CCSS mathematics

1. CCSS mathematics Phil Daro

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

3. Mile wide –inch deepcauses cures

4. Mile wide –inch deep cause: too little time per conceptcure: more time per topic = less topics

5. Two ways to get less topics • Delete topics • Coherence: A little deeper, mathematics is a lot more coherent • Coherence across concepts • 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? • to get answers because Homeland Security needs them, pronto • I had to, why shouldn’t they? • so they will listen in class • 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 • Answer getting • Making sense of the problem situation • 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. Two major design principles, based on evidence: • Focus • Coherence

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 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

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

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

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.

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

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

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

28. 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.

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)

30. 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

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 other’s thinking, • Quote student work during summary at lesson’s end

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

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

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

35. 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

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

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

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.

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

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

41. Standards are a peculiar genre 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. 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. Tools for teachers…instructional and assessment…should help them manage the variety

42. Unit architecture

43. Four levels of learning • Understand well enough to explain to others • Good enough to learn the next related concepts • Can get the answers • Noise

44. Four levels of learningThe truth is triage, but all can prosper • Understand well enough to explain to others As many as possible, at least 1/3 • Good enough to learn the next related concepts Most of the rest • Can get the answers At least this much • Noise Aim for zero

45. Efficiency of embedded peer tutoring is necessaryFour levels of learningdifferent students learn at levels within same topic • Understand well enough to explain to others An asset to the others, learn deeply by explaining • Good enough to learn the next related concepts Ready to keep the momentum moving forward, a help to others and helped by others • Can get the answers Profit from tutoring • Noise Tutoring can minimize

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

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

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

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

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