Transcript Focus of the Train-the-Trainer Session – Part 3

College
Ready
Career
Ready
National Adult Education College and Career
Readiness Training Design Initiative
Mathematical Reasoning
Presenters
Bonnie Goonen - [email protected]
Susan Pittman-Shetler - [email protected]
1
Focus of the Train-the-Trainer
Session – Part 3
• Review college and career readiness standards
for mathematical content and practices
• Explore essential mathematical practices and
behaviors
• Discuss beginning strategies for the classroom
• Explore resources for leaders to use to
enhance learning with different audiences
2
“The times they are a-changing.”
Bob Dylan
Going the Next Step
“We should be educating all students
according to a common academic
expectation, one that prepares them for
both postsecondary education and the
workforce.”
(ACT, 2006)
4
Driving Questions
• How can I support my teachers’
understanding of the College and Career
Readiness Mathematical Standards
and Practices?
• In particular, where can I go to find tasks,
video, and other resources to help teachers
implement the College and Career
Readiness Mathematical Standards and
Practices in their classrooms?
5
Overview of CCR High School
Standards and Practices
• Call on students to practice
applying mathematical ways of
thinking to real world issues and
challenges
• Require students to develop a
depth of understanding and ability
to apply mathematics to novel
situations
• Emphasize mathematical
modeling
• Identify the mathematics that all
students should study in order to
be college and career ready
6
Standards-Driven Curriculum
Standards/
Practices
Student
Achievement
Classroom
Instruction
7
Key Shifts in the Standards
Shift 1: Focus
• Focusing strongly where the standards focus
Shift 2: Coherence
• Designing learning around coherent progressions
level to level
Shift 3: Rigor
• Pursuing conceptual understanding, procedural
skill and fluency, and application – all with equal
intensity
8
Focus
Shift 1 – Focus: Focusing strongly where the
standards focus
• Key ideas, understandings, and skills are identified
• Deep learning of concepts is stressed
– TIMSS taught us fewer skills with greater depth
– In U.S., learning same skills repeatedly without mastery
• Focus on fewer concepts with more depth - “Teach
less/learn more”
– Determine our focus
– Give ourselves permission to get rid of the unessential
– Commit to depth over breadth
9
Coherence
Shift 2 – Coherence: Designing learning
around coherent progressions level to level
• Coherence for mastery
• All roads lead to algebraic thinking –
abstract reasoning
10
Rigor
Shift 3 – Rigor: Pursuing conceptual
understanding, procedural skill and fluency,
and application
• Approach with more rigor
• All roads lead to algebra – abstract
reasoning skills
11
Fill and Pour
http://nlvm.usu.edu/en/nav/frames_asid_273_g_3_t_4.html?fr
om=category_g_3_t_4.html
12
Design and Organization
Mathematical Standards
 Domains are larger groups of related
standards that progress across grades.
 Clusters are groups of related standards.
Clusters appear within domains.
 Content standards define what students should
understand and be able to do – part of a
cluster.
13
Design and Organization
Domain
Cluster
Standard
14
Conceptual Themes
•
•
•
•
•
•
Number and Quantity
Algebra
Functions
Modeling
Geometry
Statistics and Probability
15
Warm Up Activity
There are eight practices in the College and Career
Readiness Standards for Mathematics (Hint: the same as
those identified in the CCSS)
• Let’s take a little quiz.
• Think about the practices.
• How many can you list?
16
Mathematical Practices
17
Real-World Goal of Practices
Modeling
Problem
Formulate
Validate
Compute
Interpret
Report
18
Topic Based
Standards Based
• Facts and activities
center around specific
topic
• Objectives drive
instruction
• Focus learning and
thinking about specific
facts
• Instructional activities
use a variety of discrete
skills
• Use of facts and
activities are focused by
enduring understandings
• Essential questions drive
instruction
• Facts are learned to
understand transferable
concepts and ideas
• Instructional activities
call on complex
performances using a
variety of skills
19
Reflection
• What is the purpose of the standards for
mathematical practice?
• How should practices be integrated with
the content?
• How can constant awareness of the
CCSS mathematical practices influence
the mathematical content we teach?
• How will teaching fewer “topics” change
lesson planning?
20
Mathematical Reasoning
New Realities
21
What we know . . .
• People have a “lovehate” relationship with
mathematics
– Twice as many
people hated it as
any other school
subject
– It was also voted the
most popular subject
Associated Press Poll
22
What we know . . .
• Algebra is widely regarded as a
“gatekeeper.”
• Higher-level mathematics and
opportunities that come with it are
closed to students who do not
succeed in high school algebra
(Silver, 2000).
• Advanced math is needed:
– To boost college grades
– For career opportunities
– To improve earnings
23
What we know . . .
• Only high-school mathematics carries
significant cross-subject benefit
• Years of mathematics instruction was a
significant predictor of performance across all
college science subjects
• Advanced math courses improve earnings –
the higher the math course taken, the higher
the percentage increase.
24
What we know . . .
• To improve conceptual
understanding, make explicit
the important mathematical
relationships and ask students
to “work and wrestle.”
• To improve skill efficiency, use
rapid pacing, modeling, and
moving to error free practice.
• Balance these two approaches.
25
What’s new in the Mathematical
Reasoning domain?
• Identify absolute value of a rational number
• Determine when a numerical expression
is undefined
• Factor polynomial expressions
• Solve linear inequalities
26
What’s new in the Mathematical
Reasoning domain?
• Identify or graph the solution to a one variable
linear inequality
• Solve real-world problems involving
inequalities
• Write linear inequalities to
represent context
• Represent or identify a
function in a table or graph
27
What’s not directly assessed on the 2014
GED® Mathematical Reasoning Test?
• Select the appropriate operations to solve problems
• Relate basic arithmetic operations to one another
• Use estimation to solve problems and assess the
reasonableness of an answer
• Identify and select appropriate units of metric and
customary measures
• Read and interpret scales, meters, and gauges
• Compare and contrast different sets of data on the
basis of measures of central tendency
• Recognize and use direct and indirect variation
28
Overview of Content
2014 GED® test
HiSET™
45% - Quantitative
Problem Solving
• Number operations
• Geometric thinking
• Statistics and data
representation (also
included on GED®
Social Studies test and
GED® Science test)
55% - Algebraic Problem
Solving
25% - Numbers and
Operations on Numbers
25% - Measurement/
Geometry
25% - Data Analysis/
Probability/Statistics
25% - Algebraic Concepts
TASC
10-20% - Number and
Quantity
20-30% - Algebra
20-30% - Functions
20-30% - Geometry
10-20% - Statistics and
Probability
Integration of
mathematical practices
into each question
29
Then - 2002 GED® test
Ms. Nguyen is a real estate agent. One of her clients is considering
buying a house in the Silver Lakes area, where 6 houses have
recently sold for the following amounts: $160,000; $150,000;
$185,000; $180,000; $145,000; $190,000. What should Ms. Nguyen
report as the Median price of these houses?
1)
2)
3)
4)
5)
$160,000
$170,000
$180,000
$190,000
Not enough information is given.
Note: Method for
determining median was
provided in the test
booklet.
30
Now – 2014 GED® test
31
Now – HiSET™
Copyright © 2013 Educational Testing Service.
32
Now – HiSET™
Copyright © 2013 Educational Testing Service.
33
Now – TASC
Copyright © 2013 CTB/McGraw-Hill Proprietary
34
Now – TASC
Copyright © 2013 CTB/McGraw-Hill Proprietary
35
New Mathematical Tools
36
TI-30XS MultiView Calculator
37
Time Out for a Video Break!
38
It’s Your Turn!
"Calculators can only
calculate - they
cannot do
mathematics."
-- John A. Van de Walle
http://education.ti.com/en/us/products/calc
ulators/scientific-calculators/ti-30xsmultiview/classroom-activities/activitiesexchange
39
Let’s Take a Journey Back
40
41
What are the big ideas that I want
students to remember . . .
40 days
(the test)
40 months
(college)
40 years
(life)
42
Problem Solving In Your Classroom
What opportunities
do your students
currently have to
grapple with nonroutine complex
tasks and to…
…. reflect on their
thinking and
consolidate new
mathematical ideas
and problem solving
solutions?
43
Solution Pathways = Problem Solving
Polya’s Four Steps to Problem Solving
Understand
the problem
Look back
(reflect)
Devise a
plan
Carry out
the plan
Polya, George. How To Solve It, 2nd ed. (1957). Princeton
University Press.
44
Value of Teaching with Problems
• Places students’ attention on
mathematical ideas
• Develops “mathematical power”
• Develops students’ beliefs that they
are capable of doing mathematics
and that it makes sense
• Provides ongoing assessment data
that can be used to make
instructional decisions
• Allows an entry point for a wide range
of students
45
Let’s SOLVE a Math Problem
Even Albert Einstein said:
“Do not worry about your difficulties in
Mathematics. I can assure you mine
are still greater.”
46
SOLVE a Problem
S
O
L
V
tudy the problem (What am I trying to find?)
rganize the facts (What do I know?)
ine up a plan (What steps will I take?)
erify your plan with action (How will I carry out
my plan?)
E xamine the results (Does my answer make
sense? If not, rework.)
Always double check!
47
S = Study the problem
What is the problem
asking me to do?
Find the question.
We are going to practice
SOLVE with this one!
Each week, Bob gets paid
$20 per hour for his first 40
hours of work, plus $30 per
hour for every hour worked
over 40 hours. Last month,
Bob made an additional $240
in overtime wages. If Bob
works 55 hours this week,
how much will he earn?
48
O = Organize the Facts
What facts are provided in order
for you to solve the problem?
• Identify each fact.
• Eliminate
unnecessary facts.
• List all necessary
facts.
Each week, Bob gets paid $20
per hour for his first 40 hours of
work, plus $30 per hour for
every hour worked over 40
hours. Last month, Bob made
an additional $240 in overtime
wages. If Bob works 55 hours
this week, how much will he
earn?
49
L = Line Up a Plan
• Select the operations to use.
• State the plan/strategy that you will use in words.
I will use a multi-step approach. First, I will multiply
the number of regular work hours by the regular
hourly rate. Next, I will multiply the number of hours
of overtime by the overtime rate. To obtain Bob’s total
weekly salary, I will add the total amount earned for
his regular salary plus his overtime salary.
50
A Few Problem-Solving Strategies
•
•
•
•
•
•
•
•
•
•
•
Look for patterns
Consider all possibilities
Make an organized list
Draw a picture
Guess and check
Write an equation
Construct a table or graph
Act it out
Use objects
Work backward
Solve a simpler (or similar) problem
51
V = Verify Your Plan
$20.00
x 40
$800.00
Regular Wages
$ 800.00
+ 450.00
$1250.00
$30.00
x 15
$450.00
52
E = Examine the Results
(Is it reasonable? Does it make sense?
Is it accurate?)
$1250.00 IS reasonable because it is
more than Bob’s average weekly salary.
Also, the answer is a whole number
because all of the facts were whole
numbers ending in zeros. Therefore, Bob
made $1250.00 in salary for the week.
53
It’s Your Turn to SOLVE!
Two painters can paint two rooms in
two hours. If 12 rooms have to be
painted in six hours, how many
painters do you need?
54
Answer: It’s Your Turn to SOLVE!
S
Two painters can paint two rooms in two hours. If 12 rooms have to
be painted in six hours, how many painters do you need?
O
The facts that I need to solve the problem are:
Two painters – two rooms – two hours. 12 rooms – six hours
L
I will need to set up a pattern to solve the problem, so I will set up a
chart to see how many painters will be needed.
V
Look at the pattern
2 painters; 1 hour = 1 room;
2 painters ; 2 hours = 2 rooms;
2 painters; 3 hours = 3 rooms;
2 painters; 4 hours = 4 rooms;
2 painters; 5 hours = 5 rooms;
2 painters; 6 hours = 6 rooms;
Double the number of painters and you have 4 painters who in the same 6 hours can now
paint 12 rooms
E
Does it answer the question? Does it make sense?
55
Time Out for a Video Break!
56
Let’s Solve!
S
O
L
V
E
57
Let’s SOLVE!
S
O
L
V
E
58
Quantitative
Problem Solving Skills
59
Geometric Reasoning
• Seeking relationships
• Checking effects of transformations
• Generalizing geometric ideas
– Conjecturing about the “always” & “every”
– Testing the conjecture
– Drawing a conclusion about the conjecture
– Making a convincing argument
• Balancing exploration with deduction
– Exploring structured by one or more explicit
limitation/restriction
– Taking stock of what is being learned through
the exploration
60
Focus on Geometric Reasoning
Van Hiele Theory
• Level 1: Visualization
• Level 2: Analyze
• Level 3: Informal Deduction
• Level 4: Formal Deduction
• Level 5: Rigor
61
Visualization
• Recognize and name shapes
by appearance
• Do not recognize properties
or if they do, do not use them
for sorting or recognition
• May not recognize shape in
different orientation (e.g.,
shape at right not recognized
as square)
62
Visualization
63
Visualization
64
Visualization
65
Implications for Instruction Visualization
• Provide activities that have students
sort shapes, identify and describe
shapes (e.g., Venn diagrams)
• Have students use manipulatives
• Build and draw shapes
• Put together and take apart shapes
• Make sure students see shapes in
different orientations
• Make sure students see different
sizes of each shape
66
Analysis
• Can identify some
properties of shapes
• Use appropriate
vocabulary
• Cannot explain
relationship between
shape and properties
(e.g., why is second shape
not a rectangle?)
67
Analysis
Description 1
The design looks like a bird with
• a hexagon body;
• a square for the head;
• triangles for the beak and tail; and
• triangles for the feet.
68
Analysis
Description 2
69
Implications for Instruction Analysis
• Work with manipulatives
• Define properties, make
measurements, and look for
patterns
• Explore what happens if a
measurement or property is
changed
• Discuss what defines a shape
• Use activities that emphasize
classes of shapes and their
properties
• Classify shapes based on lists of
properties
70
Informal Deduction
• Can see relationships of
properties within shapes
• Can recognize
interrelationships among
shapes or classes of shapes
(e.g., where does a rhombus
fit among all quadrilaterals?)
• Can follow informal proofs
(e.g., every square is a
rhombus because all sides
are congruent)
71
Deduction
B
• Usually not reached before
high school; maybe not until
college
• Can construct proofs
E
• Understand the importance of
deduction
• Understand how postulates,
axioms, and definitions are
used in proofs
C
A
D
F
A
72
What do you think?
• Is it possible to draw a quadrilateral that has exactly 2
right angles and no parallel lines?
• Try it. While you’re working, ask yourself . . .
– What happens if…?
– What did that action tell me?
– What will be the next step?
73
Right Angle
74
Let’s SOLVE!
S
O
L
V
E
75
Algebraic Reasoning Skills
76
Algebraic Thinking in Adult Education
• Create opportunities for algebraic thinking
as a part of regular instruction
• Integrate elements of algebraic thinking
into arithmetic instruction
– Acquiring symbolic language
– Recognizing patterns and making
generalizations
• Reorganize formal algebra instruction to
emphasize its applications
Adapted from National Institute for Literacy, Algebraic Thinking in Adult Education,
Washington, DC 20006
77
Let’s SOLVE One More Time!
S
O
L
V
E
78
Some Big Ideas in Algebra
•
•
•
•
•
•
•
Variable
Symbolic Notation
Equality
Ratio and Proportion
Pattern Generalization
Equations and Inequalities
Multiple Representations of Functions
79
Symbolic Notation
A Few Examples
Sign
Arithmetic
= (equal)
. . . And the
answer is
+
Addition
operation
Subtraction
operation
-
Algebra
Equivalence
between two
quantities
Positive number
Negative
number
80
Which Is Larger?
23 or 32
34 or 43
62 or 26
89 or 98
81
Patterns – Thinking Algebraically
• Finding patterns
• Describing patterns
• Explaining patterns
• Predicting with patterns
82
Teaching Patterns
Banquet Tables
Arrangement 1
Arrangement 2
Arrangement 3
• Arrangement 1 seats four people. How many people
can be seated at Arrangement 100?
83
Teaching Patterns
Here is a sequence of squares with sides measuring
1 toothpick, 2 toothpicks, 3 toothpicks, etc.
• Make the next two squares of the pattern.
• Complete the table to show the perimeter of each
of the squares.
• Challenge: Using the rule, find the perimeter of a square with a
side length of 79 toothpicks.
84
Time Out for a Video Break!
85
1. u  r  z
2. t  w  t
3. r  r  r  r  z
4. x  y  q
5. s  v  s
6. x 2  q
7. r  r  u
8. x  u  s
Figure Out the Digits
In this number puzzle, each
letter (q - z) represents a
different digit from 0-9.
Find the correspondence
between the letters and the
digits. Be prepared to
explain where you started,
and the order in which you
solved the puzzle.
y
x
9.

z
u
86
1. u  r  z
2. t  w  t
1. 4 x 2 = 8
3. r  r  r  r  z
4. x  y  q
2. 5 + 0 = 5
5. s  v  s
6. x 2  q
7. r  r  u
8. x  u  s
y
x
9.

z
u
3. 2 + 2 + 2 + 2 = 8
4. 3 + 6 = 9
5. 7 x 1 = 7
6. 32 = 9
7. 2 + 2 = 4
8. 3 + 4 = 7
9. 6/8 =3/4
87
Use Multiple Representations
• Represent
problems using
symbols,
expressions, and
equations, tables,
and graphs
• Model real-world
situations
• Complete problems
different ways
(flexibility in
problem solving)
88
Research-Based Teaching Strategies
• Effective questioning
• Teacher responses
• Use of manipulatives
• Conceptually-based
teaching
89
Effective Questions Techniques
Ask challenging, well-crafted,
open-ended questions, such as:
– What would happen if . . . ?
– What would have to happen for
...?
– What happens when . . . ?
– How could you . . . ?
– Can you explain why you
decided . . . ?
90
Teacher Responses
• Phrases to Use
– I’m not sure I understand, could you show me an
example of ... ?
– What do you think the next step should be?
– Where would you use ... ?
– Could ____ be an answer?
– How do you know you are correct?
• Phrases to Avoid
– Let me show you how to do this.
– That’s not correct.
– I’m not sure you want to do that.
91
Math journals help students to . . .
• Be aware of what they do and
do not know
• Make use of prior knowledge
• Identify their mathematical questions
• Develop their ability to problem solve
• Monitor their own progress
• Make connections
• Communicate more precisely
92
Getting Started . . .
My Teaching Reflections
• One secret I have about teaching algebra is . . .
• My worst experience with teaching algebra was when .
..
• My best experience with teaching algebra was when . .
.
93
Mathematics is like a video game;
If you just sit and watch,
You’re wasting your time.
94
Algebraic thinking . . .
Involves the connection between all
learning levels.
• Concrete
• Representational (semi-concrete)
• Abstract
95
Time Out for a Video Break!
96
Algebra Manipulatives (the “C” of CRA)
• Students with access to virtual
manipulatives achieved higher
gains than those students taught
without manipulatives.
• Students using hands-on and
manipulatives were able to
explain the how and why of
algebraic problem solving.
97
Introduction to Algebra Tiles
Positive Tiles
x2
x
1
Negative Tiles
Remember, they could be called x, y, b, t, etc.
98
Introduction to Algebra Tiles
Each tile represents an area.
x
Area of large square = x (x) = x2
x
1
Area of rectangle = 1 (x) = x
x
1
1
Area of small square = 1 (1) = 1
Note: Tiles are not to scale. 10 little tiles don’t equal 1 big tile!
99
What’s My Polynomial?
100
Simon says show me . . .
•
•
•
•
•
•
•
•
•
2x2
4x
- x2
3
2x + 3
- x2 + 4
2x2 + 6x + 5
-2x2 - 6x - 5
x2 - 2x + 3
101
Zero Pairs (Remember Additive Inverses?)
• A combined positive and negative of the same area
(number) produces a zero pair.
• I have $1.00. I spend $1.00. I have $0 left.)
Example: 2 – 2 + 2x – 2x = 0
102
Use Algebra Tiles to Model
Addition of Integers
(+3) + (+1) =
(-2) + (-1) =
Addition of Integers
•
(+3) + (-1) =
•
(+4) + (-4) =
• Don’t forget that a positive and a negative “cancel”
each other out!
Use Algebra Tiles to Model
Integer Subtraction
• Subtraction can be
interpreted as “take-away.”
• Subtraction can also be
thought of as “adding
the opposite.”
Use Algebra Tiles to Model
Subtraction of Integers
(+5) – (+2) =
(-4) – (-3) =
Subtracting Integers – It’s Your Turn!
• (+3) – (-5)
• (-4) – (+1)
• (+3) – (-3)
Use Algebra Tiles to Combine
Polynomials
“Simplify” means to combine like terms and
complete all operations.
Terms in an expression are like terms if they have
identical variable parts.
You can combine terms that are alike.
You cannot combine terms that are unalike.
Combining Like Terms
How much
do I have
here?
I have 5x + 4.
Combining Like Terms
How much
do I have
here?
I have x2 + 2x + 6
Show Me! Now Simplify!
• x + 3 + 2x
3x + 3
• 3x + 2 + x + 4
4x + 6
• 2x2 +2x + 3 + x2 + 1
3x2 + 2x + 4
• 1 + 3x - 3x2 + 1 + x2
- 2x2 + 3x + 2
• -4 + 2x + 3x2 - x - 3x
3x2 - x -4
Big Ideas Using Algebra Tiles
• Add and subtract integers
• Model linear expressions
• Solve linear equations
• Simplify polynomials
• Solve equations for unknown variable
• Multiply and divide polynomials
• Complete the square
• Investigate
112
Teaching Beyond the Facts
“Trying to teach in the 21st century
without conceptual schema for
knowledge is like trying to build a
house without a blueprint.”
H. Lynn Erickson
Concept-Based Curriculum and Instruction
113
Conceptual Teaching
What is conceptual teaching?
• Using schema to organize new knowledge
• Developing units around concepts to help students
learn
• Providing schema based on students’ prior knowledge
or experiences
• Teaching knowledge/skill/concept in context
What it’s not!
• Worksheets
• Drill
• Memorization of discrete facts
114
What Is a Great Math Task?
A great task:
• Revolves around an interesting problem – offering several
methods of solution
• Is directed at essential mathematical content as specified in
the standards.
• Requires examination and perseverance – challenging
students
• Begs for discussion – offering rich discourse on the
mathematics involved
• Builds student understanding – following a clear set of
learning expectations
• Warrants a summary look back – with reflection and
extension opportunities
115
Think about your classrooms
Based on the criteria, think of a
specific activity you use that might
qualify as a great task.
• Which of the practices does that
activity allow students to
demonstrate?
• How do you find these activities?
Meaning does not exist on paper. The only thing you
will ever find on paper are black marks. Meaning is
in people. People give, assign, or ascribe meanings
which they already have in their experiences.
116
Guiding Questions
• Is there evidence in the student work that
demonstrates the use of the practices?
• How did your task expose student thinking about the
topic?
• What might a teacher plan as next steps as a result of
the task?
• How do we develop and support the use of the
mathematical practices with our students?
• As leaders, consider ways that these tasks can be
used with teachers and what would you consider to be
the intended outcome?
117
Real-World Algebra
My Ford Bronco was fitted at the factory with 30 inch
diameter tires. That means its speedometer is calibrated
for 30 inch diameter tires. I "enhanced" the vehicle with All
Terrain tires that have a 31 inch diameter. How will this
change the speedometer readings? Specifically, assuming
the speedometer was accurate in the first place, what
should I make the speedometer read as I drive with my 31
inch tires so that the actual speed is 55 mph?
CTL Resources for Algebra. The
Department of Mathematics. Education
University of Georgia
http://jwilson.coe.uga.edu/ctl/ctl/resources/
Algebra/Algebra.html
118
It all leads to connecting mathematical
concepts with effective mathematical
practices/problem solving
119
A Few Strategies to Get Started
• Model, explain, and provide guided
assistance, but move towards self-regulation.
• Provide opportunities for algebraic thinking.
• Keep it real – demonstrate how
skills/concepts are used in real-world
situations.
• Teach often to the whole class, in small
groups, and with individual students.
• Set high expectations.
120
Best Practices Review
Instructional
Element
Curriculum
Design
Professional
Development
for Teachers
Technology
Recommended Practices
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Manipulatives 
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Instructional 
Strategies
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Assessment
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Ensure mathematics curriculum is based on challenging content
Ensure curriculum is standards based
Clearly identify skills, concepts and knowledge to be mastered
Ensure that the mathematics curriculum is vertically and horizontally articulated
Provide professional development which focuses on:
Knowing/understanding standards
Using standards as a basis for instructional planning
Teaching using best practices
Multiple approaches to assessment
Develop/provide instructional support materials such as curriculum maps and pacing
guides and provide math coaches
Provide professional development on the use of instructional technology tools
Provide student access to a variety of technology tools
Integrate the use of technology across all mathematics curricula and courses
Use manipulatives to develop understanding of mathematical concepts
Use manipulatives to demonstrate word problems
Ensure use of manipulatives is aligned with underlying math concepts
Focus lessons on specific
concept/skills that are standards based
Differentiate instruction through flexible grouping, individualizing lessons, compacting,
using tiered assignments, and varying question levels
Ensure that instructional activities are learner-centered and emphasize
inquiry/problem-solving
Use experience and prior knowledge as a basis for building new knowledge
Use cooperative learning strategies and make real life connections
Use scaffolding to make connections to concepts, procedures and understanding
Ask probing questions which require students to justify their responses
Emphasize the development of basic computational skills
Ensure assessment strategies are aligned with standards/concepts being taught
Evaluate both student progress/performance and teacher effectiveness
Utilize student self-monitoring techniques
Provide guided practice with feedback
Conduct error analyses of student work
Utilize both traditional and alternative assessment strategies
Ensure the inclusion of diagnostic, formative and summative strategies
Increase use of open-ended assessment techniques
• Curriculum Design
• Professional
Development
• Technology
• Manipulatives
• Instructional
Strategies
• Assessment
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Real-World Math
The Futures Channel
http://www.thefutureschannel.com/algebra/algebra_real_world
_movies.php
Real-World Math
http://www.realworldmath.org/
Get the Math
http://www.thirteen.org/get-the-math/
Math in the News
http://www.media4math.com/MathInTheNews.asp
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Don’t Forget Today’s Learner!
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“High achievement always
occurs in the framework of
high expectation.”
Charles F. Kettering (1876-1958)
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QUESTIONS, INSIGHTS, SUGGESTIONS
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Presenters
Bonnie Goonen
Trainer/Consultant
[email protected]
Susan Pittman-Shetler
Trainer/Consultant
[email protected]
This workshop developed courtesy of GED Testing Service®.
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Note: This presentation may be used and reproduced
in its entirety for educational purposes in preparation
for the 2014 GED® test by including the following
attribution text:
Copyright © 2013 GED Testing Service LLC. All rights reserved. Used by
permission.
GED® and GED Testing Service® are registered trademarks of the
American Council on Education (ACE). They may not be used or
reproduced without the express written permission of ACE or GED Testing
Service. The GED® and GED Testing Service® brands are administered by
GED Testing Service LLC under license from the American Council on
Education.
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