Evidence, not Politics High performing countries like Japan
Research Lessons learned
Slide 3
2011 New Leaders | 3 Mile wide inch deep causes cures
Slide 4
2011 New Leaders | 4 Mile wide inch deep cause: too little time
per concept cure: more time per topic = less topics
Slide 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
Slide 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.
Slide 7
Why do students have to do math problems? a)to get answers
because Homeland Security needs them, pronto b)I had to, why
shouldnt they? c)so they will listen in class d)to learn
mathematics
Slide 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 students mathematical knowledge and know-how. The
correctness of answers is also part of the process. Yes, an
important part.
Slide 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
Slide 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
Slide 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?
Slide 12
Butterfly method
Slide 13
Slide 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
Slide 15
Problem Jason ran 40 meters in 4.5 seconds
Slide 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 6 th and 7
th grade, and to prepare for the start of algebra in 8th
Slide 17
Given 40 meters in 4.5 seconds Pose a question that prompts
students to formulate a function
Slide 18
Functions vs. solving How is work with functions different from
solving equations?
Slide 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?
Slide 20
Two major design principles, based on evidence: Focus
Coherence
Slide 21
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 Kongs 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
Slide 22
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
Slide 23
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
Slide 24
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.
Slide 25
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
Slide 26
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
Slide 27
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
Slide 28
motivation Mathematical practices develop character: the pluck
and persistence needed to learn difficult content. We need a
classroom culture that focuses on learninga 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.
Slide 29
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)
Slide 30
What questions do you ask When you really want to understand
someone elses 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 others thinking for students Being
listened to is critical for learning
Slide 31
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
others thinking, Quote student work during summary at lessons
end
Slide 32
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
Slide 33
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
Slide 34
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
Slide 35
What to look for Students are talking about each others
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
Slide 36
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
Slide 37
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
Slide 38
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.
Slide 39
Personalization and Differences among students The tension:
personal (unique) vs. standard (same)
Slide 40
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
Slide 41
2011 New Leaders | 41 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 days lesson is the condition of teaching, not a breakdown
in the system. We need to teach accordingly. 3.Tools for
teachersinstructional and assessmentshould help them manage the
variety
Slide 42
Unit architecture
Slide 43
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
Slide 44
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
Slide 45
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
Slide 46
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
Slide 47
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
Slide 48
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?
Slide 49
so others can understand Prepare an explanation that others
will understand Understand others ways of thinking
Slide 50
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
Slide 51
Variety of prior knowledge in every classroom; I - WE - YOU
Student A Student B Student C Student D Student E Planned time
Needed time Lesson START Level CCSS Target Level
Slide 52
Student A Student B Student C Student D Student E Variety of
prior knowledge in every classroom; I - WE - YOU Lesson START Level
CCSS Target Level
Slide 53
Student A Student B Student C Student D Student E Variety of
prior knowledge in every classroom; I - WE - YOU Lesson START Level
CCSS Target Answer-Getting
Slide 54
You - we I designs better for content that depends on prior
knowledge Student A Student B Student C Student D Student E Lesson
START Level Day 1 Attainment Day 2 Target
Slide 55
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
Slide 56
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.
Slide 57
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.
Slide 58
Variety across students of prior knowledge is key to the
solution, it is not the problem
Slide 59
15 3 =
Slide 60
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
Slide 61
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
Slide 62
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
Slide 63
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
Slide 64
Next lesson Start all over again Each day brings its
differences, they never go away
Slide 65
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 lessons purpose and position within the
sequence
Slide 66
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
Slide 67
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 elses way of
thinking Learn questions that that help the explainer make sense to
you
Slide 68
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
Slide 69
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
Slide 70
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
Slide 71
Language, Mathematics and Prior Knowledge
Slide 72
Develop language, dont 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
Slide 73
Daro problems Fraction videos at:
http://www.illustrativemathematics.
org/pages/fractions_progression
Slide 74
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?
Slide 75
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 x while keeping y constant.
Slide 76
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.
Slide 77
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.
Slide 78
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.
Slide 79
How long did it take for the second train to overtake the
first?
Slide 80
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
Slide 81
Test item We are pouring water into a water tank. 5/6 liter of
water is being poured every 2/3 minute. How many liters of water
will have been poured after one minute?
Slide 82
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.
Slide 83
Numbers: given, implied or asked about The number of liters
poured The number of minutes spent pouring The rate of pouring
(which relates liters to minutes)
Slide 84
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.
Slide 85
Slide 86
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.
Slide 87
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 ( does not mean ounce, it means of whatever
was in the pail) 3.Techniques for solving equations with fraction
in them
Slide 88
Make up a word problem for which the following equation is the
answer y =.03x + 1
Slide 89
Equivalence 4 + [ ] = 5 + 2 Write four fractions equivalent to
the number 5 Write a product equivalent to the sum: 3x + 6
Slide 90
Slide 91
Slide 92
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
Slide 93
Example item from new tests: Write four fractions equivalent to
the number 5
Slide 94
Problem from elementary to middle school Jason ran 40 meters in
4.5 seconds
Slide 95
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 6 th and 7
th grade, and to prepare for the start of algebra in 8th
Slide 96
A dozen eggs cost $3.00 96
Slide 97
A dozen eggs cost $3.00 Whoops, 3 are broken. How much do 9
eggs cost? How would you convince a cashier who wasnt sure you
answer is right? 97
Slide 98
problem Tanya said, Lets put our shoelaces end to end. Ill 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. Tanyas laces were 15 inches, DeeDees were 12, and Marias
were 18. Brent wore loafers.
Slide 99
How much longer? Use half your heights as the girls heights.
Round to the nearest inch.
Slide 100
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 160- pound 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.
Slide 101
3 + = 10 What goes in the box?
Slide 102
3 + x = 10 What does x refer to? What does 3 + x refer to in
this equation?
Slide 103
3 + x = y What does x refer to? What does y refer to in this
equation? Express y 2 in terms of x.
Slide 104
Let What does equal?
Slide 105
Placeon a number line. Explain what you know about the
intervals between the three fractions.
Slide 106
Write two word problems (see 1. and 2.) in which the following
expression plays a key role: 40 - 6x 2 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
Slide 107
Explain the different purpose served by the expression When the
work is solving equations When the work is formulating and
analyzing functions
Slide 108
On poster paper, prepare a presentation that your classmates
will understand explaining your reasoning with words, pictures, and
numbers. Raquels Idea
Slide 109
How does finding common denominators make it easy to compare
fractions?
Slide 110
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. Exploring
Playgrounds
Slide 111
The area of the blacktop is in denominations of 1/20. 1/20 of
what? Explain what 1/20 refers to in this situation.
Slide 112
Slide 113
Re-teaching vs. Re-engagement Cognitive level is usually lower.
`Revisit same task Revisit student thinking about the task practice
sets of similar problems or easier problems repeat Teach the unit
again. Address basic skills that are missing. Practice more to make
sure student learn the procedures. Focus mostly on underachievers.
Address conceptual understanding. Examine task from different
perspective. make connections across student approaches and
representations: highlight correspondences. The entire class is
engaged in the math. Cognitive level is usually higher.
Slide 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 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.
Slide 115
Two expressions refer to same quantity: Where x = ounces before
the spill Ounces after the spill Ounces spilled OR
Slide 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).
Slide 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
