Math with Meaning

A Language Based

Approach to Math

Goals for the day

Revisit Number Sense Foundations

Revisit Model Drawing Foundations

Connect Language/Speaking/Listening to CCS

Focus on Vocabulary Building Blocks

Build Language Around Manipulatives

Context for Vocabulary – Geometry

Use Mathematical Discourse

Activity for Purposeful Practice

I am…you are

Two questions

Who/What am I?

Who/What are you?

Make a circle

Common Core Framework

Conceptual Understanding: Comprehension of mathematical concepts, operations, and relations Procedural Fluency: Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately Strategic Competence: Ability to formulate, represent, and solve mathematical problems Adaptive Reasoning: Capacity for logical thought, reflection, explanation, and justification Productive Disposition: To see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

Costas’ Method

Intellectual Functioning

Level 1 - focus on Gathering and recalling

– Level 2- focus on Processing and making sense of gathered information

– Level 3 - focus on Applying and evaluating information

The Eight Mathematical Practices

1. Make sense of problems & persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments & critique others reasoning.

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.

The Octa-Habits

1. Get it and stick with it!

2. Think about it and number it!

3. Defend and critique the thinking.

4. Play with manipulatives and models.

5. Use the tools.

6. Sweat the details.

7. Bag it and Tag it!

8. Find and use the patterns.

Learning Begins with Questioning

What is the difference between

good questions and great questions?

Language Frames

Listen and Record

Questions by Practice 1. Get it…Stick with it!

How would you describe this problem in your own words?

How would you describe what you are trying to find?

What do you notice about…?

Describe the relationship between the quantities

Describe what you have already tried?

Talk me through the steps you have used at this point.

What steps are you most confident of?

Questions by Practice 2. Think about it…Number it!

What do the numbers in the problem represent?

What is the relationship of the quantities?

How is ____ related to ____ ?

What does ____ (symbol, diagram, etc) mean to you?

What properties might we use to find the solution?

How did you decide in this task that you needed to use___?

Could we use another operation/property to solve this?

Why or why not?

Questions by Practice 3. Defend…Critique the thinking!

What mathematical evidence would support your solution?

How can we be sure that …?

How could you prove that …?

Will it still work if …?

How did you test whether your approach worked?

How did you decide on the unknown?

Did you try a method that did not work? Why didn’t it?

Could you demonstrate a counter-example?

Questions by Practice 4. Play with Models & Manipulatives

What number model could you construct to represent this?

What are some ways to represent the quantities?

What’s an equation/expression that matches the diagram?

Can you use a number line, chart, table, etc?

What are some ways to visually represent …?

Is there a real world situation that is similar to this?

What formula connects with this situation?

Questions by Practice 5. Use the Tools

What mathematical tools could we use to visualize this?

What information do you have?

What do you know that is not stated in the text?

Are there different terms, measures, symbols needed?

Would it be helpful to use a protractor, ruler, compass?

Can manipulatives, diagrams, number-lines or graphs help?

What would a reasonable estimate be for a solution? Why?

What can a calculator do or not do here?

Questions by Practice 6. Sweat the Details

What are mathematical terms that apply here

How do you know that your solution was reasonable?

How could you demonstrate that your solution solves this?

Is there a more efficient strategy?

How are you showing the meaning of the quantities?

What symbols/notations are important to use here?

What language/properties can you use to explain …?

Questions by Practice 7. Bag it and Tag it!

What observations do you make about …?

What parts of the problem can you eliminate?

Can you simplify the problem?

What patterns do you find in …?

What have we learned before that we can use here?

Is this similar to other problems?

How does this relate to …?

How does this relate to other mathematical concepts?

Questions by Practice 8. Look for patterns!

Will the same strategy work in other situations?

Is this always true, sometimes true, or never true?

How would we prove that …?

What do you notice here that you have noticed before?

What would happen if …?

Is there a mathematical rule for …?

What predictions/generalizations can this pattern support?

Is this consistent with other properties of mathematics?

Teaching with the Brain in Mind

85% of students

ARE NOT

Auditory Learners

Then why do we do it?

Teaching with the Brain in Mind

MODALITY BITS PER SECOND

AUDITORY 300,000 bps

KINESTHETIC 10,000,000 bps

VISUAL 100,000,000 bps

Sousa’s Brain Based Research on Learning

Foundations of Number Sense

Concrete-Pictorial-Abstract approach

(Jerome Bruner)

Relational understanding

(Richard Skemp)

Multiple Models & Varied Practice/Experiences

(Zoltan Dienes)

Foundations of Number Sense C-P-A

Instructional strategies build understanding

through activities that move students in a

sequence from the Concrete (manipulatives) to

the Pictorial (visual models) and finally to the

Abstract (symbolic) level. - Bruner-

Counting Change

A student wanted to buy a $1 soda at a vending machine, but did not have the correct change. What is the greatest amount of change he could have and still not be able to make the exact change for the purchase?

Counting Change

Reasoning and Proof

Even wrong answers have reasons behind them

Communication

Being precise means that we understand that

coins and change are two different things. The

problem is not asking about coins, but change.

Foundations of Number Sense Perceptual Variability

Conceptual learning is maximized when

children are exposed to a concept through a

variety of physical contexts. Using a variety of

materials and experiences leads to an

understanding of mathematical abstraction.

Short, varied practice on a regular basis.

Perceptual Variability

Base 10 blocks

Place value disks

Proportional everyday materials

Perceptual Variability

What would a

progressing

sequence of

difficulty look like

with number

bracelets?

Foundations of Number Sense Relational vs Instrumental

Know the difference between Relational

Understanding (knowing what and why) and

Instrumental Understanding (procedural what

and how).

The Veddic Algorithm vs. Model

600

30

40

2

How vs. Why

Can you explain “How” it works?

Can you explain “Why” it works?

Instrumental

Relational

Model Drawing

1. Read and Reflect on the Problem

2. Rewrite the question in answer form

3. Determine who/what the problem is about

4. Draw the Unit Bar

5. Adjust, modify and place Question Mark

6. Calculate

7. Answer the question

Model Drawing

Model Drawing Sequence

TeHan has 3 balls and Holly has 2 balls. How many balls do they have altogether?

3 balls 2 balls

?

3 balls 2 balls

?

3 balls 2 balls

?

3+2=

3 balls 2 balls

? 3 2

Part Whole Practice Problem

If of a number is 12, what is the number?

23

Additive Comparison

Sean collected some clams at Nana June’s

Beach. Elliot collected 15 more clams than

Sean. If they collect 105 clams altogether, how

many clams did Sean collect?

Additive Comparison

Rain is 8 years older than Jade and 2

years younger than Ramel. The total of

their ages is 63 years. How old is Ramel?

From Model to Equation

1.

3

87

8

40

2.

Multiplicative Comparison

Xavier had 4 times as many stamps in his collection as Shannon did. If they had 345 stamps altogether, how many more stamps did Xavier have than Shannon?

Modeling Comparisons

A has 4 units. B is 2 more than A.

A

B

Modeling Comparisons

A

B

A is 3 times B.

Modeling Comparisons

B is 6 units and A is of B

A

B

1

2

Modeling Comparisons

A is as big as B and C is twice as big as B.

A B C

2

3

Fractions

Mary found of the flowers in her garden

are red, of the remainder are yellow and the

rest are pink. There are 18 pink flowers. How

many flowers are there altogether?

2 5

1 4

Vocabulary Panels

Three components

WORD

DEFINITION

GRAPHIC OR EXAMPLE

Vocabulary Panels

MEDIAN

The number that is exactly in the middle of an ordered set of values

2 3 3 3 4 7 7

Student Work

Student Work

Student Work

Vocabulary Panels

In your grade level teams

Generate Target Vocabulary

Generate Support Vocabulary

Generate a Word Panel

Word Wall = Vocabulary Panels + Word Files

Cuisenaire Rods

Comparisons

Matching

Patterns

Part Wholes

Place Value

Commutative/Associative

Fractions & Applying Properties

Factors

Cuisenaire Rods & Language Frames

Comparison

Which rod is smaller?

Which rod is larger?

Are the rods the same?

Cuisenaire Rods & Language Frames

Matching

How many smaller rods are needed to match

the length of the larger rod?

Cuisenaire Rods & Language Frames

Sequencing

What comes next?

Cuisenaire Rods & Language Frames

Part Wholes

If = 1 then ____ + ____ = ____ Explain.

If = 1 then ____ + ____ = ____ Explain.

Cuisenaire Rods & Language Frames

Part Wholes

If = 1 then how many ways can I name

light green? Say it, and write it.

If =1 then how many ways can I name

purple? Say it, and write it.

Cuisenaire Rods & Language Frames

Applying Operations

+ = or

If = 1 then 2 + 3 = 5 and …

What if =1 then what

changes? Why? How? Say it and write it.

Geometry in Two Dimensions

Geometry–Two Dimensional Scope

All Polygons

Group of Quadrilaterals

Group of Triangles

Group of Regular Polygons

Non-Polygons

Geometric Language

What is the language of polygons?

What is the language of these shapes?

What is different?

If this was a rubber stamp how many

different shapes could I stamp? Explain.

How many of each does this solid have?

What are the properties of each shape?

How do these individual stamp faces come

together to form this shape?

Geometry From Two to Three

Geometry – Attending to Precision

What:

…Is the vocabulary of shapes used?

…Is the vocabulary of 2 dimensional shapes?

…Is the vocabulary of 3 dimensional shapes?

Geometry - 300,000 bps

Without looking in the bag;

Reach in and feel the shape

Provide one unique description of the shape

Pass the bag to the next person

They will add their own unique description

Geometry - 300,000 bps

Without looking in the bag;

Reach in and feel the solid

Provide one unique description of the solid

Pass the bag to the next person

They will add their own unique description

Geometry – Stretching Language

Criteria for reference

Shape of faces

Number of faces

Variety of faces

Variety of nets

Geometry – Find a Solid

Find a solid;

Identify any 2 Dimensional components

Which of these are edged?

Which of these are curved?

Geometry – Disequilibrium

How do we use 2 Dimensional learning

to build 3 Dimensional concepts?

What elements do we need to add to

our dimensional framework?

Geometry – Disequilibrium

Whether in Geometry or other content

areas, build on what the students know,

and have them explore new content on

their terms.

How would you apply this to what you are

teaching now to generate new discussions?

Mathematical Discourse

AKA

•Number Talks

•Math Talks

Research – 8 + 17

Above Average

Students

30% used known

facts

61% used number

sense

9% used a counting

on strategy

Below Average Students

6% used known facts

0% used number

sense

72% used a counting

on strategy

22% used a counting

all strategy

Properties

CCSS K-8 math standards mention properties 66 times.

Properties are ways that numbers and operations

interact with one another.

Traditionally, these have been taught to students by

learning the definitions – this is changing!

Students need to be USING the properties in order to

appreciate them.

Using properties in math allows for flexible thinking,

which is linked to proficiency.

Number Talk General Goals

Increased discourse among students

Greater conceptual learning

Greater problem solving skills

Number Talk Progressive Goals

Goal: Improve computational fluency by

encouraging students to:

Use any strategy that makes sense to them

Then to be able to use more efficient strategies

that they have learned from their peers

And to finally progress to “just knowing it” or

using most efficient strategies for computation

Achieving Mathematical Discourse

1. Increase opportunities for students to question, explore, and challenge each other’s thinking

2. Discuss relationships between numbers and operations

3. Discuss many different strategies for solving different problems

Number Talks

High Trust Community

Inclusive/Value Based Class Discussion

Teacher-Neutral/Respectful/Encouraging Role

Role of Mental Math

Purposeful Problems

05 to 15 minutes

How many unit squares?

Number Talks - 12 x 15