ism sep 2013
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Why Has Mathematics
Instruction Changed?
Parent Session ISM 2013
Why isn’t math taught the way I learned it?
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I TAUGHT STRIPE HOW TO WHISTLE
I DON’T HEAR HIM WHISTLING
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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
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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
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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:
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Jobs of the Future
The TOP 10 jobs in 2015 are not yet invented.
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Mathematical thinking . . .
A gateway to higher mathematics?
OR
A wall blocking path for
students?
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Thinking and Learning Skills
• Critical Thinking & Problem Solving Skills• Creativity & Innovation Skills• Communication & Information Skills• Collaboration Skills
21st Century Skills Framework
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A thought
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“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.
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20th Century teaching cannot meet 21st Century demands.
Almost everyone want schools to be better …………….but almost no one wants them to be different.
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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)
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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
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“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
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The Bridge To Understanding
Representation
“SEEING” Stage
Concrete Abstract
“DOING” Stage “SYMBOLIC”Stage
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Building Mathematical Concepts
Concrete Manipulativ
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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
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Value Multiple Representations…
concrete or pictorial
tabular
verbal
symbolic
graphical Parent Session ISM 2013
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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)
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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
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Students Can Do Basics, ...
Source: NAEP 2009
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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
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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)?
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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?
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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?
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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?
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RepresentationTables
EquationsGraphs
WordsPicturesor
Models
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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?
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Use appropriate tools strategically
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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
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Attend to precision
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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
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Look for and make use of structure
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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?
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Look for and express regularity in repeated reasoning
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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?
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MATHEMATICALLY PROFICIENT STUDENTS…
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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?
?:??
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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.
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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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Parent Session ISM 2013
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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!
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