content deepening 6 th grade math january 24, 2014 jeanne simpson amsti math specialist

Content Deepening6th Grade Math

January 24, 2014

Jeanne Simpson

AMSTI Math Specialist

2

NameSchoolWhat are you hoping to learn today?

Welcome

3

He who dares to teach

must never cease to

learn.John Cotton Dana

Goals for Today

Implementation of the Standards of Mathematical Practices in daily lessons

Understanding of what the CCRS expect students to learn blended with how they expect students to learn.

Student-engaged learning around high-cognitive-demand tasks used in every classroom.

Agenda Statistics and Probability

Progression

Standards Analysis

Resources

High-Cognitive Demand Tasks Expressions and Equations

Inequalities

Resources

Standards of Mathematical Practice Fractions

acos2010.wikispaces.com Electronic version of handouts Links to web resources

http://acos2010.wikispaces.com/Home

Statistics and Probability

THE STRUCTURE IS THE STANDARDS

The natural distribution of prior knowledge in classrooms should not prompt abandoning instruction in grade level content, but should prompt explicit attention to connecting grade level content to content from prior learning. To do this, instruction should reflect the progressions on which the CCSSM are built. For example, the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning with respect to place value. Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking. http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-

422

KNOWLEDGE GAPS

This is a basic condition of teaching and should not be ignored in the name of standards. Nearly every student has more to learn about the mathematics referenced by standards from earlier grades. Indeed, it is the nature of mathematics that much new learning is about extending knowledge from prior learning to new situations. For this reason, teachers need to understand the progressions in the standards so they can see where individual students and groups of students are coming from, and where they are heading. But progressions disappear when standards are torn out of context and taught as isolated events.

http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422

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Learning Progression Jigsaw

Read your assigned section K-5 Data, pages 1-5

6-8 Overview, pages 6-7

Grade 6, pages 8-10

Grade 7, pages 11-14

Chart paper Summarize what needs to be learned.

How can this document help you in your classroom?

Be prepared to share

Analysis Tool

Content Standard

Cluster

Which Standards in the Cluster

Are Familiar?

What’s New or

Challenging in These

Standards?

Which Standards in the cluster

Need Unpacking

or Emphasizing

?

How Is This Cluster

Connected to the Other 6-8 Domains

and Mathematical Practice?

Develop understanding of statistical variability.

6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

What resources do you already have for teaching statistics?

TextbookAMSTI Units

How do these match the standards?

SP ResourcesRaisin Activity MARS – Mean, Median, ModeCMP

Data About UsCommon Core Investigations

Lessons for Learning How MAD are You?Shakespeare vs. Rowling

Raisin Activity

Count the number of raisins in your box. Make a box plot for the number of raisins in

each brand’s box. Find the median, range, and interquartile

range for each brand. Make a dot plot of the data. Find the mean

and the mean absolute deviation.

Mean, Median, Mode, and Range

Computer Games: Ratings

P-16

Imagine rating a popular computer game.

You can give the game a score of between 1 and 6.

Computer Games: Ratings

P-17

Rate the game Candy Crush with a score between 1 and 6.

Bar Chart from a Frequency Table

P-18

Mean score

Median score

Mode score

Range of scores

Matching Cards1. Each time you match a pair of cards,

explain your thinking clearly and carefully.

2. Partners should either agree with the explanation or challenge it if it is unclear or incomplete.

3. Once agreed stick the cards onto the poster and write a justification next to the cards.

4. Some of the statistics tables have gaps in them and one of the bar charts is blank. You will need to complete these cards.

P-19

Sharing Posters1. One person from each group visit a

different group and look carefully at their matched cards.

2. Check the cards and point out any cards you think are incorrect. You must give a reason why you think the card is incorrectly matched or completed, but do not make changes to the card.

3. Return to your original group, review your own matches and make any necessary changes using arrows to show if card needs to move.

P-20

AMSTI Connected Math Unit

How MAD are You?(Mean Absolute Deviation)

Fist to Five…How much do you know about Mean Absolute Deviation?

0 = No Knowledge

5 = Master Knowledge

Create a distribution of nine data points on your number line that would yield a mean of 5.

Card Sort

Which data set seems to differ the least from the mean?

Which data set seems to differ the most from the mean?

Put all of the data sets in order from “Differs Least” from the mean to “Differs Most” from the mean.

The mean in each set equals 5.

333 32

11 4 6

Find the distance (deviation) of each point from the mean.

Use the absolute value of each distance.

Find the mean of the absolute deviations.

How could we arrange the nine points in our data to decrease the MAD?

How could we arrange the nine points in our data to increase the MAD?

How MAD are you?

Shakespeare vs. Rowling

High Cognitive

vs. Low Cognitive

28

An effective mathematical task is needed to

challenge and engage students intellectually.

29

Read an

excerpt from

the article:

Comparing Two Mathematical Tasks

Solve Two Tasks:

• Martha’s Carpeting Task

• The Fencing Task

31

How are Martha’s Carpeting Task and the Fencing Task the same and how are they different?


32

Similarities and Differences

Similarities

• Both are “area” problems

• Both require prior knowledge of area

Differences

• The amount of thinking and reasoning required

• The number of ways the problem can be solved

• Way in which the area formula is used

• The need to generalize

• The range of ways to enter the problem

33

Do the differences between the Fencing Task and Martha’s Carpeting Task matter?

Why or Why not?


34

Does Maintaining Cognitive Demand Matter?

YES

Criteria for low cognitive demand tasks

•Recall

•Memorization

•Low on Bloom’s Taxonomy

Criteria for high cognitive demand tasks

•Requires generalizations

•Requires creativity

•Requires multiple representations

•Requires explanations (must be “worth explaining”)

What causes high-level cognitive demand tasks to decline?

Stein & Lane, 2012

A.

B.

C.

High High

Low Low

High LowModerate

High

Low

Task Set Up Task Implementation

Student Learning

Patterns of Set up, Implementation, and Student Learning

44

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands

• Routinizing problematic aspects of the task

• Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer

• Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behavior

• Engaging in high-level cognitive activities is prevented due to classroom management problems

• Selecting a task that is inappropriate for a given group of students

• Failing to hold students accountable for high-level products or processes

(Stein, Grover & Henningsen, 2012)

45

• Scaffolding of student thinking and reasoning

• Providing a means by which students can monitor their own progress

• Modeling of high-level performance by teacher or capable students

• Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback

• Selecting tasks that build on students’ prior knowledge

• Drawing frequent conceptual connections

• Providing sufficient time to explore

(Stein, Grover & Henningsen, 2012)

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands

46

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

(Stein, Smith, Henningsen, & Silver, 2011)

“The level and kind of thinking in which students engage determines what they will learn.”

(Hiebert et al., 2011)

Expressions and Equations

Activities

Inequalities Look at standards. What is required?

Illustrative Mathematics tasks

MARS – Laws of Arithmetic lesson Arithmetic with whole-number exponents

Order of operations

Finding area of compound rectangles by evaluating expressions

Math-Magic Using variables

What are students asked to do with inequalities?

6.EE.5 – Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.6 – Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.EE.7 – Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

6.EE.8 – Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

http://ccssmath.org/?page_id=560




Log Ride (6.EE.5)

A theme park has a log ride that can hold 12 people. They also have a weight limit of 1500 lbs per log for safety reasons. If the average adult weights 100 lbs and the log itself weights 200, the ride can operate safely if the inequality

150A + 100C + 200 < 1500is satisfied (A is the number of adults and C is the number of children in the log ride together). There are several groups of children of differing numbers waiting to ride. If 4 adults are already seated in the log, which groups of children can safely ride with them?

Group 1: 4 children

Group 2: 3 children

Group 3: 9 children

Group 4: 6 children

Group 5: 5 children

Fishing Adventures (6.EE.8)

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can hold at most eight people. Additionally, each boat can only carry 900 pounds of weight for safety reasons.

a. Let p represent the total number of people. Write an inequality to describe the number of people that a boat can hold. Draw a number line diagram that shows all possible solutions.

b. Let w represent the total weight of a group of people wishing to rent a boat. Write an inequality that describes all total weights allowed in a boat. Draw a number line diagram that shows all possible solutions.

Laws of ArithmeticProjector Resources

Laws of Arithmetic


Pre-Assessment Task


Teacher Notes


Writing Expressions

P-58

3

4 5

Write an expression to represent the total area of this diagram


Compound Area Diagrams

P-59

4 51

2

12

5 4

Area A

Area C

21

4 5

Area B

Which compound area diagram represents the expression:

5 + 4 x 2?


Matching Cards

P-60

1. Take turns at matching pairs of cards that you think belong together. For each Area card there are at least two Expressions cards.

2. Each time you do this, explain your thinking clearly and carefully. Your partner should either explain that reasoning again in his/her own words or challenge the reasons you gave.

3. If you think there is no suitable card that matches, write one of your own on a blank card.

4. Once agreed, stick the matched cards onto the poster paper writing any relevant calculations and explanations next to the cards.

You both need to be able to agree on and to be able to explain the placement of every card.


Sharing Work

P-61

1. If you are staying at your desk, be ready to explain the reasons for your group’s matches.

2. If you are visiting another group:– Copy your matches onto your paper. – Go to another group’s desk and check to see

which matches are different from your own.

3. If there are differences, ask for an explanation. If you still don’t agree, explain your own thinking.

4. Return to your original group, review your own matches and make any necessary changes using arrows to show that a card needs to move.

Variables and Expressions

6.EE.2 – Write, read, and evaluate expressions in which letters stand for numbers

6.EE.2a – Write expressions that record operations with numbers and with letters standing for numbers

6.NS.6 – Understand a rational number as a point on a number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

Math Magic – Trick #1 Pick a number… any number! (keep it a secret

though) Add 1 to that number Multiply by 3 Subtract your ‘secret’ number Add 5 Divide by 2 Subtract your secret number

Math Magic – Trick #1Now map the digit you got to a letter in

the alphabet.For example:

A =1 B=2 C=3 D=4

Pick a name of a country in Europe that starts with that letter.

Countries in Europe Albania

Andorra

Austria

Belarus

Belgium

Bosnia and Herzegovina

Bulgaria

Croatia

Cyprus

Czech Republic

Denmark

Estonia

Finland France Germany Greece Hungary Iceland Ireland Italy Latvia Liechtenstein Lithuania Luxembourg Macedonia

Malta Moldova Monaco Netherlands Norway Poland Portugal Romania Russia San Marino Serbia and

Montenegro

Slovakia (Slovak Republic)

Slovenia Spain Sweden Switzerland Turkey Ukraine United

Kingdom Vatican City

Math Magic – Trick #1 Take the second letter in the country's name, and pick

and animal that starts with that letter. Think of the color of that animal.

Directions How The Numbers Change

Algebraic Expressions

a. Think of a number 2

b. Add 5

c. Multiply by 3

d. Subtract 3

e. Divide by 3

f. Subtract your original number




b. Add 5 7

c. Multiply by 3

d. Subtract 3

e. Divide by 3





b. Add 5 7

c. Multiply by 3 21

d. Subtract 3

e. Divide by 3





b. Add 5 7

c. Multiply by 3 21

d. Subtract 3 18

e. Divide by 3





b. Add 5 7

c. Multiply by 3 21

d. Subtract 3 18

e. Divide by 3 6





b. Add 5 7

c. Multiply by 3 21

d. Subtract 3 18

e. Divide by 3 6

f. Subtract your original number 4



a. Think of a number 2 7

b. Add 5 7

c. Multiply by 3 21

d. Subtract 3 18

e. Divide by 3 6

f. Subtract your original number 4



a. Think of a number 2 7

b. Add 5 7 12

c. Multiply by 3 21 36

d. Subtract 3 18 33

e. Divide by 3 6 11

f. Subtract your original number 4 4



a. Think of a number 2 7 15

b. Add 5 7 12

c. Multiply by 3 21 36

d. Subtract 3 18 33

e. Divide by 3 6 11

f. Subtract your original number 4 4



a. Think of a number 2 7 15

b. Add 5 7 12 20

c. Multiply by 3 21 36 60

d. Subtract 3 18 33 57

e. Divide by 3 6 11 19

f. Subtract your original number 4 4 4



a. Think of a number 2 7 15 n

b. Add 5 7 12 20








b. Add 5 7 12 20 n + 5








b. Add 5 7 12 20 n + 5

c. Multiply by 3 21 36 60 3n + 15







b. Add 5 7 12 20 n + 5

c. Multiply by 3 21 36 60 3n + 15

d. Subtract 3 18 33 57 3n + 12






b. Add 5 7 12 20 n + 5

c. Multiply by 3 21 36 60 3n + 15

d. Subtract 3 18 33 57 3n + 12

e. Divide by 3 6 11 19 n + 4





b. Add 5 7 12 20 n + 5

c. Multiply by 3 21 36 60 3n + 15

d. Subtract 3 18 33 57 3n + 12

e. Divide by 3 6 11 19 n + 4

f. Subtract your original number 4 4 4 4

General MacArthur's # Game

Write down the number of the month you were born in

Double it Add 5 Multiply by 50 Add in your age Subtract 365 What is your number?

Your Game! Create your own magic with math to share

with your friends and family. Complete the worksheet with directions you

make up. Try three different numbers. Use algebraic expressions to show why the

magic works.

Exit Question Ricardo has 8 pet mice. He keeps them in two cages

that are connected so that the mice can go back and forth between the cages. One of the cages is blue, and the other is green. Show all the ways that 8 mice can be in two cages.

Standards of Mathematical Practice

Students: (I) Initial (IN) Intermediate (A) Advanced1a Make sense of

problems

Explain their thought processes in solving a problem one way.

Explain their thought processes in solving a problem and representing it in several ways.

Discuss, explain, and demonstrate solving a problem with multiple representations and in multiple ways.

1b Persevere in solving them

Stay with a challenging problem for more than one attempt.

Try several approaches in finding a solution, and only seek hints if stuck.

Struggle with various attempts over time, and learn from previous solution attempts.

2 Reason abstractly andquantitatively

Reason with models or pictorial representations to solve problems.

Are able to translate situations into symbols for solving problems.

Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations.

3a Construct viablearguments

Explain their thinking for the solution they found.

Explain their own thinking and thinking of others with accurate vocabulary.

Justify and explain, with accurate language and vocabulary, why their solution is correct.

3b Critique the reasoning of others.

Understand and discuss other ideas and approaches.

Explain other students’ solutions and identify strengths and weaknesses of the solution.

Compare and contrast various solution strategies and explain the reasoning of others.

4 Model withMathematics

Use models to represent and solve a problem, and translate the solution to mathematical symbols.

Use models and symbols to represent and solve a problem, and accurately explain the solution representation.

Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem.

5 Use appropriate toolsstrategically

Use the appropriate tool to find a solution. Select from a variety of tools the ones that can be used to solve a problem, and explain their reasoning for the selection.

Combine various tools, including technology, explore and solve a problem as well as justify their tool selection and problem solution.

6 Attend to precision

Communicate their reasoning and solution to others.

Incorporate appropriate vocabulary and symbols when communicating with others.

Use appropriate symbols, vocabulary, and labeling to effectively communicate and exchange ideas.

7 Look for and make useof structure

Look for structure within mathematics to help them solve problems efficiently (such as 2 x 7 x 5 has the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x 7) x 5, the student can mentally calculate 10 x 7.

Compose and decompose number situations and relationships through observed patterns in order to simplify solutions.

See complex and complicated mathematical expressions as component parts.

8 Look for and expressregularity in repeatedreasoning

Look for obvious patterns, and use if/ then reasoning strategies for obvious patterns.

Find and explain subtle patterns.

Discover deep, underlying relationships, i.e. uncover a model or equation that unifies the various aspects of a problem such asdiscovering an underlying function.

SMP Proficiency Matrix

SMP Instructional Implementation Sequence

1. Think-Pair-Share (1, 3)

2. Showing thinking in classrooms (3, 6)

3. Questioning and wait time (1, 3)

4. Grouping and engaging problems (1, 2, 3, 4, 5, 8)

5. Using questions and prompts with groups (4, 7)

6. Allowing students to struggle (1, 4, 5, 6, 7, 8)

7. Encouraging reasoning (2, 6, 7, 8)

Students: (I) Initial (IN) Intermediate (A) Advanced1a Make sense of

problems

Explain their thought processes in solving a problem one way.

Explain their thought processes in solving a problem and representing it in several ways.

Discuss, explain, and demonstrate solving a problem with multiple representations and in multiple ways.

1b Persevere in solving them

Stay with a challenging problem for more than one attempt.

Try several approaches in finding a solution, and only seek hints if stuck.

Struggle with various attempts over time, and learn from previous solution attempts.

2 Reason abstractly andquantitatively

Reason with models or pictorial representations to solve problems.

Are able to translate situations into symbols for solving problems.

Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations.

3a Construct viablearguments

Explain their thinking for the solution they found.

Explain their own thinking and thinking of others with accurate vocabulary.

Justify and explain, with accurate language and vocabulary, why their solution is correct.

3b Critique the reasoning of others.

Understand and discuss other ideas and approaches.

Explain other students’ solutions and identify strengths and weaknesses of the solution.

Compare and contrast various solution strategies and explain the reasoning of others.

4 Model withMathematics

Use models to represent and solve a problem, and translate the solution to mathematical symbols.

Use models and symbols to represent and solve a problem, and accurately explain the solution representation.

Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem.

5 Use appropriate toolsstrategically

Use the appropriate tool to find a solution. Select from a variety of tools the ones that can be used to solve a problem, and explain their reasoning for the selection.

Combine various tools, including technology, explore and solve a problem as well as justify their tool selection and problem solution.

6 Attend to precision

Communicate their reasoning and solution to others.

Incorporate appropriate vocabulary and symbols when communicating with others.

Use appropriate symbols, vocabulary, and labeling to effectively communicate and exchange ideas.

7 Look for and make useof structure

Look for structure within mathematics to help them solve problems efficiently (such as 2 x 7 x 5 has the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x 7) x 5, the student can mentally calculate 10 x 7.

Compose and decompose number situations and relationships through observed patterns in order to simplify solutions.

See complex and complicated mathematical expressions as component parts.

8 Look for and expressregularity in repeatedreasoning

Look for obvious patterns, and use if/ then reasoning strategies for obvious patterns.

Find and explain subtle patterns.

Discover deep, underlying relationships, i.e. uncover a model or equation that unifies the various aspects of a problem such asdiscovering an underlying function.

SMP Proficiency Matrix

Grouping/Engaging Problems



Pair-Share

Showing Thinking

Showing Thinking

Questioning/Wait Time



Questions/Prompts for Groups

Questions/Prompts for Groups

Pair-Share






Allowing Struggle

Allowing Struggle

Allowing Struggle


Showing Thinking

Encourage Reasoning



Showing Thinking

Showing Thinking

Encourage Reasoning

Encourage Reasoning

Encourage Reasoning

Fractions

FINDING THE MISSING PIECES

Middle Grades FractionsJeanne Simpson

NCTM 2014

Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It has also been linked to difficulties in adulthood, such as failure to understand medication regimens.

National Mathematics Panel Report, 2008

WHY ARE FRACTIONS SO DIFFICULT?

There are many meanings of fractions (part-whole, measurement, division, operator, ratio).

Fractions are written in an unusual way.Instruction does not focus on a conceptual understanding of fractions.

Students overgeneralize their whole-number knowledge. (McNamara & Shaughnessy, 2010)

Van de Walle, Karp, & Bay-Williams, 2013

KEY IDEAS NEEDED FOR CONCEPTUAL UNDERSTANDING

The meaning of fractions PartitioningUnit fractionsModels

Number linesEquivalent fractionsComparing fractions

PARTITIONING

PARTITIONING IS KEY TO UNDERSTANDING AND GENERALIZING CONCEPTS RELATED TO

FRACTIONS SUCH AS:

Identifying “fair shares” Identifying fractional parts of an object Identifying fractional parts of sets of objects Comparing and ordering fractions Locating fractions on number lines Understanding the density of rational numbers Evaluating whether two fractions are equivalent or finding equivalent fractions

Operating with fractions Measuring

STAGES OF PARTITIONING

Sharing – two equal partsAlgorithmic halving – equal parts that are powers of two

Evenness – even numbers that have odd factorsOddness – partitioning into an odd number of equal parts involves thinking about the relative size of each part to the whole before partitioning

Composition – using rows and columns (multiplicative)

DO I TEACH THESE STRATEGIES? NO!

Teachers should make intentional choices about which fractions they use to teach, reinforce, and strengthen concepts that can be built on understanding the impact of partitioning.Provide students with a variety of modelsStudents should partition the models into a variety of fractional parts, starting with powers of two

Have students share their strategies so that all students are exposed to a variety of ways of thinking.

Over time, students will take on other strategies as they are ready.

UNIT FRACTIONS

CCSS 3.NF.A.1

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

(Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, 8.)

MODELS FOR FRACTIONS

Area Length Set

Whole = area of defined regionParts = equal areasFraction = part of area covered

Whole = Unit of distance or lengthParts = equal distances or lengthsFraction = location in relation to other points on the line

Whole = one setParts = equal number of objectsFraction = the count of objects in the subset

THREE TYPES OF MODELS Area models (regions,

part to whole relationships)

Se

Set models (fractional part of a set of objects)

•Number lines (distance traveled or location)

MODELS

Learning is facilitated when students interact with multiple models that differ in perceptual features causing students to continuously rethink and ultimately generalize the concept.

Rectangles are better than circles. They are easier to partition equally, and they allow multiplicative reasoning.

MODELS DIFFER IN CHALLENGES

How the whole is defined

How “equal parts” are defined

What the fraction indicates

Model Whole Equal Parts Fraction Indicates

Area Defined region

Equal area The part covered of whole unit of area

Set What is in the set

Equal number of objects

The count of objects in the subset of the defined set of objects

Number line

Unit of distance or length (continuous)

Equal distance

The location of a point in relation to the distance from zero with regard to the defined unit

TO PREVENT OVER-RELIANCE…

Let students become comfortable with model.Then give them a problem where the model is cumbersome. Vary the model so students do not over-generalize.The ultimate goal is a mental model.

RESEARCHERS SAY….

Over time, students should move from the need always to construct or use physical models to carrying the mental image of the model, while still being able to make a model as they learn new concepts or encounter a difficult problem.

Petit, Laird, Marsden (2010)

“Students who are asked to practice the algorithm over and over…stop thinking. They sacrifice the relationships in order to treat the numbers simply as digits.”

Imm, Fosnot, Uittenbogaard (2012)

112

Jeanne Simpson

UAHuntsville AMSTI

[email protected]

[email protected]

Contact Information

113

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content deepening 6 th grade math january 24, 2014 jeanne simpson amsti math specialist

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th grade math