algebraic concepts for middle school teachers of english language learners: a professional...

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 OndrusThe University of Arizona

Department of Mathematics

Institute for Mathematics and EducationMarch, 2007

Center for the Mathematics Education of Latinos/as

2Center for the Mathematics Education of Latinos/as

“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


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.


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


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

• 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


Selected Course ReadingsGreenes, 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

Discussions of Readings

Discussions of teachers’ everyday teaching experiences:Relating content and readings to classroom context

Mathematics Investigations


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

a

a

b

b

b

a

Little quadrilaterals with (implied) dimensions


Algebra Tiles Support the Area Model of Multiplication

b

b

Area = b2

b

Area = ab

a


Common Use of Algebra Tiles

a

a

b

b

a + 2b

a + b

))(2( baba ++ 22 23 baba ++=


What We Did

a

b

( )babbaa 37)(2 −++++

What is the perimeter of this shape?

a+b

a – 3b


What We Did

a

b

b

ba

P

66

mess big a

+==

What is the perimeter of this thing?

Asked teachers to invent similar problems


Two Interesting Examples

P = 3a + 8b + (a – 2b)

=4a + 6b

P = 3a + 10b + (a – 2b)

=4a + 8b


A Discovery Made by the Class

Filling in a missing corner does not change the perimeter.

P = 4a + 6b

a bb

a

b


Applying our Theorem

a

b

b

ba

babaP

66

)2(2)2(2

+=+++=

What is the perimeter of this thing?


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*Taken from: Driscoll, Mark. Fostering Algebraic Thinking: A guide for Teachers Grades 6-10. Portsmouth, NH: Heinemann, 1999, p. 79.


Teachers’ Strategies

• Common algebraic problem solving strategy6+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 strategyx + (x + 1) + (x + 2) + (x + 3) = 4x + 6 = 4(x + 1) + 2

= 2 more than a multiple of 4

Sum / 4

2nd integer

1st integer

3rd integer

4th integer

9876


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


Pedagogical Themes that Arose

• Use of manipulatives

• “Discovery” of formula or “Justification” of formula

• Language


Issues of Language

Mathematics Lessons in Chinese

• Two lessons on area and perimeter of rectangles

• Lecture and limited representations

• Use of multiple representations

Anhalt, C., Ondrus, M., & Horak, V. (in press) Mathematics Teaching in the Middle School, National Council of Teachers of Mathematics (NCTM), Reston, VA.


Issues of LanguageTeachers’ 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• 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.”


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)– Mathematically– Linguistically

• 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.


http://math.arizona.edu/~cemela

Center for the Mathematics Education of Latinos/as

National Science Foundation Award No. ESI-0424983

algebraic concepts for middle school teachers of english language learners: a professional...

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