MAE Midterm

Problem Solving strategy

guess and check

Productive struggle

Encouraging perseverance through challenging tasks.

Problem Solving strategy

Dramatization

Problem Solving strategy

Estimation

Problem Solving strategy

Solve a simpler problem

Problem Solving strategy

Create a table or chart

Problem Solving strategy

Work backwards

Process for Engaging in Mathematical Practices

Make sense of a problem and persevere in solving

Process for Engaging in Mathematical Practices

Reason abstractly and quantitatively

Process for Engaging in Mathematical Practices

Create visible arguments and critique the reasonings of others

Process for Engaging in Mathematical Practices

Model with mathematics

Process for Engaging in Mathematical Practices

Use appropriate tools strategically

Process for Engaging in Mathematical Practices

Attend precision

Process for Engaging in Mathematical Practices

Look for and make use of structure

Process for Engaging in Mathematical Practices

Look for and express regularity in repeated reasonings

Levels of Abstraction

Concrete: Actual Manipulatives

Levels of Abstraction

Semi Concrete: Pictures

Levels of Abstraction

Semi Abstract: Tallies or Symbols

Levels of Abstraction

Abstract: Numbers and Numerals

Piaget Stages of Cognitive development

• Sensorimotor Stage

• Pre-operational Stage

• Concrete Operational Stage

• Formal Operational Stage

Sensorimotor Stage

• Birth- 2 years

• Infant knows the world through movement and sensations

• Children learn about the world through basic actions such as sucking, grasping, looking, and listening

• Infants learn that things continue to exist even though they cannot be seen

• They realize that their actions can cause things to happen in the world around them

Object Permanence

Objects continue to exist even when they cannot be seen, heard, touched, smelled or sensed in any way

Pre-operational Stage

• 2-7 years

• Children begin to think symbolically and learn to use words and the pictures to represent objects

• Children at this stage tend to be egocentric and struggle to see things from the perspective of others

• While they are getting better with language and thinking, they still tend to think about things in very concrete terms

Concrete Operational Stage

• 7-11 years

• During this stage, children begin to think logically about concrete events

• they begin to understand the concept of conservation; that the amount of liquid in a short, wide cup is equal to that in a tall, skinny glass

• Their thinking becomes more logical and organized, but still very concrete

• Children begin using inductive logic, or reasoning from specific information to a general principle

Formal operational Stage

• 12 and up

• children develop the ability to reason more abstractly, systematically, and reflectively.

• They can now manipulate ideas mentally, consider hypothetical scenarios, and engage in deductive reasoning without needing physical objects as referents.

• This new cognitive capacity allows them to use logic to reason out the possible consequences of each action before carrying it out.

Constructivism

we allow our students to actively explore and learn

Cooperative learning

structured learning groups (interdependence)

Collaborative learning

loosely structured groups

Peer tutoring

choose groups, manipulate

Simulations

depicting real life situations

Instructional games

make it engaging

contextual learning

authentic real life situations

educational scaffolding

provides guidance and support

• Provides direction

• use think alouds to model thought processes

• clarify the purpose

• keep students on task

• provide assessments (rubrics to clarify expectations)

• extra resources

• Foster thinking to Bloom’s Taxonomy

Bloom’s Taxonomy

measurable verbs to help us describe and classify observable knowledge, skills, attitudes, behaviors and abilities. The theory is based upon the idea that there are levels of observable actions that indicate something is happening in the brain (cognitive activity.)

Differentiation

choosing the right tactic for the right group

Types of differentiation

whole group instruction

Types of differentiation

small groups

Types of differentiation

math workshops: focuses on writing

Types of differentiation

math centers

Types of differentiation

learning partners/buddies

Analyzing/ applying appropriate math concepts

• subitize: knowing the number by looking at its representation

• transitivity: a>b b>c a>c

• iteration: repeating steps over and over

• tiling: using an item to represent abstract to concrete (tiling/ array interchangeable)

• inventive strategies: to solve complex problems

Partitioning

taking large numbers and splitting them into smaller parts

Compensation

Borrow one piece and add it to another

Standard algorithm

• area

• volume

• exchanging

• long division

• averaging

• regrouping

• long multiplication

Place value

involves how numbers are grouped into ones (units), tens (rod/long), hundreds (flat)

Set

group of objects, things or ideas that is precisely described

Addition

is a binary performed on a pair of addends in order to obtain a unique sum

Number families

look at the whole thing ex. 0+4, 4+0, 1+3, 3+1, 2+2

Fact families

all meanings for the operations ex. 1+4=5, 4+1=5, 5-1=4, 5-4=1

Subtraction

an operation performed in a sum and one addend to obtain the missing addend

One to one correspondence

numbers correspond to specific quantities (dice)

Binary operation

performed on a pair of factors to obtain a unique product factor X factor = product

Repeated addition

3+3+3=9 = 3×3=9

Arrays

arrange in rows x colums

Properties

• Commutative

• Zero property

• Identity property

• Associative property

• Distributive property

Commutative

axb=c bxc=a

Zero property

nx0=0 0xn=0

Identity property

ax1=a 1xa=a

Associative property

(axb)xc=ax(bxc)

Distributive property

ax(b+c)= (axb)+(axc)

Division

operation performed on a product and one factor to obtain the missing factor

Multiples

the set of products resulting from the multiplication of n times any whole number

Factor

are multiplied together to get a product, a factor of some number n divides evenly

Product Expression

a multiplication problem without the product

Least Common Multiple (LCM)

the smallest (least) non zero number that is a multiple of two numbers

Greatest Common Factor (GCF)

the greatest whole number that is a factor of each of the two numbers

Prime numbers

have exactly two product expression

Composite numbers

a number has more than two product expression

Mean

add the set of numbers together then divide the sum by how many numbers in the set

median

arrange the numbers in the set from the least to greatest then identify the middle number

Mode

number that appears the most often (sometimes there will be none)

Range

subtract the smallest number from the largest number