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Algebraic concepts for middle school teachers of English language learners: A professional development course taught by a mathematician and a mathematics educator

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Title: Algebraic concepts for middle school teachers of English language learners: A professional development course taught by a mathematician and a mathematics educator


1
Algebraic concepts for middle school teachers of
English language learners A professional
development course taught by a mathematician and
a mathematics educator
Center for the Mathematics Education of Latinos/as
  • Cynthia O. Anhalt Matthew Ondrus
  • The University of Arizona
  • Department of Mathematics
  • Institute for Mathematics and Education
  • March, 2007

2
I think we (as teachers) have more power than we
give ourselves credit for in terms of helping our
students achieve success, and its easy to say,
Oh, they (Latino students) come from a poverty
background or they dont know enough English. a
lot of us use that as a reason why were not
getting the results we would like to see.
CEMELA Cohort Middle School Mathematics Teacher
3
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 Centers 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.

4
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

5
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.

6
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

7
Selected Course Readings
  • Greenes, C. (2004). Algebra Its 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.

8
Typical Day of Class
9
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

10
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

11
Selected Topics
  • Perimeter with Algebra Tiles (Blocks)
  • A new activity
  • Summing Consecutive Numbers
  • A borrowed activity

12
Algebra Tiles
b
a
a
b
b
a
Little quadrilaterals with (implied) dimensions
13
Algebra Tiles Support the Area Model of
Multiplication
a
b
b
b
Area b2
Area ab
14
Common Use of Algebra Tiles
a 2b
a b
15
What We Did
b
a
a 3b
ab
What is the perimeter of this shape?
16
What We Did
Asked teachers to invent similar problems
What is the perimeter of this thing?
17
Two Interesting Examples
P 3a 8b (a 2b) 4a 6b
P 3a 10b (a 2b) 4a 8b
18
A Discovery Made by the Class
Filling in a missing corner does not change the
perimeter.
P 4a 6b
19
Applying our Theorem
What is the perimeter of this thing?
b
a
b
20
How Perimeter Activity Developed
  • CA Lets do perimeter with algebra tiles.
  • MO Huh?!
  • MO Hmmmheres what we can do.

21
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.
22
Teachers Strategies
  • Common algebraic problem solving strategy
  • 678910 688810 88888 5(8)
  • 6789 7788 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

23
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 Teachers Work ?

24
Pedagogical Themes that Arose
  • Use of manipulatives
  • Discovery of formula or Justification of
    formula
  • Language

25
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.
26
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)

27
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 didnt 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.

28
Teacher Reflections on the Instructors (Mathematic
ian 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 dont have the same views on issues.
  • the methods of teaching complemented each
    otherthe mathematics content and the mathematics
    pedagogical issues.

29
Things to Think about Next Time
  • How to get participants to really think deeply
    about the math?
  • Why learn something that they wont 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 cant
    do
  • (Deficit model)
  • Mathematically
  • Linguistically
  • Theoretical research articles versus
    less-theoretical articles
  • Perspectives on the point of manipulatives?

30
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
    CEMELAs partnering middle school teachers.

31
Center for the Mathematics Education of Latinos/as
  • http//math.arizona.edu/cemela

National Science Foundation Award No. ESI-0424983
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