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 |

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 CenterÕs 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 teachersÕ 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 studentsÕ 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 Readings |

Greenes, C. (2004). Algebra: ItÕs 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). ÒUnderstanding the needs of Latino students in reform-oriented mathematics classrooms.Ó 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 teachersÕ 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 ÒnewÓ activity | ||

Summing Consecutive Numbers | ||

A ÒborrowedÓ 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: LetÕs do perimeter with algebra tiles. | |

MO: Huh?! | |

MO: HmmmÉhereÕs what we can do. |

Summing Consecutive
Integers |

Goals: Finding and understanding patterns | |||

*Questions such asÉ | |||

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 |

TeachersÕ 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 |

Description | ||

What is the sum of all the integers from 1 to 7,399? | ||

Approach of Gauss: | ||

1 + 2 + É + 8 + 9 | ||

= (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5 | ||

= 4(10) + 5 | ||

Sample of TeacherÕs Work ˆ |

Pedagogical Themes that
Arose |

Use of manipulatives | |

ÒDiscoveryÓ of formula or ÒJustificationÓ 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 |

Issues of Language |

TeachersÕ Insights on . . . | ||

Their focus on numbers (not concepts) during lesson | ||

ÒI 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.Ó | ||

Need for ELL studentsÕ Òsilent periodÓ (Video Clip) | ||

Placement policies for ELL students (VideoClip) | ||

Teacher Perspectives on the
Course |

Balance | ||

ÒI liked the balance between the Ôpure mathÕ and the educational strategies and issues.Ó | ||

ÒI wish we could have spent more time on issues relating to strategies for teaching Latino students.Ó | ||

New perspectives on teaching algebra | ||

ÒI enjoyed the content and new methods for looking at algebraic concepts, especially the visuals and the algebra tiles.Ó | ||

ÒI loved the algebra tiles; it was the first time that I had seen algebra tiles. I didnÕt think that algebra could be seen this way.Ó | ||

Variety of activities | ||

ÒI liked working out the problems, using manipulatives, working in groups, discussing articles, and discussing issues of teaching ELLs.Ó | ||

ÒThe class tried to cover too much.Ó | ||

New knowledge | ||

ÒI can honestly say that I left every night with new knowledge or more in-depth knowledge in a specific area.Ó | ||

Teacher Reflections on the
Instructors(Mathematician & Mathematics Educator) |

ÒExcellent idea to have two instructors with different backgrounds because each one brought different points of view on how to teach the mathematics.Ó | |

ÒWhen discussing issues, we got two views, which helps open new ideas because we can see that the instructors donÕt have the same views on issues.Ó | |

ÒÉthe methods of teaching complemented each otherÉthe mathematics content and the mathematics pedagogical issues.Ó |

Things to Think about Next
Time |

How to get participants to really think deeply about the math? | ||

Why learn something that they wonÕt directly use with our kids? (Have this discussion) | ||

Éespecially 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 canÕt do | ||

(Deficit model) | ||

Mathematically | ||

Linguistically | ||

Theoretical research articles versus Òless-theoretical articlesÓ | ||

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 CEMELAÕs partnering middle school teachers. |

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

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