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

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

Mile wide –inch deep cause: too little time per concept cure: more time per topic = less topics 2011 © New Leaders | 4

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

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?

Butterflymethod

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?

SOLO PARTNER SMALL GROUP WHOLE CLASS

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 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

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 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

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 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

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 2011 © New Leaders | 42

Unit architecture

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

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

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

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 true 3. 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 A Student B Student C Student D Student E Lesson START Level CCSS Target Level

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

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

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

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

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. 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

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

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

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 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

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 thinking 3. Express your way of thinking as a number equation 4. 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?

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.

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 How many liters of water will have been poured after one minute? is being poured every 2/3 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 minutesspent 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 5 Write 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 given b) When y and x are given c) 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 97

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

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 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.

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.

Let What does equal?

• Place on 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 equation 2. 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

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

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

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.

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: Where x = ounces before the spill Ounces after the spill OR Ounces spilled

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).

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

A Progression

Number and Operations—Fractions, 3– 5

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

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