Transcript CALLA Mathematics - Claremont Graduate University

Issues in Addressing The
Needs of
English Language Learners in
Context of Mathematics
Yolanda De La Cruz
Arizona State University
Presented at
Claremont Graduate University
November 17, 2007
The Big Picture
What are you are striving to accomplish?
 Academic gains in mathematics
 English language proficiency
 Enrich Ells mathematics academic language in
the native language
 Close the Achievement Gap
 Increase the percentage of Ells that obtain a
High School deploma
2
High School Exit Exam
 Nearly 40,000
seniors from the
Class of 2006 did not
pass the exit exam.
.
Numbers of Teachers in the California Workforce
350,000
300,000
272,459
283,975 292,012
301,361 306,940 309,773 305,855 306,548 307,864
Number of teachers
250,527
250,000
200,000
150,000
100,000
50,000
0
1996-97
1997-98
1998-99 1999-2000 2000-01
2001-02
2002-03
2003-04
2004-05
2005-06
Public School Enrollment, 1990 to 2015
3,000,000
2,500,000
2,000,000
1,500,000
1,000,000
500,000
-
19
90
-9
1
19
92
-9
3
19
94
-9
5
19
96
-9
7
19
98
-9
9
20
00
-0
1
20
02
-0
3
20
04
-0
5
20
06
-0
7
20
08
-0
9
20
10
-1
1
20
12
-1
3
20
14
-1
5
Number of K-12 students
3,500,000
K-5 enrollment
6-8 enrollment
9-12 enrollment
Number of teachers without full credentials
Number of Underprepared Teachers
by Credential Type
50,000
40,587
42,427
41,739
40,000
37,309
30,000
28,139
20,399
20,000
17,839
10,000
0
1999-2000
2000-01
2001-02
2002-03
2003-04
2004-05
2005-06
More than one underprepared credential type or missing credential information
University or district intern credential
Emergency permit, pre-intern certificate, or waiver
.
.
Persistent
Inequities
.
Distribution of Interns,
by School-Level Percentage of
Minority Students, 2005-06


Intern teachers are
maldistributed– 75% of
interns are assigned to
high minority schools.
7%
18%
44%
Only 25% of interns are
assigned to low minority
schools.
31%
Lowest minority quartile
Second minority quartile
Third minority quartile
Highest minority quartile
Students in the lowest
performing schools are the
most likely to get novice
and underprepared
teachers.
Underprepared First- and Second-Year
Mathematics and Science Teachers,
2005-06
Percent of novice teachers without full credentials
45
40
40
35
35
29
30
25
29
23
20
15
10
5
0
All teachers
Middle school
mathematics
High school
mathematics
Middle school
science
High school
science
What Do These Graphs Mean?

Persistent gap in academic achievement
between Caucasian students and those from
culturally and linguistically diverse groups:
Many teachers are underprepared to make content
comprehensible for ELs.
Few teachers trained to teach initial literacy or content-area
literacy to secondary ELs.
ELs are tested in mathematics and reading under No Child
Left Behind; and in 2007-08, tests in science have been
added to the battery of assessments they must take.
11
Sections that Promote Mathematics
Excellence For English Learners
1. Learning Atmosphere & Physical
Environment
2. Instructional Practices
3. Mathematics Content & Curriculum
4. Language Practices
5. Family & Community Involvement
6. Assessment of Student Learning
12
Overview

Mathematics is taught as a problem-solving
approach focuses on the use of oral; and
written language to communicate problemsolving procedures and mathematics
reasoning. Student learn computation skills
in context of meaningful problem-solving
applications.
13
Areas that Require Work





Scaffolds for entry, engagement, and
extension in mathematics and academic
language
Enrichment of curriculum
Native language mathematics instructional
materials
Teacher fluency with mathematics, math
pedagogy, and discourse in English and
language (for bilingual teachers)
Understanding student work
14
ELLs and Mathematics


Mathemtics is not langauge-neutral
Language issues:



Medication among native language, English,
and academic mathematical English
Specialized vocabulary and syntax
Everyday vocabulary used differently
15
Understanding the Complexity of ELLs’
Culture, Language, and Knowledge






We need to build on background
knowledge
Identify cognates(Spanish-English) in
mathematics
Language production patterns
Develop language within mathematics
Vocabulary building (enrich)
Clarify the academic language
16
What is Presently Being Taught




Math in Primary grades consist of numbers, place
value, addition, subtraction, and multiplication with
whole numbers.
Intent of instruction is to make the operations
conceptually familiar, automatic, and accurate.
Math in Intermediate grades consists of
multiplication, division, decimals, fractions, ratios,
proportions, and percents. Some measurement
and geometry.
Graphing, probability, and statistics and use of
calculators are introduced.
17
What is Presently Being Taught
…….continued




The math curriculum develops students’ ability of
understand concepts through a variety of problem-solving
experiences with manipulative materials or pictorial models
prior to working with abstractions.
A foundation of basic facts and skills as in addition and
subtraction, through understanding a concrete model of the
mathematical operation.
Computational skill is developed through practicing the
procedure under a variety of circumstances.
Problem solving strategies are developed by applying
factual information to realistic problems in a variety of
different settings.
18
Actual Class Instruction


Middle school teachers spent 70% to
75% teaching skills in addition,
subtraction, multiplication, and division.
Of the time is spent on skills
development, half was spent developing
conceptual teaching and the other half
was spent on teaching problem-solving
using story problems.
19
1. Learning Atmosphere &
Physical Environment

A caring classroom atmosphere of mutual respect and
support is facilitated by the teacher who:
 Knows each child as an individual,
 Embraces languages, customs, and cultures of ELL
students,
 Provides culturally rich learning materials,
 Encourages self-expression and provides positive
recognition,
 Builds student confidence and esteem,
 Fosters an emotionally safe environment that allows
students to feel secure and to take risks.
20
1. Learning Atmosphere &
Physical Environment cont…..


The classroom is visually rich to support student learning
 Incorporates displays of student produced work,
whenever possible,
 Is colorful and thought stimulating,
 Contains pertinent, real-world information and
applications,
 Reinforces math-specific vocabulary and concepts,
 Provides color-coded learning supports when
appropriate.
Room arrangement facilitates student interaction and group
work.
21
2. Instructional Practices



Instructional practices foster cooperation
and collaboration.
Concepts are presented accurately,
logically, and in engaging ways.
Multiple representations incorporate
mathematics learning levels: concrete,
semi-concrete, and abstract.
22
2. Instructional Practices
cont.

The teacher employs student-centered instructional
practices.
 Approaches content from a concept-oriented
constructivist method,
 Surrounds students with different modalities,
 Connects new concepts to prior learning or prior
knowledge,
 Encourages students to refine and reflect about their
own work and verbalize concept understanding “in their
own words”,
 Chooses homework to optimize individual content
development,
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 Provides extra help and resources on an individual basis.
2. Instructional Practices
cont.

Students are frequently partnered with peer
learners to enhance learning opportunities.




To develop math content,
To aid English language development,
To insure sustained active participation in the
class,
To welcome new students into an established
learning community.
24
2. Instructional Practices
cont.

Instructional activities are varied and support
diverse learning styles and multiple intelligences,
including for instance:






Frequent use of models and/or manipulatives,
Music as a motivator and anchor,
Mind maps, poster-walks, and word walls
Key vocabulary and cognates presented in different
forms,
Vivid adjectives,
Graphic organizers.
25
Graphic Organizers
Description-test describes or defines information
Organizers-webs, features charts, comparison charts
Enumeration-text lists information about several related items, (e.g.,
events, characters, objects) and provides supporting evidence or
details
Organizers-tree diagrams, branch diagrams, webs, outlines,
comparison charts
Comparison-contrast-text comments on similarities and differences
among facts, people, events, and uses comparative adjectives and
transitional markers (e.g, “on the one hand…on the other,” “both…only
one”)
Organizers-Venn diagrams, comparisons
Chronological or sequential-text organized in a time sequence and
uses temporal markers, such as dates, prepositional phrases of time,
sequence words (e.g., first, next, then)
Organizers-timelines, story summaries
Cause-effect-text describes cause-effect reactions, how one thing
occurs as the result of another and uses causative words (e.g., so, as a
result, therefore)
Organizers-flow charts, sequence chains, and cycles
Problem-solution-text presents a problem, and one or more solutions,
word choice relates to options, alternatives, consequences, and results
Organizers-decision-making diagrams, semantic maps
Four Square for
Mathematical Story Problems
Decide on the Operation
Which do you choose and why?
Compute
Add, subtract, multiply or divide.
Survey the question.
Key words:___________________________
Numbers:____________________________
What are you looking for:________________
Check your work.
Does your answer make sense?
Check it by doing the opposite
operation.
Write to explain what you did and
why.
What’s Different in Mathematics
for English Learners?




Discourse structure may be vary different from
students’ previous English experience.
Grammatical forms and structures in textbooks
becomes increasingly complex.
All four academic language skills are required.
The language of mathematics is highly
specific, misunderstandings are remarkably
persistent.
28
Importance of Addressing The
Needs of English Learners
Most ELLs need 4-7 years to learn
English before they reach average
academic performance levels.
 As ELLs, they are by definition not
proficient.
 But they are tested before they are
proficient in English.

29
Specialized Vocabulary



English students may become confused with unique
terms such as addend and quotient, and terms with
specialized meanings, such as altogether, round,
table.
English learners have few cues to the meaning of
words or phrases apart from the limited and often
abstract context that is provided in the words and
symbols in the problem statement.
There are also combinations that have special
meanings, such as square root, multiplication table,
and least common denominator.
30
Specific words in mathematics single the
use of certain mathematics operations.

Addition is suggested by
the following words:

Subtractions is suggested
by the following words:
minus less less than
add and plus
difference
sum total combine
more than
decreased by
31
Differences in Math Symbols
U.S System
4,232
1,258,125
4-1/2 = 4.5
Spanish-Speaking Countries
4.232
1.258.125
4,5
32
Contextualize to Subject
Matter Scaffolds Activity
Task: Think-Pair-Share
(bridging, schema building)
Write the following sentence stem for all to see.
Ask students to take two minutes to jot down a few
notes in response:
When I see or hear the word “geometry,”
I think….
Ask students to share their responses with a partner.
33
3. Mathematics Content &
Curriculum

Glossary of mathematical terms is always
available for reference.
English-Spanish Dictionary or Math Glossary
http://www.mathnotes.com/aw_span_gloss.html
http://math2.org/math/spanish/eng-spa.htm



Content is aligned to appropriate grade-level,
mathematics TEKS and professional
standards.
Content is based on diagnosed student
needs.
34
3. Mathematics Content &
Curriculum cont….

Content is systematically designed to incorporate
sound learning principles.





To incorporate increased complexity,
To present a cohesive big-picture through chunking,
To connect concepts through bridging and scaffolding,
To emphasize multidisciplinary understandings,
To reflect on inherent patterns by comparing and
contrasting concepts.
35
3. Mathematics Content &
Curriculum cont….

Curriculum is challenging, relevant,
age-appropriate, and well-paced
 To include contextually-based problems,
 To incorporate student realities,
 To involve interactive problem solving.
36
4. Language Practices




Language support is offered without supplanting
English instruction.
Support is aligned with student’s diagnosed
language needs.
Language used is appropriate to age and grade
level and presented in a socially meaningful
context.
Mathematics-specific vocabulary is explicitly and
implicitly taught and reinforced through repetition.
37
4. Language Practices cont….



Teachers are knowledgeable about the second
language acquisition theories and best practices.
Ideally, dual language instructional support should be
offered.
When dual language teachers are not available,
sheltered instruction should be utilized to provide
strong language support by addressing content
through ESL.
38
5. Family & Community
Involvement



Schools connect to student’s family-life by
embedding contextual experiences and skills in
teaching and curriculum.
Projects are relevant and promote family
interaction.
Opportunities are available for English-speaking
higher grade-level students to mentor ELL lower
grade-level students either in an in-school or afterschool program, as appropriate.
39
5. Family & Community
Involvement cont….



Teacher engages in frequent communication with
families
 About activities and events in which parents can
participate,
 About student progress.
Teacher utilizes services provided by a community
liaison and is knowledgeable about community
resources.
Parents are informed about the benefits of using their
most cognitively advanced language at home.
40
6. Assessment of Student
Learning



Classroom assessment is designed to foster
student success.
Assessment methods allow students frequent
opportunities to demonstrate mastery in a
variety of ways.
Various assessment techniques are used to
measure student understandings.
41
6. Assessment of Student
Learning cont….



Grades are oriented to promote and
emphasize valid step-by-step logical
reasoning processes.
Assessment data and results shape
instructional planning.
Flexible time allotments are given to
demonstrate concept mastery.
42
How to Teach Problem-Solving Steps





Understand the question-teach students to understand
the problem through elaboration and imagery.
Find the needed information-Help students use
selective attention to find needed information.
Make a plan-have students identify the operation and
what the problem calls for, then choose a plan (write a
number sentence, identify parts of the problem, work
with a peer, make a table, make a list).
Solve a problem-Students write out the steps of the
problem and solve it, using cooperation to review the
steps they have taken.
Check the answer-Students use a variety of
approached o verify that their answer makes sense.
Geometry Activity

California State Board of Education Geometry
Content Standards Grades Eight Through Twelve
http://www.cde.ca.gov/be/st/ss/mthgeometry.asp
This activity will help students learn to
identify, compare, and analyze attributes of
two- and three-dimensional shapes and
develop vocabulary to describe the attributes.
44
Vocabulary
Flip
Net
Slide
Edge
Turn
Face
Finding a Suitable Definition
Slide 1
1. These are pentominoes.
(a)
(b)
2. These are not pentominoes.
(d)
(e)
(c)
(f)
3. Which of these are pentominoes?
(g)
(h)
(i)
4. How can you define a pentomino?
(j)
Finding a Suitable Definition
Activity 1
1. Show Slide 1 and have students find a
suitable definition for a pentomino.
2. Write their definition on the board or on chart paper.
The formal definition means little to English
learners; however, students can develop their own
definitions using Slide 1 through group discussions
and consensus. This helps in giving ownership of
the definition to the students while disclosing only
two of the twelve pentominoes.
A Suitable Definition
They should come up with
something that includes the
following information:
A pentomino is a two-dimentional
shape made from five congruent
squares such that each square has
at least one of its sides in common
with another square.
Finding the Twelve Pentominoes
Activity 2
1. Have students work in small groups to find the
twelve different pentominoes.
2. The new shapes should be made with five square
tiles to test their definition of a pentomino.
3. Each whole new shape should be cut out of
squared paper.
4. Each group should check to make sure they do not
have duplicate shapes by flipping, turning or
rotating similar shapes.
5. Each group will need a set of twelve pentominoes
for the activities that follow.
The Twelve Pentominoes
F
U
L
I
P
V
W
X
N
Y
T
Z
Fold into an Open Box
Activity 3
1. Have students guess which of their
pentominoes form a network for a
cube without a top.
2. Fold the pentominoes to check their
conjectures.
The Pentominoes that form an Open Box
F, L, N, T, W, X, Y, and Z
form such an open box.
Transferring between the second and
third dimensions is important in many
occupations such as architecture,
drafting, and design.
This activity facilitates the student’s
understanding of the process.
Exploring Symmetry
Activity 4
1. Have students place a Mira on all or some of the
pentominoes so that reflection is the same as the
behind the Mira. This can also be done by folding
the shapes.
2. Ask student if all pentominoes have a line of
symmetry?
3. Have them identify those that have a line of
symmetry and those that do not.
4. Have the identify those that have multiple lines of
symmetry.
Exploring Tessellations
Activity 5
1. A tessellations is created when a shape is
repeated over and over again covering a
plane without overlaps.
2. Ask students if they can find the pentominoes
that tessellate.
3. Have them identify those that tessellate and
those that do not.
What can you do with these 12 puzzle pieces?
You would be amazed!
(Remember, each piece contains 5 squares, so there are 60 cubes total.)
Using all 12 pieces, you can make:
1. A 6 x 10 rectangle
2. A 5 x 12 rectangle
3. A 4 x 15 rectangle
4. A 3 x 20 rectangle
5. An 8 x 8 square with 4 pieces missing in the middle
6. An 8 x 8 square with 4 pieces missing in the corners
7. An 8 x 8 square with 4 pieces missing almost anywhere
8. A 3 x 4 x 5 cube
9. A 2 x 5 x 6 cube
10. A 2 x 3 x 10 cube
11. A 2D replica of each piece, only three times larger
12. A 5 x 13 rectangle with the shape of 1 pentomino piece missing in the
middle
13. Shapes with jagged edges
14. Tessellations using a pentomino
15. Hundreds of other shapes!
http://www.xs4all.nl/~gp/PolyominoSolver/Polyomino.html
What’s Difficult in Math for
English Learners?




Language Dependence in Mathematics-Specialized
terms and terms used in unique ways; syntactic
features of word problems.
Non-Linguistic Difficulties-Students must process
complex concepts and different used of otherwise
familiar terms.
Cultural differences-Cultural differences in the use of
decimals and comas, in fractions, or in the strategies
students use to solve word problems.
Instructional implications-Special procedures are
needed that enable students to test hypotheses about
language use and other potential problem areas.
Cooperative learning activities are necessary to give
students practice in verbalizing new concepts.
What Can We do To Help Our English
Learners in Mathematics





Glossary of mathematical terms is always available for reference
Content is aligned to appropriate grade-level professional
standards
Content is based on diagnosed student needs
Mathematics-specific vocabulary is explicitly and implicitly taught
and reinforced through repetition.
Content is systematically designed to incorporate sound learning
principles.
 To incorporate increased complexity,
 To present a cohesive big-picture through chunking,
 To connect concepts through bridging and scaffolding,
 To reflect on inherent patterns by comparing and contrasting
concepts.
57
Websites
•http://www.math.clemson.edu/~simms/java/pentomin
oes/
•http://www.coolmagnetman.com/pent.htm
•http://www.johnrausch.com/PuzzlingWorld/
•http://www.mathleague.com/help/geometry/geometry
.htm
•http://www.math.com/homeworkhelp/Geometry.html
•http://www.coolmath4kids.com/geometrystuff.html
Websites
California High School Math Standards
http://www.cde.ca.gov/re/pn/fd/documents/mathstnd.pdf
 Further Information
http://www.tsusmell.org
 AAA Math-Activites to review basic skills
http://aaamath.com/index.html
 Algebra Story and Word Problems
http://www.hawaii.edu/suremath/intro_algebra.html

59
Websites cont….
A+Math-Interactive flashcards
http://www.aplusmath.com/
 Arithmetic-Interactive math lessons
http://hometown.aol.com/iongoal/mathlessons.html
 Ask Dr, Math-math questions & answers
http://mathforum.org/dr.math/
 Ask ERIC-math lessons in specific topics
http://www.askeric.org/cgibin/lessons.cgi/Mathematics

60
Websites cont….
Basket Math Interactive-in Spanish & English
http://www.scienceacademy.com
 CanTeach-Math activities and lessons
http://www.canteach.ca/elementary/math.html
 Ministerio Espanol de Educacion y Ciencia
http://descartes.cnice.mecd.es/
 EdHelper.com-database of lessons
http://www.edhelper.com/

61
Websites cont….
A to Z teacher Stuff-Lesson Plans
http://www.atozteachstuff.com/Lesson
Plans/Mathatics/index.shtml
 Matemática Interactiva-Spanish math lessons
http://www.eduteka.org.MI/master/interactive/
 PBS teacher Source-Math lessons
http://www.pbs.org.teachersource/math.html
 The Educator’s Reference desk-math lessons
http://www.eduref.org/cgi-bin/lessons.cgi/Mathematics

62
Websites cont….
Lesson plans and resources
http://www.cloudnet.com/~edrbsass/edmath.htm
 Math Power Points
http://www.pppst.com/index.html

63
Questions?
I’m still unclear about….
64