Helping ELLs

Reach Their Potentials:  Using Thinking Maps to Assess 

Conceptual Understanding

Lolita GerardoHS Math Teacher & Dept. Chair Pharr‐San Juan‐Alamo  ISD

Sylvia TaubeMath Education Professor

Sam Houston State University

Mathematics  for English Language Learners [MELL] Annual Conference

Texas State University, San Marcos, TXJuly 31‐August 2, 2008

Objectives of MELL Session

• Share activities and experiences in teaching  mathematics to English Language Learners;

• Explore alternative tools that can assess the ELL’s conceptual understanding in math;

• Construct different types of maps, and design rubric  to score students’

concept maps.

WARM‐UP: Can you come up with different 

representations?

Problem:    In many parades, flowers are used to  decorate the floats.  The list below shows the 

number of flowers used in each row of a  parade float.

{ (1, 54), (2, 58), (3, 62), (4, 66) }

Summarize the different  representations of functions

• Use graphic organizers

Recent State Data on ELLsand Achievement in Algebra

End-of Course Algebra 1Testing Date: Spring 2008

Number of students tested = 52,462

Commended Performance = 11%

Met standards = 56%

Current LEP = 17% (# of students tested=1,104)

Low SES = 43% (# of students tested=21,932)

Hispanic = 47% (# of students tested=20,270)

White = 66% (# of students tested = 22,020)

21

1 <1

22

<12 2 1

<1

19

8

1

5

<11 1 1 1

8

5

0

5

10

15

20

25

%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Region

ELL Students Tested by Region (Grade 3-11)

1=Edinburg; 4=Houston; 10=Dallas; 11= Fort Worth;19=El Paso;20=San Antonio

TAKS DATA 2006

A Kick‐off Summer Program for ELLs

Thinking Maps in Algebra 1

Multiple Representations Why?

• Using and understanding different forms of  representations are critical to learning math;

• Implementation and flexibility in using varied  representations provide a strong evidence of deeper 

understanding of mathematics;

• This understanding is rich with connections and  relationships.

Bring in student’s culture

Promote equity

What can representations include?• Physical objects

• Drawings, graphs

• Symbols, numbers

• Verbal

• Contextual

These are used to organize 

and record student’s 

thinking about math ideas 

and problem solving.

Predominant

Representations• Numeric

• Geometric

• Algebraic

Adapted from: An Interactive Model for Using Representational Systemsby: Behr, Lesh, Post, & Silver

Written Symbols Spoken

Symbols

Real-WorldSituations Manipulative

Aides

Pictures

Assessment Strategies

If multiple representations is being valued, then the teacher should include questions or problems that involve a variety of representations.

Will the ELLs

have advantages in  this type of assessment?

Visual Representations

Example: Taking Stock Problem

Father

has 19 animals on his farm‐some chicken  and some cows. He told me that he counted 

62 legs altogether. How many of each animal  were there?

Can you come up with at least 3 different  representations?

Using physical (human) graphs

Concepts Maps & Thinking Maps

•Arrows indicate the directions of the  relationships between concepts;

•Lines are labeled with linking words to  specify relationship;

•Concepts are placed in ovals or any shape

Concept map –an instrument for explicitly describing concepts and the relationship among them. It is a way to organize a learner’s knowledge.

Graphic organizers

Venn Diagram

Chart

Web

Timeline

Combine both linguistic and  nonlinguistic information (e.g., circles, 

lines that show relationship)

Type: Hierarchical  Purpose: Promote mathematical connection

Quadrilateral

includes

rectanglerhombus

Trapezoid

square

includes

includes

parallelogram

is also a

Isosceles 

trapezoid

TYPE: Web 

Linear Equations

Graph shows a straight line

Form: y=mx+ b

has slope

ordered pairs

Crosses either x or y-axis

“first - degree” functions

domain/range

m is the slope

b is where line crosses y-axis

zero or undefined

That is + or -

What teachers must do to facilitate student’s  use of Multiple Representations?

•By listening, questioning, and making a sincere effort to understand what they are trying to communicate with their drawings and writings, especially when they are using personal [use of invented strategies] representations. [Tarlow, L. D. (2008), MTMS, Vol 13, #8.]

•Step back to give students freedom to test their ideas; observe students’ work and decide when to intervene or pose questions to clarify ideas.[Tarlow, L. D. (2008), MTMS, Vol 13, #8.]

•Give clear, comprehensible, useful, relevant feedback.[Hill, J.D. & Flynn, K. M (2006). Classroom instruction that works with ELLs, ASCD publication]

How to assess thinking mapsTask

Draw a concept map relating the following  concepts:

Parabolas

Domain

Standard equation

Range

Apex

Symmetry

Quadratic functionTranslate/Slide

Maximum point

Minimum point

Participants will create their own  concept map

Then . . .design a rubric for scoring the concept map

Recommendations from Research

Findings mostly from science, psychology, reading, and college math

J. Novak (1984) – early researcher in science education

A scoring rubric should be able to assess the breadth and depth of students’ knowledge transformation as they progress from “novice” to “expert”. (Edmonson, 2000)

Scoring can be done by comparing expert’s (criterion) with students’ maps

Findings:Maps that focus heavily on concept relationship has strong correlations with scores in standardized tests.

The rubric should be able to measure  student progress (understanding) over 

time.

Novice 

Expert

Eventually,  “learners do come to think like 

teachers”

THE RUBRIC MUST BE ABLE TO INFORM  INSTRUCTION

Recommended use during formative  assessment or for diagnostic purpose

Learning Strategies for Math ELLs

• Must be engaged with varied collaborative  activities

• Use multiple representations

• Use activities that promote communication

• Challenge students in solving problems with  open‐ended questions

• Use graphic organizers

Learning Strategies 

• Integration of manipulative materials and  graphing technology

• Hands‐on approach of developing conceptual  understanding

• Establish home connection

• Questioning strategies• Rich connections in context

References• Bartels, B. (1995). Promoting mathematics connections with concept 

mapping. Mathematics Teaching in the Middle School, 1 542‐549.

• Bolte, L. (Jan 1999). Using concept maps and interpretive essays for 

assessment in mathematics. School Science and Mathematics,

99(1).

• Hill, J.D. & Flynn, K. M (2006). Classroom instruction that works with ELLs. 

ASCD publication.

• Novak, J. (1984). Learning how to learn. New York: Cambridge University 

Press.

• Rye, J. (Jan 2002). Scoring concepts maps: An expert map‐based scheme 

weighted for relationship. School Science and Mathematics.

• Focus issue (April 2008) Mathematics Teaching in the Middle School‐

published articles on: Developing Mathematical Understanding through 

Representations.