Teaching Elementary Math 1
Can be represented in two ways; measurement and partition.
Division
Line segment
Part of a line with two endpoints.
Return (RET)
An output instruction that causes the scan to return to the main program. It can be a conditional, or unconditional output depending on the desired operation of the subroutine.
What does the geometry standard call for students to do?
Analyze characteristics of geometric shapes and make math arguments about their relationship and use it to visualize, and for spatial reasoning and geometric modeling for problems.
Number multiplied by the sum of two other numbers can be handed out, or distributed, to both numbers, multiplied by each of them separately, products added together. Ex: a(b+c) = (a 3 b) + (a 3 c)
Distributive
reflecting on the design
8. reflect and refine
Natural numbers
counting numbers (1,2,3,4,5)
Which of the following skills would student develop first?
Counting objects
3-dimensional solid figure. Ex: 6 faces, 12 edges, + 8 vertices. F + V = E + 2
Cube
Associative Property
+/x are _________:; the order that numbers are grouped in +/x does not affect the result. ex: 2x(10x6)=(2x10)x6=120->a(bc)=(ab)c
What are the three facets involved in a math lesson?
Content Level Concept Development Class Coherence
what should be assessed
conceptual understanding, procedural fluency, strategic competence, adaptive reasoning
objective=
concrete
Teach _____ to _____
concrete, abstract
Isoceles Triangle
Has at least two sides that are exactly the same length. This forces two of their angles to also be acute angles of exactly the same size.
Range
The difference between the highest and lowest values in a data set.
When is a student's understanding of math deeper and more lasting?
When they connect mathematical ideas and see the coherent whole.
1. ex. 190^0 = 1.
When you have a number that is being raised to zero it always equals ______.
What is classification?
Where a child can name and identify sets of objects according to appearance, size, or other characteristic.
What is decentering?
Where a child can take into account multiple aspects of a problem to solve it.
What is elimination of egocentrism?
Where a child can view things from another person's perspective.
What is seriation?
Where the child is able to arrange objects in order according to size, shape, or any other attribute.
Spatial temporal knowledge
How children think about space and time, is another area that can be strengthened using puzzles such as working with a Möbius strip
Rate
How quickly computations are made
In the number 1984 the number 9 would be in what place value?
Hundred: 900
TYPES OF REASONING -
INDUCTIVE VS. DEDUCTIVE
What is reversibility?
Where the child understands that objects can be changed and returned to their original state.
common fraction
a number in the form a/b, such that a and b are whole numbers
Prime numbers
a number with exactly two whole- number factors (1 and the number itself) 2,3,5,7,11,13
Commutative property of addition and multiplication
+/x are commutative: switching the order of two numbers beging added or multiplied does not change result. ex: 100+8=8+100->a+b=b+a 100x8=8x100->ab=ba
zero property of multiplication
a number x 0 = 0
multiplicative identity element
a number x 1 = the number
Assessment types and applications ( error patterns, task analysis, student verbalizations )
-
Connecting an objective to standards and research based norms
-
RIDE ( Intervention )
- Read the problem correctly - identify the relevant information - -
a cost
a price
guess and check
a problem-solving strategy in which a student identifies the best solution they can and then checks its accuracy
Major operations on rational numbers
addition subtraction multiplication division
Scatterplot
A graph showing paired data values. A scatter plot can suggest various kinds of correlations between variables with a certain confidence interval
Histogram
A graph showing the results of tabulating the number of items found in defined categories and shown using vertical bars; often referred to as a bar chart.
Polygon
A simple closed figure with any number of sides (square, triangle, hexagon etc.).
Protractor
A square, circular or semicircular tool, typically in transparent perspex, for measuring angles. The units of measurement utilized are usually degrees
Algorithm
A step by step procedure or formula for solving a problem
Common Factor
A number that is a factor of two or more numbers. A common factor of 9 and 6 is 3.
Principle
Generalizations, develop rules
Equity
High expectations and strong support for all students
Concrete
Manipulatives --algebra tiles
Distance around the object. Ex: Around fence
Perimeter
Common multiple
A number that is a multiple of two or more numbers. A common multiple of 2 and 3 is 6.
Addend
A number that is added. in 5 + 8 = 13, the addends are 5 and 8
Dividend
A number that is divided by another number. For example, in 36/4 = 9, 36 is the dividend.
Association
Words or symbols
pictorial models
a visual representation which models the essential characteristic of a mathematical concept
Multiplicand/multiplier
A number that is multiplied by another number. In 7 x 4 =28 the multiplicand is 7 and the multiplier is 4.
proportion
equation in which a fraction is set equal to another
confidence interval
estimated range of values, likely to include a particular population parameter
Concrete
hands on manipulatives
irrational number
have a decimal, aren't whole
parallel postulates
if 2 lines make perfect right angles, they are parallel
india was primary source of developments
in 6th-12th century
What measure could be used to report the distance traveled in walking around a track?
kilometers
Modeling the Operations: Student matches the elements of a given group with abstract numbers. Ex: 3 rabbits eating 4 carrots daily would be set up like: 3 x 4
Abstract method
slope
rate of change
Grades K-1 Manipulatives
Beads, straws, tiles, macaroni, buttons, paper clips, toothpicks, legos, board games, pictographs
Teach _____ rather than just facts.
Concepts
Math should be taught _____
Conceptually
Modeling the Operations: This method the teacher allows the students to use real objects.
Concreate method
Sequence of instructional tools
Concrete, pictorial, abstract
Operations are what?
systematic
Inverse operations
Operations that are the opposite of each other and cancel each other out. Addition and subtractions are inverse operations, as are multiplication and division
Whole numbers
natural numbers and zero
Learning Logs
1. Communication of student understanding 2. For writing and drawing about concepts and solutions 3. Also for reflective responses on divergent questions --drawing solutions
Write down lesson plans how long in advance?
a week before
strategic competence
ability to formulate, represent, and solve mathematical problems
Alternative assessment
non traditional options to assess students' learning. portfolios, journals, notebooks, projects, and presentations.
3 common probability distributions
normal, binomial, geometric
*study skill 17.2
page 176
Exponent
A number that tells how many times the base is to be used as a factor or to be multiplied by itself. in 2 to the 3rd power, 2 is the base and 3 is the exponent meaning 2x2x2
Divergent Questions
1. Extending students' thinking beyond basic facts (predicting)(More than 1 answer/solution) Benefits: -Instigates discovery -Probes for deeper understanding -Uses upper levels of Bloom's -Encourages creative thinking and reflecting
Think pair share
1. For peer discussion 2. Limited to brief periods (15-30 seconds) 3. Good for less verbal students 4. Partner talk or shoulder talk
Wait time
1. Pause time (at least 3-5 seconds) between question and response Benefits: -Student responses increase dramatically over time -Confidence is strengthened -Struggling students contribute more -More reflective thinking is exercised
Factor
A number to be multiplied or a number that divides evenly into a given second number is a factor of that number. In 2x3 =6, 2 and 3 are factors of 6
To perform Prime Factorization you can use?
A number tree
Irrational Number
A number whose decimal form is nonterminating and nonrepeating. Irrational numbers can't be expressed as fractions.
Ordered Pair
A pair of numbers used to locate a point on a coordinate plane is called an ordered pair. An ordered pair is written in the form (x, y) where x is the x-coordinate and y is the y-coordinate.
Rhombus
A parallelogram with four congruent sides.
Convergent Questions
1. Questions that focus on basic knowledge and comprehension (factual answer) Benefits: -Directs attention to parts of an activity -Recalls order of info or procedures -Recalls needed facts -Ensure understanding ex. 7x8 -- only one answer
worthwhile tasks
1. the students have no prescribed or memorized rules or methods to sole 2. there is not a perception that there is one correct solution
How much time is spent teaching a new concept?
1/3
What is pre operational thought?
A pattern of thinking that is egocentric, centered, irreversibly and non transformational and dominates thinking and reasoning of children.
Collaboration
A philosophy about hot to relate to others--how to learn and work.
Cognitive guided instruction (instructional practises )
- involves ____________________ of process, activation of prior knowledge, and explicit teaching - helps students become more efficient at selecting and using effective strategies - sequence: read math problems, paraphrase, draw a diagram, establish a plan, predict correct answer, solve the problem, evaluate for correctness
Formal Operational Stage
(11-adult) The stage during which the individual can think hypothetically, can consider future possibilities, can use deductive logic, and can think abstractly
Preoperational Stage
(2-7) The stage during which a child learns to use language but does not yet comprehend the mental operations of concrete logic Responses are based on their own perception and not on logical reasoning
associative property of addition
(4+2) + 3 = 4 + (2+3)
associatie property of multiplication
(6x8) x 2= 6 x (8x2)
Concrete Operational Stage
(7-11) The stage during which children gain the mental operations that enable them to think logically about concrete events
Sensorimotor Stage
(Birth-2 years) Infants are busy discovering relationships between their bodies and the environment
Jump (JMP)
) instruction is used to bypass portions of the ladder program the jump out will leave every output in their state
factors that contribute to difficulties with problem solving
- difficulty in reading makes understanding the math problem almost impossible - difficulty with logical reasoning, which is the basis of many story problems - insufficient instruction in math - mathematics education has focused on operations and not on understanding the reasons for operations or even a thorough understanding of numbers that are involved in operations
Purposes of assessment
- instructional design. - student placement. - monitoring student progress. - summative evaluation of student. - accountability. - validating student achievement. - true/false - worked out problems. - essays. - fill in the blank. - matching. - multiple choice. - program evaluation.
Research
- it supports conceptually based teaching or teaching concepts rather than just memorizing facts. Ex: students should learn the concept of multiplication before they learning multiplication itself. - it supports the use of communication, problem solving, and working in cooperative groups to develop a broader base of understanding. - it shows that teachers should promote the idea that math is used as a thinking tool for answering important questions, not just a set of rules and procedures to memorize.
Effective use of Formative Assessment in Math includes
- learning progressions, learning goals, descriptive feedback, self-peer assessment, collaboration
Guidelines for implementing and scoring a math CBM
- multiple examples of a variety of math problems - standardized directions, 2 minutes of time
Concrete, semi concrete, and abstract( instructional practises )
- process for teaching mathematical computational and problem solving skills - sequence: hands on, tactile elements, illustrations or drawings - complete the problem without tactile elements or illstrations
Peer tutors ( intervention )
- the use of tutors is beneficial both in math and other subject areas - the use of __________ ___________ is necessary in effective peer tutoring programs
Goal setting ( intervention )
- used to address poor attitudes in math - self regulation and goal setting were linked to higher levels of achievement in math
At what degree does the F and C tempertures meet?
-40 degrees
Number Sequence
-Each number being raised to an exponent: 1,3,9,27...(3*0,3*1,3*2,3*3) -Each number may be a prime number in order (2,3,5,7...) or skipping every other prime (2,5,11,13...) -Each number might be the sum of that number plus the number before it (2,4,6,10,16)
Minuend
...
standard numerals
...
Piaget stages of development
1- Birth to 2 Years of Age: (Sensorimotor Stage) First stage of child's mental development which mainly involves sensation and motor skills such as hearing, seeing, feeling, tasting, moving, manipulating, biting, chewing, etc. In this stage the child does not know that physical objects remain in existence when out of sight. 2- 2 to 7 years of age (preoperational Stage) In this stage children use their mental ability to represent events and objects in various ways like using symbols, gestures and communication ...they are not yet able to conceptualize abstractly and need concrete physical situations to help with understanding. 3- 7-11 years of age (Concrete-Operational Stage) At this stage the child starts to conceptualize, creating logical structures that explain physical experiences. Abstract problem solving is also possible at this stage. Math problems can be solved with numbers and not just objects. 4- 11 years to adulthood (Formal-Operational Stage) Children become more systematic and reasonable; they reason tangibly and are also capable of reasoning and thinking in more abstract, hypothetical and idealistic terms.
Types of learning
1- association: words or symbols Example: students associate the word triangle by its symbol. 2- concept: relational or concrete attributes Example: the corresponding angles are equal and the ratios of corresponding sides are equal. 3- principle: generalization, developed rules Example: the area of trapezoid is developed from the concept of a trapezoid and the area of triangles, rectangles and/or parallelograms. 4- problem solving: putting together concepts and principles to solve a problem new to the learner Example: given a composite figure the student determines the area using the areas of triangles and rectangles.
Types of Assessment
1- formal vs informal. 2- diagnostic / formative / summative. 3- standardized vs criterion referenced.
Types of reasoning
1- inductive (informal): This reasoning goes from the specific to general. It uses observations and patterns to infer a generalization. 2- deductive (formal): This reasoning process reaches conclusions based on accepted truths and logical reasoning. And it goes from general to specific.
Piaget stages of development
1- sensorimotor (birth-2 years of age) First stage of child's mental development which mainly involves sensations and motor skills, such as: hearing, seeing, feeling, tasting, moving, manipulating, biting, chewing, etc. In this stage the child does not know that physical objects remain in existence when out of sight. 2- pre-operational stage (2-7 years of age) Chicken use their mental ability to represent events and objects in various ways like using symbols, gestures and communication. 3- concrete-operational stage (7-11 years of age) The child start to conceptualize, creating logical structure that explain physical activities. Abstract problem solving is also possible at this stage. Math problems can be solved with numbers not just objects. 4- formal-operational stage (11 years +) Children become more systematic and reasonabl; they reason tangibly and are also capable of reasoning and thinking in more abstract, hypothetical and idealistic terms.
A general problem solving method that can be applied to many types of problems
1- understand. 2- plan. 3- solve. 4- check.
Performance based assessments
1. Authentic- real use of skill concept, student generated. (Students show what they know) 2. Students demonstrate while teachers assess 3. Assessment is semi-concrete/concrete 4. Individual or group
commutative property of multiplication
7 X 9 = 9 x 7
What are the five types of questions teachers should ask to elicit justification and reasoning?
1. Questions that help students learn how to make math meaningful. 2. Questions that help students become self-relient and determine if something is mathematically correct? 3. Questions that help students learn to reason mathematically. 4. Questions that help students learn to solve problems. 5. Questions that relate to helping students connect mathematics, it's ideas, and application.
Group Think Lab
1. Teacher designs problems for students to solve. 2. Students are put into groups of 2 to 4 (solving problems with the use of materials) 3. Students via math reasoning and use of appropriate math manipulatives solve the problem and share results with whole group.
5 Developmental math goals
1. The students will become math problem solvers 2. The students will learn to communicate mathematically 3. The students will learn to reason mathematically 4. The students will learn to value mathematics 5. The students will become confident in one's ability to do/use math
Multi-sensory approach
1. Using more than one sense for learning a skill/concept (based on brain research of left/right hemisphere use) 2. Using a visual/auditory/kin-esthetic process for teaching/learning.
Math Word Wall
1. Word 2. Definition 3. Drawing 4. Personal Connection
the before phase of a lesson
1. activate prior knowledge 2. be sure the task is understood 3. establish clear expectations
5 sources of numbers that may be useful for teaching place value to older students
1. an odometer 2. numbers from students science or s.s. text 3. numbers from population of school 4. population data from town, country, state, or county 5. financial data page from newspaper
three phases of learning
1. before: getting ready 2. during: students work 3. after: Class discussion
what are several numeration concepts
1. cardinality 2. grouping pattern 3. place value 4. place value (base 10) 5. one digit per place 6. places- linear/ordered 7. decimal point 8. place relation/regrouping 9.implied zeros 10. face times place 11. implied addition 12. order 13. verbal names (0-9) 14. verbal names with places 15. periods and names 16. naming in the ones period 17. naming multi digit numbers 18. decimal places and their verbal names
nrc five interwoven strands that compose proficiency and definitions
1. conceptual understanding - refers to understanding mathematics concepts and operations 2. procedural fluency - being able to accurately and efficiently conduct operations and math practices 3. strategic competence - ability to formulate and conduct math problems 4. adaptive reasoning - refers to thinking about, explaining and justifying mathematical work 5. productive disposition - appreciating the useful and positive influences of understanding mathematics and how ones disposition toward mathematics influences success
preparing a lesson 3 brackets
1. content and task decisions 2. lesson plan 3. reflecting on the design
tiered lessons considerations
1. degree of assistance 2. how structured the task is 3. complexity of the task given 4. complexity of process
content and task decisions
1. determine the learning goals 2. consider your students' needs 3. select, design, or adapt a worthwhile task 4. design lesson assessments
during phase of lesson
1. let go: don't give too much guidance, embrace struggle 2. note students mathematical thinking; what ideas they are using, approaches, interactions 3. provide appropriate support: support student thinking without taking away their responsibility, solving in a way that makes sense 4. provide worthwhile extensions: preparation for students who finish quickly
conversion of units
1. multiply by conversion factor 2. cancel the miles units 3. solve
how is math performance assessed and progress monitored
1. need to see if teacher has time to use an individually administered assessment or to use a group administered measure 2. needs to determine whether measure is designed for students in age range of students he is teaching progress monitoring - use CBM (curriculum based measurement) - documenting the extent to which the student is learning the critical elements in the curriculum that you have targeted
prenumber skills students need to progress in math
1. one-to-one correspondence-matching one object with another 2. classification - ability to group or sort objects on the basis of one or more common properties 3. Seriation - depends on recognition of common attributes or properties of objects, ordering depends on the degree to which the object possesses the attribute 4. algebraic principles
maximize test
1. permit students to use calculators 2. use manipulatives and drawings 3. include opportunities for explanations 4. use open ended questions
create a classroom environment for doing mathematics
1. persistance, effort, and concentration are important in learning math 2. students share their ideas 3. errors or strategies that didn't work are opportunities for learning 4. students listen to each other 5. students look for and discuss connections
after phase of lesson
1. promote a mathematical community of learners: productive discussion, helping students learn together 2. listen actively without evaluation: find out how they approve the problem 3. summarize main ideas and identify future problems: make connections, lay groundwork for future tasks
factors that influence math ability
1. psychological factors - intelligence/cognitive ability, distractibility, and cognitive learning strategies 2. education factors- quality and amount of instructional intervention across the range of areas of math 3. personality factors- persistence, self-concept, and attitudes toward mathematics 4. neuropsychological factors- perception and neurological trauma
how can RTI be used in math
1. screening - to determine if they have math problems in numeracy, math calculations, and problem solving 2. evidence-based math - instruction based on best research available 3. interventions - additional instruction through short-term interventions when students have difficulties that aren't addressed in math program 4. progress monitoring - students progress is documented to make sure they are staying on track and meeting benchmarks
All of the following are measurements of obtuse angles
110 degrees, 135 degrees, 91 degrees
The mass of a cookie is closest to:
15 grams
divisibility rule
1: divisible by 2 if even 2: divisble by there if sum of numbers is divided by 3: divisible by 4 if last 2 digits are divisible by four
Parallelogram
2 parings of parallel lines, 2 pares of congruent angles, opposite angles are congruent , diagonals bisect
Distributive Property
2(4 + 3) = 2 x 4 + 2 x 3 = 8 + 6 = 14
net
2-dimensional figure that can be cut out and folded up to make a 3-dimensional solid
Review is what part of the lesson?
2/3
What number comes next in this pattern? 3, 8, 13, 18
23
Learning Environment - Teaching Math
4 categories: 1. task - projects, questions, problems that students engage 2. discourse - manner of representing, thinking, talking, agreeing, disagreeing that teachers and students use to engage 3. environment - setting for learning 4. analysis - systematic reflection in which teachers engage
Quadrilateral
4 sides sum of angles =360
four point rubric
4. excellent; full accomplishment; met consistent, minor errors 3. proficient; substantial accomplishment; errors minor, needs minimal feedback 2. marginal; partial accomplishment; lack of evidence of understanding, further teaching is required 1. unsatisfactory; little accomplishment; fragments are accomplished, little or no success
The typical lesson lasts how long?
45 min
commutative property of addition
5 + 3 = 3 + 5
Which is a good example to explain the idea of "doubles plus one" to students?
5 + 6 = 11
polyhedra
5 regular solids- cube, tetrahedron, octahedron, icosahedron, dodecahedron
lesson plan
5. plan the before phase of the lesson 6. plan the during phase of the lesson 7. plan the after phase of the lesson
At what age do children tend to fix their attention on a single aspect of a relationship?
6 years old/pre operational stage.
What is the smallest multiple off 12,15 and 20?
60 Start with the number 20 and go out by 10s.
Dividend, divisor and quotient? 80/8=10
80=dividend 8=divisor 10= quotient
Plane
A 2-sided surface
Geometry
A branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Considered to have its formal beginning in about 300 BC, when the Greek mathematician Euclid gathered what was known at the time, added original work of his own, and arranged 465 propositions into 13 books, called 'Elements'.
What is centration?
A child focuses only on one aspect of a situation or problem.
Guided participation
A child's performance, with guidance and support, of an activity in the adult world.
Common Denominator
A common multiple of two or more denominators for 1/6 and 5/8, it is 24.
Ratio
A comparison of two or more values( 1/2, 4/6 or 1:2, 4:6) or ( 1 is to 2; 4 is to 6).
Iteration
A computational process in which a cycle of actions or operations is repeated, generally to get closer to a final answer.
Iteration
A computational process in which the same steps are repeated until the final answer is found.
Factor tree
A diagram showing how a composite number breaks down into its prime factors.
Factor
A factor of a number divides the number evenly. This means the quotient is a whole number and the remainder is 0. ex: 24->2,3,4,6,8,12
Face
A flat surface of a solid figure
Fractions
A fraction is part of an entire object. ex: 1/5,3/7,12/50 1=Numerator _ 5=Denominator parts of integers and therefore fit between then, when comparing size. In order to compare fractions, their DENOMINATORS must be equal.
Polyhedron
A geometric figure solid with flat faces and straight edges.
line graph
A graph in which a line shows changes in data, often over time
Coordinate system
A graph with a horizontal number line (x-axis) and a vertical number line (y-axis) that are perpendicular to each other. The point of intersection is called the origin and labeled 0 on the graph. An ordered pair (x,y) is used to name a point on a coordinate system.
Performance based assessment ex.
A group project that designs a seating arrangement for a certain amount of people
Expression
A mathematical phrase that combines numbers and/or variables using mathematical operations. An expression is a representation of a value; for example, variables and/or numerals that appear alone or in combination with operators are expressions. Examples: 5 + 6 - (3 + 2)/18 a + b - c
Mass
A measure of how much matter is in an object. Commonly measured by how much something weighs. But weight can change depending on where you are (such as on the moon) while this stays the same.
Inductive Reasoning
A method of drawing a probable conclusion from an emerging configuration of data. It occurs by analyzing observations and discovering common patterns. When patterns repeat for an extended period of time, an analyst can logically predict that those patterns will continue to repeat.
Multiples
A multiple of a number is the product of that number and any whole number. ex: 3->6,9,12,15...
Pi
A name given to the ratio of the circumference of a circle to the diameter. That means, for any circle, you can divide the circumference (the distance around the circle) by the diameter and always get exactly the same number. It doesn't matter how big or small the circle is, Pi remains the same. While Pi has a decimal that never ends, it is usually estimated as 3.14.
Inverse Relationship x/division & +/-
A number fact is made up of 3 numbers. -These 3 numbers can be used to make up other number facts. 3,4,7 Addition->3+4=7, 4+3=7 Subtraction->7-3=4, 7-4=3 Multiplication/Division->Fact families 4x4, 5x5, 6x6
Power (of a number)
A number found by multiplying the number by itself one or more times.
Rules of Divisibility
A number is divisible by another number if the quotient is a counting number and the remainder is 0. 2: The number is even. 3: The sum of the digits is divisible by 3. 5: The last digit is 0 or 5. 6: The number is even and divisible by 3. 9: The sum of the digits is divisible by 9.
Average
A number obtained by dividing the sum of tow or more addends by the number of addends (2+4+6 = 12/3 = 4
Ordered pair
A number pair, such as (2,3), in which the 2 (x-axis) is the first number and the 3 (y-axis) is the second number.
Prime Number
A number that can be divided evenly only by 1 or itself. And it must be a whole number greater than 1.
Divisor
A number that divides another number. In the example 36/4 = 9, the 4 is the divisor.
Understanding Word Problems
A problem solving process can help students make sense of the problems. They can do this by reading the problem more than once, annotating words and numbers, visualizing the situation etc.
Mathematical Proofs
A proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid.
Parallelogram
A quadrilateral (any four-sided polygon) with opposite sides parallel and congruent.
Trapezoid
A quadrilateral with one pair of parallel sides. These sides are called the upper and lower bases.
Function
A relation that uniquely associates members of one set with members of another set. There will always be three main parts: The input, The relationship, The output. For instance, 4(input) X 2(function) = 8(output)
Circle Graph
A round graph that uses different-sized wedges to show how portions of a set of data compare with the whole set.
Heterogeneous ability grouping
A strategy that groups students of varied ability instead of by grade/age level.
Array
A systematic arrangement of objects or numbers, generally in rows and columns
Acute triangle
A triangle that contains acute angles (<0 and > 90 degrees
Scalene Triangle
A triangle where all three sides are different in length.
Equilateral Triangle
A triangle with all three sides the same length. All equilateral triangles are also isoceles triangles. All three internal angles are also congruent to each other and are each 60°.
Pictograph
A visual representation used to make comparisons. A key always appears at the bottom of a pictograph or picture graph showing how many each object represents
Visual Representations
A way for students to access abstract math ideas. Drawing a situation, graphing lists of data, or placing numbers on the number line. This strategy gives students tools and ways of thinking that they can use as they advance their learning.
What is mathematical communication?
A way of sharing ideas and clarifying understanding.
Algorithms
A way to solve problems without visual models. Algorithms are standard step by step procedures for solving mathematical problems.
Set
A well-defined collection of objects or numbers.
Composite number
A whole number greater than 1 that is not a prime number (e.g 4, 6, 9, 10, 12 etc).
2 pairs of angles of 2 triangles are = in measurement + pair of corresponding sides are equal in length = congruent
AAS (Angle-Angle-Side)
2 triangles are = measurement, included sides = then = congruent
ASA (Angle-Side Angle)
What are numbers and operations curriculum for sixth graders?
Able to write prime factorizations using exponents, compares and orders non-negative rational numbers.
Number's distance from zero on the number line. Ignores + or - sign of a number. |x| used on calculator. Ex: |-5| = 5
Absolute Value
Components of math fluency
Accuracy, Automaticity, Rate, Flexibility.
Learning
Actively build new knowledge from experience and previous knowledge
Service learning
Activity that promotes learning and development through participation in a meaningful community service project.
The Zero Property of Addition/Multiplication
Adding 0 to a number leaves it unchanged. We call 0 the additive identity ex:88+0=88->a+0=a Multiplying any number by 0 gives 0 ex: 88x0=0->0x1003=0
Associative Property
Addition: a(b+c)= (a+b)+c Multiplication: a(bc)= (ab)c
Commutative Property
Addition: a+b= b+a Multiplication: ab=ba
Identity
Additive : a + 0= a Multiplicative: a x1=a
The number 0 added to any natural number yields a sum that is the same as the other natural number is?
Additive identity, or identity element of addition
Inverse Property Additive/ Multiplicative
Additive-> a+(-a)=0 Multiplicative->ax(1/a)=1
Inverse
Additive: a+ (-a)= 0 Multiplicative: a x (1/a)=1
Requires knowledge of the concepts of a variable, function, and equation.
Algebraic Expressions
A statement that is written using one or more variables and constants that shows a greater than or less than relationship. Ex: 2x+ 8 > 24
Algebraic Inequality
fenced zone,
All rungs that exist between the two MCR instructions are part of a
Interior Angles
An angle inside a shape. When you add up the Interior Angle and Exterior Angle you get a straight line, 180°. The sum of the measures of the interior angles of a triangle is 180 degrees. All the interior angles of a square are right angles -- that means that they are all 90 degrees.
Obtuse angle
An angle that measures greater than 90 degrees and less than 180 degrees.
Acute Angle
An angle that measures greater that 0 degrees and less than 90 degrees
Peer Tutoring
An instructional grouping practice where pairs of students work on assigned skills.
Manipulatives
An object which is designed so that a learner can perceive some mathematical concept by manipulating it. The use of manipulatives provides a way for children to learn concepts in a developmentally appropriate, hands-on and an experiencing way. Mathematical manipulatives are used in the first step of teaching mathematical concepts, that of concrete representation.
What opportunities do digital videos provide for students?
An opportunity to evaluate their own explanation of problem solutions.
Jump to Subroutine
An output instruction that causes the scan to jump to a specified subroutine file (U3 through U255) when input conditions are true.
Ratio
Another way to write a fraction -if the ratio is 2:3, it means two out of 3 or 2/3
Some/Sum
Answer to addition problem
Rational number
Any number that can be expressed as a fraction a/b where a and b are integers and b does not equal 0, such as 3, 3/1, 1/4, .34 and 56 percent.
Math Scene Investigator
Application used to solve mathematics word problems. Involves inspecting and finding clues, plan and solve, and retrace.
Expository instruction
Approach to instruction in which information is presented in more or less the same form in which students are expected to learn.
Peer tutoring
Approach to instruction in which one student provides instruction to help another student master a classroom topic.
Discovery learning
Approach to instruction in which students develop an understanding of a topic through firsthand interaction with the environment.
Learner-directed instruction
Approach to instruction in which students have considerable say in the issues they address and how to address them.
Convergent thinking
Approach to instruction in which students work with a small group of peers to achieve a common goal and help one another learn.
Cooperative learning
Approach to instruction in which students work with a small group of peers to achieve a common goal and help one another learn.
Teacher-directed instruction
Approach to instruction in which the teacher is largely in control of the content and course of the lesson.
Authentic activity
Approach to instruction similar to one students might encounter in the outside world.
Learner-centered instruction
Approach to teaching in which instructional strategies are chosen largely on the basis of students' existing abilities, predispositions, and needs.
Reciprocal teaching
Approach to teaching reading and listening comprehension in which students take turns asking teacher-like questions of classmates.
What are numbers and operations curriculum for Pre-K
Count by ones to 10 or higher, by fives or higher, and combine, separate, and name concrete objects. Begin to recognize the concept of zero, identify first and last in a series, and compare concrete objects using same, equal, more than,and less than.
Squaring the measure of the side of the square. Rep as: A = s2
Area of a square
Found by multiplying the measure of the length by width of the rectangle. Rep as: A = 2xw
Aread of a rectangle
Logical-mathematical knowledge/skills to work on
Arithmetic, classifying, ordering, number concdepts
Can model a multiplication problem. First number is vertical and second number is horizontal.
Array
How is algebra 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, function, and generalizations.
At what age do children develop a basic understanding of numbers?
As early as two.
Types of learning
Association, concept, principle, problem solving
Property: grouping three or more addends or factors in a different way does not change the sum or product. Ex: (3+7)+5 results the same sum as 3 + (7+5)
Associative
Grade 6 Manipulatives
Attribute logic blocks, algebra tiles, video games, math software games, iPods, balanced metric eight scale, computer/internet, smart boards, digital cameras, computer templates
Research Basis:
Ausabel, Bloom, Brownell, Gagne, Piaget, Vygotsky, etc. Keep in mind that it is not so much the name, but what their research tells us. The Competency does not specify the researcher.
Mean
Average of a set of values
Grade 3 Manipulatives
Base-ten blocks, tangrams, patterns, scales: customary and metric, spinners, dice, games, measuring tape, 0-99 charts, bar graphs
Flexibility
Being able to solve problems in more than one way and selecting the most appropriate method.
"A job worth doing, is worth doing right."
Ben Franklin
Grade 2 Manipulatives
Blocks, cubes, chips, measuring cups, money, number lines, dominos, unified cubes, ruler, balance scales, linear graphs
Abstract is _____
Brains only
"Every pole has to be laid in the right place..."
C.T. Studd
Square Root (SQR)
Calculates the square root of the source and stores the value in a specified destination.
Logical mathematical knowledge
Can be developed by using arithmetic, classifying and ordering and using number concepts activities.
Tiling
Can be used to relate to calculating the area of rectangles wherein a rectangle is divided into unit squares and counted to find the area.
Discrete Models
Counters
NCTM standards proposed 5 categories of questions that teachers should ask to elicit justification and reasoning
Category 1: questions that hep students learn how to make learning math meaningful "Do you agree (disagree) with the correct answer?" Category 2: questions that help students become self-reliant and determine whether something is mathematically correct "Does that make sense?" Category 3: questions that help students learn to reason mathematically "How could you explain this is your own words?" Category 4: questions that help students learn to solve problems "What would happen if (if not)....?" Category 5: questions that relate to helping students connect mathematics, its ideas, and its applications "Can you write another problem like this one?"
Negate (NEG):
Changes the sign of the source and stores the value in a specified destination.
Physical knowledge
Children get by observing their physical environment, seems basic. However, good listening and observing skills must be sharpened through teaching these skills directly and practicing them.
Problem-based learning
Classroom activity in which students acquire new knowledge and skills while working on a complex problem similar to those in the outside world.
Project-based learning
Classroom activity in which students acquire new knowledge and skills while working on a complex, multifaceted project that yields a concrete end project.
Ausabel
Cognitive frameworks
What must the curriculum be like in mathematics instruction?
Coherent, focused on important mathematics, and well-articulated concepts across the grades.
Inductive reasoning
Collecting data to draw a conclusion that may or may not be true.
A whole number that is a multiple of two or more given numbers. Ex: 2, 3, and 4 are 12, 24, 36, 48....
Common Multiple
Property: Order of adding addends or multiplying factors does not determine the sum or product. Ex: 6x9 gives the same product as 9x6. Division and subtraction can not use this.
Commutative
Inner Limit:
Compares a test value to an upper and lower limit. If the test value is between upper and lower limit, the instruction is true.
Outer Limit
Compares a test value to an upper and lower limit. If the test value is outside the two limits, the instruction is true
Focal points: 6th grade
Compares and order nonnegative rational numbers, generates equivalent forms of rational numbers, including who numbers, fractions, and decimals Uses integers to represent real-life situations Is able to write prime factorizations using exponents; identifies facts of a positive integer, coachman facts, and the GCF of a set of positive integers
Less Than or Equal:
Compares two words, or one word to a preset value. The instruction is true if A <= B.
Greater Than or Equal
Compares two words, or one word to a preset value. The instruction is true if A >= B.
Not Equal:
Compares two words, or one word to a preset value. The instruction is true if A != B.
Less Than
Compares two words, or one word to a preset value. The instruction is true if A < B.
Greater Than
Compares two words, or one word to a preset value. The instruction is true if A > B.
Equal:
Compares two words, or one word to a preset value. The instruction is true if A = B.
Working memory
Component of memory that holds and actively thinks about and processes a limited amount of information.
Long-term memory
Component of memory that holds knowledge and skills for a relatively long time.
Number greater than zero which is divisible by at least one other number besides one (1) and itself resulting in an integer (has at least 3 factors). Ex: 9....1,3, and 9 are all?
Composite Number
Other counting numbers, composed of several counting number factors. Number 1 is neither prime nor composite, has only one counting number factor: 1.
Composite numbers
COMPREHENSION LEVEL
Comprehension questions have students represent information in their own words or in a different way = What are some ways to express a sum of 10? = What other problems could you write that show how 7, 3, and 10 are related to each other? Key words to aid in determining whether questions and activities are at this level include: = Tell in your own words = Describe in your own words = interpret in your own words = compare and contrast = explain in your own words = what does it mean.
Children in Preoperational stage experience problems with at least these 2 perceptual concepts
Concentration and Conservation
Grade 5 Manipulatives
Cuisenaire rods, virtual manipulative, graduated cylinders, timers, metric beaker for volume, metric trundle wheel, fraction tower cubes, connecting cubes, board games, reaction tiles, pie charts and graphs
What are the process in the concrete operational stage?
Decentering, reversibility, conservation, serialization, classification, and elimination of egocentrism.
Characteristics of children in the Concrete Operational Stage
Decentering: child can form a conclusion based on reason rather that perception Reversibility: child understands that objects can be changed and then returned to their original state Conservation: child understands that size of an object is unrelated to the arrangement or appearance of it Serialization: able to arrange objects in the order of size or shape Classification: child can name and identify sets of objects according ro appearance, size or there characteristics Elimination of Egocentrism: child is able to view things from another person's perspective
Decimals
Decimals are a method of writing fractional numbers without writing a fraction having a numerator and denominator. The fraction 7/10 could be written as the decimal 0.7. The period or decimal point indicates that this is a decimal.
Deductive Reasoning
Deductive reasoning is one of the two basic forms of valid reasoning. It begins with a general hypothesis or known fact and creates a specific conclusion from that generalization. The basic idea of deductive reasoning is that if something is true of a class of things in general, this truth applies to all members of that class.
What are numbers and operations curriculum for kindergarten?
Demonstrate part of and whole with real objects, uses patterns, and model and create addition and subtraction problems, uses whole number concepts to describe how many objects are in a set up to 20.
Modeling
Demonstrating a behavior for another; also, observing and imitating another's behavior.
2- diagnostic/formative/summative
Diagnostic - Used to pre-determine stu- dents' knowledge. Example: Pre-test before instruction on a unit has begun. • Formative - Used to determine students' ongoing learning and retention, whether or not additional instruction is needed. Example: Giving a quiz within a unit. • Summative - Used to determine mastery of material. Example: A semester exam or a project incorporating all objectives from a unit.
Diagnostic/ formative / summative
Diagnostic: used to pre-determine students knowledge. Example: pr-test before instruction on a unit has begun. Formative: used to determine students ongoing learning and retention, whether or not additional instruction is needed. Example: giving a quiz within a unit. Summative: used to determine mastery of material. Example: a semester exam or a project incorporating all objectives from a unit.
Is a line segment where two faces of a three-dimensional figure meet.
Edge
Types of problems for Multiplication/Division
Equal groups or repeated addition Area and array Combination Multiplicative Comparison
The NCTM identified principles that should guide mathematics instruction
Equity Curriculum Teaching Learning Assessment Technology
What are the six principles that should guide mathematics instruction?
Equity, curriculum, teaching, learning, assessment and technology.
Technology
Essential in teaching and learning math; it influences the teaching of math while enhancing and facilitating students' learning
Rounding numbers to the nearest decimal place for accuracy.
Estimating
Compute (CPT):
Evaluates an expression, and then stores the value in a specified destination. This instruction takes more processing power/scan time than those previously discussed.
Equivalent expression without parentheses. Ex: 263= 200 + 60 + 3
Expanded Form
Showing place value by multiplying each digit in a number by the appropriate power of 10.
Expanded Notation
Focal points: Pre-K
Explore concrete models and materials Counts to 10 or higher Names "how many" concrete objects Begins to recognize the concept of zero "same", "equal", "one more", "less than"
Symbolic way of showing how many times a number or variable is used as a factor. Exponent expressed out: 5x5x5=125
Exponential Notation
Expanded form
Expressing a number as factors [325 = (3x100) + (2x10) + (5x1)]
Rounding
Expressing a number to the nearest thousandth, hundredth, tenth, one, ten, hundred, thousand, and so on as directed.
Each of the plain regions of a geometric body is a face.
Faces
Any of the numbers, or symbols in mathematics that, when multiplied together, form a product.
Factors
1. Recalls information
Facts definitions, terms, properties, rules, procedures, algorithm, rote responses.
Estimating
Finding a number that is close enough to the right answer. You are not trying to get the exact right answer. What you want is something that is close enough. Also, involves the concept of predicting, or making an educated guess
Prime Factorization
Finding which prime numbers multiply together to make the original number.
Standard System of Measurement
Fluid Ounces Cups Pints Quarts Gallons Ounces Pounds Tons Inches Feet Yards Miles
Concentration
Focusing on only one aspect of a situation of problem
One is considered the what of multiplication?
Identity element
1- formal vs informal
Formal - An end-product testing. Example: A unit test, a final exam • Informal-The teacher observes, listens, and questions students in order to gather information regarding student learning. Example: When conducting guided prac- tice, the teacher observed that three stu- dents were "counting" using their fingers. • Formal assessments are often the familiar paper-and-pencil tests such as teacher- made tests or quizzes, but can also be such tests as diagnostic screenings, stan- dardized tests, spelling tests or final exams. Informal assessments may take the form of teacher observation, graded daily work, or student answers to oral questions. Teachers, who often need to know before the final grading period if their students are having difficulty, should use both.
Right
Forming a 90 degree angle
What must students do in data analysis and probability standard?
Formulate questions and collect, organize, and display relevant data.
Equivalent Fractions
Fractions that may look different, but are equal to each other. Two equivalent fractions may have a different numerator and a different denominator. For instance, The fractions 2/3 and 4/6 are equivalent. (A fraction is also equivalent to itself. In this case, the numerator and denominator would be the same.)
Congruent Triangles
If two triangles are congruent they will have exactly the same three sides and exactly the same three angles. The equal sides and angles may not be in the same position (if there is a turn or a flip), but they will be there.
Venn Diagram
Illustrates the similarities and differences between two sets of concepts
Grade 4 Manipulatives
Geometric solids, geoboards, calculators, protractors, tangrams, pentominoes, graphing paper, fraction kit, platform scale, graphic boards, scatter plot graphs
four step problem solving
George polya, 1. understand the problem (What is the problem?) 2. Devising a plan (Think about how to solve it) 3. carrying out the plan (implementation) 4. looking back (reflection, does it make sense, is it understood)
Accuracy
Getting the correct answer
What do tangrams do?
Give tactile experience with triangles.
Focal points: 1st grade
Has ability to create sets of 10 Reads and writes numbers up to 99 Separates a whole into two, three, or four equal parts and uses appropriate language Medels and creata addition and subtraction problems Identifies individual coins by name and value
Diagonal
In a polygon, a segment that connects one vertex to another vertex but is not a side of the polygon.
The largest positive integer that divides into the numbers without producing a remainder.
Greatest Common Divisor (GCD)
70%
Group discussion
NTCM 6 principles
Guide math instruction 1. Equity - high expectation and strong support for all students 2. curriculum - coherent, focused on important and well articulated math concepts across grades 3. teaching - understanding what students know and need to learn while challenging/supporting students 4. learning - learn math with understanding, actively building new knowledge from experience and previous knowledge 5. assessment - support the learning of math concepts & give useful information to both teachers and students 6. technology - influences the teaching of math while enhancing and facilitating student learning
A fourth-grade student with a specific learning disability in mathematics has just failed a mathematics quiz. When the student shows the quiz to his special education teacher, she sees that he has made the same error repeatedly, as illustrated by the following example: 239 +446=6715. The special education teacher arranges to meet later that afternoon with the student's fourth-grade teacher. The special education teacher's best recommendation regarding an instructional accommodation for the student would be to suggest that the fourth-grade teacher:
Have the student solve problems that reinforce the concept of place value
Which of the following experiences demonstrates a meaningful way to show math integrated with another subject area?
Having students read a short story and record how many times they come across a given set of sight words, then working with the class to construct a bar graph representing the data
How does the use of manipulatives help in mathemastics instruction?
Helps teachers move students from concrete through abstract stages of reasoning necessary for learning higher-level concepts.
ANALYSIS LEVEL
Here is where outcomes are = thought through, = analyzed, and = alternatives are considered. Jake now has more than $12. He had $5 at the beginning of the day and completed several jobs in his neighborhood for which he was paid. = what is the minimum amount Jake could have earned working today? = How did you arrive at your answer? = Are other answers possible? Keywords in questions and activities might include: = reason = why? = what are the causes? = what are the consequences? = what are the steps in the process? = how did you start? = what are some examples? = list all solutions possible? = what problems might arise?
What is equity in mathematics instruction?
High expectations and strong support for all students.
Cognitive apprenticeship
Mentorship in which a teacher and a student work together on a challenging task and the teacher gives guidance about how to think about the task.
Pythagorean Theorem
In any right triangle, the are of the square whose side in the the hypotenuse (longest side and opposite from the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the sides that meet at the right angle). Where c is the length of the hypotenuse and a and b are the lengths of the two sides, this may be expressed as a squared + b squared = c squared.
Place Value
In our decimal number system, the value of a digit depends on its place, or position, in the number. Each place has a value of 10 times the place to its right.
Rate
In tracking how many exercises were correctly done in a fixed amount of time.
Explicit and systematic instruction
Includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.
Computer-based instruction (CBI)
Instruction provided via computer technology.
Development of Learning:
Instructional Moves: Linked between and among the following using modeling, describing and recording by both teacher and student. • Concrete—manipulatives, models, hands-on • Pictorial—pictures, diagrams, graphs, technology • Abstract—symbols, words Properties and Concepts: • Learner centered • Mathematically correct • Developmentally appropriate
Development of learning
Instructional moves, properties and concepts
Whole number that includes all positive and negative numbers, including zero.
Integers
Learning strategy
Intentional use of one or more cognitive processes for a particular learning task.
What are the three main segments of thematic units?
Introduction to unit, presentation of content, and closing activity.
Combination
Involve different combinations that can be made from two sets. ex: shirts and pants Multiplication->Oscar 17apples/Tom 5 times as many 5x17=__ apples as Oscar, how many does Tim have? Division-> Tom has 85 apples, 5 times as many as Oscar. 85 divided by 5=__
A number that cannot be represented as an exact ratio or two integers.
Irrational Numbers
Concrete instruction
Is connected to students real experiences, and uses activities that students can see, hear, taste and touch.
Basic strategy for inequalities and equations is to?
Isolate x on one side
What are the challenges students face with math nomenclature?
It abounds assumptions concerning students' prior knowledge of special terms. Terms that have a meaning in one subject can assume a different meaning in the vocabulary of math, and the vocabulary tends to encompass a variety of homophones.
Individual education plan (IEP)
It's a plan with legal requirements for assisting students with physical or learning disabilities
constructivism
Jean Piagets work, developed in the 1930s, notion that learners are not blank slates but rather creators or constructors of their own learning. integrated networks, cognitive schemas
All basic mathematics skills are introduced in grades?
K-2ed
Bloom's Taxonomy?
Knowledge Comprehension Application Analysis Synthesis Evaluation
Properties and concepts must be
Learner centered, correct, developmentally appropriate
What is the deductive approach to teaching?
Learning step by step.
Inductive teaching
Learning through examples or Learning step by step
What is inductive teaching?
Learning through examples.
sociocultural theory
Lev Vygotsky, mental processes exist between and among people in social learning settings and from this social settings the learner moves ideas into his or her own psychological realm information is accessible if the learner has support from peers or more knowledgeable others
Gagne
Levels or learning, each requires different types of instruction
When graph of pairs of numbers results in points lying on a straight line, a function is said to be?
Linear
Parallel lines
Lines in the same plane that do not intersect.
Perpendicular lines
Lines that intersect at right angles.
First find the greatest common denominator. Start with the biggest common denominator listed. Thats 10. Not all denominators fit into 10. So go up by 10's until you find a number that fits. 40. 5x8=40, 3x8= 24 giving you 24/40. Do the same to the rest. 20/40, 24/40, 30/40, 36/40 aka 2/4, 3/5, 6/8, 9/10.
List 3/5, 9/10, 6/8, and 2/4 in least to greatest order.
Physical knowledge/skills to work on
Listening, observation
Manipulatives
Manipulatives are any objects that can be touched and moved (manipulated) to assist understanding. Manipulatives can take the form of counters such as small toys or buttons used to help a kindergarten student add, 100 noodles glued onto construction paper to show place value, or a circle folded and cut into eighths to assist with understanding fractions.
What trains the intellect?
Math
MOVE FROM CONCRETE TO ABSTRACT
Mathematics should be taught conceptually. Children should learn the "big ideas" behind what they do before they learn to do it. Concrete instruction is connected to students' real experiences, and uses activities that students can see, hear, taste and touch. Abstract ideas require children to use their imaginations or their brains only, without help from pictures or real objects. With each new idea taught, good teachers start with easier, more reality-based thinking and move to more imaginary ideas.
Quadrilaterals
Means "four sides. "Any four-sided shape is a Quadrilateral. But the sides have to be straight, and it has to be 2-dimensional.
Students know how many in each group (set) but do not know how many sets.
Measurement division
Mnemonic
Memory aid or trick designed to help students learn and remember a specific piece of information.
Metric System of Measurement
Milliliters Liters Grams Kilograms Tonnes Millimeters Centimeters Meters Kilometers
PEMDAS
Mnemonic code for students to remember the orders of operations.
Of any counting number are the results of multiplying that continues number by all the counting numbers. Ex: 7: 7,14, 21,28....
Multiples
When a factor is multiplied by 0, then the product is 0.
Multiplication property of 0
Multiplication and Division of Fractions
Multiplication->Denominators do not have to be the same. Multiply across both n/d. ex: 3/7x3/5=9/35 or 4/6 x5/8=20/48 Division-> Denominators do not need to be the same BUT need to KCF=keep change flip aka: INVERSE
Combination
Multiplication: How many combinations of shirts and pants can be made out of 5 shirts and 17 pants? Division: If you have 5 shirts, how many pants are needed to make 85 combinations of pants and shirts?
Multiplicative Comparison
Multiplication: Oscar has 17 apples and Tom has 5 times as many apples as oscar does? How Many apples Does Tom have? Division: Tom has 85 apples. This is 5 times as many as what Oscar has. How many apples does Oscar have?
Equal Groups or repeated addition
Multiplication: Oscar has 5 bags of apples with 17 apples in each bag. How many apples does Oscar have altogether. Partition or sharing division: Oscar has 85 apples. He Arranges the apples into 5 bags with the same amount of apples in each bag. How many apples are in each bag? Measurement or subtractive division: Oscar has 85 apples. He arranges the apples into bags of 17 apples each. How many bags of apples did he make?
Area and Array
Multiplication: Oscar has a farm of apple trees planted in 5 rows of 17 apple trees in each row. How many apple trees does he have on his farm? Division: Oscar planted 85 palm trees on his farm. He want's to plant the trees in 5 equal rows of palm trees. How many palm trees will he need to plant in each row?
Any number multiplied by 1 remains the same... 34 x 1 = 34.
Multiplicative identity property of 1.
Scientific Notation
Multiply out the scientific notation. -Multiplied by 10(to a positive number such as 2), the decimal is moved right. 6.89x10=689. -Multiplied by 10( to a negative number such as -2, you move the decimal place to the left ex: 5367.x10=53.67 ex: 3.4567x10=3456.7 (BEFORE COMPARING), 26543X10*-3=26.543
Curriculum
Must be coherent, focused on important and well-articulated math concepts across the grades
Positive integer or non-negative integer
Natural Numbers
Equivalent to the set of counting numbers.
Natural numbers
Parities Division
Needed when students know how many groups there needs to be but not how many objects will be in each group.
Measurement Division
Needed when students know how many objects are in each group but do not know how many groups there are.
Creativity (creative thinking)
New and original behavior that yields a productive and culturally appropriate result.
When graph of pair numbers is not on a line the function is?
Non-Linear
What are the five strands of content students should learn in mathematics?
Number and operations, algebra, geometry, measurement, and data analysis and probability.
Continuous Models
Number lines that children might choose for solving problems
Name three types of conservation in math?
Numbers Mass/ Energy Volume Time Length Area
Even
Numbers divisible by 2
Three Biblical Considerations When Teaching Math:
Numbers reveal God's order Math embodies truth Math builds character
Symbolic references
Objects or experiences
Jean Piaget
Observed and recorded the intellectual abilities of infants, children, and adolescents Stages of intellectual development are related to brain growth and led him to conclude that the thinking and reasoning of children were dominated by preparation thought (i.e., a pattern of thinking that is egocentric, centered, irreversible, and non transformational)
Array
One way to model multiplication visually. One factor is shown vertically and the other is horizontally. 2x4 **** ****
With two points this is exactly one straight line, straight lines are considered?
One-dimensional
In the number 1984 the number 4 would be in what place value?
One: 4
Organizing
Organizing information in a problem and help students better understand the patterns needed to solve the problem. Organize info in the problem.
Bivariate data
Pairs of linked numerical observations. Ex. a list of heights and weights for each player on a football team. Box plot. A method of visually displaying a distribution of data values by using the median, quartiles, and extremes of the data set. A box shows the middle 50% of the data.
Students know the number of groups (sets) but they do not know the number of objects in each set.
Partitive division
A sequence governed by a rule that can be expressed in words or symbols.
Patterns
P = 22 + 2w
Perimeter
P = 4s
Perimeter
By adding twice the length of the rectangle to twice the width. Ex: l=10m w=5m P = 2(10m) + 2(5m) = 30m
Perimeter of a rectangle
Multiplying four times the measure of a side of the square.
Perimeter of a square
Found by adding the measures of 3 sides of the triangle. Rep as: P = s1 + s2 + s3
Perimeter of a triangle
Model
Person who demonstrates a behavior for someone else.
Significant Number (or Figures)
The digits in a value that are known with some degree of confidence. As the number of these increases, the more certain the measurement. They are especially important in rounding.
Who came up with the stages of development?
Piaget
Pictorial
Picture or mental image. Visualize
Basic Foundation for understanding mathematic computation.
Place Values
Flat surfaces without edges are?
Planes
A third grade student with dyscalculia is likely to have the most difficulty with which of the following tasks?
Playing a game involving quantity and place value
Apply the order of operations
Please excuse my dear aunt sally P: Parenthese E: Exponents(work from left to right) M/D: Multiplication/Division (work from left to right) A/D: Addition/Subtraction (work from left to right)
Is a specific location, taking up no space, having no area, no dimensions and frequently represented by a dot.
Point
A special education teacher and a general education teacher co-teach a kindergarten class that includes several students with disabilities. The teachers are designing a mathematics learning center to give students practice with shapes and patterns. The general education teacher feels may not be very accessible to the students with disabilities. Which of the following strategies would likely be most effective for the special education teacher to use to promote communication and collaboration between the teachers?
Pointing out specific aspects of the materials that may be challenging for the students with disabilities and suggesting possible alternative materials.
Differentiated instruction
Practice of individualizing instructional methods, and possibly also individualizing specific content and instructional goals, to align with each student's existing knowledge, skills, and needs.
What does PANIC stand for?
Preparation Attention Need Information Closing
Methods of Mathematics and Engagement?
Presentation Organization Presenter
Every composite number can be written as a product of prime numbers and is called?
Prime Factorization
Number with only two counting number factors - 1 and the number itself. Ex: 2,3,5,7,11,13, and 17.
Prime number
Does not matter which factor pair you start with in prime factorization as long as you continue factoring until you only have?
Prime numbers
Probability
Probability (or likelihood) is a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a value between 0 (0% chance or will not happen) and 1 (100% chance or will happen). The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen.
Deductive reasoning
Process of drawing a logical inference about something that must be true, given other information that has already been presented as true.
Critical thinking
Process of evaluating the accuracy and worth of information and lines of reasoning.
Divergent thinking
Process of moving mentally in a variety of directions from a single idea.
Modeling
Promote mathematical thinking and facilitate and understanding of key concepts and mathematical structure. Students engage their senses to better understand and reason with abstract concepts.
Manipulatives
Provide "hands-on-learning" Excellent way for students to develop self-verbalization learning strategies
What do geoboards do?
Provide the tactile experience of creating right triangles.
The Readiness Rule
Provide young students with readiness of math abstractions one to two years (time frame)before the proficient use of the math is performed in paper-and-pencil or mental math formats. Move from concrete to semi-concrete to abstract math performance
A third grade student with a physical disability has difficulty manipulating objects. The student's teacher often has students work in small groups, using manipulatives to solve mathematics problems. Which of the following instructional strategies would most effectively promote the student's participation in these learning activities?
Providing the student and his group with adapted versions of the manipulatives used by the other groups in the class.
Problem Solving
Putting together concepts and principles to solve a problem new to the learner
Volume
Quantity of liquid
KNOWLEDGE LEVEL
Questions and activities at this level are simple memory and recall questions. = How much is 2 + 3? = How many sides does a triangle have? Activities and questions ask the students to tell list describe state define identify ask who what when where
Zone of proximal development (ZPD)
Range of tasks that child can perform with the help and guidance of others but cannot yet perform independently.
Can be expressed as a ratio or quotient of two non-zero integers.
Rational Numbers
Any number that is positive, negative, or zero, and is used to measure continuous quantities.
Real Numbers
Brownell
Real-life connections
Inductive Reasoning
Reasoning in which conclusions are based on observation.
Deductive Reasoning
Reasoning in which conclusions are based on the logical synthesis of prior knowledge of facts and truths.
Mathematical Reasoning
Reasoning refers to students ability to hypothesize, test their theories, and draw conclusions. Three main types: Inductive, Deductive, Adaptive.
Reciprocals
Reciprocals are two numbers which multiply together and make 1. They are also called multiplicative inverses of each other. For example: 3 and 1/3 are reciprocals because 3 × 1/3 = 1 5/6 and 6/5 are reciprocals because 5/6 × 6/5 = 1 -0.2 and -5 are reciprocals because -0.2 × -5 = 1
Indicates the renaming of a number from one place value to another.
Regrouping
Concept
Relational or concrete attributes
What promotes diligence?
Repetition
Forms of assessment
Reports, applications, models, lab investigations, projects, always/sometimes/never.
Abstract ideas
Require children to use their imaginations or their brain only, without help from pictures or real objects.
Teaching
Requires understanding what students know and need to learn while challenging and supporting students to learn it well
Two pairs of sides of two triangles are equal in length + included angles are equal in a measurement = congruent
SAS (Side-Angle Side)
3 pairs of sides of 2 triangles equal in length = congruent
SSS (side-side-side)
Norm referenced
a standardized test that focuses on a comparison of a students score to average of a norm group.
Form of writing a number as the product of a power of 10 and decimal number greater or equal to 1 and less than 10. Ex: 2,400,000 = 2.4x 10 to the 6 power.
Scientific Notation
What is the process for planning a theme?
Select a broad theme, use state curriculum to find main principles or generalizations and key objectives to involve each content area, gather materials.
The basic steps to planning and organizing thematic units
Selecting a theme Designing the integrated curriculum Gathering materials for the unit Arranging thematic activities
Automaticity
Selecting problem solving methods and performing computations without requiring much time to think the process through
Modeling the Operations: This method students work with visual representations (pictures) instead of actual objects.
Semi concrete method
Modeling the Operations: Students work with one symbol (tally marks, x's, y's, etc) to represent objects instead of actual objects, pictures, or abstract (numerical) representations. Students use one symbol.
Semiabstract method
What are the 4 general stages of development?
Sensorimotor (0-2) Pre-Operational (3-7) Concrete Operational (8-11) Formal Operational (12-15)
What are the four main developmental stages that children acquire information?
Sensorimotor stage (birth - 2 years) Preoperational stage ( Years 2-7) Concrete operational stage (years 7-11) Formal operational stage (years 11 - adult)
Subset
Set A is a _______ of B if and only if each element of A is also an element of B.
Proper Subset
Set A is a ________ _______ of set B if and only if A is a subset of B, and A is not equal to B. (C)
Equal Sets
Sets with exactly the SAME elements.
Collective self-efficacy
Shared beleif of members of a group that they can be successful when they work together on a task.
Assessment
Should support the learning of important math concepts, and furnish useful information to both teachers and students
What should teachers do to help try and get kids interested in math?
Show them real world professions that use math.
Hierarchy Diagram
Shows all the functions of a particular concept. Which is broken down to subordinate relationship and super-ordinate concepts.
Why should talking about their work be encouraged?
So students learn to use mathematical vocabulary.
Spatial-temporal knowledge
Space and time thinking
Mobius Strip good for
Spatial-temporal reasoning
3- standardized vs criterion referenced
Standardized-written to general content, and the performance on the test is based on a comparison to other similar students who took the test. Example: SAT, ACT, GRE, National Assessment of Educational Progress. • Criterion Referenced-written to measure specific content and the criteria for passing the test is pre-specified. Examples: Well- designed teacher unit test, STAAR, Texas Examination of Educator Standards
Standardized vs criterion referenced
Standardized: written to general content, and the performance on the test is based on a comparison to other similar students who took the test Ex: SAT, GRE, ACT. Criterion referenced: written to measure specific content and the criteria for passing the test is pre-specified. Ex: well designed teacher unit test, STAAR, TEXES
Commutative Property
States that changing the order of addends does not change the sum. That is, a + b = b + a. Commutative Property of Multiplication: It states that changing the order of factors does not change the product. That is, a × b = b × a.
Zero - Product Property
States that if the product of two factors is zero, then at least one of the factors must be zero
Distributive Property
States that the product of a number and a sum is equal to the sum of the individual products of addends and the number. That is: a(b + c) = ab + ac.
80%
Student activity-experience by doing
EVALUATION LEVEL
Students are asked to = make judgments based on set criteria and to = defend or support their thinking Evaluation is the highest level of thinking in Bloom's Taxonomy In the purest sense, the Evaluation stage is rarely reached at the elementary level simply because of the nature of cognitive development present until later However, students can be involved in many activities where they are asked to make choices and support their choices. Activities and questions where students are asked to explain their thinking, give students the opportunity to support choices they have made. keywords include = which is the best choice and why = what do you think about = rate from good to poor = what is the problem = are all of the solutions the same = will all the solutions work = decide which = justify your choice
Math curriculum for grades 3-5
Students continue developing number concepts to include multiplication, division, fraction and decimal representation, geometric principles, and algebraic reasoning
Math curriculum for grades 1-2
Students continue exploring number concepts and begin learning basic computation skills
Which of the following best promotes understanding of math operations?
Students verbalizing their reasoning and solutions to problems
Addition and Subtraction of Fractions
The denominators must be equal the add or subtract and leave the denominator the same. -5/7-(-3/7)=5-3/7 -2/7 or 4/5+1/5=4+1/5=5/5 Unlike Denominator->You have to look at the denominators and find the lowest common denominator both share.
Supplementary Angles
Supplementary angles are two angles that add up to give a straight angle, 180°.
Scaffolding
Support mechanism that helps a learner successfully perform a task within his or her zone of proximal development.
Face
Surface of a geometric solid
What processes are included in the pre operational stage of development?
Symbolic functioning, centration, intuitive thought, egocentrism, and inability to conserve.
The Professional Standards for Teaching Mathematics (NCTM) presents standards for the teaching of math, organized under four categories
Tasks Discourse Environment Analysis *These strands are integrated and interdependent, and are crucial in shaping what does on in math classrooms
Presenter?
Teacher Student
50%
Teacher modeling
What are advantages of collaborative or small-group learning?
Teachers reduce their workloads and the amount of time spent on helping students and planning lessons
Peer Interaction
Teachers should pair students carefully, model effective ways to interact, provide students with relevant tools, and offer specific advice. Struggling students can benefit from peer explanations, clarify a process and ask and answer questions.
Unfamiliar Structures for ELLs
Technical terms, problem solving verses computation and notation, math symbols, multiple words indicating the same operation, ...."if"....."then" statements, etc.
Nomenclature of Math
Technical vocabulary
In the number 1984 the number 8 would be in what place value?
Ten: 80
What must young children understand about counting objects?
That the idea of an amount is inclusive of all perviously counted things, not just the same as the last block.
label (LBL)
The JMP instruction is always paired with a
Subitizing
The ability to instantly "see" the number of objects in a small set without having to count them.
Adaptive Reasoning
The ability to think logically about the relationships between concepts and to adapt when problems and situations change.
What are characteristics of the formal operational stage?
The ability to use symbols and think abstractly.
Absolute Value
The absolute value of a real number is equal to the numeric value of the number without regard to its sign (e.g -3 is 3) Absolute value is often thought of as the distance a number is from zero on the number line
Volume
The amount of 3-dimensional space an object occupies. For a rectangle the formula would be length times width time depth or height. Since there are three dimensions it is expressed as cubics (9 cubic inches).
Area
The amount of space inside the boundary of a flat (2-dimensional) object such as a triangle, rectangle or circle. Different objects have different formulas to determine area. For a rectangle it is length X width. Area of a triangle is 1/2 of the base times the height. The area would be expressed as a square (10 square feet, etc.)
Mean
The average of a set of numbers; the sum of the numbers divided by how many number there are; 2 + 5 +5 =12, then 12/3 = an average/mean of 4
Percents
The best way to compare a percent to other number expression is either to _convert it to a decimal and leave it as that. _convert it to a fraction (depending on how the other numbers are expressed) ex: 76%->move 2 decimal places to the left .76. ex: .76-> 76/100->can be simplified to 16/25
Associative Property
The change in grouping of three or more addends or factors does not change their sum or product. holds good for both addition and multiplication, but not for subtraction and division. Addition: (2 + 3) + 5 = 2 + (3 + 5) Multiplication: (4 X 5) X 10 = 4 X (5 X 10)
Vertex
The common endpoints of two rays that form an angle, or the point of intersection of two sides of a polygon of polyhedron
Natural Numbers
The counting numbers 1,2,3,4,5...
Diameter
The distance across a circle through its center point. It is twice the radius of the circle.
Circumference
The distance around a circle (C). C=pi times the diameter or d X pi. Or, 2radius X pi. (Pi - 3.14).
Perimeter
The distance around a two-dimensional shape, such as a triangle or rectangle.
Girth
The distance around something; the circumference.
Radius
The distance from the center of a circle to the edge of the circle. It is also half the diameter of a circle.
Distributive Property
The distributive property of x over +multiplucation may be distributed over addition ex:10x(50+3)=(10x50)+(10x3) 3x(12+99)=(3x12)+(3x99( a(b+c)=ab+ac
Subroutine (SBR)
The first input condition entered in the subroutine. It serves to identify the file as a subroutine. It is always true, and is actually optional, though recommended.
Greatest Common Factor (GFC)
The greatest number that is a factor of each of two or more given numbers. Examples: The greatest common factor of 24 and 15 is 3. The greatest common factor of 40, 50, and 25 is 5.
Translation
The image of a figure that has been "slid" to a new position without flipping of turning.
Least Common Multiple (LCM)
The least number that is a common multiple of two or more numbers. Find the LCM of 30 and 20. 30: 30,60,90,120 20: 20,40,60,80,100,120
Discourse (strands)
The manner of representing, thinking, talking, agreeing, and disagreeing that teachers and students use to engage in these tasks
Probability
The measure of how likely an event is. It is usually determined by dividing the number of ways something can happen by the total number of outcomes.
Median
The middle number of a set of numbers after they have been placed in numerical order. In the set [2,3,4], 3 is the median. If there are an even number of numbers, the median is the average of the two middle numbers.
Frequency
The number of times a score appears in a list of data.
Mode
The number that occurs the most frequently in a set of data. In the set [2,4,4,3,5] 4 is the mode.
Additive Inverse
The opposite of the number. A number and its opposite add up to give zero. They are called inverse additives of each other.
Thematic instruction
The organization of curriculum content based on themes or tops Integrates basic disciplines (e.g., reading, math, music, art, and science) with the exploration of broad subjects
Endpoint
The point at the end of a line segment
CONNECTIONS
The process of making connections should be modeled and reinforced. Students should understand how mathematics can be part of everything they see, do and know. They should be guided to an awareness of how their knowledge of mathematics can help them with other school subjects, such as using a population graph in social studies. They should also appreciate how knowing mathematics helps lead to better jobs and future opportunities. People who know arithmetic can make good decisions about how to spend allowance money, or whether or not they can afford a new bike or a vacation to Mexico.
Tasks (strands)
The projects, questions, problems, constructions, applications, and exercises in which students engage
Fact Families
The related number sentences for addition and subtraction or multiplication and division that contains all the same numbers (e.g. 2+3 =5; 3+2 =5; 5-3 =2; and 5-2 =3)
Environment (strands)
The setting for learning
A fourth grade student receives special education services due to a specific learning disability in mathematics. The student's special education teacher is scaffolding instruction to help her complete several addition problems involving decimals. First, the teacher recites aloud each step for completing an addition problem while the student listens and watches the teacher complete the problem. Next, the student recites the same steps aloud as she and the teacher complete a second addition problem together. Which of the following approaches would be most appropriate for the student and the teacher to use to complete a third addition problem?
The student quietly says the steps to herself as she completes the problem, and the teacher helps as needed.
What is the importance of a placing patterns activity?
The students are practicing patterns, which will help build a foundation for algebraic thinking
Perimeter
The sum of the lengths of the sides of a polygon (p = 2l x 2w where l =length and w = width).
Analysis (strands)
The systematic reflection in which teachers engage
Competency
The teacher understands how students learn mathematical skills and uses that knowledge to plan, organize and implement instruction and assess learning.
Objectives
The teacher understands how students learn mathematical skills and uses that knowledge to plan, organize, and implement instruction and assess learning.
Rote counting
The verbal repetition of numbers
Integers
The whole numbers and their negatives (e.g. -2, -1, 0, 1, 2).
Whole Numbers
The whole numbers are the counting numbers and 0. The whole numbers are 0,1,2,3,4,5... Cannot be negative
What are the mathematical process standards and what do they do?
They highlight ways of acquiring and applying content knowledge. Problem solving, reasoning and proof, communication, connections, and representations.
Think Pair Share ex.
Think: Students think independently about the question that has been posed, forming ideas of their own. Pair: Students are grouped in pairs to discuss their thoughts. This step allows students to articulate their ideas and to consider those of others. Share: Student pairs share their ideas with a larger group, such as the whole class. Often, students are more comfortable presenting ideas to a group with the support of a partner. In addition, students' ideas have become more refined through this three-step process.
APPLICATION LEVEL
This is where most problem-solving will occur in the elementary classroom. If Jake had $7 and his father gave him an additional $3 how much money does he have now? Keywords in questions and activities at this level include: = demonstrate = solve = what are the next steps = how can this be used = contrast = separate into parts = use = change = apply = estimate = determine
SYNTHESIS LEVEL
This level is where = divergent thinking, = originality, and = imagination occur At this level, concepts that have been learned = are taken apart and thought about in a totally different way. In upper-level mathematics, synthesis is a major component of algebra, geometry, trigonometry, and calculus Although synthesis can certainly occur at the elementary level, it is less frequent Keywords include = create = design = how many hypotheses can you suggest = how many different ways = how else = what would happen if = how many ways are possible = compose = develop = suppose
1. Inductive (Informal):
This reasoning goes from the specific to the general. It uses observations and patterns to infer a generalization.
2. Deductive (Formal):
This reasoning process reaches conclusions based on accepted truths and logical reasoning. This reasoning goes from the general to the specific, and uses a general rule or statement to draw a reasonable conclusion.
In the number 1984 the number 1 would be in what place value?
Thousand: 1000
How should a teacher of math create a learning environment that fosters development of each student's ability?
Through structuring time necessary to explore math and grapple with problems, using physical space and materials, providing context, and respecting students' ideas and ways of thinking.
doing mathematics
engaging in the science of pattern and order, posing worthwhile tasks and then creating an environment where students take risks, share, and defend mathematical ideas
Symmetry
To discover what symmetry is, take a piece of paper, fold it, and cut out a shape along the fold. Unfold the shape that you cut out. This figure is symmetric. That means it is exactly the same on both sides of the crease. The simplest symmetry is Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry). It is easy to recognise, because one half is the reflection of the other half.
Bisect
To divide into two congruent parts
What does thematic planning provide an opportunity of for students?
To hear similar information in various instructional segments and from a variety of sources.
The NCTM standards includes 5 major shifts in the environment of mathematics classrooms from current practice to teaching for the empowerment of students
Treating classrooms s math communities rather than simply a collection of individuals Students using logic and math evidence as verification rather than students relying solely on the teacher for correct answers Students employing math reasoning rather than early memorizing answer-finding Emphasizing problem solving rather than an emphasis on mechanistic answer-finding Students connecting math, its ideas, and its applications rather than viewing math as a body of isolated concepts and procedures
Complementary Angles
Two Angles that add up to 90 degrees (a Right Angle). They don't have to be next to each other, just so long as the total is 90 degrees.
Congruent angles
Two angles that have the same degree of measurement.
Complementary angle
Two angles whose sum is equal to 90 degrees.
Adjacent angles
Two angles with a common vertex, a common ray, and not common interior points
Additive Inverses
Two numbers whose sum is 0 are additive inverses of one another. Ex. 3/4 & -3/4 are additive inverses of one another because 3/4 + (-3/4) = (-3/4) + 3/4 = 0.
Planes are infinite length and breadth but no depth are considered to be?
Two-dimensional
Cultural differences in coherence?
USA 1.9 topics Germany 1.6 topics Japan 1.3 topics
What were the cultural differences in grade level between the USA, Germany, and Japan?
USA 7.4 Germany 8.7 Japan 9.1
Cultural differences in development in the US, Germany, and Japan.
USA 80% stated Germany 25% stated Japan 20% stated
Which schools are interrupted more then any other country?
USA schools
Problem-solving method UPSC
Understand, plan, solve, check
What does effective mathematics teaching require?
Understanding of what students know and need to learn and then challenging and supporting students to learn it well.
Conservation
Understanding that quantity, length, or number of items is unrelated to the arrangement or appearance of the object or items
What is conservation?
Understanding that quantity, length, or number of items is unrelated to the arrangement or appearance of the object or items.
What does the measurement standard include?
Understanding unit systems and process of measurement and applying techniques and formulas to determine measurements.
Semi abstract model
Use a single symbol (such as an x or a tally mark) to represent numbers of objects while performing operations.
What are numbers and operations curriculum for second grade.
Use concrete models of hundreds, tens, and ones to represent a given whole number. Use place values to read, write, and describe numbers, uses concrete models to represent and name fractional parts of a whole object, models addition and subtraction of two-digit numbers with objects, pictures, and words, can recall and apply basic addition and subtraction facts, determine value of collection of coins up to a dollar.
What are number and operations curriculum for first grade.
Use concrete models to represent and name fractional parts of a whole object, model addition and subtraction of two-digit numbers with objects, separates a whole into two, three, and four equal parts, identify individual coins by name and value and describe relationship, create sets of tens and ones using concrete objects and reads and writes numbers up to 99.
Concrete Model
Use objects to demonstrate operations
Semi Concrete Model
Use pictures (instead of actual objects) to demonstrate operations
What are numbers and operations curriculum for third grade?
Use place values to describe numbers through 9,999, uses fraction names to describe fractional parts of whole objects and compares parts of whole objects in a problem situation, Uses operations to solve problems involving whole numbers up to 999, applies multiplication facts through 12 by using concrete models and objects.
Focal points: 2nd grade
Uses concrete models of hundreds, tens, and ones Read and write the value of whole numbers up to 999 Uses concrete models to represent and name fractional parts Models addition and subtraction of 2-digit numbers with objects, pictures, words, and numbers Determines the value of a collection of coins up to one dollar
Focal points: 3rd grade
Uses place value to read, write, and describe the value of whole numbers, compares and orders who numbers through 9,999 Uses fraction names and symbols to describe fractional parts Selects and uses addition or subtraction to solve problems involving whole numbers through 999 Applies multiplication facts through 12 using concrete models and objects Uses models to solve division problems
Focal points: 4th grade
Uses place value to read, write, compare, and order whole numbers through 999,999 Uses concrete objects and pictorial models to generate equivalent fractions Uses multiplication and division to solve problems (no more than 2 digits times 2 digits) Uses strategies, including rounding and compatible number, to estimate solution to +,-,x,/
Focal points: 5th grade
Uses place value to read, write, compare, and order whole numbers through 999,999,999,999 and decimals through the 1000th place Identifies common factor of a set of whole numbers; uses multiplication and division to solve problems (no more than 3 digits times 3 digits)
What are numbers and operations curriculum for fifth graders?
Uses place value to read, write, compare, and order whole numbers through 999,999,999,999 and decimals through thousandths, identifies common factors of a set of whole numbers, uses multiplication up to three digits x two digits.
What are numbers and operations curriculum for fourth graders?
Uses place values to read, write, compare, and order numbers through 999,999,999 and decimals involving tenths and hundredths, can use concrete objects to generate equivalent fractions, uses rounding and compatible numbers to estimate solutions to problems, multiplication of 2 digits by 2 digits.
Focal points: K
Uses whole number concepts Uses sets of concrete objects to represent quantities given in verbal and symbolic descriptions Begins to demonstrate the concept of "part of" and "whole" Sorts to explore number; uses patterns
Abstract model
Using numbers only to perform operations
Union of two segments or point of intersection of two sides of a polygon.
Vertex
Learning Modalities:
Visual, Auditory, Kinesthetic
Learning modalities
Visual, auditory and kinesthetic.
Mapping Diagrams
Visually illustrates relationship between two or more concepts and are linked by a term.
Which of the following are basic assumptions of the criteria for teaching problem solving in mathematics?
a teacher can expect students to approach problems with varied strategies, problem solving can be applied to subject areas other than math, and students can help each other investigate possible problem-solving strategies
How much space is inside of a 3-dimensional closed containers.
Volume
The Multiplicative Identity
We call 1 the multiplicative identity multiplying any number by 1 leaves the number unchanged ex: 88x1=88-> ax1=a
Read
We remember 10%
Hear
We remember 20%
See
We remember 30%
See and Hear
We remember 50%
Discuss with others
We remember 70% --discussion with peers
Experience by doing
We remember 80%
Teach
We remember 95%
$197.95
What is 25% off of $264?
PROBLEM SOLVING
When encountering different kinds of prob- lems, students should be able to choose the best problem-solving method from several that they know. A general problem-solving method that can be applied to many types of problems is: 1. Understand 2. Plan 3. Solve 4. Check For any word or logic problem, students must read and understand the problem care- fully (including removing any unnecessary information), plan how to find the answer, solve the problem, and check their work. Students may also choose to use other strategy tools such as drawing a picture, making a list, chart or table, estimating, guessing and checking, working backwards, or using objects.
When does math because more meaningful and easier to internalize?
When it is used in real-life situations.
Efficiency
When students do not get caught up in too many steps or get confused with the logic of the problem or strategy or conceptual meaning. An efficient algorithm is carried out easily with out confusion
Thinking Aloud
When students verbalize what they know, it helps them reflect upon and clarify the problem and focus one step at a time. "Thinking Aloud require talking through the details, decisions, and reasoning behind those decisions. Gives time to fully comprehend the problem.
Order of operations
When there is more than one operation and parentheses are used, first do what is inside the parentheses, then the exponents. Next, multiply or divide from left to right. Then add of subtract from left to right (PEMDAS of Please Excuse My Dear Aunt Sally).
How should addition problems be posed when moving from quantification to computation?
With concrete objects
How must student's learn mathematics?
With understanding, actively building new knowledge from experience and previous knowledge.
differentiating instruction
a teachers lesson plan includes strategies to support the range of different academic backgrounds found in classrooms that are academically, culturally, and linguistically diverse what content, process, product can be diverse physical environment
MATH INSTRUCTION
Young children learn mathematics best when they DO mathematics. Conceptual development, rather than reliance on teaching a set of step by step procedures, or algorithms, is the goal. Constance Kamii, a specialist in early childhood math education, tells of observing a class of first graders. When presented with a math problem similar to 3 + 5 would write: 3 + 5 = 5 To determine what the problem was, she had them model the problem with counters. She began to watch what they were doing and discovered that they would put out three counters to model the 3 then place two more counters with the three to make five counters in all. So their answer to 3 + 5 = 5 made sense to them. Kamii used this example to point out the importance of not rushing too quickly to the abstract representation of a concept, in this case the concept of addition. Addition is much more complex than one might think initially.
Integers
______ are called whole numbers, negative whole numbers, and zero.
Which is an example of the additive identity?
a + 0 = a
Which type of graphs would best be used to represent the number of students who like red, green, or yellow best?
a bar graph or pictograph
Numbers provide what?
a complete continuum
Geoboards
a mathematical manipulative used to explore basic concepts in plane geometry such as perimeter, area and the characteristics of triangles and other polygons. It consists of a physical board with a certain number of nails half driven in, around which are wrapped rubber bands.
ratio notation
a method of notating fractions for the purpose of comparison
additive identity element
a number + 0 = the number
Arithmetic Sequence
a sequence in which each term is obtained by adding a constant value to the preceding term. The constant value is called the common difference (d).
Geometric Sequence
a sequence in which the ratio of each term to its predecessor is the same for all terms. or each term is obtained by multiplying the preceding term by a constant value called the common ratio (r).
Empty Set
a set containing no elements
Concrete operational stage (7-11 years)
ability to think logically about concrete objects or relationships. Decentering - form a conclusion based on reason rather than perception reversibility - objects can be changed and then returned to their original state ex) 4+4 = 8 . . . 8-4 = 4 conservation - quantity, length, number of items is unrelated to the arrangement or appearance of the object. serialization - arrange objects in an order according to size, shape or any other attribute classification - name/identify sets of objects according to appearance, size, or other characteristics elimination of egocentrism - view things from another's perspective
Three things that need to progress in higher math levels?
abstract thought complexity problem length
Which angle would measure less than 90 degrees?
acute
Diagnostic tests
are used with the diagnostic-prescriptive teaching of mathematics. This process is an instructional model that consists of diagnosis, prescription, instructions, and ongoing assessment. can be used to help identify specific problem areas. can be teacher made or commercially developed
What types of manipulatives should be used in grade 6
algebra tiles, video games, ipods, math games software, smart boards, digital cameras.
real number
all numbers, rational or irrational
regular polygon
all sides and angles are equal
engage students in productive struggle
aloo students to have disequilibrium in learning, it is part of the process and developing concepts, must have tools and prior knowledge
Authentic assessment
alternative assessment that incorporates real-life functions and applications.
A "shape set" is:
assortment of triangles, rectangles, squares, and circles in various colors and sizes
Numerical values are always ________________
ambiguous
dispersion
amount of spread
Using scissors to cut a piece of paper represents which of the following spatial concepts?
an angle intersecting a place
Know your class.....
and teach to the middle
the same problem will always give you the same __________
answer
Help them to develop what?
answers
tool
any object, picture, or drawing that can be used to explore a concept. includes calculators and manipulatives
expository instruction
approach to instruction in which info is presented in essentially the same form in which students are expected to learn it
learner-directed instruction
approach to instruction in which students have considerable control regarding the issues they address and the ways they address them
teacher-directed instruction
approach to instruction in which the teacher is largely in control of the content and course of the lesson
backward design
approach to instructional planning in which a teacher first determines the desired end result (i.e.. what skills a students should acquire), then identifies appropriates assessments, and finally determines appropriate instructional strategies
Integers
are also called whole numbers negative whole numbers and zero. ex: 560,-35,2,0,-197744
Number Properties
are another important aspect of number theory.
Composite
are numbers composed of several whole number factors. 30 is composed of several whole-numbers
manipulatives
are physical objects that students and teachers can use to illustrate and discover mathematical concepts, whether made specifically for mathematics or for other purposes. Choices abound from common objects such as lima beans to commercially produced objects
Subroutines
are short programs that are used by the main program to perform a specific function. are stored in program files numbered LAD 3 through LAD 255 subroutines can be nest
Base Ten Blocks
area, classification, comparing, computations (whole numbers and decimals), decimal fractional- percent equivalences, metric measurement, number concepts, ordering, percent, perimeter, place value, polynomials, sorting, square, and cubic numbers.
Which represents shape set activities?
arrange pieces in repeating pattern, sorting the pieces in the set by shape, and using the set's pieces to make a model of a real-life object
tessellation
arrangement of closed shapes that completely covers a plane without overlapping or leaving gaps
deduction
arrive at conclusion based on statements known to be true
assessment
as a way to collect evidence about students content knowledge, flexibility in applying that knowledge, and disposition or attitudes toward mathematics
inquiry students
ask questions, determine solutions, use math tools, make conjectures, seek patterns, communicate, reflect, make connections
rephrasing
asking students to restate someone else's ideas in their own words will ensure that ideas are stated in a variety of ways
reasoning
asks the student what they think of the idea proposed by another student
formative assessment
assessment conducted before or during instruction to facilitate instructional planning and enhance student's learning
you can find a child's aptitude in math from this?
assessment test
variance
average sqaured distance from each value to mean
when doing a diagnostic interview
avoid revealing whether a student is right or wrong, wait silently for an answer, shouldn't interject clues or teach, do not interrupt the student
symbolic representation
basic language of math
be sure the task is understood
be able to explain what the task is asking them to do, not explain how
Open-ended questions, portfolios, and writing activities can:
be used as assessment of mathematical skill and knowledge
What are types of manipulatives to be used in grades K - 1
beads, strings, sewing cards, color tiles, macaroni, paper clips, legos, board games, pictographs, buttons.
start and jump numbers
begin with a number(start) and add(jump) a fixed amount. for example start with 3 and then jump by 5s.
What types of manipulatives should be used in grade 2
blocks, cubes, chips, measuring cups, money models, number lines, dominoes, clock faces, ruler, yard stick, balance scales, linear graphs.
implications for teaching mathematics
build new knowledge from prior knowledge provide opportunities to communicate about mathematics create opportunities for reflective thought encourage multiple approaches engage students in productive struggle treat errors as an opportunity for learning scaffold new content honor diversity create a classroom environment for doing mathematics
technology principle
calculators, computers, and other emerging technology are essential for learning mathematics
What types of manipulatives should be used in grade 5
calculators, timers, metric beaker sets ,connecting cubes, board games, fraction tiles, pie charts and graphs
adaptive reasoning
capacity for logical thought, reflection, explanation, and justification
Manipulatives
enhance student understanding, enable students to have conversations that are grounded in a common model, help students recognize and correct any misconceptions Games - children learn best when they can manipulate materials to check their understanding and link what they are learning to real life situations
Accuracy
careful recording of the computational algorithm, memorizing basic facts, knowing number relationships and place value, and checking reasonableness or correctness of the results
scaffold new content
carefully structured content, new concepts require more structure or assistance
when did math start
cave paintings prior to 20,000 bce in africa and france
central tendency
center of data set
Equal groups or repeated addition
involves making a certain numbers of equal sized groups repeated 3x6=6+6+6 3 numbers involved: numbers of groups (factor), size of the group (factor), total number of the objects (product) ex:3x6 3 groups of 6 objects
Presentation Organization?
class work seat work
Attribute blocks
classifying, geomtry, logical reasoning
authentic activity
classroom activity similar to an activity that students are apt to encounter in the outside world
how can math interventions used to improve math performance
cognitive approaches behavioral approaches alternative instructional delivery systems which includes cooperative learning, computer assisted instruction and interactive video games
verbs that engage mathematics
collaborate, describe, justify, predict, compare, create, invent, explain, formulate, use, and verify, these lead to higher level thinking
Number Sequence
collection of numbers, called terms arranged in order. Sequences are arranged from left to right, the numbers are separated by commas.
Set
collection of objects
stanine "standard nine"
combine the understandability of percentages with the properties of normal curve of probability
scoring
compares students work by established criteria set in advance, many times collected by a rubric
Achievement test battery
composed of subtests of math concepts and skills and usually includes technical aspects of math.
conceptual understanding
comprehension of mathematical concepts, operations, and relations flexible web of connections and relationships within and between ideas, interpretations and images of mathematical concepts, will allow students to connect what they know
Math helps a child develop ___________?
concepts
Concrete instruction is _____
connected, tactile
learning progressions
contain clearly articulate subgoals of the ultimate learning goal
Finite Set
contains zero elements or a number of elements that can be stated as a specific natural number.
Standardized test
content areas and provide useful information about students' math skills. their validity on three basic assumptions: students have been equally exposed to test content, students know the languages of the the directions and responses, and students just like those taking the test have been included in the standardization samples to establish norms and make infrequence.
Compare
involves no action, but involve comparision between 2 different sets- how much or how much less is one/than another. Variation: Difference unknown: 56+__=85 or 85-56=__ Larger unknown: 56+29=__ Smaller unknown: 29+__=85 or 85-29=__
continued --- Concrete Stage (7 - 11) Brings the ability to think logically and classify based on multiple attributes other than simply visual ones. children at this stage are = able to think and reason in 2 and 3 dimensions = can understand problems and problem solving approaches in ways other than their own. = They can see and understand different points of view and perspectives. This addition of different perspectives opens up the idea that there is often more than one way to approach and solve a problem
continued ---- Formal Stage (11 - 16 - adult) Marks a movement to a better understanding of abstract mathematics. It should be noted here that not all children arrive at this stage at the same age (this is also true of the other stages as well) and this is one reason why not all children are ready for upper-level mathematics at age 12 or 13. Developmental age is very important in determining readiness for abstract mathematics. Frustration will ensue if students are forced into abstract thinking before they are ready Students at this stage are capable of = abstract thinking such as limits, = areas under a curve, = infinity, = a formal geometric proof or = complex numbers
continued ------- This is also a good time to discuss the importance of pre-assessment Pre-assessment is the process of determining where the students are with respect to their understanding of the prerequisites necessary to move into the new unit. Example: = It would not be wise to move into the exploration of multiplication if students do not have a firm understanding of addition They mastery of addition is a prerequisite to the study of multiplication
continued ----- Regardless of the grade level you teach, it is important to know the state outlined curriculum for your grade level as well as the ones for the grade level below and above your grade level These curriculum guidelines will give you a better idea of the depth of understanding required for the next grade. For instance, = A 6th grade teacher may be very frustrated that her 6th grade students have no understanding of how to convert a fraction to a decimal. When she looks at the 5th grade curriculum, it becomes evident that converting fractions to decimals in NOT a 5th grade curriculum. In the 5th grade students are required to convert a decimal representation to a fraction with denominators of 10, 100, and 1000
continued ---- Developmentally, children begin by counting 1, 2, 3, 4, 5...10. They have no understanding of numbers or one-to-one correspondence or relative size. They simply count. Much like they recite their ABCs before they actually recognize letters or sounds or words. Before children can begin putting numbers together to add, they must develop an understanding of one-to-one correspondence. As they count the objects with their finger, the finger moves from object to object, increasing the count by one with each object. The problem 3 + 5 can then be understood to mean: "If one set has three objects and another set has five objects, how many objects are there when the two sets are put together? The child will build a three counter set and a five counter set, put the sets together, and count the total in the combined set.
continued ----- Piaget studied the stages of cognitive development in children. His research determined that there were 4 general stages of development 1. Sensorimotor 2. Preoperational 3. Concrete 4. Formal
continued ------ One final area to address as units are being planned is the level of thinking that the teacher wants the students to achieve. One way to analyze the level of the activities you are planning is to use Bloom's Taxonomy. Bloom's Taxonomy consists of 6 levels of understanding, all of which are necessary for long-term retention.
continued ----- 6 levels Knowledge Comprehension Application Analysis Synthesis Evaluation
continued ------- Students do not learn in isolation It is important to incorporate time for = individual work, = pair work, = small group work and = whole class work If children can explain their thinking to others, they have truly mastered the concept, and often children, even young ones, have a way of explaining concepts to their peers that enhances understanding
continued ------ In elementary school, thematic units are often used. Thematic planning lends itself to the integration of = music = science = social studies = reading = writing = and mathematics into a cohesive, interdependent unit
continued ------- Lesson planning should begin with the desired end or outcome in mind The teacher must first understand what end result is desired The she should clearly decide how mastery is to be measured and develop the assessment to be used first This is similar to planning a vacation. You first choose the destination then plan how to get there. Otherwise, you could wander aimlessly, never to reach the destination, or not even realize when you arrive The same is true for planning a unit of study So lets suppose the teacher wants to plan a unit using this 1st grade math state outlined curriculum (3) Number, operation, and quantitative reasoning. The student recognizes and solves problems in addition and subtraction situations. The students are expected to: (A) model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences; and (B) use concrete and pictorial models to apply basic addition and subtraction facts { up to 9 + 9 = 18 and 18 - 9 = 9 } { up to sums of 9 + 1 = 10, teacher decision }
continued ------- As the teacher considers these state outlined curriculum that are to be taught to mastery at 1st grade, they must first decide what the final assessment will be. Example: Let's suppose that this is the first time in the school year these areas in the curriculum have been addressed. The teacher might first decide that her goal is for students to understand through pictures and model sums of ten. The assessment could be pictorial. It could have pictures of objects where students must circle a number that represents the total, and all the sums will be ten or less. There will be a section on the final assessment where students will use objects to build addition problems and draw their representations in spaces provided where sums will be ten or less.
continued ---- Sensorimotor Stage (birth-2yrs.) Children experience their world through their own senses = touch = taste = smell = sight = hearing Children in this stage of development are = egocentric and can only experience the world from their own perspective With respects to math, children in this stage begin to develop the idea on = one to one correspondence Children who cannot count to ten and match that number to the counting of ten objects, are not yet ready to move to the next stage of development and will need additional practice in the development of counting and one to one correspondence
continued ------- Preoperational Stage (2 - 7) Children begin to use their understanding of one to one correspondence to develop a concrete (hands on/Manipulative based) understanding of addition and subtraction They can recognize = patterns = give the next 2 or 3 members of a simple sequence Children at this stage generally operate in only = one dimension Children at this stage are unable to understand = two dimensional mathematics If two objects look different they are assumed actually to be different. This means that while the rectangles are actually identical because they have different orientations, they will be perceived as being different Students at this stage are still limited to what they can experience in their world. Therefore concrete, manipulative tools are an absolute necessity. It is also a great idea to incorporate objects from the students everyday world to demonstrate mathematical concepts
continued ------- Notice that the assessment described in the scenario preceding the discussion about Bloom's Taxonomy only addresses a portion of the state outlined curriculum: = addition = pictorial representations = concrete models = sums of ten Now begins the process of developing carefully crafted lessons that will guide the students to mastery of this portion of the state outlined curriculum delineated above One of the biggest issues that beginning teachers face is = Pacing = choosing enough material to challenge the students but not frustrate them. These portions of the curriculum will be addressed multiple times in the school year with each new segment building on what has already been mastered. As you plan, be creative and incorporate as many higher level questions and activities as possible
continued -------- Assessments fall into 2 general categories: = formative = summative Formative: = are ongoing and can be formal or informal Formal Formative = would be a = quiz = a paper = tangible student work that a teacher would grade Informal Formative = might be = teacher observation = questioning the students orally and checking for understanding The purpose for Formative assessments is the = gathering of data in order to inform instruction Informed instruction is = the ability to analyze where students are in their understanding of the learning, = what misconceptions they might have, and = adjusting the instruction accordingly Summative assessments = may take many forms but are in essence, = the final assessment for a unit of study
continued ------- Elementary students generally fall into the = Pre-operational Stage and = Concrete Operations Stage Mathematical planning for these students should involve the use of a wide variety of manipulative tools = pattern blocks = base ten blocks = tangrams = geoboards = counters = cm cubes = fraction bars = fraction circles = rulers = meter sticks = scales and weights = cylinders for liquid measure in metric and standard units In general, commercial manipulatives are either = proportional or = non - proportional Proportional materials consist of objects that are proportional to each other with respect to shape and size Example of Proportional manipulative materials: = base ten blocks = Cuisenaire rods = tangrams = fraction tiles = fraction circles Non - Proportional materials are materials like color counters where the color is the only thing that distinguishes one from another
continued -------- Measurement can be either formal or informal. Formal measurement = uses traditional measurement tools. = rulers = tape measures = meter sticks = yardsticks = cups = gallons = pounds = grams Informal measurement = uses readily available objects at hand = footsteps = arm lengths = book lengths = pitcher-fills and so on Informal measurement precedes formal measurement and is necessary and useful in the conceptual development of measurement
assessment principle
continuously gathering data and looking at a variety of techniques
discrete models
counters
formative assessment (alternative assessment)
create response rather than select answer
learning goals
criteria for success are clearly identified and communicated to students
summative assessments
cumulative evaluations that might generate a single score, such as an end of unit test or standardized test that is used in your state or school district
Separate
involves removing elements 3quanities/ variations result unknown->85-29=__?__ change unknown->85-__?__=29 start unknown->__?__-56=29
simple event
describes a single outcome
instructional goals
desired long-term outcome of instruction
instructional objectives
desired outcome of a lesson or unit
National Council of Teachers of Math (NCTM)
development and improvement of math education 6 principles // 10 standards that children K-12 should master
Rhombus
diagonals are right angles, all sides are congruent, diagnols bisect
Multiplicative Comparison
involves the comparison of 2 quantities manipulatively, involves finding "how many times as much" of 1 quantity is compared in another quantity, or "stretching" the original by a certain quantity.
apothem
distance from center of polygon to one side
distributive properties
distribute numbers outside of parenthesis
Multiplication of the reciprocal
division of fractions
instrumental understanding
doing something without understanding introduced by Richard Skemp in 1978, a student who only knows the procedure for simplifying a fraction for example
honor diversity
each learner is unique, with a different collection of prior knowledge and cultural experiences. lessons begin with eliciting prior experiences and understandings and contexts for the lessons selected based on students knowledge and experiences
Recursive Pattern
each successive term of the sequence is obtained from the previous term(s), at least after the first few. to reveal a pattern, the sequence should contain a minimum of 3 terms.
fractal
endlessly repeating pattern that varies according to a set formula
family in mathematics
engage them, talk about their learning, describe their role, use cooperative groups, advocate technology, provide support
provide opportunities to communicate about mathematics
engagement with other learners, reflective thinking, expanding on different ideas and networks
equation solves to y=any number
equation is horizontal line, slope is 0
equation solves to x= any number
equation is vertical line, slope is undefined
six principles
equity, curriculum, teaching, learning, assessment, technology
One-to-One-Correspondence
exactly one element for each element of the other set.
The equity principle
excellence in mathematics education requires high expectations and strong support for all students
communication
explains ideas in writing using words, pictures, numbers/equations, graphs, tables analyzes the thinking of others uses precise language, units, and labeling to clearly communicate ideas
Line of Symmetry
f you can reflect (or flip) a figure over a line and the figure appears unchanged, then the figure has reflection symmetry or line symmetry. The line that you reflect over is called the line of symmetry. A line of symmetry divides a figure into two mirror-image halves. The dashed lines below are lines of symmetry.
problem translation
factual knowledge - one meter equals 100 centimeters linguistic knowledge- floor tiles and tiles refer to the same thing
induction
find a pattern from group of examples
drill limitations
focus on a singular method and an exclusion of flexible alternatives -false appearance of student understanding -a rule oriented or procedural view of mathematics
problem based tasks
focus on important mathematics concepts, stimulate connection of students previous knowledge, allow multiple solutions methods, offer opportunities for correction, encourage students to explain, create opportunities to observe, and generate data
procedures WITH connection tasks
focus students attentions on the use of procedures for the purpose of developing deeper understanding, general procedures, are usually represented in multiple ways, require engagement
cartesian (rectangular coordinate system)
formed by 2 perpendicular axes (x and y)
decimal numbers
fractions written in respect to base 10
standards
general statements regarding the knowledge and skills that students should gain and the characteristics that their accomplishments should reflect
Types of learning: Principle -
generalizations, developed rules Example: The area of a trapezoid is developed from the concept of a trapezoid and the area of triangles, rectangles, and/or parallelograms.
What types of manipulatives should be used in grade 4
geometric solids, geoboards, calculators, protracters, pentominoes, graphing paper, fraction kit, weight set, scatter plot graphs
translation task
give a computational task, ask to write a word problem, ask for an illustration, and explain their process of arriving at an answer or the meaning of the operation
productive disposition
habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy
Part-Part_Whole
involve no action or change over time as happens with join and separate problems. Focus on the relationship between a set and its two subsets (or a whole and 2 parts) Variations involve situations the whole is unknown-> 29+56=__?__ part of the whole is unknown -> 29+__?__=85
Denominator
he bottom number of a fraction, telling in how many parts the whole is divided. in 1/3 the 3 is the denominator.
Least Common Multiple
he multiples of a number are what you get when you multiply it by other numbers. When you list the multiples of two (or more) numbers, and find the same value in both lists, then that is a common multiple of those numbers. The "Least Common Multiple" is simply the smallest of the common multiples.
Equity principle
high expectations and intentional ways to support all students. all need the opportunity, support, to learn no matter what
classroom discussion
how did you decide what to do? what did you do that helped you make sense of the problem? did you find any numbers or information you did not need? did you try something that didn't work?
Exponents
how many times a base number is used. to compare number with exponents it is best to multiply them out or estimate what they might be. ex: 2*2=4, if you had 2*-2 the answer would be 1/4 or (1 over 2*2) ex: when you have a number that is being raised to zero it always equals 1. (195*0=1)
Naturalistic assessment
involves evaluations that is based on the natural setting of the classroom. It involves the observations of students' performance and behavior in an informal context.
Equivalent Set
if and only if there exists a one-to-one correspondence between the sets. (A~B)
inverse
if not p, then not q
contrapostive
if not q, then not p
conditional
if p, then q
converse
if q, then p
Where can a teacher expect students to find items representing the shape of a rectangle?
in the classroom, in their room at home, in a picture book
establish clear expectations
includes both how they will be working and what they what students to demonstrate with their learning
procedural knowledge
includes information about how to perform specific tasks includes basic action sequences as well as more complex knowledge needed to carry out complex cognitive processing of information
4. Extended thinking and complex reasoning
incorporate demands from other content ares in the development and support of real world mathematical arguments.
Master Control Reset
instruction belongs to a group of instructions referred to as program control instructions. These instructions are used to enable/disable blocks of program logic, or move the program scan from one location to another.
Fill (FIL)
instruction can be used when working with a word-to-file data transfer.
File Arithmetic and Logic (FAL)
instruction is more complex than the previous to file-level data transfer instructions.
bit distribute (BTD)
instruction is similar to the move instruction. It differs in that it can be used to move data within a single word.
evidence-based practice
instructional method or other classroom strategy that research has consistently shown to bring about significant gains in student's development and/or academic achievement
Data Compare
instructions are used to compare the numerical data stored in specified words. They do not move, or change the data stored in the specified words.
logarithms
inverse of exponential
Join
involve adding or joining elements to a set. 3 quantities involved: the starting amount__?__ +56=85 the changing amount 29+__?__=85 the resulting amount 29+56=__?__ variations of the join problem include situations when the result is unknown, the change is unknown, or the starting amount is unknown.
Area and Array
involve finding the area of a rectangular area or arrangement. 5 rows of 17 apple trees ex:Multiplication->5x17=__ 85 palm trees/5rows Division->85 divided by 5=__
Variable
is a letter or other symbol that stands for any number within a specified set of numbers.
Expression
is a meaningful string of numbers or variables, or both and operation symbols, and possibly also grouping symbols.
Equation
is a statement of the equality of mathematical expressions; it is a sentence in which the verb "is" represents = to solve an equation, or find the solutions for an equation, means to find all replacements for the variables that make the equation true.
Prime Numbers
is a whole number greater than 1 that has exactly two factors, 1 and itself. 2-the only even prime number, 3,5,7,11,13,17,19,23,31,41,43,47,53,59,61,71,73,79,83,89,97 1 IS NOT PRIME
Composite Number
is a whole number greater than 1 that has more than two factors 4,6,8,9 1 IS NOT COMPOSITE
1
is not considered prime or composite
drill and practice
is present to at least some degree
Copy (COP) instruction
is used to duplicate the data in file, to a different file specified by the user program.
Scale (SCL):
is used to enlarge or reduce very small or very large numbers by the rate value. This is utilized for converting values generated by a 4-20mA
What is true of multiplication?
it represents repeated addition, is commutative and associative, can be modeled with area diagrams
A pictograph of results is appropriate for young children because:
it shows them how math is used in their own lives, it gives them a chance to participate directly in the lesson, and it provides the opportunity to quantify results and compare values
Addition and Subtraction: 4 types of problems
join part-part-whole separate compare
reasoning
justifies solution methods and results recognizes and uses counterexamples makes conjectures and/or constructs logical progressions of statements based on reasoning
relational understanding
knowing what to do and why, introduced by Richard Skemp in 1978, a child that can draw diagrams, give examples, find equivalencies, and tell the approximate size
recalling simple facts?
knowledge
ELL and Math
math is not universal use Arabic numbers does not present problems for ELL's can't comprehend the explanation of the process in a language that ELL's have not mastered. Inductive teaching - learning through examples deductive teaching - learning step by step
law of large numbers
larger the sample size, closer sample mean will be to population mean
greatest common factor
largest number that a factor of all numbers given in a problem
learning principle
learning mathematics with understanding is essential, it not only requires computational skills but also the ability to think and reason and solve problems
metric system
length: meter weight: gram volume: liter
problem solving develops
mathematical processes, student confidence, provides a context to help students, allows extensions and elaborations, engages students so that there were fewer problems
theorums
mathematical statements that can be proven by geometry
problem solving strategies
look for patterns, predict and check, justify claims, create a list, create a table, create a chart, simplify the problem, write an equation
questioning technique
make questions to clarify problem, eliminate solutions, and simplify the problem-solving process
standards for mathematical practice
make sense of problems and persevere in solving them reason abstractly and quantitatively construct viable arguments and critique the reasoning of others model with mathematics use appropriate tools strategically attend to precision look for and make use of structure look for and express regularity in repeated reasoning
manipulatives (concrete)
materials students can physically handle and move
1. Nomenclature of Math for ELL
math classrooms assume students know specific terms such as denominator, subtraction, minuend, divisor, subtrahend based off of prior knowledge. ELL's do not have this prior knowledge
z- score
measure of distance in standard deviation of a sample from the mean
low level of cognitive demand
memorization tasks, procedures without connection tasks
create opportunities for reflective thought
mentally engaged, see how concepts are connected to each other, interconnected rich web of interrelated ideas
early math started...
mesopotamia, egypt, greece, rome
generative learning
must provide the students with an opportunity to mentally 'play with' information to create a personal understanding of the subject to be learned
continuous model
number lines that students might choose for solving problems
central limit theorem
number of sample increases, distribution sample means approaches normal distribution
integers
numbers preceded by either a + or -
Tessellation
ometimes referred to as tiling of the plane. A tessellation is a collection of plane figures that fills the plane with no overlaps or gaps. Tiles on a kitchen floor can be thought of as a simple form of tessellation.
permutation
one of a number of possible selections of items, without repetition, where order of selection is important
combination
one of a numer of possible selections, without repetition, where order of selection is not important
Student passes out 15 napkins for 18 students and determines he needs more by
one-to-one correspondence and pre-subtraction skills
function
only one x corresponds to one y
linear programming
optimization of linear quantity that is subject to constraints expressed as linear equations or inequalities
What is the progression of a lesson?
oral review written review new concept review of new concept
Thematic instruction
organization of curriculum content based on themes or topics integrates basic disciples (reading, math, music, art, science) with the exploration of broad subjects provides opportunity for students to hear similar information in various instructional segments and form a variety of sources.
Thematic activity
organized into 3 main segments 1. introduction to the unit 2. presentation of the content 3. closing activity
Abstract
paper and pencil/mental math
alternate interior angle theorem
parellel lines cut by transversal, alternate interior angles are congruent (corresponding angles are equal
semiotic meditation
part of sociocultural theory, refers to the use of language and other tools, such as diagrams, pictures, and actions they are exchanged between and among people when this occurs
frieze
pattern that repeats in one direction; seven possible patterns
Use __________ to ___________ influence
peer to peer
study us system
pg 186
fundamental counting principles
pg 201
4 basic tessellations
translation, rotation, reflection, glide reflection
Quadrilaterals
trapezoids, kites, parallelograms, rectangles, rhombuses, and squares.
differentiated instruction
practice of individualizing instructional methods- and possibly also individualizing specific content and instructional goals- to align with each student's existing knowledge, skills, and needs
lesson plan
predetermined guide for a lesson that identifies instructional goals or objectives, necessary materials, instructional strategies, and one or more assessment methods
constructivism
prior knowledge greatly influences learning math
Students use a balance and cubes to determine which toy is heavier, they are using:
problem solving and number sense
five process standards
problem solving, reasoning and proof, communication, connections, representation
Solving 20-15 with p + 15 = 20 shows that:
problems can be solved in multiple ways, addition and subtraction are inverse operations, and a letter can stand in the place of an unknown number when solving a problem
Open-ended problems
problems for which students are asked to find more than one answer ours more than one method. Students may also be asked to design an extension to the problem.
solution execution
procedural knowledge - 7.2 x 5.4 = 38.88 0.3x 0.3 = .09 38.88 x .09 =432 432 x $0.72 = $311.04
high level cognitive demand
procedures with connection tasks, doing mathematics tasks
task analysis
process of identifying the specific behaviors, knowledge, or cognitive processes necessary to master a particular topic or skill
Deductive Reasoning
process of reaching a necessary conclusion solely from a set of facts or hypotheses. These facts are called the assumptions or premises.
Preoperational Stage (2-7 years)
process of symbolic functioning, centration, intuitive thought, egocentrism, inability to conserve. child experience problems with two perceptual concepts (centration and conservation) 1. centration - focusing on only one aspect of a situation or problem 2. understanding that quantity, length, or number of items is unrelated to the arrangement or appearance of the object or item.
Formative Assessment
process used by teachers and students during instruction that provides feedback to adjust on going teaching and learning to improve students achievement of intended instructional outcomes
memorization tasks
producing already learned facts, are routine, have no connection to related concepts
When introducing students to a new math topic, a teacher should:
provide opportunities for students to investigate and explore the new material in small groups and individually
self and peer assessment
providing student an opportunity to think meta cognitively about their learning
to simplify
put in simplest form
Types of learning: Problem Solving—
putting together concepts and principles to solve a problem new to the learner Example: Given a composite figure the student determines the area using the areas of triangles and rectangles.
Without math science is what?
qualitative
subjective and relative?
qualitative
Objective and absolute?
quantitative
With math science is what?
quantitative
waiting
quiet time should not be uncomfortable, but should feel like thinking time
treat errors as opportunities for learning
rarely give random answers, so their wrong answers are an insight to what they may have not understood correctly
scale factor
ratio of any 2 corresponding measurements of similar solids
3. Strategic thinking and complex reasoning
reasoning,planning, using evidence, and a higher level of thinking than the previous two levels; making conjectures is also at the this level.
Is 1 divided by that number.
reciprocal
Product of any number (except 0) multiplied by its __________ is 1.
reciprocal
practice
refers to different tasks or experiences, spread over numerous class periods, each addressing the same basic ideas increased opportunity to develop conceptual ideas and useful connections to develop alternative and flexible strategies a greater chance for all students to understand a clear message that math is about figuring things out
drill
refers to repetitive exercises designed to improve skills or procedures already acquired -an increased facility with a procedure -review of facts or procedures so they are not forgotten
Reliability
refers to the consistency of a measure. A test is considered reliable if we get the same result repeatedly. For example, if a person is administered the same test repeatedly his/her results on the test should be approximately the same each time, if the test is reliable.
number sense
refers to whether a students understanding of a number and of its use and meaning is flexible and fully developed
Types of learning: Concept—
relational or concrete attributes. Example: Similar figures have relational attributes. The corresponding angles are equal and the ratios of corresponding sides are equal.
compensation
replacing numbers when guessing and checking
models
represent mathematicals concepts by relating concepts to real-world situations
elaborating
request for students to challenge, add on, or give an example. intended to get more participation, deepen understanding
doing mathematical tasks
require complex and non algorithmic thinking, exploration, demand self monitoring, require cognitive effort, access to relevant knowledge, and analyzation of the tasks
2.Basic application of concepts and skills
requires engagement of some mental processing beyond a habitual response, and making some decisions as to how to approach the problem or activity, following a defined series of steps.
anchored instruction
requires putting the students in the context of a problem-based story. The students "play" an authentic role while investigating the problem, identifying gaps to their knowledge, researching the information needed to solve the problem, and developing solutions. For example, the students play the role of a pilot to learn about aeronautics subject matter such as gravity, airflow, weather concepts, and basic flight dynamics. The teacher facilitates and coaches the students through the process. Learning and teaching activities should be designed around an "anchor" which is based on a contextualized case study or problem situation. Curriculum materials should allow exploration by the learner (e.g., interactive sites) to allow active manipulation, questioning, and involvement in the situation.
quantity discrimination
requires students to name which of two numbers is larger or smaller
Performance assessment
requires that completion of a task, project, or investigations; communicates information ; or constructs a response that demonstrates knowledge or understanding of a skill or concept.
revoicing
restating the statement as a question in order to clarify, apply appropriate language, and involve more students
one up one down
results in an answer that is one less than the original problem. 7x7=49 8x6=48
congruent figure
same size, same shape
postulates
sas (side angle side) sss (side side side) asa (angle side angle)
problem integration
schematic knowledge - the tile problem requires the formula area = length x width
Math is the language of what?
science
Mathematics is the language of _________?
science
geometric sequence
series of numbers in which a common ratio can be multipled by a term to yield the next term =
Domain
set
Intersection
set of all elements common to both sets A and B.
Universal Set
set of all the numbers being considered
Coordinates
set of numbers called an ordered pair (x,y) indicating the horizontal and vertical location of a point in space.
arithmetic sequence
set of numbers with specific difference between terms = a + (n-1)d
common core state standards
set of standards that most U.S states have adopted to guide instruction and assessment in English-language arts and math
y>or= 2x + 2
shaded top left
curriculum principle
should be coherent and built around 'big ideas' in the curriculum and in daily classroom instruction
tiered lessons
similar problems focused on the same mathematical goals, but adapted to meet the strange of learners, with different groups of students working on different tasks
polygon
simple closed figure composed of line segments
Range
single element in a set.
teachable moment
situation or event (often unplanned) in which students might be especially predisposed to acquire particular knowledge or skills
procedural fluency
skill in carrying out procedures flexibly, accurately, efficiently, and appropriately includes flexibility and ability to choose an appropriate strategy
Using a number line is appropriate to introduce
skip counting and counting backward
least common multipler
smallest number that all given numbers will divide into
Which are behaviors demonstrating concrete understanding of mathematics?
sorting objects into equally sized groups, giving one pencil to each student in the class, and counting the number of books on each shelf
activate prior knowledge
specific mathematical learning goals, what has been previously learned
procedures without connections tasks
specifically call for use of procedure, are straightforward, have no connection to related concepts, require no explanations, are focused on producing correct answers
standard deviation
square root of variance
solution planning and monitoring
strategic knowledge - step by step procedure
Traditional math workouts ______________ the mind?
strengthen
Reading a story out loud about two children playing hide-and-seek would
strengthen the children's understanding of spatial concepts such as behind, under, and inside
Alternative assessment in math may include:
student explanation of reasoning behind the answer, analysis of data, and multimedia
build new knowledge from prior
students apply their knowledge, test ideas, make connections, compare and make conjectures. The more students see the connections among problems and among mathematical concepts, the more deeply they understand mathematics
missing number
students are provided with a string of numbers and are asked to identify which number is missing
computation
students asked to complete computations that are representative of their grade level - have 2 minutes to complete as many problems as possible
number writing
students asked to write number when given a number orally between 1 and 20
Math Language
students must build a shared understanding of mathematical terms to successfully share and refine their ideas. Differentiated instruction when learning terms and symbols.
number identification
students must orally identify numbers between 0 and 20 when presented randomly on a piece of paper
descriptive feedback
students receive evidence based feedback linked to the intended instructional outcomes and criteria for success
Learning Log ex.
students record the process they go through in learning something new. This allows students to make connections to what they have learned, set goals, and reflect upon their learning process.
grading
summarizing a students performance through the accumulation of a variety of scores and data about their understanding of important skills and concepts
What types of manipulatives should be used in grade 3
tangrams, pattern blocks, playing cards, scales, magnetic numbers, chalk boards, spinners, dice, calendar, games, measuring tape, charts, bar graphs.
bloom's taxonomy
taxonomy of 6 cognitive processes, varying in complexity, that lessons might be designed to foster
collaboration
teachers and students are partners in learning.
Quality Assessment
teachers gather student work and materials in order to gauge and advance student learning
teaching principle
teachers must understand deeply the mathematics content, understand how students learn it, and select meaningful instructional tasks and generalizable strategies that will enhance learning
Progress Monitoring
teachers should be monitoring students learning progress by asking these questions before, during and after math instructions: - is the student making progress? - what does the student need to learn next? - how solid is the students understanding? - does the student need more work with a specific content? -etc.
2. Nomenclature of Math for ELL
terms have one meaning in one subject and another meaning in math which can confuse ELL's. quarter, column, product, rational, even, table.
summative assesment
test what has been learned; like a test or final project
Objective assessment
testing that requires the selections of one item from a list of choices provided with the question. This type of assessment includes true false responses, yes- no answers and questions with multiple- choice answers.
Complement
the __________ of set A is the set of all elements in the universal set U that are not in A
Greatest Common Factor (GCF)
the greatest factor that two or more numbers have in common. ex: 6 is the greatest common factor of 18 and 30. Use the factor tree, include 1
associative property
the grouping of numbers does not matter
Elements
the objects in a set
communiatvie properties
the order of numbers does not matter
Inductive Reasoning
the process of making a generalization based on a limited number of observations or examples.
Ability grouping
the process of placing students of similar abilities into groups and attempting to match instruction to the needs of these groups.
Natural Numbers
the set composed of the natural numbers and zero
whole numbers
the set containing number 0, 1, 2, 3, ...
counting numbers
the set containing numbers 1, 2, 3, ....
Union
the set of all elements in set A or B or both.
word names
the specific names given to the concept of a specific quantitative unit
operations
the standard process through which a qualitative entity is manipulated
Jean Piaget
thinking and reasoning of children were dominated by preoperational thought (patter of thinking that is egocentric, centered, irreversible, and nontransformational). 4 schemes of children: sensorimotor stage, preoperational stage, concrete operational stage, formal operational stage
to multiply
times
plus
to add
to increase by
to add
Which of the following is an effective math teaching strategy?
to be clear about instructional goals, to progress experiences and understanding from the concrete to the abstract stage, and to communicate to students what is expected of them and why
to solve
to find the answer
to solve for "x"
to find what the value of "x" is
A good way to create a concrete learning experience for students to learn how to count money is:
to give students an assortment of coins to sort and count
to round a number
to make a number simpler
to enlarge
to make bigger
to reduce
to make smaller
to label
to mark
times
to multiply
to double
to multiply by 2
to triple
to multiply by 3
to square a number
to multiply the number by itself
Why are manipulative, models, and technology used by math teachers?
to promote interest, to address diverse learning needs, to give hands-on math experience
to box answer
to put a square around answer
to explain
to put into words
to add
to put together
What is the main purpose of having K students count by twos?
to recognize patterns in numbers
to cost
to sell for
to complete
to solve
to divide
to split into equal parts
minus
to subtract
to decrease by
to subtract
to subtract
to take away
probability strategies
tree diagrams, grids
Correct steps yield to correct answers
true
Math provides a way to measure
true
conditional probability
two events that are not independent are dependent
compound event
two or more simple events
Competency 001-Math Instruction
understand how students learn math
Infinite Set
unlimited number of elements.
solve problem with overlapping lists
use a venn diagram
encourage multiple approaches
use of strategies that make sense to students, allow them to use their own strategy to find the answer or solution
formative assessment
used to check status of students development during instructional activities, to pre assess, or to attempt to identify students naive understandings or misconceptions 1. identify where learners are 2. identify goals for learners 3. identify path to reach the goal
visualize
using manipulatives, acting it out, drawings, helps understandings
Abstract
using numbers or numerals
cooperative learning
usually small groups of students (3-5) can be used to have students work together to solve problems
rote counting
verbal repetition of numbers begins around the age of 2-3 years foundation for building and understanding of the number concepts of combining, separating, naming amounts
Semi-concrete
visuals, no manipulatives --Using pictures to add
3. Nomenclature of Math for ELL
vocab tends to encompass tons of homophones (same pronunciation but different meanings) which can confuse ELL's. Even, face, plane, mean, right, some/sum, volume
Disjoint
when A and B sets = O
assimilation
when a new concept fits with prior knowledge and the new information expands an existing network
accommodation
when the new concept does not fit with the existing network causing a disequilibrium
Flexibility
when the students are able to understand more than one computational algorithm for a particular exercise. The students are able to choose the most appropriate approach for a given exercise.
rational number
whole numbers, integers, and fractions (with no decimals)
Types of learning: Association—
words or symbols Example: Even very young children associate the word "triangle" with the ∆, without knowing the attributes and properties of a triangle.
problem solving
works to make sense of and fully understand problems before beginning perseveres to demonstrate a variety of strategies assesses the reasonableness of answers
National Council of Teachers of Mathematics
worlds largest mathematics education organization. has an emphasis on what is best for students and learners
3rd grade students are recording the length of the hallway. What unit should they use?
yards
