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NCTM (2000) identified six principles that guide mathematics instruction. These are
>Equity-high expectations and strong support for all students
>Curriculum-coherent, focused on important mathematics, and well-articulated concepts across the grades
>Teaching-understanding of what students know and need to learn and then challenge and support to learn well
>Learning-learn with understanding, actively building new knowledge from exp. and prev knowledge
>Assessment-support the learning of important mathematics concepts/
>Technology-learning mathematics; influences the teaching of mathematics and enhances student's learning.
Content standards describe the five strands of content which are:
>Number and Operations
>Algebra
>Geometry
>Measurement
>Data Analysis and Probability
Process standards describe five strands of content which are:
>Problem Solving
>Reasoning and Proof
>Communication
>Connections
>Representations
This is a content standard strand that deals with understanding numbers developing meanings of operations and computing fluently.
Number and Operations
This is a content standard strand that it is best learned as a set of concepts and techniques tied to the representation of quantitative relations and as a style of mathematical thinking for formalizing patterns, functions and generalizations:
Algebra
As a mathematics teacher you must understand
and use numbers, number systems and their structure, operations and algorithms, quantitative reasoning and technology appropriate to teach (TEKS) Preperations.
TEKS Standards
>Standard 1-Number Concepts
>Standard 2-Patterns and Algebra
>Standard 3-Geometry and Measurement
>Standard 4- Probabiltiy and Statistics
>Standard 5- Mathematical Processes
>Standard 6- Mathematical Perspective
>Standard 7- Mathematical Learning and Instructions
>Standard 8 - Professional Developement
Learning through example is:
inductive teaching
Learning step by step
deductive approach
The most troublesome aspect of math to ESL is the language:
Math assumes you have prior knowledge of words such as denominator, subtraction, etc, or terms that can have different meanings like quarter, column, product, etc, finally, homonphones, were words pronounced the same way but have different meaning (Even).
Thinking and reasoning of children is dominated by preoperational thought -
this is egocentric, centered, irreversible and nontrasformational.
How do children aquire new information:
>Sensorimotor stage (birth - 2 years)
>Preoperational stage (2-7)
>Concrete Operational stage (7-11)
>Formal Operational stage (11-adult)
At this age children tend to fix their attention on a single aspect of a relationship. Two row of the same number of coins are lined up, one-to-0ne, equally spaced, and the student is asked if the rows are the same they will say yes, but when the appearance of the rows are changed they will judge them to be unequal, although they represent the same amount as before.
Preoperational stage (2-7)
Children in the Pre-operational stage experience problems with 3 problems:
>Centration
>Conservation
when the student focus on only one aspect of a situation or problem>
Centration

Example, take two 5 * 8 cards and roll each into a tube, one the short way and the other the long way. Tape to make a cylinder and fill with beans to compare how much each holds. The preschool student might judge the quantity of beans in the tall cylinder to be more than the shorter based on their perception of tall and short.
When understanding that quantity, length, or number of items is unrelated to the arrangement or appearance of the object or items.
Conservation

Example: Understanding the value of money for pre-schoolers. They may think the value of a nickle is more than one dime because the nickle is larger.
Second grade to 7th graders belong to what stage of learning:
Concrete Operational Stage

>This stage is characterized by the ability to think logically about concrete objects or relationships.
Characteristics of the Concrete Operational Stage of cognitive development are:
>Decentering
>Reversibility
>Conservation
>Seriation
>Classification
>Elimination of Egocentism
When a student can take into account multiple aspects of a problem to solve it, based on reason rather than perception.
Decentering
Understanding that objects can be change, then returned to their original state:
Reversibility
The Professional Standards for Teaching Mathematics presents six standards, organized under four categories.
4 categories
>tasks
>discourse,
>environment
>analyses
>provide and structure the time necessary to explore sound mathematics and grapple with significant ideas and problems
>use physical space and materials in ways that facilitate the learning of math
>provide a context that encourages the development of mathematical skill and proficiency
>Respect and value student's ideas, ways of thinking and mathematical disposition.
Creating a leaning environment that fosters development.
Review the materials used in math classrooms according to grade
Go to page 108
TEKS standards that are meant to provide early experiences and exploration of number concepts using concrete objects to learn about one-to-one correspondence is used for what grade
Kindergarten
At these grades students continue exploring number concepts and begin learning basic computation skills
1st and 2nd
Students (according to TEKS) are taught to develop number concepts to include multiplication, division, fraction and decimal representations, geometric principles, and algebraic reasoning.
3rd through 6th
Pre-K learning:
>Explore concrete models and materials, begins to arrange sets of concrete objects in one-to-once correspondence, cont by ones to 10 or higher, by fives or higher, and combine, seperate,and name "how many" concrete ojects."
>Begin to recognize the concept of zero, identify first and last in a series, to compare the number of concrete objects using language
Kindergarten
>Use whole number concepts to describe how many objects are in a set (through 20) using verbal and symbolic descriptions.
>Begin to demonstrate part of and whole with real objects
>Sort to explore numbers, uses patterns, and able to model and create addition and subtraction problems in real situation with concrete objects
First
>create sets of tens and ones using concrete objects to describe, compare and order whole numbers, reads and writes numbers to 99 to describe sets of concrete objects, compares and orders whole numbers up to 99 (less than, greater than, or equal)
>Seperates a whole into two, three, or four equal parts
>Models and creates addition and subtraction problem situations with concrete objects and writes corresponding number sentences
>Identifies individual coins by name and value and describes relationships among them
Second
>Use concrete models of 100's, tens and ones to represent a given whole number up to (999) in various ways. Use place value to read, write and describe the value of whole numbers to 999, and records the comparisons using numbers and symbols (<=>)
>use concrete models to represent and name fractional parts of a whole object
>model addition and subtraction of two-digit numbers with objects, pictures, words and numbers, solve problems with and without regrouping and able to recall and apply basic addition and subtraction facts up to (18)
>Determines the value of a collection of coins up to one dollar and describe how the cent symbol, dollar symbol and the decimal point are to used to name the value of a collection of coins.
Third
>use place value to read, write (in symbols and words), and describes the value of whole numbers and compares and orders whole numbers through 9999.
>Use fraction names and symbols to describe fractional parts of whole objects or sets of objects using concret models
>selects addition or subtraction and uses the operation to solve problems involving the whole numbers through 999. Use problem solving strategies, use rouding and compatible numbers to estimate solutions to addition and subtraction problems
>Apply multiplication facts through 12 by using concrete models and objects, solve division problems
Fourth
>use place value to read, write, compare and order: whole numbers through 999,999,999 and decimals involving tenths and hundreths.
>Use concrete objects and pictorial models to generate equivalent fractions.
>Uses multiplication to solve problems (nor more than two digits, times two digits) and sue division to solve problems.
>Uses strategies, including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication and division problems.
Fifth
>Use place value to read, write and compare, and order whole numbers through 999,999,999,999 and decimals through the thousands place.
>Identify common factors of a set of whole numbers, uses multiplication to solve problems involving whole numbers (no more than 3 digits times 2 digits) and uses division to solve problems involving whole numbers (no more than two digit divisors and 3digit dividends),
Sixth
>Compares and orders non-negative rational numbers, generates equivalent form of rational numbers including whole numbers, fractions, and decimals, uses integers to represent real life sit.
>able to write prime factorizations using exponents, identifies factors of a positive integer, common factors, and the greatest common factor of a set of positive integers.
The organization of curriculum content based on themes or topics:
Thematic instruction
The first step of thematic units is
Identify a Theme

>take into account the interest to children, relevance of the topic and the connection to the sate curriculum. Broad enough to allow integration of content areas. Example (topic like "our solar system" is broad enough to cover every content area of curriculum
The second step of thematic units based on the state curriculum (TEKS) the guiding force in the organization of the integrated curriculum is:
Gathering Materials for the Unit

>Should focus on the same set of generalizations/principles and each of the content areas.
>Materials should include narratives, expositions, drama, poems and a variety of materials and resources books, magazines, newspapers, movies, etc.
>should represent a range of thinking abilities and literacy development
>represent the home as well as the school culture and language of the student.
Thematic Units are organized into 3 main segments
>introduction to the unit
>presentation of the content
>closing activity
This step of thematic unit, is designed to motivate children and to them them interested in the them:
Introduction to the unit
This step exposes the goals and the steps in the presentation of the theme; that is done through various coordinated lessons and activities:
Presentation of the content
This step of the thematic process review and pull together the concepts learned and to celebrate the mastery of the content.
Closing Activity
The five categories for questioning strategies for mathematics discourse are on:
Page 114, table 2-6
An established and well-defined step-by-step problem solving method used to achieve mathematical results.
Algorithm
-10,-8,-7,-5,-3,0,1,3,5,7,9,10 are examples of
Integers

>Integers do not include decimals or fractions.
1,2,3,4....to infinititey
Positive integers
0,1,2,3,4,,,, to infinity
Natural Numbers

>Do not include negative numbers, fractions or decimals
3/5 or 0.6 are:
Rational Numbers

>number that can be expressed as a ratio or quotient of two non-zero integers. Include finite decimals, repeating decimals, mixed numbers, whole numbers.
(2 square root of 2) is Pi
Irrational Numbers

>a number that cannot be represented as an exact ratio of two integers. The decimal form of the number never terminates and never repeats.
-4,0,4....4.25....Pi are all:
Real Numbers
5 to the 3rd (5*5*5) is an example of:
Exponential Notation

>Symbolic way of showing how many times a number or variable is used as a factor.
2,400,000 = 2.4 * 10(to 6th power), 0.0024=2.4*10(to -3rd power)
Scientific Notation
>a form of writing a number as the product of a power of 10 and decimal number greater than or equal to 1 and less than 10.
[-5}=5 or [5]=5

-5,0,5 (The absolute value from 0 to 5 and 0 to -5) is the same 5.
Absolute Value

>the numbers distance from zero on the number line. This ignores the + or -.
(a+b)(2ndPower) is a(2ndPower)+2ab+b(2ndPower)

or

263=200+60+3
Expanded form
534 = 5*100+2*10+381 or 5*10{2ndP}+2*10{1stP}+3*10{0P}
Showing place value by multiplying each digit in a number by the appropriate power of 10.
1984

Thousand or 1000 = to number 1 in 1984
Hundred or 900 = 9
Ten or 80 = 8
One or 4=4
Place Values

>Basic foundation for understanding math computation, were a single number can be explained based on position.
Adding and Subtracting Homogeneous Fractions:
>When the Denominator (bottom number) is the same.