Why Has Mathematics Instruction Changed? · Why Has Mathematics Instruction Changed? Why isnʼt...

Why Has Mathematics Instruction Changed?

Why isn’t math taught the way I learned it?

Parent Session ISK 2013

I TAUGHT STRIPE HOW TO WHISTLE

I DON’T HEAR HIM WHISTLING


I SAID I TAUGHT HIM. I DIDN’T SAY HE LEARNED IT

At The Crossroads—

Meeting the Challenges of a Changing World —


The world we know is changing

�  75 % of jobs will be in STEM

�  Not just STEM careers,

it is STEM in every job

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

�  Business and industry want employees with these skills! OECD


Forces changing skill demands

"   Automation

"   Globalization

"   Workplace change

"   Demographic change

"   Personal risk and responsibility


What Are Our Challenges?

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


Nearly two-thirds of new jobs will require postsecondary education

Source: Bureau of Labor Statistics. (2008, February). Occupational projections and training data: 2008-9 edition. Washington, DC: U.S. Department of Labor. (p. 4, Table I-3)

New jobs, 2006-2016:


Jobs of the Future

The TOP 10 jobs in 2015 are not yet invented.


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


Thinking and Learning Skills

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

21st Century Skills Framework


Almost everyone wants schools to be better,

"  but almost no one wants them to be different.


A thought


“If we teach today as we taught yesterday, we rob our children of tomorrow.”

John Dewey

Mathematical thinking . . .

A gateway to higher mathematics?

OR

A wall blocking path for

students? Parent Session ISK 2013

Problem ��Solving

Computational ��& Procedural ��Skills

DOING MATH

Conceptual ��Understanding

“WHERE” THE MATHEMATICS WORKS

“HOW” THE

MATHEMATICS WORKS

“WHY” THE

MATHEMATICS WORKS


Students Can Do Basics, ...

Source: NAEP 2009


347 + 453 90% 73% 864 – 38

… But Students Cannot Solve Problems

Ms. Yost’s class has read 174 books, and Mr. Smith’s class has read 90 books. How many more books do they need to read to reach the goal of reading 575 books? 33%

Critical Thinking & Problem Solving: Important

"   Nearly 60% of employers rate critical thinking and problem solving as “very important” for entering the workforce … yet 70% of employers rate them “deficient” in those skills.

"   While 73% of school superintendents think h.s. grads meet expectations for “problem solving,” only 45% percent of colleges and employers think so.

"   78% of employers expect critical thinking/problem solving to become even more important in the near future.

Sources: 1) Conference Board. (2006, October). Are they really ready to work? New York: Author. (p. 21, Table 3 and p. 32, Table 6) 2) Conference Board. (2008, March). Ready to innovate: Are educators and executives aligned on the creative readiness of the U.S. workforce? New York: Author. Parent Session ISK 2013

What is Problem Solving? "   “Problem solving means engaging in a task for which

the solution method is not known in advance.” "   Principles and Standards for School Mathematics

"   It encompasses exploring, reasoning, strategizing, estimating, conjecturing, testing, explaining, and proving.

"   "We only think when confronted with a problem." -- John Dewey


Learning Mathematics

"   For all students to become mathematically proficient, major changes must be made in instruction, assessments, teacher education, and the broader educational system.

"   Adding It Up (NRC)


How Students Learn

"   “Can engage in instructional activities but teaching has not occurred until student learning has occurred“

"   “…covering the material and explaining it well is NOT the same as the student learning it.”

NRC


International Benchmarking "   identify and describe exemplary practices in the APEC region with

respect to different key features of a mathematics education delivery system, including standards, assessments, teachers, and low-performing students and schools.

" http://hrd.apec.org/index.php/Mathematics_Standards


The Bridge To Understanding

Representation

“SEEING” Stage

Concrete Abstract

“DOING” Stage “SYMBOLIC” Stage


Building Mathematical Concepts

Concrete Manipulatives

Pictorial Representation

I I I I

I I I I

Abstract Symbols

4 + 4 = 8

2 x 4 = 8

*Significant time must be spent working with concrete materials

and constructing pictorial representations in order for abstract symbol and

operational understanding to occur Parent Session ISK 2013

Value Multiple Representations… concrete or pictorial

tabular

verbal

symbolic

graphical


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

Grade Priori)es in Support of Rich Instruc)on and Expecta)ons of Fluency and Conceptual Understanding

K–2 Addi)on and subtrac)on, measurement using whole number quan))es

3–5 Mul)plica)on and division of whole numbers and frac)ons

6 Ra)os and propor)onal reasoning; early expressions and equa)ons

7 Ra)os and propor)onal reasoning; arithme)c of ra)onal numbers

8 Linear algebra

Priorities in Mathematics

5/29/12

Key Fluencies Grade Required Fluency

K Add/subtract within 5 1 Add/subtract within 10

2 Add/subtract within 20 Add/subtract within 100 (pencil and paper)

3 Mul<ply/divide within 100 Add/subtract within 1000

4 Add/subtract within 1,000,000 5 Mul<-‐digit mul<plica<on 6

Mul<-‐digit division Mul<-‐digit decimal opera<ons

7 Solve px + q = r, p(x + q) = r 8

Solve simple 22 systems by inspec<on 29

Reading Fluency

FLUENCY Accuracy

Prosody

Efficiency

Fluency is the ability to read with sufficient ease and accuracy that one can focus attention on the meaning and message of text.”

Adams, 2002


MATH FLUENCY (Russell, 1999)

Efficiency: Student does not get

bogged down into too many steps or lose track of logic or strategy.

(WORKING MEMORY)

Flexibility: Knowledge of more than

one approach to problem solve. Allows student to choose appropriate strategy and to double check work.

(EXECUTIVE FUNCTIONING)

Accuracy: A working knowledge

of number facts, combinations, and other important number

relationships. (AUTOMATIC RETRIEVAL)

FLUENCY


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 33

Distributive Property & Area Models

34 www.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 =__

14 x 25: An Area Model

*Sketch is not drawn to scale.

35 www.JennyRay.net

20 + 5

10 + 4 80 20

200 50

Partial Products (Area Model) 62

x 18

60 2 600

480

10 600 20 20

16

8 480 16 1116

Algebra 1: Multiplying Binomials

*Sketch is not drawn to scale.

37 www.JennyRay.net

x + 5

x + 4 4x 20

x2 5x

Decimals

3 x 0.24 3(.20 + .04)= .60+.12 = .72

Mixed Numbers, too!

8 x 3 ¾

8 x 3 = 24

6424

438 ==x

24 + 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

Draw a picture that shows

43

32 �

Array

2 of 3 rows

3 of 4 in each row

21

126

4332 ===

ofrowsofrows

areatotalareashaded

Solve these problems. Use a model to record and then discuss your thinking with your group. Write an equation for each problem

.

"  You have 3/4 of a pizza left. If you give 1/3 of the left-over pizza to your brother, how much of a whole pizza will your brother get?

"   Frankie had 2/3 of the lawn left to cut. After lunch, she cut 3/ 4 of the lawn she had left. How much of the whole lawn did Frankie cut after lunch?

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)


“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

1

Reason abstractly and quantitatively

2 Construct viable arguments and critique the reasoning of others

3

Model with mathematics

4

Use appropriate tools strategically.

5

Look for and make use of structure

6

Attend to precision

7 Look for and express regularity in repeated reasoning

8


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?


Shift in Mathematics #1 Deeper 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 #2 Focus on Strong Number Sense and Problem Solving


Students 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 #3 Focus on Communication of Thinking and Language Rich Classrooms


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 math Talk about math


Shift in Mathematics #4 Perseverance and Grappling with Mathematics


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?

"   ?:??


"   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.�


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


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