Working Group Session # 6  Marjorie Enneking and Laurie Burton

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,
18005758130.)
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 nonmajors.
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 493593 Geometry for Middle School Teachers
Mth 494594 Arithmetic and algebraic Structures for Middle School Teachers
Mth 495595 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 teachersinresidence 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–inresidence 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 inclass 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 overall 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 quarterlong mathematics classes for preservice
Elementary Education majors [10 semester hours].
Early Childhood/Elementary [Age 3  Grade 8, Elementary School]
Academic Requirements General breadth in all areas including five quarterlong mathematics classes for preservice
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 quarterlong mathematics classes for preservice
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 quarterlong mathematics classes for
preservice 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 quarterlong mathematics classes for preservice
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 PRESERVICE MATHEMATICS COURSES
All preservice 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
realworld 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 preservice teachers at WOU take this course. Math focus Early Childhood/Elementary preservice
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 preservice teachers may take this course. All math
focus Elementary/Middle School preservice 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 20002001 offering of this course.
The 20002001 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)
0  1  2  3  4  
0  
1  
2  
3  
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 preservice teachers may take this course. All math
focus Elementary/Middle School preservice 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 20002001 course
taught by Sandy Kralovec worked though a set of problemsolving oriented geometry materials for Middle School teachers developed
by
Michael Arcidiaciano for Portland State University. The students in 20002001 course offering were challenged by working with
geometry in a problemsolving 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 preservice teachers may take this course. All math
focus
Elementary/Middle School preservice 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 followup 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, handson 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 writeups due from the previous week, others may be presentations to be given during class, or a longerterm 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 5point 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 TI30 calculators, but you will probably be happier with a graphing calculator (TI83 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, MarchApril 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 problemsolving setting using geometric models and further refined in weekly
writeups 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 writeups 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 opennotes 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 writeups, an inclass midterm and a final, instructor perception of class
contribution (attendance, class participation, etc.) and a student selfassessment in which you reflect back over the course
and discuss your growth, participation, etc. I anticipate something like the following: Weekly writeups 60%, midterm 10%,
final 10%, instructor perception 10%, student perception 10%. The Holistic Scoring Guide has cutoffs established for the weekly
writeups. 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 writeup 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
Deemphasis 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
daytime 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