Domain III: Mathematics
Competency 016
(Mathematics Instruction)
The teacher understands how
children learn mathematical skills
and uses this knowledge to plan,
organize, and implement
instruction and assess learning.
Principles of Mathematics
The NCTM identified six principles for school mathematics—
equity, curriculum, teaching, learning, assessment and
technology (2000).
1. Equity. Excellence in mathematics education requires
equity-high expectations and strong support for all
students.
2. Curriculum. A curriculum must be coherent, focused on
important mathematics, and well articulated across grades.
3. Teaching. Effective mathematics teaching requires
understanding of what Students know and need to learn,
and then challenging and supporting them to learn it well.
4. Learning. Students must learn mathematics with
understanding, actively building new knowledge from
experience and previous knowledge.
5. Assessment. Assessment should support the learn- of
important mathematics concepts, and furnish useful
information to both teachers and Students.
6. Technology. Technology is essential in teaching and
learning mathematics it influences the teaching of
mathematics and enhances student's learning.
Principles of Mathematics
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Use various materials to teach skills and
concepts.
Use different instructional techniques.
Have students explain skills and concepts.
Allow students to see their progress (e.g.,
records of performance).
Teach the language of mathematics (e.g., word
walls, flash cards, etc.).
Use a variety of cues to check for understanding
(e.g., thumbs up, color coded cards, happy face
cards, etc.).
Include concrete, representational, and abstract
activities.
Avoid over reliance on workbooks for dictating
curriculum and providing practice opportunities.
Principles of Mathematics
(cont.)
 Use instructional approaches that will ensure
comprehension and mastery of skills and
concepts, such as cooperative learning, graphic
organizers, use of manipulatives, etc…
 Avoid excessive paper-and-pencil drill that serves
merely as busy work rather than as a meaningful
practice experience.
 Link new instructional knowledge to present
knowledge.
 Show students -and have students explain -how
mathematics is part of daily living.
 Teach mathematical skills and concepts within a
problem-solving context.
Remember…. We learn:
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10% of what we read
20% of what we hear
30% of what we see
50% of what we both hear and see
70% of what we discuss with others
80% of what we experience personally
95% of what we teach to someone else
William Glasser
How Do Children Learn
Mathematics?
Research indicates that children must develop
higher level thinking abilities in order to
interpret certain mathematical concepts.
Because the development of some of these
abilities is so natural, teachers often fail to
consider that children in certain stages of
development may not have acquired them. To
be an effective teacher you must know when
children can be introduced to a given concept
and at what level of abstraction they can deal
with the concept. Basically there are three
levels at which concepts can be represented:
concrete, pictorial, and symbolic.
Levels of math development
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Concrete Level. A concept can be
represented by the appropriate
manipulation of objects. Example:
Placing three beans into each of four
margarine tubs and finding the total
number of beans illustrates that 4 X 3
= 12.
Pictorial Level. A concept can be
represented by appropriate pictures.
Example: A picture of four groups of
three to illustrate 4 X 3 = 12.
Symbolic Level. A concept can be
represented by symbols. Example: 4 X
3 = 12.
Standard for Mathematics
The standards for mathematics are
divided into two
sections- content and process.
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Content standards
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Process standards
Standard for Mathematics
Content Standards
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1. Number and operations. These components
include the concept of number, and fraction, and
basic computation.
2. Algebra. This component includes elementary
algebraic reasoning involving ing patterns and
sets of numbers
3. Geometry. This encompasses the Study of
geometric shapes and spatial reasoning,
4. Measurement. This component includes the
units of measurements (standard and metric)
and the process for measurement in general.
5. Data analysis and probability. These two
components cover the collection, analysis, and
display of mathematics Information.
Standard for Mathematics
Content Standards
Number and operations.
Pre-K-1st Grade. Draw, read, and write values to 99. Count
objects by grouping tens. Compare and order numbers to 99
using models/pictures. Identify coins and value (pennies. nickels
and dimes)
2nd Grade. Draw, read, and write values to 999. Compare and
order numbers to 999. Count values of all coins. Recognize
fractions using models. Skip Count. Recognize odd and even
numbers.
3rd Grade. Draw, read, and write values to 99,999. Compare and
order numbers to 99,999. Show bills and coins to equal a given
number.
4th Grade. Read, write, and determine place value of numbers up
to one million. Draw pictures of numbers to millions. Draw, read,
and write values to hundredths. Compare and order decimals
using models to hundredths. Round numbers to ten and
hundred. Compare fractions using pictures and patterns.
Standard for Mathematics
Content Standards
Numbers and Operations (con’t): (Addition and Subtraction)
Pre-K-1st Grade. Use objects to act out and subtraction stories. Addition
and subtraction to ten. Act out or use objects to describe addition or
subtraction situations including: comparing, missing parts, how many
left. Represent addition and subtraction situations with a number
sentence.
2nd Grade. Estimate sums and differences to 99. Select correct operation
and solve real life problems involving addition and subtraction. Use
addition to solve problems to 999. Addition and subtraction with
money: cents to 99, dollars to 999. Use subtraction to solve problems
with minuends through 99. Basic Fact recall addition and subtraction
to 18, 3 – 5 seconds average recall per fact.
3rd Grade. Estimate sums and differences to 9,999. Use addition to solve
problems with numbers through 9,999
4th Grade. Use addition and subtraction of decimals to solve problems
(tenths and hundredths).
Standard for Mathematics
Content Standards
Numbers and Operations (con’t): (multiplication
and division)
Pre-K-1st Grade. Use a multiplication or division number
sentence to describe a modeled situation. Draw a picture
for a given multiplication or division word problem.
2nd Grade. Use a multiplication or division number sentence to
describe a modeled situation. Draw a picture for a given
multiplication or division word problem.
3rd Grade. Multiply number with factors through 10. Select
correct operation to solve real-life problems involving
multiplication and division.
4th Grade. Estimate products of 2 digit by 2 digit factors. Use
multiplication and division to solve problems: multiplication
3 digit by 2 digit, division one digit quotients with and
without remainders. Basic fact recall, multiplication and
division, 3 to 5 second average recall per fact.
Standard for Mathematics
Content Standards
Algebra.
Pre-K-1st Grade. Identify and extend pattern using
objects. Demonstrate the relation between addition
and subtraction.
2nd Grade. Identify and extend pattern. Determine
missing elements. Write number families for
addition and subtraction. Use operation properties:
addition and subtraction with zero, addends orders
does not matter (commutative property).
3rd Grade. Find the relationship of number pairs
(function) and extend the pattern. Write number
families for multiplication and division.
4th Grade. Use operation properties: multiplication
and division by one, multiplication commutative and
distributive. Use patterns to solve problems.
Standard for Mathematics
Content Standards
Geometry.
Pre-K-1st Grade. Construct squares, rectangles, triangles on
geoboards and with pattern blocks. Locate interior and
exterior points (locations) on plane figures.
2nd Grade. Identify and sort real objects by shape: cube, cone,
sphere, cylinder. Construct congruent shapes on a geoboard
and on dot paper.
3rd Grade. Classify (sort) polygons. Identify figures having
symmetry. Identify line of symmetry. Identify a figure
congruent to a sample figure.
4th Grade. List characteristics of polygons: quadrilaterals,
parallelograms, rectangle, rhombus, square, trapezoid,
pentagons, hexagons, octagons. Classify 3 dimension
figures and faces: cube, sphere, cone, cylinder, prisms,
pyramids
Standard for Mathematics
Content Standards
Measurement.
Pre-K-1st Grade. Compare length and weight. Read a
calendar. Tell time to hour and half-hour.
2nd Grade. Estimate and measure: customary
inches/feet/yards/pounds, metric
centimeters/meters/kilograms. Tell time to 5 minutes.
3rd Grade. Estimate and measure: customary
inches/feet/yards/miles/kilometers/ounces/ pounds, metric
centimeters/meters/grams/ kilograms. Temperature
4th Grade. Find areas of rectangles. Find perimeters. Estimate
and measure capacity: customary cup/pint/quart/gallon,
metric milliliter/liter. Solve problems involving elapse time.
Standard for Mathematics
Content Standards
Data analysis and probability.
Pre-K-1st Grade. Make real graphs. Identify events
that are sure to happen, sure not to happen, or
unsure of outcome.
2nd Grade. Make and interpret picture graphs and
bar graphs. Interpret and use charts.
3rd Grade. Make picture and bar graphs where
each cell represent multiple units. Use bar graphs
to solve application and non routine problems.
4th Grade. List possible outcomes in a given
situation. Interpret line graphs. Plot points in a
coordinate plane. Collect record and organize
data into tables, charts, bar graphs and line
graphs.
Standard for Mathematics
Process Standards
1. Problem solving. In this component students are guided to
formulate and solve mathematics problems that can be
used in real life situations.
2. Reasoning and proof. These two components allow
student opportunities to examine problems, find solutions,
and justify them using logical and mathematics principles.
3. Communication. This component teaches children to use
precise and appropriate mathematics vocabulary to explain
processes and outcomes.
4. Connections. This component emphasizes the importance
of mathematics and the connection to other content areas
and life in general.
5. Representations. This component teaches how
mathematics information can be presented in various
ways- numbers, letters, tables graphs and so on.
Standard for Mathematics
Process Standards
Problem solving.
Pre-K-1st Grade. Act out and draw pictures to represent addition
and subtraction including: how many left, missing parts, comparing.
2nd Grade. Draw a pictures. Use patterns. Act out and draw pictures
to represent 'word problems‘: multiplication and division, non-routine
problems.
3rd Grade. Use strategies: make an organized list, make a table.
Write a number sentence to describe word problems involving
addition and subtraction, and multiplication and division.
4th Grade. Solve word problems with extra information and
determining missing information. Solve problems working backwards.
Standard for Mathematics
Process Standards
Problem solving. (con’t)
In grades k-4, the study of mathematics should emphasize
problem solving so that students can:
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Use problem solving approaches to investigate and
understand mathematical content.
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Formulate problems from everyday mathematical situations.
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Develop and apply strategies to solve a wide variety of
problems.
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Verify and interpret results with respect to the original
problem.
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Acquire confidence in using mathematical meaningfully.
Standard for Mathematics
Process Standards
Problem solving. (con’t)
The mathematics teacher understands
and uses numbers, number systems
and their structure, operations and
algorithms, quantitative reasoning, am
technology appropriate to teach the
statewide curriculum (Texas Essential
Knowledge and Skills [TEKS]) in order
to prepare students to use
mathematics..
Standard for Mathematics
Process Standards
Reasoning and proof.
In grades k-4, the study of mathematics should
emphasize reasoning so that students can:
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Draw logical conclusions about mathematics.
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Use models, known facts, properties, and
relationships to explain their thinking.
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Justify their answers and solution processes.
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Use patterns and relationships to analyze
mathematical situations.
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Believe that mathematical make sense.
Standard for Mathematics
Process Standards
Communication.
For one step problems, students can be asked the following
questions as a way to discuss their work:
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What are you trying to find?
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Which data in the story were needed to find the solution?
Were there unnecessary data?
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What action in the story suggested the operation you used
to find the answer?
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Can you give the answer in a complete sentence?
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Have you checked your work and your answer?
Standard for Mathematics
Process Standards
Connections.
Students should learn how math is
connected to the other content areas,
such as, language arts, science, social
studies, health, physical education, art,
music, etc.
For example, if they are studying
“temperature changes,” they should
know numbers, algebraic computations,
basic operations, etc, to read the
thermometer and use this information
in graphs.
Standard for Mathematics
Process Standards
Representations.
Students should learn the different
ways to represent the numerical
information they gathered from a
problem by using algorithms,
numbers, tables, charts, graphs,
etc.
The Texas Essential Knowledge and
Skills
– TEKS –
The Texas Essential knowledge and Skills (TEKS) requires a
well-balanced curriculum beginning in kindergarten and
continuing through grade12. (TEA, 2006)
It incorporates the principles and standards of the NCTM and
introduces these in a sequential manner that reflects the
cognitive development of children.
Children in kindergarten begin exploring number concepts
using concrete objects and mastering one-to-one
correspondence.
In first and second grades they continue exploring number
concepts and begin studying basic computations skills.
In third through fifth grades they continue expanding number
concepts to include multiplication, division fractions,
decimal representations, geometric principles and algebraic
reasoning.
Mathematics curriculum for K-4
Kindergarten
Whole-number concepts and using, patterns and sorting to explore
numbers, data and shapes
First
Adding and subtracting whole numbers and organizing and analyzing
data
Second
Comparing and ordering whole numbers, applying additional and
subtraction, and using measurement processes
Third
Multiplying and dividing whole numbers connecting fraction symbols to
fractional quantities and standardizing language and procedures in
geometry and measurement
Fourth
Comparing and ordering fractions and decimals, applying multiplication
mid division and developing ideas related to congruence and
symmetry
Cognitive Development and
Mathematics
The cognitive development of children in PreKindergarten (Pre-K) through grade 4 represents a
special challenge when attempting to learn the
symbolic and abstract representations used in
mathematics.
Piaget classified students in Pre-K through grade 4
into two broad stages:
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Preoperational (2 - 7 years), and
Concrete Operational (2-1 1 years) (cited in
Sperry Smith, 2001)
Children in the Preoperational stage of cognitive
development (2 - 7 years, Pre-K through grade 2)
experience problems with at least two perceptual
concepts - centration and conservation (Sperry
Smith, 2001).
Cognitive Development and
Mathematics
Centration
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Four and five-year-old children focus attention on
one characteristic of an object and ignore the
others.
They might notice that the objects are round, but
fail to notice that they have different colors and
texture.
Children at this age play with blocks in a very
simplistic and systematic way-linear fashion.
They do not become more creative because they
are emphasizing one feature at time.
Based on cognitive development, children at this
age generally experience problems developing
and recognizing patterns in mathematics.
Cognitive Development and
Mathematics
Conservation
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Four and five-years-olds might not understand
that changes in the appearance do not
necessarily change characteristics of the object
(Conservation).
This limitation can affect children's ability to
measure volume and to understand the value of
money.
That is, children might get confused when liquid
is moved between containers of different shapes
or when trying, to determine the value of a
quarter versus five nickels.
These perceptual limitations can affect children's
ability to understand measurement and the value
of money.
Cognitive Development and
Mathematics
During the Concrete Operational stage of cognitive
development second to seventh grades children experience
rapid growth in cognitive development.
This stage is characterized by the ability to think logically
about concrete objects or relationships.
Some of' the accomplishments of students at this stage are as
follows:
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Can form conclusions based Oil reason rather than
Perception
Can arrange objects based oil characteristics (Classification)
Can organize objects based oil multiple criteria (Ordering or
Seriation)
Call Understand that changes III appearance do not
necessarily affect the substance (Conservation)
Can conceptualize what would have happened if an action is
reversed (Reversibility)
Cognitive Development and
Mathematics
Despite the cognitive growth of this stage teachers
have to structure lessons to provide students
with a concrete foundation to support their
thinking.
For example, a teacher can introduce the concept of
graphing by asking students to follow the growth
of a plant for a period of time and document the
growth using a Iine graph or to compare how one
plant grows versus another plant using a bar
graph.
Teachers should also break down a task into
manageable components with the use of graphic
organizers (charts, diagrams, webs, time lines,
etc.)
Piaget’s Levels of Cognitive
Development
Children evolve through specific stages in which cognitive
structures become progressively more complex.
Cognitive development refers to the changes that occur in
an individual’s cognitive structures, abilities, and
processes.
Cognitive development is the transformation of the child’s
undifferentiated, unspecialized cognitive abilities into
the adult’s conceptual competence and problem-solving
skill.
Piaget believed children’s schemes, or logical mental
structures, change with age and are initially actionbased (sensorimotor) and later move to a mental
(operational) level.
Piaget’s Levels of Cognitive
Development
Sensorimotor Stage (0-2 years)
Intelligence develops through sensory experiences and movement.
During the sensorimotor stage, infants and toddlers "think" with
their eyes, ears, hands, and other sensorimotor equipment.
Piaget said that a child’s cognitive system is limited to motor
reflexes at birth, but the child builds on these reflexes to
develop more sophisticated procedures.
They learn to generalize their activities to a wider range of
situations and coordinate them into increasingly lengthy chains
of behavior.
Learning involves pulling pushing, turning, twisting, rolling, poking,
and interacting with many different properties of objects.
Piaget’s Levels of Cognitive
Development
Preoperational Stage (2-6/7 years)
Intelligence includes the use of symbols such as
pictures and words to represent ideas and objects.
At this age, according to Piaget, children acquire
representational skills in the area of mental
imagery, and especially language.
They are very self-oriented, and have an egocentric
view; that is, preoperational children can use these
representational skills only to view the world from
their own perspective.
Learning involves discovering distinct properties and
functions of objects as they compare, sort, stack,
roll, distinguish triangles from squares, and begin to
use abstractions to communicate.
Piaget’s Levels of Cognitive
Development
Concrete Operational Stage (6/7-11/12 years)
Cognitive development includes logic but requires physical
examples to which the logic can be applied.
They require experiences with touching, smelling, seeing,
hearing, and performing.
They must use hands-on tools to investigate.
As opposed to preoperational children, children in the
concrete operations stage are able to take into account
another person’s point of view and consider more than
one perspective simultaneously, with their thought
process being more logical, flexible, and organized than
in early childhood.
Piaget’s Levels of Cognitive
Development
Concrete Operational Stage (cont.)
Children can also represent transformations as well
as static situations.
Although they can understand concrete problems,
Piaget would argue that they cannot yet
contemplate or solve abstract problems, and that
they are not yet able to consider all of the
logically possible outcomes.
Children at this stage would have the ability to pass
conservation (numerical), classification, seriation,
and spatial reasoning tasks.
Piaget’s Levels of Cognitive
Development
Formal Operational Stage (11/12+ years)
Thinking includes abstract concepts. This allows analytical
and logical thought without requiring references to
concrete applications.
Persons who reach the formal operation stage are capable
of thinking logically and abstractly. They can also
reason theoretically.
Piaget considered this the ultimate stage of development,
and stated that although the children would still have to
revise their knowledge base, their way of thinking was
as powerful as it would get.
Piaget’s Levels of Cognitive
Development
Piaget suggest that there are four broad factors that are necessary and that affect
the progression through these stages of cognitive development. They are
(1)
(2)
(3)
(4)
maturation
physical experience,
social interaction, and
equilibration.
Clearly, learning experiences for children through the age of 12 must involve
objects, tools, interaction, reflection, and social interaction with materials for
optimal cognitive growth.
The EC-4 teacher knows that mathematical concepts are best learned by children
by manipulating materials and observing what happened-individually and
collaboratively.
A key to EC-4 mathematics is planning concrete experiences that facilitate learning.
A sound mathematics classroom learning environment and curriculum reflect
this cognitive approach to learning.
Manipulatives to implement TEKS
KINDERGARTEN
MUST HAVE
 Pattern blocks; Unifix
cubes; sorting
materials
(vehicles, buttons,
colored pasta, toy
animals);
 counters; geometric
solids; rocker or pan
balance scales;
containers (cans, jars,
boxes);
 sand, rice, or beans.
NICE TO HAVE
 Attribute blocks;
Relation- shapes; oval
links;
 geoboards; 1”
wooden cubes; floor
graphing
mats; literature books
to introduce/teach
mathematical skills
Manipulatives to implement TEKS
FIRST GRADE
MUST HAVE
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Pattern blocks; Unifix
cubes; counters; two
color counters; sorting
materials (vehicles,
buttons, colored pasta,
toy animals);
attribute blocks;
geometric solids;
tangrams; individual
clocks; coins; rocker or
pan balance scales;
containers (cans, jars,
boxes-no standard
measuring utensils);
sand, rice, or beans.
NICE TO HAVE
 Base 10 blocks (units &
rods); Cuisenaire Rods;
color tiles; geoboards;
1” wooden cubes;
 floor graphing mats;
number cubes;
literature books to
introduce/teach
mathematical skills
Manipulatives to implement TEKS
SECOND GRADE
MUST HAVE
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Hundreds boards; base-10
blocks (flats, rods & units);
Unifix cubes; counters; twocolor counters; attribute
blocks; tangrams; geometric
solids; inch worm clocks;
rocker or pan balance scales
containers (cans, jars, boxesno standard measuring
utensils)
sand, rice, or beans
eye droppers
1” wooden cubes; centimeter
cubes
thermometers
fraction bars or circles
coins
oval links
NICE TO HAVE
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Relation shapes; color tiles;
geoboards
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stop watches; kitchen timers
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number cubes
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literature books to
introduce/teach mathematical
skills
Manipulatives to implement TEKS
THIRD GRADE
MUST HAVE
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Base 10 blocks;
calculators; attribute
blocks;
Cuisenaire Rods; coins and
bills; fraction
manipulatives; pattern
blocks;
Individual thermometers;
individual clocks;
Tape measures; tangrams
1” wooden cubes;
geometric solids;
toothpicks; two-color
counters; rulers;
meter/yard sticks; color
tiles
NICE TO HAVE
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Teddy bear counters
Power Shapes; oval
links
Centimeter grid paper
Number cubes
Dot paper
Literature books to
introduce/ teach
mathematical skills
Manipulatives to implement TEKS
FOURTH GRADE
MUST HAVE
 Base 10 blocks; Cuisenaire
Rods; pattern blocks;
fraction manipulatives;
calculators
 geoboards/rubber bands;
mirrors
 floor scale; platform scale
 color tiles
 Miras
 pan balance with
ounce/pound and
grams/kilogram weights;
 geometric solids;
tangrams
 Measuring cylinders, jars,
cups, spoons, and
pitchers;
 sand, beans, or rice
 toothpicks
NICE TO HAVE
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Two-color counters;
centimeter grid paper;
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Dot paper
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Literature books to
introduce/teach
mathematical skills
ESL MODIFICATIONS
Listed below are ESL modifications teachers should
consider using to better meet the needs of their
students with limited English proficiency.
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Use mixed-level groups or partners
Use same-language partner for beginning students
Emphasize oral language development
Use picture cues, video support, real objects (make
concepts concrete)
Use writing frames
Simplify oral or written language
Provide oral tests
Give short answer tests
Give modified tests
Provide highlighted texts
ESL MODIFICATIONS
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Use visual aids
Provide additional instructions
Provide advanced organizers-webbing, outlining,
graphing
Extend time for assignment completion
Shorten assignments
Use assignment notebooks and prompts
Teach in small group
Provide repeated reviews and drills-vary teaching
strategies
Allow for peer teaching
Reduce paper/pencil tasks
ESL MODIFICATIONS
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Provide manipulatives
Seat at front of the classroom
Help student build a card file of vocabulary words
Read to the students
Encourage student to underline key words or
facts
Use language experience activities
Allow students an opportunity to express key
concepts in their own words
Provide a time and place for projects to be
completed at school rather than at home
Provide before and after school time to complete
homework with teacher assistance
ESL MODIFICATIONS
Teacher should:
 Modify "teacher talk" .
 Slower rate
 Clearer articulation
 More use of high frequency vocabulary,
less slang, fewer idioms
 Shorter sentences
 Simpler syntactic structures
 Use redundancy
ESL MODIFICATIONS
Teacher should: (con’t)
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Use direct questioning techniques on previously presented
materials
Check comprehension and retention of information through
direct questions that elicit previously presented
information; also ask as appropriate, "Do you understand?"
Reformulate misunderstood messages
Use gestures, visual, real objects, dramatics
Teach the students how to ask such questions or make
such demands
Make it clear to students that it is perfectly legitimate and
desirable to seek such help in ways that you present to
them
ESL MODIFICATIONS
Teacher should: (con’t)
 Explain new or unfamiliar concepts that are
part of an instructional unit and that might
cause confusion for the learners if they are
not clarified before instruction begins
 Analyze instructional content and language
of each lesson in terms of the learners'
conceptual and linguistic needs
 Provide definitions (explanation) of new or
unfamiliar words
 Teach or clarify new or unfamiliar language
concepts, forms prior to presentation of the
unit or during instruction
ESL MODIFICATIONS
Teacher should: (con’t)
• Provide contextual support
 Use non-verbal frame of reference, such
as physical objects, real life, or
experiences familiar to the students
 Present information that is essential for
learning but which may be unfamiliar to
the students prior to or during the course
of instruction
ESL MODIFICATIONS
Teacher should: (con’t)
 Develop sensitivity to non-verbal
feedback
 Watch for non-verbal feedback that
indicates understanding or lack of
understanding and confusion
 Find out what culturally-determined
non-verbal gestures your students
automatically use to indicate lack of
comprehension; let them know that you
understand those gestures and accept
and appreciate them
ESL MODIFICATIONS
Teacher should: (con’t)
 Learn to detect and interpret
feedback from the learners that
may be culturally different from
what you are used to; silence may
denote confusion in one culture
but comprehension in another.