La Cosecha 2012
Santa Fe, New Mexico
DEVELOPING BI-LITERACY
THROUGH MATHEMATICS IN
KINDERGARTEN CLASSROOMS
Sylvia Celedón-Pattichis, Ph.D.
University of New Mexico
National Science Foundation Award No. ESI-0424983
OVERVIEW OF THE SESSION
Center for the Mathematics Education of
Latinas/os (CEMELA)
 Background of Kindergarten Study
 Cognitively Guided Instruction (CGI)
 Activities and video clips that engage
participants in understanding children’s
mathematical thinking and ways to support
discursive habits such as listening, speaking,
reading, writing, and representing solutions.

BACKGROUND OF CEMELA
AND KINDERGARTEN STUDY
OVERVIEW OF THE SESSION
Center for the Mathematics Education of
Latinas/os (CEMELA)
 Background on young children and Latina/o
students
 Cognitively Guided Instruction (CGI)
 Professional development work with bilingual
kindergarten teachers
 Activities and video clips that engage
participants in understanding children’s
mathematical thinking and ways to support
discursive habits such as listening, speaking,
reading, writing, and representing solutions.

CEMELA
 University
of Arizona: Marta Civil,
Virginia Horak, and Luis Moll
 University of California at Santa Cruz:
Judit Moschkovich and Kip Téllez
 University of Illinois at Chicago: Lena
Khisty and Aria Razfar
 University of New Mexico: Rick Kitchen
and Sylvia Celedón-Pattichis
CEMELA GOALS
To develop an integrated knowledge model that
connects mathematics teaching and learning to
the cultural, social, political, and linguistic
context of Latina/o children and
 To increase the number of mathematics
educators with this integrated knowledge to
ultimately improve the mathematics education of
low-income Latina/o children.

CONTEXT: THE SCHOOL
 An
urban elementary bilingual school in
New Mexico
 Promotes bilingualism and biliteracy
 Reform-based mathematics curriculum
 Mathematics taught in Spanish
 86% Latina/o student population (mostly
Mexican immigrants)
 100% free or reduced meals
CONTEXT: THE TEACHERS
 Larger
study included 7 teachers. The
focus is on two teachers.
 Ms. Arenas--Kinder--Experienced-Guatemala
 Ms. Carrera--Kinder--Novice--Mexico
 Both teachers had attended CEMELA
Summer Institutes, had in-class support,
and participated in 3 workshops/semester
PORTRAIT OF INSTRUCTION
 Problem
solving lessons
conducted twice a week,
for about 30 minutes
 Average of 3 problems
per lesson
 Both whole group and
small group formats used
 Students had access to a
range of tools
TEACHING MATHEMATICS TO EMERGENT
BILINGUAL (EB) STUDENTS
To foster mathematics academic literacy in the bilingual
classroom, we need:
 High expectations for students’ academic achievement
and maintaining the native language (while developing
English)
 Understanding language as a resource instead of a
deficiency
 Fostering EB participation in mathematics
conversations besides vocabulary (Moschkovich, 2007, 2010)
COGNITIVELY GUIDED
INSTRUCTION (CGI)
(Carpenter et al., 1999)
Predict the percentage of
kindergartners who can solve the
different type of word problems by
the end of the school year.
WHAT THIS RESEARCH
 There
TELLS US…
is often an underestimated
problem solving capacity of young
children (CGI Studies, Carpenter et al., 1999).
 Young children can engage in
problem solving BEFORE they
master basic mathematics facts.
COGNITIVELY GUIDED INSTRUCTION (CGI) AS
A FRAMEWORK (CARPENTER ET AL.,1999)

It is a framework for understanding children’s
mathematical thinking.

Children enter school with a great deal of informal and
intuitive knowledge of mathematics.

Bridging students’ experiential knowledge with formal
school mathematics is critical.

Use of context-rich word problems is based on
knowledge of students’ communities and the
mathematical practices in which their families engage
(González, Moll, & Amanti, 2005).
PROBLEM TYPES
Join
(Result Unknown)
Connie had 5 marbles.
Juan gave her 8 more.
How many does Connie
have altogether?
(Change unknown)
Connie has 5 marbles.
How many does she need
to have 13 altogether?
(Start Unknown)
Connie had some marbles.
Juan gave her 5 more.
Now she has 13. How
many marbles did Connie
have to start with?
Separate
(Result Unknown)
Connie had 13 marbles.
She gave 5 to Juan. How
many marbles does
Connie have left?
(Change unknown)
Connie had 13 marbles.
She gave some to Juan.
Now she has 5 marbles
left. How many marbles
did Connie give to Juan?
(Start Unknown)
Connie had some marbles.
She gave 5 to Juan. Now
she has 8 marbles left.
How many marbles did
Connie have to start with?
Part-Part
Whole
(Whole Unknown)
Connie has 5 red marbles and 8 blue
marbles. How many marbles does she
have?
Compare
(Difference Unknown)
Connie has 13 marbles.
Juan has 5 marbles. How
many more does Connie
have than Juan?
(Part Unknown)
Connie has 13 marbles. 8 are blue. How
many red marbles does Connie have?
(Compare Quantity
Unknown)
Juan has 5 marbles. Connie
has 8 more.
How many does she have?
(Referent Unknown)
Connie has 13 marbles.
She has 5 more than
Juan. How many marbles
does Juan have?
TYPES OF WORD PROBLEMS
Kinder
Mastered in
2nd
WHAT WE LEARNED FROM THE
KINDERGARTEN STUDY ABOUT
SUPPORING DISCURSIVE
MATHEMATICAL HABITS
(Sfard, 2000, 2001)
1. MATHEMATICS ACADEMIC LITERACY:
CONTEXTUALIZING THROUGH STORYTELLING

Introducing problem solving through
“storytelling” conversations.
 Problems that reflect familiar contexts invite students
to draw upon lived experiences to make sense of
mathematical ideas.
 The narrative structure of the problems scaffolds
students’ explanations.
 Problems in the form of stories help students learn to
represent mathematical ideas and connect multiple
representations (e.g., drawings, symbols, objects).
PRACTICE #1: VIDEO CASE
 Ms.
Arenas’ Class, April of 2006
 Typically
began problem solving session
with “Fíjense amorcitos pues, les voy a
contar una historia”
 Video
clip illustrates conversation that
prompted division problem (9÷3)
VIDEO CASE
What opportunities are students
afforded to represent their
mathematical thinking (i.e., by
listening, speaking, reading, and
writing) in your own classrooms?
2. MATHEMATICS ACADEMIC LITERACY:
MULTIMODAL REPRESENTATION



Fostering the use of multimodal
approaches (i.e., pictorial, symbolic,
and written) to communicate the
mathematical thinking.
Representing information in nonlinguistic ways is also an important
consideration for mathematics
academic literacy development.
Representing problem solving
strategies in different ways:
 explaining,
 direct modeling,
 drawing,
 number sentence
Video of students solving a
subtraction problem
(Separate Result Unknown)
MULTIPLICATION PROBLEM
Students
learn to
represent
solutions
pictorially.
MATHEMATICS ACADEMIC LITERACY:
DEVELOPING AND COMMUNICATING EFFECTIVE
PROBLEM SOLVING STRATEGIES
Type
Students develop more
effective and
sophisticated problem
High-frequency
solving strategies by
vocabulary
listening and using oral
/phrases
and written explanations
to explain their
General
mathematical thinking.
vocabulary
 Students must draw
/phrases
from all vocabulary types
participating in
Specialized
mathematical
vocabulary
conversations.

/phrases
Spanish
English
Mas
mismo
Hay
More
Same
There is/are
Igual
Contar
Es Mayor que
Es Menor que
Equal
Count
Is More than
Is Less than
Sumar
Restar
Contar de
cinco en cinco
Oración
numérica
Add
Subtract
Count by fives
Numeric
sentence
3. MATHEMATICS ACADEMIC LITERACY:
DEVELOPING AND COMMUNICATING
EFFICIENT PROBLEM SOLVING STRATEGIES
Video: Multiplication
Comparing strategies


Which strategy is
more efficient?
I had three boxes. In
each box I had five
lollipops. How many
lollipops did I have?
3. MATHEMATICS ACADEMIC LITERACY:
DEVELOPING AND COMMUNICATING EFFICIENT
PROBLEM SOLVING STRATEGIES
Consistently, teachers used scaffolding strategies
and provided multiple opportunities to engage in
mathematics conversations.

Modeling mathematical ways of talking
 Revoicing students’ explanations:
 “So you’re saying that he counted by
fives?”
 “Oh, so you are saying that you counted
on, you started at 4 and then counted on, 5,
6, 7, 8.)”
3. MATHEMATICS ACADEMIC LITERACY:
DEVELOPING AND COMMUNICATING EFFICIENT
PROBLEM SOLVING STRATEGIES
Teachers used questioning to:
 Make sense of the problem and search for a solution
How would you describe the problem in your own
words?
 How would you describe what you are trying to find?
 What do you notice about...?
 What information is given in the problem?


Construct arguments and explain reasoning




Would you explain to me how you figured this out?
How did you count?
Which way to solve the problem is faster? Why?
How can we be sure that...? / How could you prove
that...?
3. MATHEMATICS ACADEMIC LITERACY: DEVELOPING
AND COMMUNICATING EFFECTIVE PROBLEM SOLVING
STRATEGIES

Validating and generalizing mathematics procedures.
“We were already generalizing and they were abbreviating the
long addition process into multiplication process. (…) We try to
validate and generalize algorithms in my class(…) we generalize and
validate procedures. I let the students use the procedure that they
feel better with, as soon as it is valid. You might ask me what
happens when the algorithms are not valid? It doesn’t matter. It is
good learning for them. When the algorithms are not valid, we try
them with the whole class or the whole group, sometimes I do it
with small groups, and then we validate it once and they learn by
their errors and they’ll never use that algorithm again. And the
advantage is that they have several different algorithms to use, so
when they have to solve it quickly (…) they go with the one that is
easier to them, and then they can perform more accurate and more
quickly.”
What are ways to support
students in making a shift
from problem solving to
problem posing?
4. MATHEMATICS ACADEMIC LITERACY: FROM
PROBLEM SOLVERS TO PROBLEM POSERS
First,
students develop and use their
own strategies to solve problems:
Direct modeling (using concrete objects
or manipulatives)
 Counting strategies (counting up or
down, counting on from, etc.)

Finally, students construct their own
“story” problem becoming “problem
posers.”
 Students develop ownership and
confidence as mathematics learners.

MATH JOURNALS
PROMOTE ACADEMIC
LITERACY
DEVELOPMENT
Thinking:
Requires in depth
understanding of
the structure of a
problem
Drawing:
Representing the
problem and
solution
Writing:
Providing the
story line for
their own number
stories
Creating their own problems, students
learn to pose problems in writing,
representing the solution pictorially and
symbolically.
PARTITIVE DIVISION PROBLEMS
Yo tenía ocho galletas y
les di a mis amigas
cuatro.
The drawing shows
how she distributed
four cookies to each
friend.
The algorithm shows
that she didn’t have
any left.
Clear understanding
of what zero
represents.
DEVELOPING MATHEMATICS ACADEMIC
LITERACY AND BEING PART OF A MATHEMATICS
DISCOURSE COMMUNITY (CELEDÓN-PATTICHIS & TURNER, 2012).


Students were afforded opportunities to hear and
use the language needed for learning mathematics,
necessary for appropriation (Chval & Khisty, 2009).
Students progressively incorporated more accurate
ways of explaining their ideas and strategies.
ACKNOWLEDGEMENTS
Dr.
Erin Turner
Dr. Sandra Musanti
Dr. Mary Marshall
UNM CEMELA Research Team
Kindergarten Teachers
CONTACT INFORMATION
CEMELA
Website:
http://cemela.math.arizona.edu
Sylvia
Celedón-Pattichis
sceledon@unm.edu
REFERENCES
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