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Common Core creates potential for collaborative groups in states and districts to get more economic mileage for ____, ____, and ____.
 curriculum development
 assessment
 professional development

Intent of the Common Core: ____, ____, ____, and ____.
 the same goals for all students
 coherence
 focus
 clarity and specificity

____ and ____ are emphasized equally in the Common Core.
 Conceptual understanding
 procedural skills

NCTM states that coherence also means that ____, ____, and ____ are aligned.
 instruction
 assessment
 curriculum

Key ideas, understandings, and skills are identified in the Common Core. ____ of concepts is stressed, which means that time is spent on a topic and on learning the topic well.
Deep learning

____ and ____ are clearly defined in Common Core.

Common Core sets the expectation that students are able to apply ____ and ____ to new situations.

CCSSM stands for ____.
Common Core State Standards for Mathematics

The Common Core proposes a set of Mathematical Practices that all teachers should develop in their students. These practices are similar to NCTM’s ____ ____ from the Principles and Standards for School Mathematics.
Mathematical Processes

CCSSM Mathematical Practices:
1) Make sense of problems and ____ in solving them.
persevere

CCSSM Mathematical Practices:
2) Reason ____ and ____.

CCSSM Mathematical Practices:
3) Construct viable ____ and critique the ____ of others.

CCSSM Mathematical Practices:
4) ____ with Mathematics.
Model

CCSSM Mathematical Practices:
5) Use appropriate ____ strategically.
tools

CCSSM Mathematical Practices:
6) Attend to ____.
precision

CCSSM Mathematical Practices:
7) Look for and make use of ____.
structure

CCSSM Mathematical Practices:
8) Look for and express regularity in ____ ____.
repeated reasoning

Common Core Format: ____ are larger groups of related standards. Standards from different  may sometimes be closely related. Overarching big ideas that connect topics across grades.
Domains

Common Core Format: ____ are groups of related standards. Standards from different  may sometimes be closely related, because mathematics is a connected subject.
Clusters

Common Core Format: ____ define
what students should be able to understand and be able to do.
Standards

The format of the Common Core for Grades K8 is 1)____, 2)____, and 3)____.
 Grade
 Domain
 Cluster
 Standards

____ are content statements.
Standards

For grades preK8, a model of implementation can be found in NCTM’s ____ ____ ____.
Curriculum Focal Points

Build new knowledge from ____ ____.
prior knowledge

Provide opportunities to talk about ____.
mathematics

Build on opportunities for ____ ____.
reflective thought

Encourage ____ ____.
multiple approaches

Treat errors as ____ ____ ____.
opportunities for learning

____ new content.
Scaffold


You must have a ____ and a ____ to have mathematics proficiency.
 conceptual understanding
 procedural understanding

____ ____ is knowledge about the relationship or foundational ideas of a topic.
Conceptual understanding

____ ____ is knowledge of the rules and procedures in carrying out mathematical processes and also the symbolism used to represent mathematics.
Procedural understanding

What are the five strands of mathematical proficiency?
 1) conceptual understanding
 2) procedural fluency
 3) strategic competence
 4) adaptive reasoning
 5) productive disposition

Comprehension of mathematical concepts, operations, and relations
conceptual understanding

Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
procedural fluency

Ability to formulate, represent, and solve mathematical problems
strategic competence

Capacity for logical thought, reflection, explanation, and justification
adaptive reasoning

Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.
productive disposition

The most critical period for the development of whole number place value is grades ____ ____ ____.
prek to 3

Children are exposed to
patterns in grades ____ and ____.

Children formally learn that groups of ten are connected to our placevalue system in grade ____.
2

Understanding of patterns and place value are extended and decimals are introduced in grades ____ and ____.

In grades ____ and ____, ideas with whole numbers are extended to decimals.

____ ____ requires an integration of new and difficulttoconstruct concepts of groupings by tens with procedural knowledge of how groups are recorded in our placevalue scheme, how numbers are written, and how they are spoken.
Placevalue understanding

What are three different way sin which children begin counting?
 1) counting by ones
 2) counting by groups and singles
 3) counting by tens and ones

This is the only way children can be convinced that quantities are equal before baseten ideas develop. (onetoone correspondence)
counting by ones

This counting must be coordinated with a count by ones before it can be a means of telling “how many”.
counting by groups and singles

An idea or abstraction that represents a quantity
number

A symbol representing numbers
numeral

Used to describe how many elements are in a finite set
cardinal number

Numbers used to denote order, i.e., “second in line”
ordinal number

What five principles must students understand before they can engage in real counting?
 1) Any collection of real or imagines objects can be counted
 2) Counting numbers are arranged in a sequence that does not change
 3) Onetoone correspondence
 4) Orderirrelevance
 5) Cardinality principle

One and only one number is used for each item counted, and each item is counted only once
onetoone correspondence

The order in which items are counted is irrelevant
orderirrelevance

There is a special significance to the last number counted. It is not only associated with
the last item but also represents the total number of items in the set
cardinality principle

For the most basic strategies, children use physical objects (counters) or fingers to ____ ____ the action or relationships described in each problem.
directly model

Over time, children’s strategies become more abstract and efficient. ____ ____ strategies are replaced by more abstract
StudentInvented Strategies/Counting Strategies, which in turn are replaced with ____ ____.
 Direct Modeling
 Number Facts

This strategy is distinguished by a child’s explicit physical representation of each quantity in the problem and the action or relationship involving those quantities before counting the resulting set.
direct modeling

A child essentially recognizes that it is not necessary to actually construct and count sets. The answer can be figured out by focusing on the counting sequence itself.
counting

Physical objects are used to represent objects in a problem.
direct modeling

Physical objects are used to keep track of counts.
counting

Children learn ____ before other number combinations.
doubles

Children learn sums of ____ relatively early.
ten

____ ____ solutions are based on understanding relations between numbers.
Derived Fact

Even without specific instruction, most children use ____ ____ before they have mastered all their number facts at a recall level.
Derived Facts

When children have the opportunity to discuss
alternative strategies, the use of ____ ____ becomes even more prevalent.
Derived Facts

Children appear to “____ ____” Direct Modeling, Counting, and Derived Fact strategies when number choices/kinds of quantities are consistent across problems.
move through

____ ____ strategies are not easily used with some problem types.
Direct Modeling

____ ____ are extensions of modeling strategies.
Mental strategies

Children learn ____ ____ and apply this knowledge to solve problems.
number facts

Children learn certain ____ ____ before others.
number combinations

Children often use a small set of memorized facts to derive solutions for problems involving other ____ ____.
number combinations

What are the three primary strategies used by children to solve math problems?
 1) modeling strategy
 2) split strategy
 3) increment/jump strategy

Involves drawing pictures of tens and ones and
counting, first by tens and then by ones
modeling strategy

Involves splitting the numbers in the strategy into smaller numbers that are easier to work with and then adding the totals together to get the answer
split strategy

Primarily through the use of an empty number line
increment/jump strategy

Mental strategies involve decomposing numbers by tens and ones
modeling strategy

Representational strategies involve modeling and carrying out operations using tens sticks and units and documenting using drawings of tens sticks and ones with ‘actions.’
modeling strategy

Notational strategies involve recording work with numbers as tens and ones
split strategy

Mental strategies involve holding one number in memory and then decomposing the second number, mentally joining or separating
increment/jump strategy

Representational strategies involve modeling and carrying out operation using an empty number line (which also provides written documentation)
increment/jump strategy

Classroom instruction is generally organized and orchestrated around ____ ____
mathematical tasks

The ____ with which students engage determine what they learn about mathematics and how they learn it
tasks

The inability to enact ____ ____ well is what distinguished teaching in the U. S. from teaching in other countries that had better student performance on TIMSS
challenging tasks

“Not all tasks are____ ____, and different tasks will provoke different levels and kinds of student thinking.”
created equal

TaskFocused Activities: ____ between high and low cognitive demand mathematics tasks
Distinguishing

TaskFocused Activities: ____ the cognitive demands of highlevel tasks during instruction
Maintaining

A ____ ____ ____ task begins where the students are (zone of proximal development; scaffolding)
high cognitive demand

The problematic or engaging aspect of a ____ ____ ____ task is due to mathematics that students are to learn.
high cognitive demand

A ____ ____ ____ task requires justifications and explanations for answers and methods.
high cognitive demand

____ ____ ____ tasks or activities are the vehicle through which the curriculum can be developed. Maintaining the cognitive demands of ________ tasks during instruction affects the learning that occurs.
 high cognitive demand
 highlevel

First step of the Mathematical Tasks Framework:
TASKS as they appear in curricular/instructional materials or are designed by teachers

Second step of the Mathematical Tasks Framework:
TASKS as they are set up by the teacher

Third step of the Mathematical Tasks Framework:
TASKS as they are implemented by students

Fourth step of the Mathematical Tasks Framework:
TASKS as they are summarized by teacher and students

Fifth step of the Mathematical Tasks Framework:
Student learning

SUBITIZING CARDS:
Recognizing a number instantly in a pattern
perceptual

SUBITIZING CARDS:
Uses colors to show separate patterns together so that you can recognize two different patterns make a whole
conceptual

Subitizing cards are both ____ and ____.

What are the five process standards of NCTM?
 1) problem solving
 2) reasoning and proof
 3) representation
 4) communication
 5) connections

What are the five proficiency strands of NCSM?
 1) conceptual understanding
 2) procedural fluency
 3) strategic competence
 4) adaptive reasoning
 5) productive reasoning

What two things are combined to make up the eight standards in the Common Core?
 1) The five process standards of NCTM
 2) The five proficiency strands of NCSM

What are the eight standards for mathematical practice under the Common Core?
 1) Make sense of problems and persevere in solving them
 2) Reason abstractly and quantitatively
 3) Construct viable arguments and critique the reasoning of others
 4) Model with mathematics
 5) Use appropriate tools strategically
 6) Attend to precision
 7) Look for and make use of structure
 8) Look for and express regularity in repeated reasoning

Strategies become more ____ over time.
abstract

An ____ has an equals sign.
equation

An ____ does not have an equals sign.
expression

What are the three primary strategies for solving problems that are invented by students?
 1) increment/jump strategy
 2) split strategy
 3) modeling strategy

What are the three phases that are typical to mastering basic number combinations?
 1) counting strategies
 2) reasoning strategies
 3) mastery

What are some of the reasoning strategies used for addition facts?
 1) one more
 2) two more
 3) using five as an anchor
 4) make 10
 5) doubles
 6) near doubles

What are some of the reasoning strategies used for subtraction facts?
 1) subtraction as think addition
 2) down over ten
 3) take from ten

What are the four basic types of addition and subtraction problems?
 1) joining
 2) separating
 3) partpartwhole comparisons
 4) comparison situations

Using object counting or verbal counting to get an answer (counting on, direct modeling, etc.)
counting strategies

Using known information to logically figure out an answer
reasoning strategies

Developing meanings for operations
operations sense

Gaining a sense for the relationships among operations
operations sense

Determining which operation to use in a given situation
operations sense

Recognizing that the same operation can be applied in problem situations that seem quite different
operations sense

Developing a sense for the operations’ effects on numbers
operations sense

Realizing that operation effects depend upon the types of numbers involved
operations sense

There were significant positive correlations between ____ ____ of how their children solved addition and subtraction problems and their ____ ____ to solve the problem.
 teachers' knowledge
 children's ability

Teachers who believed that children bring ____ to instruction and instruction should be built on that ____, had higher levels of ____ than did teachers who did not agree as strongly with that perspective.
 knowledge
 knowledge
 achievement

Classes with the highest levels of achievement were those of teachers who believed most strongly that the teacher was not the ____ ____ ____ ____ and that instruction should be designed to help children ____ ____ ____ ____ for themselves.
 ultimate source of knowledge
 construct solutions to problems

What does CGI stand for?
Cognitively Guided Instruction

What are the basic ideas of CGI?
 1) Children bring to school informal or intuitive knowledge of mathematics.
 2) Children intuitively solve word problems by modeling the action and relations described in them.
 3) One of the most useful ways of classifying problems focuses on the types of action or relation described in the problem. Providing a framework to identify the relative difficulty of problems.

A basic idea of CGI is that children bring informal or intuitive ____ of mathematics to school.
knowledge

A basic idea of CGI is that children intuitively solve word problems by ____ the action and relations described in them.
modeling

A basic idea of CGI is that one of the most useful ways of classifying problems focuses on the types of ____ or ____ described in the problem, providing a framework to identify the relative difficulty of problems.

The teacher's role in CGI is to learn how students initially use ____ ____, ____, and how they evolve.
 concrete materials
 modeling

The teacher's role in CGI is to focus is on what children ____ do rather than on what they ____ do.

The teacher's role in CGI is to work back from ____ to find out what valid conceptions students do have.
errors

The teacher's role in CGI is to ____ problems their students can solve.
identify

The teacher's role in CGI is to shift the emphasis from personally finding ways of ____ ____ ____ to the students ____ ____ ____ representations.
 representing mathematical knowledge
 constructing their own

Lucy has 8 fish. She wants to buy 5 more fish. How many fish would Lucy have then?
8 + 5 = ?
join result unknown

Janelle has 7 trolls in her collection. How many more does she have to buy to have 11 trolls?
7 + ? = 11
join change unknown

Sandra has some pennies. George gave her 4 more. Now Sandra has 12 pennies. How many pennies did Sandra have to begin with?
? + 4 = 12
join initial unknown

TJ had 13 chocolate chip cookies. At lunch she ate 5 of them. How many cookies did TJ have left?
13  5 = ?
separate result unknown

11 children were in the sandbox. Some children went home. There were 3 children still playing in the sandbox. How many children went home?
11  ? = 3
separate change unknown

Max had some money. He spent $9.00 on a video game. Now he has $7.00 left. How much money did Max have to start with?
?  $9 = $7
separate initial unknown

Susan has 8 red apples and 5 green apples. How many apples does she have?
8 red + 5 green = ? apples
partpartwhole ~ whole unknown

Brandy has 16 Gummy Bears. 8 are red and the rest are green. How many green Gummy Bears does she have?
16 gummy bears = 8 red + ? green
partpartwhole ~ part unknown

Willy has 12 crayons. Lucy has 7 crayons. How many more crayons does Willy have than Lucy?
12  7 = ?
compare difference unknown

George has 4 more pennies than Sandra. Sandra has 8 pennies. How many pennies does George have?
8 + 4 = ?
compare larger unknown

Lydia had 4 fewer pencils than Henry. Henry has 10 pencils. How many pencils does Lydia have?
10  4 = ?
compare smaller unknown

What is another term for the borrowing method?
decomposition method

This method is based upon the idea that adding the same amount to two different numbers will not change the different between the two numbers.
Equal Additions Method

Also know as the additions method. The answer is found by directly relating the answer to additions.
Austrian Algorithm

