and activities for classroom use.
familiarity with the van Hiele theory of the development of geometric thinking and how to nurture this in a classroom

COURSE CONTENT
The course will include investigations involving spatial visualization, shapes and their properties, symmetry, geometric constructions using a variety of methods, congruence and similarity, transformations, tessellations and measurement. In general, these topics are explored in class in a group problem-solving setting using geometric models and further refined in weekly write-ups done individually.  An underlying theme of the activities is that of experiencing geometry within a dynamic context- one where objects can be constructed, compared, moved about, transformed, or put together with the intent of discovering and/or applying geometric relationships and of promoting a growing sense of spatial awareness. 

ASSESSMENT
Each week we will swap work. You will hand in your efforts on the previous week's assignment and I will give you another set of interesting problems to chew on. It will work better if you keep your write-ups in a folder, since there will be multiple pages and constructions that may accompany your work. The folder will be returned with a grade based on the Holistic Scoring Guide that I will provide you, as well as pertinent notes, comments, etc. Since the final exam will be an open-notes assessment of all the previous
assignments and course work, it is to your advantage to complete or repair any assignments that need it.

EVALUATION
Your grade will be based on several components: weekly write-ups, an in-class midterm and a final, instructor perception of class contribution (attendance, class participation, etc.) and a  student self-assessment in which you reflect back over the course and discuss your growth, participation, etc. I anticipate something like the following: Weekly write-ups- 60%, midterm- 10%, final- 10%, instructor perception- 10%, student perception- 10%. The Holistic Scoring Guide has cutoffs established for the weekly write-ups. The exams and other assessments will have their own cutoffs; as a baseline, you can think along the lines of 90, 80, 70, 60 for A-, B-, C-, D- percentage cutoffs; I suspect they will vary somewhat from this depending on the difficulty level involved.

ATTENDANCE
You are expected to be present for each class. Meeting once a week makes each class 10% of our time! Attendance is a necessary but not sufficient condition for class participation and course growth. If there is an accident, contact me by phone or email. If you do have to miss, turn in your assignment early if at all possible. See me prior to the next class! Check for handouts in the tray outside my office.

LATE WORK
If for some reason your work is not turned in at the beginning of the class when it is due, there will be a penalty of 10% per day. Work must be submitted prior to the next class meeting, preferably in time for me to read and assess it.

STUDY GROUPS
I encourage you to work together, but only after you have given sufficient thought to the problems in the assignment. Don't shortchange your growth by seeking help too soon; much satisfaction comes from engaging the material, getting insights and experiencing that AHA! Each person must submit a write-up in his or her own words. Copying is plagiarism, subject  to a failing grade and school disciplinary sanctions. Don't copy!

Marj presented a summary of some of the things the three programs have in common and some of the aspects different on each campus.

Common practices:

Problem solving activities to promote involvement through exploration and experimentation which allow students to construct and reconstruct mathematics understanding and knowledge

Visual reasoning as well as symbolic deductive modes of thought (that incorporate models, concrete materials, diagrams and sketches) 

Small group work and cooperative learning

Discussing and listening to how others think bout a concept, problem, or idea

Becoming aware of one’s own mathematical thought processes (and feelings about mathematics) and those of others

Weekly written reports and problem summaries

Reflective writing, written communication between instructors and individual students

De-emphasis on formal testing and use of other modes of assessment

Supportive class environment

Teaching that models how middle school teachers might teach

Program distinctions and comparisons:

PSU: Program designed for elementary/middle school teachers. Courses offered both summer and academic year, three courses every summer and one each term, late afternoon or evening. Students can complete program in three summers, two summers and intervening year, or two years and intervening summer. Students are one half or more practicing elementary or middle school teachers, the rest are undergraduates planning to become elementary or middle school teachers. 

WOU: Program designed for elementary/middle school teachers. Courses offered primarily during the academic year in regular day-time schedule. Students are primarily undergraduate students planning to become elementary or middle school teachers.

SOU: Program is designed for middle and secondary teachers. Courses are offered both academic year and summer. Students are primarily practicing teachers seeking middle grades or secondary mathematics endorsements for their teaching license.

The session ended with a discussion of the great importance of having programs that are designed for middle grades teachers, especially future middle grades teachers who come from the ranks of students initially preparing to become elementary teachers.  Participants were invited to contact the presenters and their colleagues for more information about any of these programs:

Dr. Marjorie Enneking (marj@mth.pdx.edu) or Dr. Michael Shaughnessy (mike@mth.pdx.edu) at PSU

Dr. Laurie Burton (burtonl@wou.edu) at WOU

Dr. John Englehardt (engelhardt@sou.edu) at SOU