1. CCSS mathematics
Phil Daro

2. Evidence, not Politics
• High performing countries like Japan
• Research
• Lessons learned

3. Mile wide –inch deep
causes
cures
2011 © New Leaders | 3

4. Mile wide –inch deep
cause:
too little time per concept
cure:
more time per topic
= less topics
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5. Two ways to get less topics
1. Delete topics
2. Coherence: A little deeper, mathematics is a
lot more coherent
a) Coherence across concepts
b) Coherence in the progression across grades

6. Silence speaks
no explicit requirement in the Standards about
simplifying fractions or putting fractions into
lowest terms.
instead a progression of concepts and skills
building to fraction equivalence.
putting a fraction into lowest terms is a special
case of generating equivalent fractions.

7. 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?
c) so they will listen in class
d) to learn mathematics

8. Why give students problems
to solve?
• 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.

9. Three Responses to a Math
Problem
1. Answer getting
2. Making sense of the problem situation
3. Making sense of the mathematics you can
learn from working on the problem

10. Answers are a black hole:
hard to escape the pull
• Answer getting short circuits
mathematics, making mathematical sense
• Very habituated in US teachers versus
Japanese teachers
• Devised methods for slowing
down, postponing answer getting

11. Answer getting vs. learning
mathematics
• USA:
• How can I teach my kids to get the
answer to this problem?
Use mathematics they already know.
Easy, reliable, works with bottom half, good
for classroom management.
• Japanese:
• How can I use this problem to teach the
mathematics of this unit?

12. Butterflymethod

14. More examples of answer getting
• “set up proportion and cross multiply”
• Invert and multiply
• FOIL method
Mnemonics can be useful, but not a substitute
for understanding the mathematics

15. Problem
Jason ran 40 meters in 4.5 seconds

16. Three kinds of questions can be
answered:
Jason ran 40 meters in 4.5 seconds
• How far in a given time
• How long to go a given distance
• How fast is he going
• A single relationship between time and distance, three
questions
• Understanding how these three questions are related
mathematically is central to the understanding of
proportionality called for by CCSS in 6th and 7th
grade, and to prepare for the start of algebra in 8th

17. Given 40 meters in 4.5 seconds
• Pose a question that prompts students to
formulate a function

18. Functions vs. solving
• How is work with functions different from
solving equations?

19. Fastest point on earth
• Mt.Chimborazo is 20,564 ft high. It sits very
near the equator. The circumfrance at sea
level at the equator is 25,000 miles.
• How much faster does the peak of Mt.
Chimborazo travel than a point at sea level on
the equator?

20. SOLO
PARTNER
SMALL GROUP
WHOLE CLASS

21. Two major design principles, based on
evidence:
–Focus
–Coherence

22. The Importance of Focus
• TIMSS and other international comparisons suggest that the
U.S. curriculum is „a mile wide and an inch deep.‟
• “On average, the U.S. curriculum omits only 17 percent of the
TIMSS grade 4 topics compared with an average omission
rate of 40 percent for the 11 comparison countries. The United
States covers all but 2 percent of the TIMSS topics through
grade 8 compared with a 25 percent non coverage rate in the
other countries. High-scoring Hong Kong’s curriculum
omits 48 percent of the TIMSS items through grade 4, and
18 percent through grade 8. Less topic coverage can be
associated with higher scores on those topics covered
because students have more time to master the content
that is taught.”
• Ginsburg et al., 2005

23. Grain size is a major issue
• Mathematics is simplest at the right grain size.
• “Strands” are too big, vague e.g. “number”
• Lessons are too small: too many small pieces
scattered over the floor, what if some are missing
or broken?
• Units or chapters are about the right size (8-12 per
year)
• Districts:
– STOP managing lessons,
– START managing units

24. What mathematics do we want
students to walk away with from
this chapter?
• Content Focus of professional learning
communities should be at the chapter level
• When working with standards, focus on
clusters. Standards are ingredients of
clusters. Coherence exists at the cluster level
across grades
• Each lesson within a chapter or unit has the
same objectives….the chapter objectives

25. What does good instruction look
like?
• The 8 standards for Mathematical Practice
describe student practices. Good instruction
bears fruit in what you see students doing.
Teachers have different ways of making this
happen.

26. Mathematical Practices Standards
1. Make sense of complex problems and persevere in solving
them.
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning.
College and Career Readiness Standards for Mathematics

27. Expertise and Character
• Development of expertise from novice to
apprentice to expert
– Schoolwide enterprise: school leadership
– Department wide enterprise: department
taking responsibility
• The Content of their mathematical
Character
– Develop character

28. What does good instruction look like?
Students explaining so others can understand
Students listening to each other, working to
understand the thinking of others
Teachers listening, working to understand
thinking of students
Teachers and students quoting and citing each
other

29. motivation
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.

30. Students Job: Explain your thinking
• Why (and how) it makes sense to you
– (MP 1,2,4,8)
• What confuses you
– (MP 1,2,3,4,5,6,7,8)
• Why you think it is true
– ( MP 3, 6, 7)
• How it relates to the thinking of others
– (MP 1,2,3,6,8)

31. What questions do you ask
• When you really want to understand someone
else’s way of thinking?
• Those are the questions that will work.
• The secret is to really want to understand
their way of thinking.
• Model this interest in other’s thinking for
students
• Being listened to is critical for learning

32. Explain the mathematics when
students are ready
•
•
•
•
Toward the end of the lesson
Prepare the 3-5 minute summary in advance,
Spend the period getting the students ready,
Get students talking about each other’s
thinking,
• Quote student work during summary at
lesson’s end

33. Students Explaining their reasoning
develops academic language and their
reasoning skills
Need to pull opinions and intuitions into the open:
make reasoning explicit
Make reasoning public
Core task: prepare explanations the other students
can understand
The more sophisticated your thinking, the more
challenging it is to explain so others understand

34. Teach at the speed of learning
•
•
•
•
•
Not faster
More time per concept
More time per problem
More time per student talking
= less problems per lesson

35. School Leaders and CCSS
• Develop the Mathematics Department as an
organizational unit that takes responsibility for
solving problems and learning more
mathematics
• Peer + observation of instruction
• Collaboration centered on student work
• Summarize the mathematics at the end of the
lesson

36. What to look for
• Students are talking about each other’s
thinking
• Students say second sentences
• Audience for student explanations: the other
students.
• Cold calls, not hands, so all prepare to explain
their thinking
• Student writing reflects student talk

37. Look for: Who participates
• EL students say second sentences
• African American males are encouraged to
argue
• Girls are encouraged to engage in productive
struggle
• Students listen to each other
• Cold calls, not hands, so no one shies away
from mathematics

38. Shift
1. From explaining to the teacher to convince her you
are paying attention
–To explaining so the others understand
2. From just answer getting
–To the mathematics students need as a
foundation for learning more
mathematics

39. Step out of the peculiar world that never worked
• This whole thing is a shift from a peculiar
world that failed large numbers of students.
We got used to something peculiar.
• To a world that is more normal, more like life
outside the mathematics classroom, more like
good teaching in other subjects.

40. Personalization and Differences
among students
The tension: personal (unique) vs.
standard (same)

41. Why Standards? Social Justice
• Main motive for standards
• Get good curriculum to all students
• Start each unit with the variety of thinking
and knowledge students bring to it
• Close each unit with on-grade learning in
the cluster of standards
• Some students will need extra time and
attention beyond classtime

42. 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
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43. Unit architecture

44. Four levels of learning
I. Understand well enough to explain to others
II. Good enough to learn the next related
concepts
III. Can get the answers
IV. Noise

45. 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. Good enough to learn the next related
concepts
Most of the rest
III. Can get the answers
At least this much
IV. Noise
Aim for zero

46. Efficiency of embedded peer tutoring is necessary
Four levels of learning
different students learn at levels within same topic
I. Understand well enough to explain to others
An asset to the others, learn deeply by explaining
II. Good enough to learn the next related
concepts
Ready to keep the momentum moving forward, a help to
others and helped by others
III. Can get the answers
Profit from tutoring
IV. Noise
Tutoring can minimize

47. When the content of the lesson is
dependent on prior mathematics
knowledge
• “I do – We do– You do” design breaks down for
many students
• Because it ignores prior knowledge
• I – we – you designs are well suited for content
that does not depend much on prior
knowledge…
• You do- we do- I do- you do

48. Classroom culture:
• ….explain well enough so others can
understand
• NOT answer so the teacher thinks you know
• Listening to other students and explaining to
other students

49. Questions that prompt explanations
Most good discussion questions are applications
of 3 basic math questions:
1. How does that make sense to you?
2. Why do you think that is true
3. How did you do it?

50. …so others can understand
• Prepare an explanation that others will
understand
• Understand others’ ways of thinking

51. Minimum Variety of prior knowledge
in every classroom; I - WE - YOU
Student A
Student B
Student C
Student D
Student E
Lesson START
Level
CCSS Target
Level

52. Variety of prior knowledge in every
classroom; I - WE - YOU
Planned time
Student A
Student B
Student C
Student D
Student E
Needed time
Lesson START
Level
CCSS Target
Level

53. Variety of prior knowledge in every
classroom; I - WE - YOU
Student A
Student B
Student C
Student D
Student E
Lesson START
Level
CCSS Target
Level

54. Variety of prior knowledge in every
classroom; I - WE - YOU
CCSS Target
Student A
Student B
Student C
Student D
Student E
Answer-Getting
Lesson START
Level

55. You - we – I designs better for content
that depends on prior knowledge
Student A
Student B
Student C
Student D
Student E
Lesson START Day 1
Day 2
Level
Attainment Target

56. Differences among students
• The first response, in the classroom: make
different ways of thinking students‟ bring to
the lesson visible to all
• Use 3 or 4 different ways of thinking that
students bring as starting points for paths
to grade level mathematics target
• All students travel all paths: robust,
clarifying

57. Prior knowledge
There are no empty shelves in the brain waiting
for new knowledge.
Learning something new ALWAYS involves
changing something old.
You must change prior knowledge to learn new
knowledge.

58. You must change a brain full of
answers
•
•
•
•
To a brain with questions.
Change prior answers into new questions.
The new knowledge answers these questions.
Teaching begins by turning students’ prior
knowledge into questions and then managing
the productive struggle to find the answers
• Direct instruction comes after this struggle to
clarify and refine the new knowledge.

59. Variety across students of prior
knowledge
is key to the solution, it is not the problem

60. 15 ÷ 3 = ☐

61. Show 15 ÷ 3 =☐
1.
2.
3.
4.
5.
6.
As a multiplication problem
Equal groups of things
An array (rows and columns of dots)
Area model
In the multiplication table
Make up a word problem

62. 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 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

63. Show 16 ÷ 3 = ☐
1.
2.
3.
4.
5.
6.
As a multiplication problem
Equal groups of things
An array (rows and columns of dots)
Area model
In the multiplication table
Make up a word problem

64. Start apart, bring together to
target
• Diagnostic: make differences visible; what are the
differences in mathematics that different students
bring to the problem
• All understand the thinking of each: from least to
most mathematically mature
• Converge on grade -level mathematics: pull
students together through the differences in their
thinking

65. Next lesson
• Start all over again
• Each day brings its differences, they never go
away

66. Design
• Mathematical Targets for a Unit make more sense
and are much more stable than targets for a
single lesson.
• Lessons have Mathematical missions that depend
on the purpose of the lesson and the role it is
designed to play in the unit.
• The Mathematical missions for a lesson depend
on the overarching goals of the Unit and the
specifics of the lesson’s purpose and position
within the sequence

67. Mathematical Targets for a Unit make more
sense and are much more stable than targets for
a single lesson.
• Invest teacher collaboration and math
expertise in: what mathematics do we want
students to keep with them from this unit?
• have teachers use the CCSS themselves, the
Progressions from the Illustrative
Mathematics Project, and the teacher guides
from the publisher that discuss the
mathematics.
• Good use of external mathematics experts

68. Concepts and explanations
• Start how students think; different ways of
thinking
• Work to understand each other:
– learn to explain so others understand
– Learn to make sense of someone else’s way of
thinking
– Learn questions that that help the explainer make
sense to you

69. Seeing is believing
and the power of abstraction
•
•
•
•
•
Learn to show your thinking with diagrams
What is a diagram?
Explain diagrams
Correspondence across representations
Drawing Things you count and groups of
things:
• Diagram of a ruler

70. Concrete to abstract every day
• What we learn is sticks to the context in which
we learn it
• Mathematics becomes powerful when liberate
thinking from the cocoon of concreteness
• The butterfly of abstraction is free to fly to
new kinds of problems

71. Make a poster that helps you explain
your way of thinking:
1. how did you make sense of the problem?
2. Include a diagram that shows your way of
thinking
3. Express your way of thinking as a number
equation
4. Show how you did the calculation

72. Language, Mathematics and
Prior Knowledge

73. Develop language, don’t work around
language
• Look for second sentences from
students, especially EL and reluctant speakers
• Students Explaining their reasoning develops
academic language and their reasoning power
• Making language more precise is a social
process, do it through discussion
• Listening stimulates thinking and talking
• Not listening stimulates daydreaming

74. Daro problems
Fraction videos at:
http://www.illustrativemathematics.
org/pages/fractions_progression

75. Consider the expression
where x and y are positive.
What happens to the value of the expression when we
increase the value of x while keeping y constant?

76. Consider the expression
where x and y are positive.
Find an equivalent expression whose structure shows
clearly whether the value of the expression
increases, decreases, or stays the same when we
increase the value of xwhile keeping y constant.

77. Shooting Hoops
• A basketball player shoots the ball with an
initial upward velocity of 20 ft/sec. The ball is
6 feet above the floor when it leaves her
hands.

78. Hoops
• A basketball player shoots the ball with an initial upward
velocity of 20 ft/sec. The ball is 6 feet above the floor when it
leaves her hands.
– A. How long will it take for the ball to reach the rim of the
basket 10 feet above the floor?
– B. Analyze what a defender could do to block the shot, if
the defender could jump with an initial velocity of 12
ft/sec. and had a reach 9 feet high when her feet are on
the ground.

79. Trains
• A train left the station and traveled at 50 mph.
Three hours later another train left the station
in the same direction traveling at 60mph.

80. • A train left the station and traveled at 50 mph.
Three hours later another train left the station
in the same direction traveling at 60mph.
• How long did it take for the second train to
overtake the first?

81. Water Tank
• We are pouring water into a water
tank. 5/6 liter of water is being
poured every 2/3 minute.
– Draw a diagram of this situation
– Make up a question that makes this a
word problem

82. Test item
• We are pouring water into a water tank. 5/6 liter of water
How many
liters of water will have been poured
after one minute?
is being poured every 2/3 minute.

83. Where are the numbers going to come
from?
• Not from water tanks. You can change to gas
tanks, swimming pools, or catfish ponds without
changing the meaning of the word problem.

84. Numbers:
given, implied or asked about
• The number of liters poured
• The number of minutesspent pouring
• The rate of pouring (which relates liters to
minutes)

85. Diagrams are reasoning tools
• A diagram should show where each of these
numbers come from. Show liters and show
minutes.
• The diagram should help us reason about the
relationship between liters and minutes in this
situation.

87. • The examples range in abstractness. The least
abstract is not a good reasoning tool because it
fails to show where the numbers come from.
The more abstract are easier to reason with, if
the student can make sense of them.

88. Learning targets
1. Expressing two different quantities that have
the same value in a problem situation as an
equation of two expressions
2. Building experience with fractions as scale
numbers in problem situations ( ½ does not
mean ½ ounce, it means ½ of whatever was
in the pail)
3. Techniques for solving equations with
fraction in them

89. Make up a word problem for which the
following equation is the answer
• y = .03x + 1

90. Equivalence
4+[]=5+2
Write four fractions equivalent to the number 5
Write a product equivalent to the sum:
3x + 6

93. Write 3 word problems for
y = rx, where r is a rate.
a) When r and x are given
b) When y and x are given
c) When y and r are given

94. Example item from new tests:
Write four fractions equivalent to the number 5

95. Problem from elementary to middle
school
Jason ran 40 meters in 4.5 seconds

96. Three kinds of questions can be
answered:
Jason ran 40 meters in 4.5 seconds
• How far in a given time
• How long to go a given distance
• How fast is he going
• A single relationship between time and distance, three
questions
• Understanding how these three questions are related
mathematically is central to the understanding of
proportionality called for by CCSS in 6th and 7th
grade, and to prepare for the start of algebra in 8th

97. A dozen eggs cost $3.00
97

98. A dozen eggs cost $3.00
• Whoops, 3 are broken.
• How much do 9 eggs cost?
How would you convince a cashier who wasn’t
sure you answer is right?
98

99. problem
• Tanya said, “Let’s put our shoelaces end to
end. I’ll bet it will be longer than we are end
to end.” Brent said, “um”.
• DeeDee said, “No. We will be longer?”
• Maria said, “How much longer?” Brent said
“um”.
• Tanya’s laces were 15 inches, DeeDee’s were
12, and Maria’s were 18. Brent wore loafers.

100. How much longer?
• Use half your heights as the girls’ heights.
Round to the nearest inch.

101. According to the Runners’ World:
On average, the human body is more than 50
percent water. Runners and other endurance
athletes average around 60 percent. This equals
about 120 soda cans’ worth of water in a 160pound runner!
• Check the Runners’ World calculation. Are there
really about 120 soda cans’ worth of water in the
body of a 160-pound runner?
– A typical soda can holds 12 fluid ounces.
– 16 fluid ounces (one pint) of water weighs one
pound.

102. 3 + = 10
• What goes in the box?

103. 3 + x = 10
• What does x refer to?
• What does 3 + x refer to in this equation?

104. 3+x=y
• What does x refer to?
• What does y refer to in this equation?
• Express y – 2 in terms of x.

105. Let
What does
equal?

106. • Place
on a number line.
Explain what you know about the intervals
between the three fractions.

107. Write two word problems (see 1. and 2.) in
which the following expression plays a key role:
40 - 6x2
1. Student constructs and solves an equation
2. Student defines a function and uses it to
answer questions about the problem
situation
Option: do the same for .04x -3

108. Explain the different purpose served
by the expression
• When the work is solving equations
• When the work is formulating and analyzing
functions

109. Raquel‟s Idea
On poster paper, prepare a
presentation that your
classmates will understand
explaining your reasoning with
words, pictures, and numbers.

110. How does finding common
denominators make it easy to
compare fractions?

111. Exploring Playgrounds
On poster paper, prepare a
presentation that your classmates will
understand explaining why your
solution to question 3 below makes
sense. Use a diagram in your
explanation.

112. The area of the blacktop is in
denominations of 1/20.
1/20 of what?
Explain what 1/20 refers
to in this situation.

114. Jack and Jill
• Jack and Jill climbed up the hill and each
fetched a full pail of water. On the way
down, Jack spilled half a pail and Jill spilled ¼
of a pail plus 10 more ounces. After the
spills, they both had the same amount of
water.
1. Write an equation with a solution that is the
number of ounces in a full pail.

115. Two expressions refer to same
quantity:
Where x = ounces before
the spill
Ounces after the spill
OR
Ounces spilled

116. Anticipated difficulties
• Equating amount of spill, but subtracting 10
ounces (thinking a spill is a minus )
• Not realizing that a full pail can be expressed
as x = ounces in a full pail, so that 1 ounce can
be subtracted from
which means, of the
ounces in a full pail (MP 2).

117. SOLVE
• 2. Show a step by step solution to the
equation:
• 3.Prepare a presentation that others will
understand that
– explains the purpose (what you wanted to
accomplish) of each step
MP 8
– justifies why it is valid (properties (page
90, CCSS), definitions & prior results).
MP 3

118. A Progression

119. Number and
Operations—Fractions, 3–
5

120. The goal is for students to see unit fractions as the
basic building blocks of fractions, in the same sense
that the number 1 is the basic building block of the
whole numbers; just as every whole number is
obtained by combining a sufficient number of
1s, every fraction is obtained by combining a
sufficient number of unit fractions.
–Number and Operations—Fractions 3-5