Leadership Summit
K-12 Mathematics
November 3, 2015
Dr. Lynda Luckie
The Learning System
Questions to consider…
• Where are we in terms of student
achievement and the systems that affect it?
• Where are we along the
rubric of the School Keys?
EXEMPLARY
NOT EVIDENT
The Silent
Epidemic:
Perspectives of
High School
Dropouts
A Research Report for
the Bill and Melinda
Gates Foundation
March 2006
The Silent Epidemic
• Bill and Melinda Gates
Foundation’s research on high
school dropouts shows that…
• 45 % of students who drop out
did not feel their previous
schooling had prepared them
for high school.
The Silent Epidemic
• Many felt behind when they left
elementary school.
• 47% said classes weren’t
interesting.
• 81% called for more “real-world”
learning opportunities.
• Two-thirds said they would have
stayed if their schools had
demanded more of them.
The Silent Epidemic
What does this have to do with us
as teachers ?
As administrators?
As support personnel?
As district – level leaders?
So Why Are Kids Having So Much
Difficulty with Math?
The Poverty of “E”s
The Poverty of “E”s
• The Poverty of Exposure
– Poverty because of low-level
questions and classroom work
– Rote learning v creativity, grit, and
strenuous mental gymnastics
– A set of destinations and a set of
rules to get there v a map and how
to read it
– Discrete algorithms for a correct
answer v how to attack a new or
different kind of equation
The Poverty of “E”s
• The Poverty of Exposure
• The Poverty of Experience
– Traditional problems v rigorous tasks
– Ritual engagement v authentic
engagement
– Textbook generated v related to their
world
The Poverty of “E”s
• The Poverty of Exposure
• The Poverty of Experience
• The Poverty of Expectations
– What do you expect them to
learn/produce/achieve?
Frustrated Math Teachers
The Research Says…
• The impact of decisions
regarding instruction made by
individual teachers is far greater
than the impact of decisions
made at the school level.
» Robert Marzano
» What Works in Schools: Translating
Research into Action, 2003
The Research Says…
• Differences in the effectiveness
of individual classroom
teachers are the single largest
contextual factor affecting the
academic growth of students.
» W. Sanders
» The School Administrtor
According to Teacher
Keys - GADOE
• It is estimated that only about 3%
of the contribution teachers make
to student learning is associated
with teacher experience,
educational level, certification
status, and other readily
observable characteristics.
According to Teacher
Keys - GADOE
• The remaining 97% of teachers’
effects on student achievement is
associated with intangible aspects
of teacher quality that defy easy
measurement, such as classroom
practices.
Teacher Efficacy
Teacher Efficacy
Teacher Efficacy
Teacher Efficacy
It’s Time to Move Our Cheese!
Best Practice
Focus on Four
1. Good Questioning
2. Critical Thinking & Number
Sense
3. The Workshop Model for
Differentiated Instruction
4. Collaboration
Best Practice
Focus on Four
1. Good Questioning
Best Practice…
…it’s all about the
questions we ask.
Too often we give our children
answers to remember rather
than problems to solve,
effectively keeping them IN the
box.
What Are Genuine
Questions?
– They are questions the teacher
asks for which s/he has no way
of knowing what the answer will
be.
• NOT, “How many angles does
a parallelogram have?”
• INSTEAD, “Tell me what you
know about a parallelogram.”
Traditional Question
• What is the name of this shape?
Application Question
• Tell me everything you know
about this shape.
OR
• If this is a hexagon, draw other
kinds of hexagons you know
about. Record some shapes that
are NOT hexagons.
OR
• Compare and contrast these two
shapes.
Traditional Question
• What is the name of this
shape?
Application Question
• Tell me everything you know about
this shape.
Or…
• If this is a pentagon inscribed in a circle,
draw other pentagons inscribed in circles
you know about. Explain how they are
different.
Or…
• Compare and contrast these two
figures.
So…
…what kinds of questions
are we asking?
Find the Answer
12 ÷ 3
4x2
How Many Ways Can You
Represent Each of These?
12 ÷ 3
4x2
Find the Answer
- 16 + (- 8)
(2x + 4) x (3x – 6)
How Many Ways Can You
Represent Each of These?
- 16 + (- 8)
(2x + 4) x (3x – 6)
Mr. Billingsley’s Trip
A taxi charges:
$2.40
for the first 1.5 miles
for every additional ¼ mile
.10
Mr. Billingsley paid $12.00 for his taxi ride from
work to home. How far is Mr. Billingsley’s work
place from home?
Mr. Billingsley’s Trip
A taxi charges:
for the first 1.5 miles
$2.40
for every additional ¼ mile
.10
Mr. Billingsley paid $12.00 for his taxi ride from work to
home. How far is Mr. Billingsley’s work place from
home?
Renee’s solution:
$12.00 - $2.40 = $9.60
$9.60 ÷ $0.10 = 9.6
9.6 x 1.5 = 14.4
Mr. Billingsley’s work place is 14.4 miles from his
home.
There is something wrong with Renee’s solution.
Show how you would solve the problem.
What Number Makes
Sense?
The following is a textbook question on the
topic of numbers and number operations.
Tickets to a concert cost $15 per adult and
$8 per child. Mr. Adams bought tickets for 4
adults and 5 children. How much did he
spend altogether?
What Number Makes
Sense?
Read the problem. Look at the numbers in the
box. Put the numbers in the blanks where you
think they fit best. Read the problem again. Do
the numbers make sense?
CONCERT TICKETS
Tickets to a concert cost ______ per adult and
______ per child. Mr. Adams paid _____ for
tickets. He bought tickets for _____ adults and
______ children.
4
5
9
$8
$15
$100
Typical
Task
Typical Task
Dollar Line Task
Function and Pattern
• Think of a situation
which could be
represented in the graph
below.
• Write a full description of
the situation (be sure to
tell what each axis
represents in your
situation.)
• What questions could be
answered by your
completed graph?
From Balanced Assessment
Typical Textbook Problem
Make It More Engaging
Even Better
Real World?
ALWAYS ask yourself…
• What mathematics do I want my
students to learn by doing this
activity/task?
Best Practice
Focus on Four
1. Good Questioning
2. Critical Thinking and Number
Sense
Frayer Models
WEhW
What is it?
Characteristics
Examples
Non-Examples
Conceptual
Understanding
• These are…
• These are not…
These are____________________.
These are not________________________.
Which of these are______________?
Explain how you know.
These are__SQUARES__.
These are not__SQUARES_.
Which of these are_SQUARES__?
Explain how you know.
These are_numbers that round to 600__.
597
633
642
563
576
These are not__numbers that round to 600___.
541
678
515
651
525
Which of these are_numbers that round to 600___?
692
Explain how you know.
555
539
588
640
679
Equivalent Fractions
Name: _____________________________________________ Date:_____________________
These are Equivalent Fractions
5/6
=
2/8 =
1/ 4
12/16 = 3/4
9/ 15
3/5 =
2/6 = 1/3
These are NOT Equivalent Fractions
1 /4
5/5 = 5/6
½ = 10/15
21 /
28
=
=
10 0
25 /
10 /
12
¾=
3/5
2
6/1
10 /
12
=
Which of these are Equivalent Fractions?
5/12 = 20/48
7/8 = 49/57
¾ = 75/100
1/3 = 2/18
6/9 = 2/3
3/5 = 5/7
Explain how to recognize an Equivalent Fraction.
5/7
Three’s a Crowd!
three-fourths
six-eighths
two-thirds
centimeter
inch
millimeter
circle
square
rectangle
What I Know about…
What’s My Rule?
WHAT’S MY RULE?
Theme: Sports
Yes
No
Strike
Stick
Split
Puck
Pin
Hoop
Gutter
Goal
Rule:
Bowling Terms
WHAT’S MY RULE?
Theme: Geometry
Yes
No
Triangle
Cube
Rectangle
Pyramid
Square
Pentagon
Quadrilateral
Octagon
Rule: Plane figures with less than 5 sides.
WHAT’S MY RULE?
Theme: _______________
Yes
Rule:
No
_________________________
Logic Puzzles
Number Sense
• an understanding that allows
students to approach concepts,
ideas, and problems with an intuitive
feel for numbers and their
relationships
• an intuitive understanding of
numbers, their magnitude,
relationships, and how they are
affected by operations
Wikipedia
Another Definition of
Number Sense
“Number sense can be described as a 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.
No substitute exists for a skillful teacher and an
environment that fosters curiosity and exploration
at all grade levels.”
From Hilde Howden, “Teaching Number Sense,”
Arithmetic Teacher, 36(6). (Feb. 1989), p. 11.
7+6=?
How did you
solve this
equation?
38 + 38 = ?
How did you
solve this
equation?
395 + 56= ?
How did you
solve this
equation?
What Do Kids Say?
6+5=
+4
What Do Kids Say?
6+5=
+4
Are we teaching
them to THINK?
Teaching Kids to
Think Algebraically
6+5=
+
Even higher level…
Teaching Kids to
Think Algebraically
6+5=
Even higher…
+++
Teaching Kids to
Think Algebraically
6x5=
x3
…with multiplication
Teaching Kids to
Think Algebraically
6x5=
…with
x
multiplication
What Do Kids Say?
.06 + .15 =
+4
Teaching Kids to
Think Algebraically
.06 + .15 =
Higher level
+
Teaching Kids to
Think Algebraically
.06 + .15 =
+
Even higher…
+
Teaching Kids to
Think Algebraically
.06 + .15 =
x3
…with multiplication
Teaching Kids to
Think Algebraically
.06 + .15 =
x
…with multiplication
Teaching Kids to
Think Algebraically
.06 + .15 =
?
Even higher…
?
My
Personal
Favorites
for
Number
Sense
Spotlight on Number Examples
Critical Thinking/Number Sense
Implications for Teaching
We need to replace the question,
“Does the student know it?”
with the question,
“How does the student understand
it?”
John Van de Walle
Copyright © Allyn and Bacon 2010
Best Practice
Focus on Four
1. Good Questioning
2. Critical Thinking and Number
Sense
3. The Workshop Model for
Differentiated Instruction
Workshop
Model for Math
A Strategic Model for
Differentiated Instruction
Turn and Talk
• What happens in each setting?
Small Groups
Whole Group
Nuts and Bolts
• What do the groups look like?
– Heterogeneous or homogeneous?
Nuts and Bolts
• What do the groups look like?
– Heterogeneous or homogeneous?
HOMOGENEOUS GROUPS!
WHY??????
Nuts and Bolts
• How do you determine your
small groups?
Nuts and Bolts
• WITH DATA!
• Tickets out the Door
–Weekly assessments
• Formative and Summative
Assessments
How Can You Determine
Small Groups?
• Observation of an assigned task
• Small group discussion of problem solving related
to the concept to be studied
• Written explanation of understanding by students
in their math journals
• Paper and pencil pretest
• Formative test results (TICKET OUT THE DOOR)
• Performance in earlier work on sequential math
concepts
• Checklists and Conferencing
Sample TOD
Sample TOD
Why Tickets out the Door?
•
•
•
•
Quick formative assessment
It’s the DRIVER !
Helps determine your groups
Does not give students
permission to forget
• Wake up call
Math Workshop Model
Teacher
Facilitated
Group
At Your
Seat
Interactive
Practice
Math Workshop Logistics
Group A: ________________________________
Group B: ________________________________
Group C: ________________________________
Teacher Interactive
Facilitated
Practice
1st Rotation
Group A
Group B
At Your
Seat
Group C
2nd Rotation
Group C
Group A
Group B
3rd Rotation
Group B
Group C
Group A
What Does This Look Like
in a 90 Minute Block?
• Mini Lesson – How long?
• Group Rotations – How long?
• Formats for small groups?
Where in the room?
ALWAYS Ask Yourself…
• What mathematics do I want
my students to learn by doing
this activity?
• Are there students who may
already be proficient in this
area?
• Are there students who will
need more time?
Students Need to Know…
• Which group are we in today?
• Where are our meeting areas?
• Do we know what materials
we need?
• Do we know our schedule?
• Can we work independently?
Make It Your Own
Make It Your Own
So…how do you keep
everyone engaged?
With QUALITY work!
Meaningful math games
Practice that reinforces
Math Journals
Pair/group activities
NOT something they’ve never
seen before!!
Interactive Practice
Examples
ALL about PRACTICE
• iPad practice apps
• Laptop interactive
activities/programs
• Scavenger Hunts
• Meaningful math partner games
Scavenger Hunts
The Power of Meaningful
Math Games
The Power of Meaningful
Math Games
…Meaningful Practice
Games for Primary
Closest to 100
My Rolls:
1. ________
1. ________
1. ________
1. ________
1. ________
1. Roll the dice exactly SIX times.
2. Decide if your roll will be a “one” or a
“ten.”
3. Fill in the grid AND record it in the box.
4. Add all six rolls. Closets to 100 wins.
1. ________
Total: ____________
Closest to 100
• Use a 10 x 10 grid.
• Students take turn rolling the dice
exactly SIX times.
• When they roll, they decide whether
they want that number of ones or tens.
Eg: Student rolls 4…they can get four
“ones” or four “tens.” Record on the 10
x 10 grid.
• After six rolls, student closest to 100
without going over wins.
Domino Drawings
• Students use large Double Six or
Double Nine dominoes and can
work with a partner.
• Work together to:
– Draw domino
– Write a number sentence to show
sums of dots.
Make Ten Concentration
Use two sets of cards
Small Ten Frame Cards
Winnipeg School Division
Numeracy Project
Fraction Building
Use colored cubes to build and
record collections.
1/3
one-third
2/4
1/8
two-fourths one-eighth
1/5
one-fifth
Race to 50
What I Rolled
Solution
Total
Race to 50
• Materials
– 10 sided dice
– +/- die or spinner
– Recording Sheet
• Directions
– Players take turns rolling two dice and
the +/- die and solve, recording what
they rolled and the solution on the
recording sheet.
– First player to 50 wins.
Games for Intermediate
Grades
Pattern Block Fraction Pizza
Pattern Block Money
Rectangular Array Game
Knock Out Three
Knock Out Three
Directions: Roll two dice. Place a marker over any box below that
describes what you rolled. First player to get three in a row
(horizontally, vertically, or diagonally) wins.
Sum = 7
Product <
9
Both
cubes
are odd
Both
Difference
cubes > 3
=4
One cube
>4
Quotient
is an
even
number
Sum = 9
Quotient
is < 2
One cube
is an even
number
Product = One cube
12
=5
Doubles
Sum = 4
Sum is
odd
Difference
= zero
Sum
= 10
One cube
is an odd
number
Product
is even
Product >
Both
10, but < cubes are
20
even
Both
cubes
are < 3
Sum = 6
Sum is a
Difference
multiple of
is < 5
3
Is It True?
More Is It True?
Games for MS/HS
Spin for Expressions
Spin for Expressions
• Materials:
– Plus/minus spinner
– A number cube
– Expression playing cards
• Pass out all the expression cards FACE DOWN.
• The dealer then rolls the number cube to determine
the value of the variable, and then spins the spinner
to determine if the value is positive or negative.
• Each player will turn over one card and evaluate
his/her expression. The player with the greatest
value for that round takes all the cards.
• The player with the most cards at the end of the
game is the winner.
Domino Cards
Spinning for Polynomials
• Materials
– Polynomial Spinner
– Dice
• Student one spins spinner 3 times
and combines the monomials.
Student two does the same.
• Roll dice for value of x, and
highest final number wins.
Spinning for Polynomials
7x
3
x
2
5x
8
4x
6
2
3x
-2x
9
2
24 GAME
Workshop Model Tips…
• Ensure that students understand directions before
dismissing them.
• Practice transitions to and from whole group
areas.
• Practice moving from station to station.
• Set routines for what to do when assignments are
complete (e.g., Anchor Packets).
• Establish positive reinforcements for meeting
expectations.
Workshop Model Tips…
• Set expectations for behavior when working
independently or with partners.
• Have back-up seat-work assignments for students
who are not on task.
• Establish a signal for redirection and transitions.
• And most of all…
Biggest Tip…
• Don’t make it
harder than it is!!
At the end of the day…
So…how do you keep
everyone engaged?
With QUALITY work!
Meaningful math games
Practice that reinforces
Math Journals
Pair/group activities
NOT something they’ve never seen
before!!
The Workshop Model
in Action
This Teacher Said…
“I know more about what my
students know and how they think in
8 days than in the entire first
semester!”
Best Practice
Focus on Four
1. Good Questioning
2. Critical Thinking & Number
Sense
3. The Workshop Model for
Differentiated Instruction
4. Collaboration
Turn and Talk
• What are you excited about?
• Where does collaboration fit in
this model?
• What are your reservations?
– How can we deal positively
with them?
Remember…it will likely NOT be perfect
the first time, or even the second.
You WILL see positive results!