Why Has Mathematics
Instruction Changed?
Why isn t math taught the way I learned it?
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
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At The Crossroads—
Meeting the Challenges of a Changing World —
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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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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
New jobs,
2006-2016:
Source: Bureau of Labor Statistics. (2008, February). Occupational projections and training data: 2008-9 edition. Washington,
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DC: U.S. Department of Labor. (p. 4, Table I-3)
Jobs of the Future
The TOP 10
jobs in 2015
are not yet
invented.
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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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21st Century Skills Framework
Thinking and Learning Skills
•  Critical Thinking & Problem Solving Skills
•  Creativity & Innovation Skills
•  Communication & Information Skills
•  Collaboration Skills
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Almost everyone wants
schools to be better,
"  but almost no one
wants them to be
different.
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A thought
“If we teach today as we taught
yesterday, we rob our children of
tomorrow.”
John Dewey
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Mathematical thinking . . .
A gateway to higher mathematics?
OR
A wall blocking path for
students?
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“WHERE”
THE
MATHEMATICS
WORKS
Computational & Procedural Skills
Problem Solving
DOING
MATH
Conceptual Understanding
“HOW”
THE
MATHEMATICS
WORKS
“WHY”
THE
MATHEMATICS
WORKS
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Students Can Do Basics, ...
347 + 453
90%
864 – 38
73%
… But Students Cannot Solve Problems
33%
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?
Source: NAEP 2009
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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.
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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
DOING Stage
Abstract
SYMBOLIC
Stage
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Building Mathematical Concepts
Concrete
Manipulatives
Pictorial
Representation
Abstract
Symbols
IIII
4+4=8
IIII
2x4=8
*Significant time must be spent
working 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
symbolic
graphical
tabular
verbal
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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
is the wrong question to ask. Both kinds of
The question of which kind of knowledge is most important
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
Priorities in Mathematics
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
7
8
Ra)os and propor)onal reasoning; early expressions and equa)ons
Ra)os and propor)onal reasoning; arithme)c of ra)onal numbers
Linear algebra 5/29/12
Key Fluencies
Grade
Required Fluency
K
Add/subtract within 5
1
Add/subtract within 10
2
Add/subtract within 100 (pencil and paper)
3
Add/subtract within 20 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 2!2 systems by inspec<on
29
Reading Fluency
Efficiency
Prosody
FLUENCY
Accuracy
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)
Accuracy:
Efficiency:
A working knowledge
of number facts, combinations,
and other important number
relationships.
(AUTOMATIC RETRIEVAL)
Student does not get
bogged down into too many steps
or lose track of logic or strategy.
(WORKING MEMORY)
FLUENCY
Flexibility:
Knowledge of more than
one approach to problem solve.
Allows student to choose appropriate
strategy and to double check work.
(EXECUTIVE FUNCTIONING
) Session ISK 2013
Parent
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
3 x 7 =__ 5
3
15
+
+
2
6
3x7=
3 x (5 + 2) =
(3 x 5) + (3 x 2)= 15 + 6 = 21
www.JennyRay.net
34
14 x 25: An Area Model
10 + 4 20
80
200
+
5
20
50
www.JennyRay.net
35
Partial Products (Area Model)
62
x 18
60
2
600
480
10
600
20
20
16
8
480
16
1116
Algebra 1: Multiplying
Binomials
x + 4 x
4x
x2
+
5
20
5x
www.JennyRay.net
37
Decimals
3 x 0.24
3(.20 + .04)= .60+.12 = .72
Mixed Numbers, too!
8x3¾
8 x 3 = 24
3 24
8x =
=6
4 4
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
2
3
3
4
Array
2 of 3
rows
3 of 4 in each row
shaded area 2 rows of 3 6 1
=
= =
total area
3 rows of 4 12 2
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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1
Make sense of
problems
and
persevere in
solving them
2
Reason
abstractly and
quantitatively
6
Look for and
make use of
structure
viable
arguments and
critique the
reasoning of
others
5
4
Model with
mathematics
3 Construct
Standards for !
Mathematical !
Practice!
7
Attend to
precision
Use
appropriate tools
strategically.
8 Look for
and express
regularity in
repeated
reasoning
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Grouping the Standards of Mathematical Practice
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.
Overarching habits of mind of a productive mathematical thinker. 7. Look for and make use of structure.
8. Look for and express regularity in
repeated reasoning.
William McCallum University of Arizona- April 1, 2011
Reasoning and
explaining
Modeling and
using tools.
Seeing structure
and generalizing.
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1. Make sense of problems and persevere in solving
Do students:
• EXPLAIN?
• ANALYZE?
• 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?
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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?
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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?Parent Session ISK 2013
"  
Use mathematical induction as technique for proof?
4. Model with mathematics
Do students:
• Apply the mathematics they know everyday?
• Analyze relationships mathematically to draw conclusions?
• 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?
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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?
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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?
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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?
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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
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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
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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?
"   ?:??
"   Which is more rigorous ?
"  
1895? 1931? 2012?
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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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Together we make a difference!
Parent Session ISK 2013