Center for the Mathematics
Education of Latinos/as
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
“I think we (as teachers) have more power than we
give ourselves credit for in terms of helping our
students achieve success, and it’s easy to say, ‘Oh,
they (Latino students) come from a poverty
background or they don’t know enough English.’
…a lot of us use that as a reason why we’re not
getting the results we would like to see.”
~ CEMELA Cohort Middle School Mathematics Teacher
Center for the Mathematics
Education of Latinos/as
2
About CEMELA
• 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.
Center for the Mathematics
Education of Latinos/as
3
Participants
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
Center for the Mathematics
Education of Latinos/as
4
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.
Center for the Mathematics
Education of Latinos/as
5
Selected Course Topics
• Functions
– 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
Center for the Mathematics
Education of Latinos/as
6
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.OrtizFranco, 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.
Center for the Mathematics
Education of Latinos/as
7
Typical Day of Class
Mathematics Investigations
Discussions of Readings
Discussions of teachers’ everyday teaching experiences:
Relating content and readings to classroom context
Center for the Mathematics
Education of Latinos/as
8
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
Center for the Mathematics
Education of Latinos/as
9
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
Center for the Mathematics
Education of Latinos/as
10
Selected Topics
• Perimeter with Algebra Tiles (Blocks)
– A “new” activity
• Summing Consecutive Numbers
– A “borrowed” activity
Center for the Mathematics
Education of Latinos/as
11
Algebra Tiles
b
a
a
b
a
b
Little quadrilaterals with (implied) dimensions
Center for the Mathematics
Education of Latinos/as
12
Algebra Tiles Support the Area
Model of Multiplication
a
b
b
b
Area = b2
Center for the Mathematics
Education of Latinos/as
Area = ab
13
Common Use of Algebra Tiles
b
a + 2b
a
a
b
a+b
(a  2b)( a  b)
Center for the Mathematics
Education of Latinos/as

a 2  3ab  2b 2
14
What We Did
b
a – 3b
a
a+b
What is the perimeter of this shape?
a  2(a  b)  7b  a  3b
Center for the Mathematics
Education of Latinos/as
15
What We Did
b
a
Asked teachers to invent
similar problems
b
What is the perimeter of this thing?
P  a big mess
 6a  6b
Center for the Mathematics
Education of Latinos/as
16
Two Interesting Examples
P = 3a + 8b + (a – 2b)
=4a + 6b
Center for the Mathematics
Education of Latinos/as
P = 3a + 10b + (a – 2b)
=4a + 8b
17
A Discovery Made by the Class
Filling in a missing corner does not change the perimeter.
P = 4a + 6b
a
b
Center for the Mathematics
Education of Latinos/as
a
b b
18
Applying our Theorem
What is the perimeter of this thing?
b
a
b
Center for the Mathematics
Education of Latinos/as
P  2(2a  b)  2(a  2b)
 6a  6b
19
How Perimeter Activity Developed
• CA: Let’s do perimeter with algebra tiles.
• MO: Huh?!
• MO: Hmmm…here’s what we can do.
Center for the Mathematics
Education of Latinos/as
20
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
*Taken from: Driscoll, Mark. Fostering Algebraic Thinking:
A guide for Teachers Grades 6-10. Portsmouth, NH:
Heinemann, 1999, p. 79.
Center for the Mathematics
Education of Latinos/as
21
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)
2 integer
3 integer
• Common visual strategy
(4 consecutive)
Sum / 4
1 integer
4 integer
nd
rd
st
th
6
7
8
9
• My favorite strategy
x + (x + 1) + (x + 2) + (x + 3) = 4x + 6 = 4(x + 1) + 2
= 2 more than a multiple of 4
Center for the Mathematics
Education of Latinos/as
22
Related Homework Assignment
Description
• 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 
Center for the Mathematics
Education of Latinos/as
23
Pedagogical Themes that Arose
• Use of manipulatives
• “Discovery” of formula or “Justification” of
formula
• Language
Center for the Mathematics
Education of Latinos/as
24
Issues of Language
Mathematics Lessons in
Chinese
• Two lessons on area
and perimeter of
rectangles
• Lecture and limited
representations
• Use of multiple
representations
Center for the Mathematics
Education of Latinos/as
Anhalt, C., Ondrus, M., & Horak, V. (in press) Mathematics Teaching in the
Middle School, National Council of Teachers of Mathematics (NCTM),
Reston, VA.
25
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)
Center for the Mathematics
Education of Latinos/as
26
Teacher Perspectives on the Course
• Balance
– “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.”
Center for the Mathematics
Education of Latinos/as
27
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.”
Center for the Mathematics
Education of Latinos/as
28
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)
– Mathematically
– Linguistically
• Theoretical research articles versus “less-theoretical articles”
• Perspectives on the point of manipulatives?
Center for the Mathematics
Education of Latinos/as
29
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.
Center for the Mathematics
Education of Latinos/as
30
Center for the Mathematics
Education of Latinos/as
http://math.arizona.edu/~cemela
National Science Foundation Award No. ESI-0424983
Center for the Mathematics
Education of Latinos/as
31