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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
Parent Session ISK 2013
Forces changing skill demands
" Automation
" Globalization
" Workplace change
" Demographic change
" Personal risk and responsibility
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What Are Our Challenges?
" Of the 20 fastest growing occupations, 15 need extensive math and science preparation
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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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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
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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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Almost everyone wants schools to be better,
" but almost no one wants them to be different.
Parent Session ISK 2013
A thought
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“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
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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 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
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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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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
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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 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
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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
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
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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
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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
Algebra 1: Multiplying Binomials
*Sketch is not drawn to scale.
37 www.JennyRay.net
x + 5
x + 4 4x 20
x2 5x
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
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)
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“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
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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
Parent Session ISK 2013
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.
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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?
Parent Session ISK 2013
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?
Parent Session ISK 2013
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?
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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?
Parent Session ISK 2013
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?
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6. Attend to precision. Do students:
• Communicate precisely with others?
• Use clear definitions?
• Use the equal sign consistently & appropriately?
• Calculate accurately & efficiently?
Parent Session ISK 2013
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?
Parent Session ISK 2013
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?
Parent Session ISK 2013
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
Parent Session ISK 2013
Shift in Mathematics #2 Focus on Strong Number Sense and Problem Solving
" How Parents Can Help Students at Home
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
Parent Session ISK 2013
Shift in Mathematics #3 Focus 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 math Talk about math
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Shift in Mathematics #4 Perseverance 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
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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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