ISM Sep 2013

Why Has Mathematics

Instruction Changed?

Parent Session ISM 2013

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


I TAUGHT STRIPE HOW TO WHISTLE

I DON’T HEAR HIM WHISTLING


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

Forces changing skill demands

Automation

Globalization

Workplace change

Demographic change

Personal risk and responsibility


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


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.


Mathematical thinking . . .

A gateway to higher mathematics?

OR

A wall blocking path for

students?


Thinking and Learning Skills

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

21st Century Skills Framework


A thought


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

John Dewey

21st Century learning requires:

An understanding of the meaning and relevance of ideas to concrete problems

An ability to apply core concepts and modes of inquiry to complex real-world tasks

A capacity to transfer knowledge and skills to new situations, to build on and use them

Abilities to communicate ideas and to collaborate in problem solving

An ongoing ability to learn to learn.


20th Century teaching cannot meet 21st Century demands.

Almost everyone want schools to be better …………….but almost no one wants them to be different.



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


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

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, &

Human, 1997

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What is Mathematics?Mathematics is:

a study of patterns and relationships.

a way of thinking.

an art, characterized by order and internal consistency.

a language that uses carefully defined terms and symbols.

a tool. Parent Session ISM 2013

Problem Solving

Computational & Procedural Skills

DOING MATH

Conceptual Understanding

“WHERE” THEMATHEMATICSWORKS

“HOW” THE

MATHEMATICSWORKS

“WHY” THE

MATHEMATICSWORKS

A Balanced Mathematics Program


The Bridge To Understanding

Representation

“SEEING” Stage

Concrete Abstract

“DOING” Stage “SYMBOLIC”Stage


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


Value Multiple Representations…

concrete or pictorial

tabular

verbal

symbolic

graphical Parent Session ISM 2013


Understanding Mathematics

CONCEPTUAL UNDERSTANDING: What a student needs to KNOW

PROCEDURAL UNDERSTANDING:What a student needs to be able to DO

REPRESENTATIONAL UNDERSTANDING:How a student SHOWS what he/she knows or can do.

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Remember

Understanding + Representations = Time; Depth

Conceptual understanding is NOT an option,

It is an expectation!

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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)


Standards for Mathematical Practice

Apply

• [1] Make sense of problems and persevere in solving them.

• [4] Model with mathematics.

• [5] Use appropriate tools strategically.

Understand

• [2] Reason abstractly and quantitatively.

• [7] Look for and make use of structure.

• [8] Look for and express regularity in repeated reasoning.

Evaluate

• [3] Construct viable arguments and critique the reasoning of others.

• [6] Attend to precision.

Standards



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 ISM 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


Make sense of problems and

persevere in solving them.

What Are You Thinking?

Does this make sense?

What does this term mean?

What do you want to know? Put it in a sentence.

Can you break the problem into simpler problems (multi-step)?



Reason abstractly and quantitatively

Quantities:

What quantities are in the problem?

What are the relationships among the quantities in the situation?

How can we label the quantities?

What inferences can we draw?



Construct viable arguments and critique the reasoning of others

Is it true?

Is it always true?

Is it never true?

When is it true? Under what conditions?

Is there a counter example?

A contradiction?

Is there another way to prove that this statement is true or not true?



Model with mathematics

Can you represent the idea in words, tables, diagrams, formulas, or graphs and explain the relationships between them?

Can you create your own visual representation of this situation?

Can you solve the problem in more that one way?


RepresentationTables

EquationsGraphs

WordsPicturesor

Models



Use appropriate tools strategically

Why did you choose to use this model (manipulative) to help you understand the task?

How does your model compare to someone else’s?

Is there an additional representation for this concept?

Was this tool the most efficient?


Use appropriate tools strategically


Proficient studentsSolve math problems arising in everyday life

Apply assumptions and approximations to simplify complicated tasks

Use tools such as diagrams, two-way tables, graphs, flowcharts and formulas to simplify

Analyze relationships mathematically to draw conclusions

Interpret results to determine whether they make sense


Attend to precision


Mathematically proficient students

Communicate precisely to others

Use clear definitions in discussion with others

State the meaning of the symbols consistently and appropriately

Calculate accurately and efficiently

Accurately label axes and measures in a problem

Attend to precisionHow would you explain this to someone who didn’t understand?

How does your statement link to what others have said?

How does what you say add to what Anna just said?

Is it a justification? A special case? A generalization?

Evidence? A supporting argument?

A logical extension? A contradiction?

A counterexample?Parent Session ISM 2013


Look for and make use of structure


Mathematically proficient students• look closely to discern a pattern or structure• step back for an overview and shift

perspective• see complicated things as single objects, or

as composed of several objects

Look for and make use of structure

Can you identify the basic component in this structure?

Can you break the problem into smaller components?

Is there a pattern?

Can you simplify the situation?



Look for and express regularity in repeated reasoning


Mathematically proficient students

• notice if calculations are repeated and look both for general methods and for shortcuts

• maintain oversight of the process while attending to the details, as they work to solve a problem

• continually evaluate the reasonableness of their intermediate results

Look for and express regularity in repeated reasoning

Do you notice a pattern?

Is there anything in this pattern that is repeating?

Is it possible to make a generalization? A rule?


MATHEMATICALLY PROFICIENT STUDENTS…


Make sense of problems and persevere in solving them

Reason abstractly and quantitativelyConstruct viable arguments and critique the

reasoning of othersModel with mathematicsUse appropriate tools strategicallyAttend to precisionLook for and make use of structureLook for and express regularity in repeated

reasoning

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





Instruction Matters!

• The expectations we have for our students are so much greater now than they’ve ever been. What was good enough yesterday, is not good enough today! Many of the jobs our children will pursue in the future haven’t been created yet. We must teach our students new ways to think because life may possibly be much different for them than it was for us.


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Together we make a difference!


Education

ISM Sep 2013