domain iii: mathematics competency 016 (mathematics instruction)

Domain III: MathematicsCompetency 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.


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.


(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:

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

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.

Content standards

Process standards

Standard for MathematicsContent Standards

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.


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.


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).


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.


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.


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


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.


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 MathematicsProcess 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.


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.


Problem solving. (con’t)

In grades k-4, the study of mathematics should emphasize problem solving so that students can:

Use problem solving approaches to investigate and understand mathematical content.

Formulate problems from everyday mathematical situations.

Develop and apply strategies to solve a wide variety of problems.

Verify and interpret results with respect to the original problem.

Acquire confidence in using mathematical meaningfully.


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..


Reasoning and proof.

In grades k-4, the study of mathematics should emphasize reasoning so that students can:

Draw logical conclusions about mathematics.

Use models, known facts, properties, and relationships to explain their thinking.

Justify their answers and solution processes.

Use patterns and relationships to analyze mathematical situations.

Believe that mathematical make sense.


Communication.

For one step problems, students can be asked the following questions as a way to discuss their work:

What are you trying to find?

Which data in the story were needed to find the solution? Were there unnecessary data?

What action in the story suggested the operation you used to find the answer?

Can you give the answer in a complete sentence?

Have you checked your work and your answer?


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.


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

KindergartenWhole‑number concepts and using, patterns and sorting to explore

numbers, data and shapes

FirstAdding and subtracting whole numbers and organizing and analyzing data

SecondComparing and ordering whole numbers, applying additional and

subtraction, and using measurement processes

ThirdMultiplying and dividing whole numbers connecting fraction symbols to

fractional quantities and standardizing language and procedures in geometry and measurement

FourthComparing 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 Pre-Kindergarten (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:

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).


Centration

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.


Conservation

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.


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:

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)


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 action-based (sensorimotor) and later move to a mental (operational) level.


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.


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.


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.


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.


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 suggest that there are four broad factors that are necessary and that affect the progression through these stages of cognitive development. They are

(1) maturation (2) physical experience, (3) social interaction, and (4) 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 TEKSKINDERGARTEN

MUST HAVE Pattern blocks; Unifix

cubes; sorting materials(vehicles, buttons, colored pasta, toy animals);

counters; geometric solids; rocker or panbalance scales; containers (cans, jars, boxes);

sand, rice, or beans.

NICE TO HAVE Attribute blocks;

Relation- shapes; oval links;

geoboards; 1” wooden cubes; floor graphingmats; literature books to introduce/teachmathematical skills

Manipulatives to implement TEKSFIRST GRADE

MUST HAVE Pattern blocks; Unifix

cubes; counters; two color counters; sorting materials (vehicles, buttons, colored pasta, toy animals);

attribute blocks; geometric solids; tangrams; individualclocks; 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 TEKSSECOND GRADE

MUST HAVE

Hundreds boards; base-10 blocks (flats, rods & units);

Unifix cubes; counters; two-color counters; attribute blocks; tangrams; geometric solids; inch worm clocks; rocker or pan balance scales

containers (cans, jars, boxes-no 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 Relation shapes; color tiles;

geoboards stop watches; kitchen timers number cubes literature books to

introduce/teach mathematical skills

Manipulatives to implement TEKSTHIRD GRADE

MUST HAVE 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

Teddy bear counters Power Shapes; oval

links Centimeter grid paper Number cubes Dot paper Literature books to

introduce/ teach mathematical skills

Manipulatives to implement TEKSFOURTH 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 Two-color counters;

centimeter grid paper; Dot paper 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.

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

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

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 MODIFICATIONSTeacher should: (con’t)

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.

domain iii: mathematics competency 016 (mathematics instruction)

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