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Fractions / Decimals /
Percents: Success for ALL
Students
Heather Lindfors-Navarro
February 2-3 2013
GSDMC Annual Conference
TODOS: Mathematics for All
The mission of TODOS: Mathematics for ALL is
to advocate for an equitable and high quality
mathematics education for all students— in
particular, Hispanic/Latino students— by
increasing the equity awareness of educators
and their ability to foster students’ proficiency in
rigorous and coherent mathematics.
Diving Deeper Into the
Common Core State Standards
How the Mathematical Practices support a language acquisition in a
classroom setting that is Culturally responsive.
What I have learned about introducing concepts related to fractions,
decimals, and percents so that ALL of my students can experience
success.
I will share with you: the importance of developing the ideas in
relation to benchmarks,and relationships to the concept of one‐half,
The importance of connecting these mathematical ideas with
students’ foundational knowledge
Questions that help guide teachers in facilitating lessons that involve
rigorous mathematics.
standards for mathematical
practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critiques the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Instructional practices that support
language acquisition
MP
Alignment
Pedagogy
Description
Culturally
Responsive
Teaching
(CRT)
Instruction that builds on the languages and cultures that children
bring from their home and community (Slaving & Cheung, 2005).
Cooperative
Learning
Instruction that involves the use of small groups as a means to
optimize students’ own and each other’s learning.
MP 3
Instructional
Conversation
Extended instructional discourse between the teacher and
students.
MPs 2,3, 8
Instruction allowing student to articulate their thinking which in turn
Cognitively Guided
provides teachers with a better understanding of how children learn
Instruction (CGI)
math.
MPs 1-8
Instruction utilizing technology to help connect learning in the
Technologyclassroom to real-life situations and allows students to access
MPs 4,5
Enriched Instruction
information
their native
language.
Almaguer, I., Diaz, Z., Esquiredo, J., Ramirez,
O. in
(2011).
Teaching
for Excellence and Equity in
Mathematics. Vol. 3 No.1, Developing Mathematics Literacy for Bilingual Learners: A framework for
Effective Learning.
Culturally Responsive
Mathematics teaching (CRMT)
CRMT
Evidence
Reflective Question for Teachers How does my
MP
lesson...
Alignment
Cognitive demand
...enable students to closely explore and analyze concepts, procedures,
and reasoning strategies?
MPs 1,2,3
Depth of knowledge &
student
understanding
...make student thinking/understanding visible and deep?
MPs 1,2,3
Mathematical
discourse
...create opportunities to discuss mathematics in meaningful and rigorous
ways?
MPs 3,6
Power and
participation
...distribute mathematics knowledge authority, value student contributions,
and address status differences among students?
Academic language
support (for ELLs)
...provide academic language support for English Language Learners?
Cultural/communitybased funds of
knowledge/social
justice
...(a.) help students connect math with relevant/authentic situations in their
lives?
...(b.)support students’ use of mathematics to understand, critique, and
change an important equity or social justice issue in their lives?
•
MPs 3, 6
Aguirre, J. (2012). Noticias de TODOS. Developing Culturally Responsive Mathematics
Teachers.
problem solving
Rich mathematical problems that promote
reasoning
Opportunities for student groupings to maximize
learning
Problems demanding justification and
communicating mathematically
•
Leiva, M. (2006). National Mathematics Panel Testimony.
Teacher questions
What are the prerequisite concepts and skills?
What is the required Mathematics AND English
vocabulary?
What do the words, phrases, and sentences
mean?
Are these part of our common experiences? If
not, how can we change the problem or use a
teachable moment to teach and reach all
students?
CCSS Fraction progression
Grade 1/2 : Describe partitions of shapes into equal shares
Grade 3: Partitioning a whole into equal parts, unit fractions, preliminary reasoning
about equivalent fractions, add decimal fractions by converting them to fractions
Grade 4: Includes wholes that are collections of objects, equivalent fractions,
comparisons, addition, subtraction(common denominators) and finite decimals,
multiplication of a whole number and a unit fraction, decimal fractions (fractions w/
denominator of 10 or 100)
Grade 5: Adding and subtractions fractions w/ unlike denominators, multiplying
fractions by whole numbers and fractions, dividing whole numbers by fractions,
scaling
Grade 6: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity
means 30/100 times the quantity); solve problems involving finding the whole,
given a part and the percent. Recognizing and describing ratios, rates, and
proportional relationship from other types of situations. (CCProgression, 2011)
Meaning of per cent
The word percent means “per 100” (cent is an abbreviation of
the Latin centum “hundred”). If 35 milliliters out of every 100
milliliters in a juice mixture are orange juice, then the juice
mixture is 35% orange juice (by volume). If a juice mixture is
viewed as made of 100 equal parts, of which 35 are orange
juice, then the juice mixture is 35% orange juice.
Foundational lessons
Introduce
fractions/decimals/p
ercents at the same
time.
Initial diagnostic
Name That Portion, Fractions, Percents, and Decimals. (2004).
Investigations in Number, Data, and Space. TERC. Scott Foresman.
Pearson Education, Inc.
Background knowledge
Name That Portion, Fractions, Percents, and Decimals. (2004). Investigations in Number,
Data, and Space. TERC. Scott Foresman. Pearson Education, Inc.
Math Pathways and Pitfalls
MPP problems require students
to analyze fictional students’
thinking, explain the thinking,
and argue one’s judgment.
Encourages students to take on
a role of “expert” and to offer
her/his thinking as a way to help
the fictional student.
Overall this validates student
contributions to the
mathematical discourse.
Kisty, L., Radosavljevic, A. (2010). A Descriptive Analysis of
Math Pathways and Pitfalls in a Latina/o Bilingual Classroom.
Math Pathways and Pitfalls
This seemly small characteristic
of MPP (i.e., that of having
students analyze fictional
students’ thinking) shifts the
focus away from a single voice
of authority (i.e., the author) to
one of community collaboration.
It is a vehicle for drawing on
students’ knowledge and
resources.
Defines students as having
“voice” which contributes to
everyone’s learning
Kisty, L., Radosavljevic, A. (2010). A Descriptive Analysis of
Math Pathways and Pitfalls in a Latina/o Bilingual Classroom.
Key concepts
100% is equivalent to 1 whole
One-half is equivalent to 50%, and we can gauge whether an amount is greater than
or less than using one-half equivalencies
1% is equivalent to 1 hundredth and 10% is equivalent to 10 hundredths and to 1
tenth
Percents are comparisons to 1 or 100%
Relate fraction, percent, and decimal amounts using an area model
Use benchmarks to compare amounts
Different names for the same amount
Compare number in different symbolic forms on a number line
Use contextual situations
Quilting investigation
Keisha’s grandma and mom have been making quilts for as
long as she can remember. After reading The Keeping Quilt
by Patricia Polacco, Keisha asks her mom if she can learn to
make quilts too.
Keisha’s mom is sewing a quilt that is 8 feet long and 6 feet
wide. Keisha is going to help her mom make this quilt into
their family’s own “keeping quilt.” Her mom asks her to sew a
rectangular piece of the quilt that is 2 feet in length and 3 feet
in width.
What fraction of the area of the whole rectangular quilt will
Keisha sew? Show how you found your solution.
This investigation is adapted from “The Area Model of Multiplication of Fractions” by J.K. Tsankova and K.
Pjanic in Mathematics Teaching in the Middle School, December 2009/January 2010.
Key Instructional practices for
ELLs
Regular and active participation in the classroom—not only
reading and listening but also discussing, explaining, writing,
representing, and presenting—is critical to the success of
ELLs in mathematics.
Research has shown that ELLs can produce explanations,
presentations, etc. and participate in classroom discussions as
they are learning English.
To develop written and oral communication skills, students
need to participate in negotiating meaning for mathematical
situations and in mathematical practices that require output
from students.
Application of Common Core State standards for English Language
Learners
Resources
NCTM SmartBrief - At the end of August 2012, NCTM began to email to its
members this daily newsletter. It is FREE to ANYONE who wants to receive it.
It is a great update on the latest news and trends in education. To subscribe
go to: https://www.smartbrief.com/nctm/index.jsp
NCTM in TWITTERThe NCTM journals Teaching Children Mathematics (TCM)
and Mathematics Teaching in the Middle School (MTMS) now have separate
Twitter accounts: https://twitter.com/TCM_at_NCTM
https://twitter.com/MTMS_at_NCTM
¿TIENES TAREAS?
Tareasplus, http://www.tareasplus.com/, an onlineeducation portal for the
Spanish-speaking world, offers thousands of video lessons on mathematics
and science subjects from kindergarten through early college.
www.todos-math.org.
References
Aguirre, J. (2012). Noticias de TODOS. Developing Culturally Responsive Mathematics Teachers.
Almaguer, I., Diaz, Z., Esquiredo, J., Ramirez, O. (2011). Teaching for Excellence and Equity in Mathematics. Vol. 3
No.1, Developing Mathematics Literacy for Bilingual Learners: A framework for Effective Learning.
Core Standards. http://www.corestandards.org/assets/CCSSI_Math%20Stan
dards.pdf
Core Standards for English Learners. http://www.corestandards.org/assets/application-for-english- learners.pdf
Kisty, L., Radosavljevic, A. (2010). A Descriptive Analysis of Math Pathways and Pitfalls in a Latina/o Bilingual
Classroom.
Leiva, M. (2006). National Mathematics Panel Testimony.
Name That Portion, Fractions, Percents, and Decimals. (2004). Investigations in Number, Data, and Space. TERC.
Scott Foresman. Pearson Education, Inc.
Progressions for the Common Core. (2011). http://commoncoretools.me/wpcontent/uploads/2012/02/ccss_progression_nf_35_2011_08_12.pdf
Ramirez, N.G. & Celedon-Pattichis, S. (2012). Second Language Development and Implications for the Mathematics
Classroom. In S. Celedón-Pattichis & N. Ramirez (Eds.), Beyond good teaching: Advancing mathematics education
for ELLs. (pp. 19-38). Reston, VA: NCTM.
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