Post on 21-Aug-2018
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.