CCSS mathematics

CCSS mathematics

Phil Daro

Evidence, not Politics

• High performing countries like Japan• Research• Lessons learned

2011 © New Leaders | 3

Mile wide –inch deep

causes

cures


Mile wide –inch deepcause:

too little time per conceptcure:

more time per topic = less topics

Two ways to get less topics

1. Delete topics2. Coherence: A little deeper, mathematics

is a lot more coherenta) Coherence across conceptsb) Coherence in the progression across grades

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.

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

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.

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

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

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?

Butterfly method

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

Problem

Jason ran 40 meters in 4.5 seconds

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

Given 40 meters in 4.5 seconds

• Pose a question that prompts students to formulate a function

Functions vs. solving

• How is work with functions different from solving equations?

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?

Two major design principles, based on evidence:

–Focus –Coherence

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

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

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

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.

Mathematical Practices Standards

1. Make sense of complex problems and persevere in solving them.

2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of

others.4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated reasoning.

College and Career Readiness Standards for Mathematics

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

What does good instruction look like?

Students explaining so others can understandStudents listening to each other, working to

understand the thinking of othersTeachers listening, working to understand

thinking of studentsTeachers and students quoting and citing each

other

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.

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)

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

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

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 publicCore task: prepare explanations the other

students can understandThe more sophisticated your thinking, the more

challenging it is to explain so others understand

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

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

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

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

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

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.

Personalization and Differences among students

The tension: personal (unique) vs. standard (same)

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


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

Unit architecture

Four levels of learning

I. Understand well enough to explain to othersII. Good enough to learn the next related

conceptsIII. Can get the answers IV. Noise

Four levels of learningThe truth is triage, but all can prosper

I. Understand well enough to explain to othersAs 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. NoiseAim for zero

Efficiency of embedded peer tutoring is necessaryFour levels of learning

different students learn at levels within same topic

I. Understand well enough to explain to othersAn 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. NoiseTutoring can minimize

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

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

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 true3. How did you do it?

…so others can understand

• Prepare an explanation that others will understand

• Understand others’ ways of thinking

Minimum Variety of prior knowledge in every classroom; I - WE - YOU

Student AStudent BStudent CStudent D Student E

Lesson START Level

CCSS Target Level

Variety of prior knowledge in every classroom; I - WE - YOU


Planned time

Needed time

Lesson START Level

CCSS Target Level



Lesson START Level

CCSS Target Level



Lesson START Level

CCSS Target

Answer-Getting

You - we – I designs better for content that depends on prior knowledge


Lesson START Level

Day 1 Attainment

Day 2Target

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

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.

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.

Variety across students of prior knowledge

is key to the solution, it is not the problem

15 ÷ 3 = ☐

Show 15 ÷ 3 =☐

1. As a multiplication problem2. Equal groups of things3. An array (rows and columns of dots)4. Area model5. In the multiplication table6. 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 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 row6. Make up a word problem

Show 16 ÷ 3 = ☐

1. As a multiplication problem2. Equal groups of things3. An array (rows and columns of dots)4. Area model5. In the multiplication table6. Make up a word problem

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

Next lesson

• Start all over again• Each day brings its differences, they never go

away

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

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

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

Seeing is believingand 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

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

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

thinking3. Express your way of thinking as a number

equation4. Show how you did the calculation

Language, Mathematics and Prior Knowledge

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

Daro problems

Fraction videos at: http://www.illustrativemathematics.

org/pages/fractions_progression

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?

x + yxy

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.

x + yxy

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.

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.

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.

• 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?

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

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?

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.

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)

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.

• 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.

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

Make up a word problem for which the following equation is the answer

• y = .03x + 1

Equivalence

4 + [ ] = 5 + 2

Write four fractions equivalent to the number 5Write a product equivalent to the sum:

3x + 6

Write 3 word problems for y = rx, where r is a rate.

a) When r and x are givenb) When y and x are givenc) When y and r are given

Example item from new tests:

Write four fractions equivalent to the number 5

Problem from elementary to middle school

Jason ran 40 meters in 4.5 seconds

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

A dozen eggs cost $3.00

96

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?

97

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.

How much longer?

• Use half your heights as the girls’ heights. Round to the nearest inch.

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.

3 + = 10

• What goes in the box?

3 + x = 10

• What does x refer to?• What does 3 + x refer to in this equation?

3 + x = y

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

y=1x

Let

What does 1

y equal?

• Place 12

,1

3and

1

4on a number line.

Explain what you know about the intervals between the three fractions.

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 equation2. Student defines a function and uses it to

answer questions about the problem situation

Option: do the same for .04x -3

Explain the different purpose served by the expression

• When the work is solving equations• When the work is formulating and analyzing

functions

On poster paper, prepare a presentation that your

classmates will understand explaining your reasoning with words, pictures, and numbers.

Raquel’s Idea

How does finding common denominators make it easy to

compare fractions?

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

The area of the blacktop is in denominations of 1/20.

1/20 of what?Explain what 1/20 refers

to in this situation.

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.

Two expressions refer to same quantity:

1

2x =

3x4

−10

Where x = ounces before the spill

Ounces after the spill

Ounces spilled

1

2x =

14x+10

OR

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).

1

2x =

14x−10

3

4x 3

4

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

1

2x =

3x4

+10

Documents

CCSS mathematics