Algebraic concepts for middle school
teachers of English language learners:
A professional development
course taught by a mathematician and a mathematics educator
Cynthia O. Anhalt & Matthew Ondrus
The University of Arizona
Department of Mathematics
Institute for Mathematics and Education
March, 2007

Slide 2
The Center for the Mathematics Education of Latinos/as is a Center for Learning and Teaching (CLT) funded by the National Science Foundation (NSF, Award No. ESI-0424983).
The Center’s main goal is to understand the interplay of mathematics education and the unique language, social, cultural, and political issues that affect Latino communities.
One focus area of the Center is teacher education, especially in the growth and professional development of middle school teachers of mathematics.
The Center has developed a series of five professional development mathematics courses for middle school teachers, and this course is the first one in the series.

Cohort of 22 middle school teachers from five CEMELA partner schools
Teachers varied in experience, ethnicity, linguistic backgrounds, age, education backgrounds
7 Latina females, 5 Latino males, 5 White females, 4 White males, 1 Chinese female
Teaching experience range from 1-28 years
21 BAs in education, 1 BS in engineering, of these, 8 MAs in education

Overview of the Course:
Algebra for Middle School Teachers
The goals of this course were:
To strengthen teachers’ understanding of algebra, particularly as it applies to expanding the vision of what algebra is in the middle school and the transition from arithmetic to algebraic thinking;
To discuss professional readings pertaining to Latino students’ learning of algebraic concepts; and
To discuss unique linguistic and cultural resources that Latino students bring to the classroom and how these can be used as assets in learning mathematics.

Selected Course Topics
Linear, Quadratic, Exponential
Algebra-Geometry Connections
Algebraic reasoning from geometric perspectives
Area of Quadrilaterals
Pythagorean Theorem
Area and Perimeter with Algebra Tiles
Completing the square and optimizing (quadratic functions)
Sums of Consecutive Integers
Algebra in the Context of Sheltered Instruction
Multiple Representations
Issues of Language

Selected Course Readings
Greenes, C. (2004). Algebra: It’s elementary!  FOCUS (Web) Magazine, August. Eisenhower National Clearing House.
Khisty, L. L. (2002).  Mathematics learning and the Latino student: Suggestions from research for classroom practice.  Teaching Children Mathematics, September, pp. 32-35.
Lager, C. (2004). Unlocking the Language of Mathematics to Ensure Our English Learners Acquire Algebra.
Moschkovich, Judit N. (1999).  “Understanding the needs of Latino students in reform-oriented mathematics classrooms.” In W. Secada, L.Ortiz-Franco, N. G. Hernandez, & Y. De La Cruz, (Eds.), Changing the faces of mathematics: Perspectives on Latinos, Reston, VA: NCTM.
Taylor, R. (1990). Teacher expectations of students enrolled in an algebra course.  In E. L. Edwards (Ed.) Algebra for everyone. Reston, VA: NCTM.

Typical Day of Class
Co-Teaching Successes
For teachers
All course content was doubly scrutinized
Greater diversity of ideas from instructors
Mathematical content
More mathematical ideas
Awareness of how ideas connect to calculus, computer science, etc.
Pedagogical mathematics content
Content at appropriate level and relevant to teachers’ curriculum
Focused & relevant article readings
For us
Co-teaching & co-planning – value in interaction/negotiation
We had to convince each other of usefulness of various topics
Credibility with teachers

Co-Teaching Challenges
Time to prepare for class
Tension: Class time spent on a given topic/problem?
Negotiation process
Choosing mathematical material
Planning the big ideas versus planning the details
Mathematically egocentric perspective
Differing theoretical perspectives
Math as tool vs. math as a study (math = useful?)
Differing approaches to planning (how this evolved)
Looking for existing activities to use
Trying to invent activities

Selected Topics
Perimeter with Algebra Tiles (Blocks)
A “new” activity
Summing Consecutive Numbers
A “borrowed” activity

Algebra Tiles
Algebra Tiles Support the Area
Model of Multiplication
Common Use of Algebra Tiles
What We Did
What We Did
Two Interesting Examples
A Discovery Made by the Class
Applying our Theorem
How Perimeter Activity Developed
CA: Let’s do perimeter with algebra tiles.
MO: Huh?!
MO: Hmmm…here’s what we can do.

Summing Consecutive Integers
Goals: Finding and understanding patterns
*Questions such as…
Is it possible to write 42 as the sum of three consecutive integers? (Yes, 42 = 13 + 14 + 15)
Which numbers can be written as the sum of four consecutive integers?
10 = 1 + 2 + 3 + 4
14 = 2 + 3 + 4 + 5
18 = 3 + 4 + 5 + 6
22 = 4 + 5 + 6 + 7

Teachers’ Strategies
Common algebraic problem solving strategy
6+7+8+9+10 = 6+8+8+8+10 = 8+8+8+8+8 = 5(8)
6+7+8+9 = 7+7+8+8 = 4(7.5)
Common visual strategy
(4 consecutive)
My favorite strategy
x + (x + 1) + (x + 2) + (x + 3)  = 4x + 6 = 4(x + 1) + 2
= 2 more than a multiple of 4

Related Homework Assignment
What is the sum of all the integers from 1 to 7,399?
Approach of Gauss:
1 + 2 + … + 8 + 9
= (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5
= 4(10) + 5
Sample of Teacher’s Work ą

Pedagogical Themes that Arose
Use of manipulatives
“Discovery” of formula or “Justification” of formula

Issues of Language
Mathematics Lessons in Chinese
Two lessons on area and perimeter of rectangles
Lecture and limited representations
Use of multiple representations

Issues of Language
Teachers’ Insights on . . .
Their focus on numbers (not concepts) during lesson
“I was trying to concentrate on the numbers, drawings and table, and I was trying to figure out a few of the Chinese characters, but anything other than that was beyond my comprehension.”
Need for ELL students’ “silent period” (Video Clip)
Placement policies for ELL students (VideoClip)

Teacher Perspectives on the Course
“I liked the balance between the ‘pure math’ and the educational strategies and issues.”
“I wish we could have spent more time on issues relating to strategies for teaching Latino students.”
New perspectives on teaching algebra
“I enjoyed the content and new methods for looking at algebraic concepts, especially the visuals and the algebra tiles.”
“I loved the algebra tiles; it was the first time that I had seen algebra tiles.  I didn’t think that algebra could be seen this way.”
Variety of activities
“I liked working out the problems, using manipulatives, working in groups, discussing articles, and discussing issues of teaching ELLs.”
“The class tried to cover too much.”
New knowledge
“I can honestly say that I left every night with new knowledge or more in-depth knowledge in a specific area.”

Teacher Reflections on the Instructors
(Mathematician & Mathematics Educator)
“Excellent idea to have two instructors with different backgrounds because each one  brought different points of view on how to teach the mathematics.”
“When discussing issues, we got two views, which helps open new ideas because we can see that the instructors don’t have the same views on issues.”
“…the methods of teaching complemented each other…the mathematics content and the mathematics pedagogical issues.”

Things to Think about Next Time
How to get participants to really think deeply about the math?
Why learn something that they won’t directly use with our kids?  (Have this discussion)
…especially when they want to know how to teach ELL students
How to look beyond language (Latino cultural resources)?
It was tempting to focus on what students can’t do
(Deficit model)
Theoretical research articles versus “less-theoretical articles”
Perspectives on the point of manipulatives?

Closing Remarks
A course such as this one that addresses the mathematics content that is aligned with middle school curriculum and addresses the issues and needs of ELLs from a cognitive perspective embedded in theoretical frameworks for teachers to ponder and reflect proved to be a critical component for the professional development of CEMELA’s partnering middle school teachers.