At The Crossroads—Meeting the Challenges of a Changing

World —

Parent Session SAS 2013

What Are Our Challenges?

Of the 20 fastest growing occupations, 15 need extensive math and science preparation

Parent Session SAS 2013

Parent Session SAS 2013

Mathematical thinking . . .

A gateway to higher mathematics?

OR

A wall blocking path for

students? Parent Session SAS 2013

5

The Future

75 %75 % of jobs will be in STEM

Not justjust STEM careers,

it is STEM in everyevery job

Technology Technology as a “global knowledge economy” is the futurefuture, and it requires different skills.

Business and industry wantwant employees with these skills!

21st Century Learning

“We are responsible for preparing students to address problems we cannot foresee with knowledge that has not yet been developed using technology not yet invented.” Ralph Wolf

Parent Session EARJ 2013

Jobs of the Future

The TOP 10 jobs in 2015 are not yet invented.

Parent Session EARJ 2013

Thinking and Learning Skills

• Critical Thinking & Problem Solving Skills• Creativity & Innovation Skills• Communication & Information Skills• Collaboration Skills

21st Century Skills Framework

Parent Session EARJ 2013

Problem Solving

Computational & Procedural Skills

DOING MATH

Conceptual Understanding

“WHERE” THEMATHEMATICSWORKS

“HOW” THE

MATHEMATICSWORKS

“WHY” THE

MATHEMATICSWORKS

Parent Session SAS 2013

What Does It Mean to Understand Mathematics?

Knowing Understanding

Understanding is the measure of quality and quantity of connections between new ideas and existing ideas

ASB MCI2 Problem Solving

“Understanding is the key to remembering what is learned and being able to use it flexibly.”

- Hiebert, in Lester & Charles,

Teaching Mathematics through

Problem Solving, 2004.

A Thought

“People who do not understand mathematics today are like those who could not read or write in the industrial age.”

Robert Moses

ASB MCI2 Problem Solving

The Bridge To Understanding

Representation

“SEEING” Stage

Concrete Abstract

“DOING” Stage “SYMBOLIC”Stage

Parent Session EARJ 2013

Building Mathematical Concepts

Concrete Manipulativ

es

Pictorial Representatio

n I I I I

I I I I

Abstract Symbols

4 + 4 = 8

2 x 4 = 8

*Significant time must be spentworking with concrete materials

and constructing pictorial representations

in order for abstract symbol and operational understanding to occur

PARENT 2013

Value Multiple Representations…

concrete or pictorial

tabular

verbal

symbolic

graphical Parent Session EARJ 2013

Conceptual vs. Procedural Knowledge

Conceptual (connected networks)Knowledge and understanding of logical relationships and representations with an ability to talk, write and give examples of these relationships.

Procedural (sequence of actions)Knowledge of rules and procedures used in carrying out routine mathematical tasks and the symbols used to represent mathematics.

-- David Allen The question of which kind of knowledge is most important is the wrong question to ask. Both kinds of knowledge are required for mathematical expertise...

Instead, we should focus on designing teaching environments that help students build internal representations of procedures that become part of larger conceptual networks.

James Heibert and Tom Carpenter, Learning and Teaching with Understanding, 1992

Phil Daro

GradePriorities in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding

K–2Addition and subtraction, measurement using whole number quantities

3–5Multiplication and division of whole numbers and fractions

6Ratios and proportional reasoning; early expressions and equations

7Ratios and proportional reasoning; arithmetic of rational numbers

8 Linear algebra

Priorities in Mathematics

5/29/12

Key FluenciesGrade Required Fluency

K Add/subtract within 5

1 Add/subtract within 10

2Add/subtract within 20

Add/subtract within 100 (pencil and paper)

3Multiply/divide within 100

Add/subtract within 1000

4 Add/subtract within 1,000,000

5 Multi-digit multiplication

6Multi-digit division

Multi-digit decimal operations

7 Solve px + q = r, p(x + q) = r

8Solve simple 22 systems by inspection 19

Number Sense …

Howden (1989) described it as “good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.”

.

Prior Understandings

2.G.2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

www.JennyRay.net 21

Distributive Property & Area Models

22www.JennyRay.net

3

5 + 2

15 6+

3 x 7 =3 x (5 + 2) = (3 x 5) + (3 x 2)= 15 + 6 = 21

3 x 7 =__

Area Models25 x 38=

23www.JennyRay.net

20 +

530 + 8

150 40

600 160

750 + 200 = 950

950

Partitioning Strategies for Multiplication

27 x 4 27 x 4

4 x 20 = 80 27 + 27 + 27 + 27

4 x 7 = 28 108

54 54

108

-----------------------------------

267 x 7

7 x 200 = 1400 1820

7 x 60 = 420 1876

7 x 8 = 56www.JennyRay.net 24

14 x 25: An Area Model

*Sketch is not drawn to scale.

25www.JennyRay.net

20 + 5

80 20

200 50

Algebra 1: Multiplying Binomials

*Sketch is not drawn to scale.

26www.JennyRay.net

x + 5

4x 20

x2 5x

Partial Products

38 78

x 19 x 54

30 x 10 300 70 x 50 3500

30 x 9 270 70 x 4 280

8 x 10 80 8 x 50 400

8 x 9 + 72 8 x 4 + 32

722 4212

Partial Products (Area Model) 62

x 18

60 2 600

480

10 600 20 20

16

8 480 16 1116

1500+350+120+28 =1998

50 4

30

7

1500

350

120

28

54 x 37

Decimals

3 x 0.24

0.3 x 0.60.3 x 0.6

0.12 + 0.60 = 0.720.12 + 0.60 = 0.72

Draw a picture that shows

4

3

3

2

Array

2 of 3 rows

3 of 4 in each row

2

1

12

6

43

32

ofrows

ofrows

areatotal

areashaded

Mixed Numbers, too!

8 x 3 ¾

8 x 3 = 248 x 3 = 24

64

24

4

38 x

24 + 6 = 3024 + 6 = 30

Where’s the Math?

Models help students explore concepts and build understanding

Models provide a context for students to solve problems and explain reasoning

Models provide opportunities for students to generalize conceptual understanding

–œ ‡°± ·þ

þþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþðþþþýþþþýþþþýþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþ þþþþþþþþþþþþ þþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþ

Multiplication (rational numbers)

What does 4 x 3 mean?

4 x 2/3 = 4 groups of 2/3 each

 

 =

4X2/3 =2/3+2/3 +2/3+2/3+2/3 = 8/3

Why is it not 8/12?

Multiplication – number line model Aoife earns €12 per hour. What would she earn in 2, 3, 4, 3/4 hours?

Notice “of “ becoming multiplication

3/4 x12 = 12 x ¾ =9

Multiplication – Area Model

Cara had 2/5 of her birthday cake left from her party. She ate ¾ of the leftover cake. How much of the original cake did she eat?

2/5 cakeDivide into quarters

¾ of 2/5

Area of 3x2 out of area of 4x5

3 2 6 3

4 5 20 10

http://www.learner.org/courses/learningmath/number/session9/part_a/try.html

Multiplication making smaller!

A muffin recipe requires 2/3 of a cup of milk. Each recipe makes 12 muffins. How many muffins can be made using 6 cups of milk?

Adapted from Multiplicative Thinking. Workshop 1. Properties of Multiplication and Division. http.nzmaths.com, 2010.

The Additive Thinker

A muffin recipe requires 2/3 of a cup of milk. Each recipe makes 12 muffins. How many muffins can be made using 6 cups of milk?

1

2

2

34

Each rectangle represents a third of a cup of milk.

A muffin recipe requires 2/3 of a cup of milk. Each recipe makes 12 muffins. How many muffins can be made using 6 cups of milk?

Content + Practices

“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education.”

(CCSS, 2010)

Parent Session EARJ 2013

“Learning happens within students, not to them. Learning is a process of making meaning that happens one student at a time.”

Carol Ann Tomlinson and Jay McTighe

Integrating Differentiated Instruction and

Understanding by Design © 2006

Standards for Mathematical Practice

Make sense of problems and persevere in solving them

11

Reason abstractly and quantitatively

22 Construct viable arguments and critique the reasoning of others

33

Model with mathematics

44

Use appropriate tools strategically.

55

Look for and make use of structure

66

Attend to precision

77Look forand expressregularity in repeated reasoning

88

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

6. Attend to precision.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Reasoning and explaining

Modeling and using tools.

Seeing structure and generalizing.

Grouping the Standards of Mathematical Practice

William McCallum University of Arizona- April 1, 2011

Overarching habits of mind of a productive mathematical thinker.

1. Make sense of problems and persevere in solving

Do students:

• EXPLAIN?

• Make CONJECTURES?

• PLAN a solution pathway?

• MULTIPLE representations?

• Use DIFFERENT METHODS to check?

• Check that it all makes sense?

• Understand other approaches?

• See connections among different approaches?

• ANALYZE?

2. Reason abstractly and quantitatively

Do students:

• Make sense of quantities & their relationships?

• Decontextualize?

• Contextualize?

• Create a coherent representation?

• Consider units involved?

• Deal with the meaning of the quantities?

3. Construct viable arguments and critique the reasoning of others.

Do students:Understand & use stated assumptions, definitions, and previous results?

Analyze situations, recognize & use counterexamples?

Justify conclusions, communicate to others & respond to arguments?

Compare the effectiveness of 2 plausible arguments?

Distinguish correct logical reasoning from flawed & articulate the flaw?

Look at an argument, decide if it makes sense,& ask useful questions to clarify or improve it?

Make conjectures& build a logical progression?

Use mathematical induction as technique for proof?

Write geometric proofs, including proofs of contradiction?

4. Model with mathematics

Do students:

• Analyze relationships mathematically to draw conclusions?

• Apply the mathematics they know everyday?

• Initially use what they know to simplify the problem?

• Identify important qualities in a practical situation?

• Interpret results In the context of the situation?

• Reflect on whether the results make sense?

5. Use appropriate tools strategically.Do students:

• Consider available tools?

• Know the tools appropriate for their grade or course?

• Make sound decisions about when tools are helpful?

• Identify & use relevant external math Sources?

• Use technology tools to explore & deepen understanding of concepts?

6. Attend to precision.

Do students:

• Communicate precisely with others?

• Use clear definitions?

• Use the equal sign consistently & appropriately?

• Calculate accurately & efficiently?

7. Look for and make use of structure.

Do students:

• Look closely to determine a pattern or structure?

• Use properties?

• Decompose & recombine numbers & expressions?

• Have the facility to shift perspectives?

8. Look for and express regularity in repeated reasoning.

Do students:

• Notice if calculations are repeated?

• Look for general methods & shortcuts?

• Maintain process while attending to details?

• Evaluate the reasonableness of intermediate results?

a% of b = b% of a

Shift in Mathematics #1Deeper Learning Fewer Concepts

•How Parents Can Help Students at Home

Students must … Parents can …Spend more time on fewer concepts

Know what the priority work is for the grade level

Represent math in multiple ways Ask, “Can you show me that in another way?”

Apply strategies, not just get answers

Focus on how the child is tackling the problem over what the answer is

Shift in Mathematics #2Focus on Strong Number Sense and Problem Solving

•How Parents Can Help Students at HomeStudents must … Parents can …Be able to apply strategies and use core math facts quickly

Ask the child’s teacher what core math facts should be practiced at home Ask students which strategies they are using

Compose and decompose numbers Help children break apart and put together numbers to make problem solving easier

Shift in Mathematics #3Focus on Communication of Thinking and Language Rich Classrooms

•How Parents Can Help Students at Home

Students must … Parents can …Understand why the math works—explain and justify

Ask questions to find out whether the child really knows why the answer is correct

Talk about why the math works—explain and justify

Ask children to explain how they solved the problem and why they chose the strategies they used

Prove that they know why and how the math works—explain and justify

Ask children to show how they know they have the correct solution Talk about alternative strategies

Use academic vocabulary to explain their reasoning and critique that of others

Expect children to use the language of mathTalk about math

Shift in Mathematics #4Perseverance and Grappling with Mathematics

•How Parents Can Help Students at Home

Students must … Parents can …See mistakes as learning opportunities Help their children use their mistakes

as windows into their thinking

Understand that there is usually more than one way to solve a problem

Celebrate and value alternative responses Ask, “Is there another way to solve this?”

Spend more time solving a single problem in a deep way

Expect fewer problems but more writing and explaining in homework

Making Sense of Mathematics?

• ?:??

PARENT 2013

• Which is more rigorous ?

• 1895? 1931? 2012?

Eighth Grade Test questions---1895 Arithmetic [Time, 1.25 hours]

• 1. Name and define the Fundamental Rules of Arithmetic.

•2. A wagon box is 2 ft. deep, 10 feet long, and 3 ft. wide. How many bushels of wheat will it hold?

•3. If a load of wheat weighs 3942 lbs., what is it worth at 50cts/bushel, deducting 1050 lbs. for tare?

•4. District No. 33 has a valuation of $35,000. What is the necessary levy to carry on a school seven months at $50 per month, and have $104 for incidentals?

•5. Find the cost of 6720 lbs. coal at $6.00 per ton.

Parent Session EARJ 2013

Eighth Grade Test

• 6. Find the interest of $512.60 for 8 months and 18 days at 7 percent.

•7. What is the cost of 40 boards 12 inches wide and 16 ft. long at $20 per metre?

•8. Find bank discount on $300 for 90 days (no grace) at 10 percent.

•9. What is the cost of a square farm at $15 per acre, the distance of which is 640 rods?

•10. Write a Bank Check, a Promissory Note, and a Receipt

Parent Session EARJ 2013

Parent Session EARJ 2013

Parent Session EARJ 2013

Parent Session EARJ 2013

Instruction Matters!

PARENT 2013

A thought…

• We can best close the achievement gap by eliminating the opportunity gap. If we,as mathematics teachers K-12, each make it our personal goal for every student to have the opportunity to learn mathematics in ways that promote the habits of mind espoused

• in the standards for mathematical practice,we will be successful in helping all students tobe successful in learning and doing mathematics.

Parent Session EARJ 2013

Together we make a difference!

PARENT 2013