# “Teaching for Understanding” NCDPI Curriculum and Instruction Mathematics

```NCDPI
Curriculum and Instruction
Mathematics
“Teaching for Understanding”
Posted on January 28, 2013 by Bill McCallum
Once every few months or so I receive a message about the
following standard:
6.G.2. Find the volume of a right rectangular prism with
fractional edge lengths by packing it with unit cubes of the
appropriate unit fraction edge lengths, and show that the
volume is the same as would be found by multiplying the edge
lengths of the prism. Apply the formulas V=lwh and V=bh to find
volumes of right rectangular prisms with fractional edge lengths
in the context of solving real-world and mathematical problems.
Guess what people think the problem is before reading
on?
http://commoncoretools.me/2013/01/28/to-b-or-not-to-b/
Norms
• Listen as an Ally
• Value Differences
http://thebenevolentcouchpotato.wordpress.com/201
1/11/30/norm-peterson-bought-the-house-next-door/
• Maintain Professionalism
• Participate Actively
Who’s in the Room?
“Teaching for Understanding”
Phil Daro
Math SCASS
February 12, 2013
Dr. Phil Daro
“In Person”
(Almost)
Problem:
Cause:
Cure:
Mile wide –inch deep curriculum
Too little time per concept
More time per topic
“LESS TOPICS”
Why do students have to do
math problems?
a. To get answers because Homeland
Security needs them, pronto.
b. I had to, why shouldn’t they?
a. So they will listen in class.
d. To learn mathematics.
To Learn Mathematics
• Answers are part of the process, they are not the
product.
• The product is the student’s mathematical knowledge
and know-how.
• The ‘correctness’ of answers is also part of the
process. Yes, an important part.
What is learning?
• Integrating new knowledge with prior
knowledge; explicit work with prior knowledge
• Prior knowledge varies across students in a class
(like fingerprints); this variety is key to the
solution, it is not the problem.
• Thinking in a way you haven’t thought before
and understanding what and how others are
thinking.
United States:
• “How can I teach my kids to get the answer to this
problem?”
Japan:
• “How can I use this problem to teach the
mathematics of this unit?”
“The Butterfly Method”
Discussion
• How might these ideas challenge teachers in
• How can we move from “answer getting” to
“learning mathematics”?
• What evidence do you have that teachers
might not know the difference?
“The Butterfly Method”
Blogstop.com
“Faster Isn’t Smarter”
by
Cathy Seeley
“Hard Arithmetic is not Deep
Mathematics”
p. 83
“Hard Arithmetic is not Deep
Mathematics”
• What issues or challenges does this message
raise for you?
• In what ways do you agree or disagree?
• What barriers might keep students from
reaching these standards, and how can you
tackle these barriers?
Let’s Do Some
Math!
Area and Perimeter
• What rectangles can be made with a
perimeter of 30 units? Which rectangle
gives you the greatest area? How do
you know?
• What do you notice about the
relationship between area and length of
the sides?
Instructions
• Discuss the following at your table
– What thinking and learning occurred as
– Would this task be considered “Deep
Mathematics”? Why or why not?
Compared to….
5
10
What is the area of this rectangle?
What is the perimeter of this
rectangle?
Traveling With Graphs
Bing.com
Carl’s bike trip takes 3.5 hours. During each hour of the trip, Carl notes
the speed at which he is traveling. The trip is over level ground, and for
most of the trip Carl maintains a constant speed, since he is a good rider.
Based on the data Carl collected, what is the total distance he covered
during his trip?
Hours
Speed (mph)
0
0
1
8
2
8
3
8
3.5
0
For these data, he plots a speed/time graph.
Compared to….
1. What is the area of a rectangle with a length of 2in.
and a height of 8in.?
2. What is the area of a triangle whose base is 3 units
and it’s height is 8 units?
3. If Carl rode a bicycle for 3 hours and traveled 40
miles, what was his average speed?
Traveling With Graphs
• What concepts are addressed in this
situation?
• What strategies could be used to
develop conceptual understanding?
• At what level could this task be used as
foundational for future concepts?
“Who’s doing the talking, and
who’s doing the math?”
Cathy Seeley, former president, NCTM
How do we move from a culture
of “answer getting” to one of
“learning mathematics”?
“Modeling in Mathematics”
by
CCSSO and Math SCASS
(Council of Chief State School Officers)
(The State Collaborative of Assessment and Student Standards)
What is modeling?
A word with different meanings
- An instructional strategy where the teacher shows step
by step actions of how to set up and solve the task
Mathematical
2 + ___ = 8
Use step by step
actions to “model”
how to solve this
What is modeling?
A word with different meanings
2. “Model with Manipulatives”
demonstrate and understand how to solve the problem.
math
Toothpicks as
a model
What is modeling?
A word with different meanings
3. “Model with Mathematics”
mathematical model to solve the task
Four birds sat on a
wire, 2 flew away.
How many birds
remain on the wire?
appropriate mathematical
e.g. writing the number
sentence 4 – 2 = 2
What is modeling?
A word with different meanings
4. “A Model with Mathematics”
What is modeling?
A word with different meanings
1.
2.
3.
4.
“Modeling with Manipulatives”
“Model with Mathematics”
“A Model with Mathematics”
What is modeling?
A word with different meanings
1.
2.
3.
4.
“Modeling with Manipulatives”
“Model with Mathematics”
“A Model with Mathematics”
What makes something a modeling
• Are there criteria for “modeling tasks”?
• What are the skills involved?
Let’s Do Some
Math!
Alysha really wants to ride her favorite ride at eh amusement park
one more time before meeting her parents for lunch at 12:20 pm.
There is a pretty long line at his ride, which Alysha joins at 11:50
am (point A in the diagram below). Alysha is nervously checking
the time as she is moving forward in the line. By 12:03 she has
made it to point B in line. What is your best estimate for how long
it will take Alysha to reach the front of the line? Can she ride one
more time before she is supposed to meet her parents?
How well posed is well enough?
• Should a student still have questions after
• Should students have to find their own
information outside of what is given in the
problem?
• Should assumptions be stated, or reasoned
differently by each individual student?
Problems to Ponder
• Painting A Barn
• The Ice Cream Van
How each problem is posed.
• Birthday Cakes
How much information is
provided and when it’s
provided?
• Sugary Soft Drinks
How much information is
needed and how will they
find it?
Painting A Barn
Alexis needs to paint the four exterior
walls of a large rectangular barn. The
length of the barn is 80 feet, the width is
50 feet, and the height is 30 feet. The
paint cost \$28 per gallon, and each
gallon covers 420 square feet. How
much will it cost Alexis to paint the barn?
Ice Cream Van
You are considering dividing ice cream van during
the summer vacation. Your friend who “knows
everything” tells you that “its easy money.” You
make a few inquiries and find that the van costs
\$600 per week to rent. Each ice cream cone costs
50 cent to make and sell for \$1.50.
Birthday Cakes
Would all the birthday cakes eaten by
all the people in Arizona in one year
fit inside the University of Phoenix
Cody Patterson Original
The SLV High School graduation started at 1:00
pm. After some speeches, the principal started
reading off the names of the students,
alphabetically by last name. When he finishes, the
Sugary Soft Drinks
How many packets of sugar are in a
20 ounce bottle of soda?
http://threeacts.mrmeyer.com/sugarpackets/
Lunch
Collecting and Selecting Information
“Modeling Information Descriptors”
All and only
relevant
information
is given
Brainstorm
what you need
and then are
given it
Told what you
need, you go
and find it
Given
information,
but you
decide what
is useful
Determine
what
information is
needed and
find the
information
yourself
Collecting and Selecting Information
“Modeling Information Descriptors”
Use the contents of the envelope on your
table to:
• Match each task with it’s aligned
“Collecting and Selecting Information”
description.
Collecting and Selecting Information
“Modeling Information Descriptors”
1. All and
only
relevant
information
is given
2. Brainstorm
what you need
and then are
given it
3. Told what you
need, you go
and find it
4. Given
information,
but you
decide what
is useful
5. Determine
what
information is
needed and
find the
information
yourself
Collecting and Selecting Information
“Modeling Information Descriptors”
1. All and
only
relevant
information
is given
2. Brainstorm 3. Told what you
what you need need, you go
and then are
and find it
given it
4. Given
information,
but you
decide what
is useful
5. Determine
what
information is
needed and
find the
information
yourself
Matching Activity
Use the contents of the envelope on your table to:
• Match each task with it’s aligned “Collecting and
Selecting Information” description.
• Order the tasks on a continuum based on
the amount of information needed?
Collecting and Selecting
Information
How much information is needed?
1
3
4
2
5
Collecting and Selecting
Information
How much information is needed?
1
3
4
2
5
How much information needs to be found?
4
1
2
3
5
“All Around the School”
A class was studying metric and customary
measurement, comparing quantities of one
unit of measure to quantities in the other.
(2003)
Question: If all the students in the school hold
hands, will they create a chain long enough to
circle the school?
Compared To……
Our school is 485 meters around. There are
535 students in the school, and the average
arm span of a child is 2 meters. Can we circle
the school if we hold hands and make a human
chain?
Blogstop.com
“Faster Isn’t Smarter”
by
Cathy Seeley
“Constructive Struggling”
p. 88
Let’s Do Some
Math!
Show 15  3 =  
1. As a multiplication problem
2. Equal groups of things
3. An array (rows and columns of dots)
4. Area model
5. In the multiplication table
6. Make up a word problem
Show 15  3 =  
1. As a multiplication problem (3 x [ ] = 15 )
2. Equal groups of things: 3 groups of how many
make 15?
3. An array (3 rows, ? columns of 3 make 15?)
4. Area model: a rectangle has one side = 3 and an
area of 15, what is the length of the other side?
5. In the multiplication table: find 15 in the 3 row
6. Make up a word problem
Show 16  3 =  
1. As a multiplication problem
2. Equal groups of things
3. An array (rows and columns of dots)
4. Area model
5. In the multiplication table
6. Make up a word problem
16 3 =  
• What concepts are addressed in this
situation?
• What strategies could be used to
develop conceptual understanding?
“Who’s doing the talking, and
who’s doing the math?”
Cathy Seeley, former president, NCTM
Blogstop.com
“Faster Isn’t Smarter”
by
Cathy Seeley
“Faster Isn’t Smarter”
p. 93
Personalization
The Tension:
personal (unique) vs. standard (same)
Why Standards?
• Social Justice
• Good curriculum for all students
students bring
• On-grade learning in the cluster of standards
Standards are a Peculiar Genre
1. We write as though students have learned approximately
100% of what is in preceding standards. This is never even
approximately true anywhere in the world.
2. Variety among students in what they bring to each day’s
lesson is the condition of teaching, not a breakdown in the
system. We need to teach accordingly.
3. Tools for teachers…instructional and assessment…should
help them manage the variety.
What is learning?
• Integrating new knowledge with prior
knowledge; explicit work with prior knowledge
• Prior knowledge varies across students in a class
(like fingerprints); this variety is key to the
solution, it is not the problem.
• Thinking in a way you haven’t thought before
and understanding what and how others are
thinking.
Minimum degree of varying prior
knowledge in the average classroom
Student A
Student B
Student C
Student D
Student E
Lesson START
Level
Degree of prior knowledge in the
average classroom
Planned time
Student A
Student B
Student C
Student D
Student E
Needed time
Lesson START
Level
CCSS Target
Level
I - WE - YOU
CCSS Target
Student A
Student B
Student C
Student D
Student E
Lesson START
Level
I - WE - YOU
CCSS Target
Student A
Student B
Student C
Student D
Student E
Lesson START
Level
You - We – I
Instruction based on prior knowledge
Student A
Student B
Student C
Student D
Student E
Lesson START Formative
Assessment
Level
Four Levels of Learning
I. Highest Standard:
Understand well enough to explain to others
II. Good enough Standard:
Understand enough to learn the next related concepts
III. Low Standard:
IV. No Standard:
Noise
Four levels of learning
The truth is triage, but all can prosper
I.
Understand well enough to explain to others
As many as possible, at least 1/3
II. Understanding enough to learn the next
related concepts
Most of the rest
III. Can get the answers without understanding
Sometimes we have to settle for low, but don’t aim low
IV. Noise
Aimless
Blogstop.com
“Faster Isn’t Smarter”
by
Cathy Seeley
“Crystal’s Calculator”
p. 159
Illustrative Mathematics
Example Problems
illustrativemathematics.org
Teach at the speed of learning
• Not faster
• More time per concept
• More time per problem
• More time per student talking
• Fewer problems per lesson
The Mathematical Practices develop
character: the pluck and persistence needed
to learn difficult content. We need a
classroom culture that focuses on learning…a
try, try again culture. We need a culture of
patience while the children learn, not
impatience for the right answer. Patience,
not haste and hurry, is the character of
mathematics and of learning.
www.ncdpi.wikispaces.net
Nominate An
Outstanding Teacher
Website: www.paemst.org
[email protected]
What questions do you
have?
DPI Mathematics Section
Kitty Rutherford
Elementary Mathematics
Consultant
919-807-3841
[email protected]
Johannah Maynor
Secondary Mathematics
Consultant
919-807-3842
[email protected]
Ashton Megson
Secondary Mathematics
Consultant
919-807-3934
[email protected]
Barbara Bissell
K – 12 Mathematics Section
Chief
919-807-3838
[email protected]
Susan Hart
Mathematics Program
Assistant
919-807-3846
[email protected]
```