CCSS mathematics

1. CCSS mathematics The chance for change… and the challenge

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

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

4. 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

5. Mile wide –inch deepcauses cures

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

7. Two ways to get less topics: • Delete topics • Coherence: when studied a little deeper, mathematics is a lot more coherent • coherence across concepts • coherence in the progression across grades

8. 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

9. 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

10. Wrong answers • Are part of the process, too • What was the student thinking? • Was it an error of haste or a stubborn misconception?

11. Three responses to a math problem • Answer getting • Making sense of the problem/situation • Making sense of the mathematics by learning to work through the problem

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

13. Answer getting vs. learning mathematics U.S.: • How can I teach my kids to get the answer to this problem? Use mathematics they already know – this is easy, reliable, works with bottom half, good for classroom management Japan: • How can I use this problem to teach the mathematics of this unit?

14. Butterflymethod

16. Use butterflies on this TIMSS item: 1/2 + 1/3 +1/4 =

17. “Set up and cross multiply” • Set up a proportion and cross multiply • It’s an equation, so say, “set up an equation” Solve it: How? Using basic tools of algebra: multiply both sides by a number, divide both sides by a number

18. Old State Standard Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals: 2.3 Solve simple problems, including ones arising in concrete situations that involve the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less), and express answers in the simplest form.

19. Use equivalent fractions as a strategy to add and subtract fractions 1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

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

21. 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

22. 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

23. Two major design principles, based on evidence: • Focus • Coherence

24. Silence speaks • No explicit requirement in the Standards about simplifying fractions or putting fractions into lowest terms • Instead a progression of concepts and skills, build to fraction equivalence • Putting a fraction into lowest terms is a special case of generating equivalent fractions

25. 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.

26. You must change a brain full of answers • To a brain with questions. Change prior answers into 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.

27. What is learning? • Integrating new knowledge with prior knowledge; explicit work with prior knowledge; prior knowledge varies across 25 students in a class; this variety is key to the solution, it is not the problem • Thinking in a way you haven’t thought before: thinking like someone else; like another student; understanding the way others think

28. 15 ÷ 3 = ☐

29. Show 15 ÷ 3 =☐ • 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

30. Show 15 ÷ 3 = ☐ • As a multiplication problem (3 x ☐ = 15 ) • Equal groups of things: 3 groups of how many make 15? • An array (3 rows, ☐ columns make 15?) • Area model: a rectangle has one side = 3 and an area of 15, what is the length of the other side? • In the multiplication table: find 15 in the 3 row • Make up a word problem

31. Show 16 ÷ 3 = ☐ • 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

32. 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

33. Attend to precision

34. Precision The process of making language precise IS the process we want students to engage in The process usually begins with imprecise language, often alternative imprecise language

35. Definition settles arguments in mathematics • Imprecise language could be using the same word withdifferent meanings • The work is making the meanings explicit • And then recognizing the difference • And then specifying a common meaning • Testing definitions with a variety of examples is a very useful process…does the definition decide whether the example is an example of the defined term? If the definition does not decide, it needs to be made more precise

36. Reference and correspondence • Another useful process is making references and correspondences explicit; for example, labeling the parts of a diagram so the quantities that the parts refer to are explicit; writing the units…inches, pounds….

37. Represent relationships explicitly • Another process is representing relationships in diagrams • Expressions represent relationships in concise way

38. Language differences and content • How knowledge, cognition, and language are threads in a single fabric of learning • inadvertent ways system unravels this fabric: silos, assessment, classification of students, instruction • Practices linked to discipline reasoning expressed in language and in multiple representations • Access to content courses • Don’t leave ELLs out from progression in text complexity or teaching for understanding

39. Discussions • How can increased discussion from CCSS benefit ELLs, rather than leave them out • Communicative stamina needed, builds intellectual stamina • Video shown to kids • How do we teach teachers to lead and manage discussions?

40. Imperfect • Imperfect language is valuable and can express precise reasoning and ideas • Progression through reality means progression through imperfections • It’s not about waiting for the precise wording, but about the use of imperfect language to express reasoning and then making the language and reasoning more precise together • Perfect teaching is unnecessary, imperfect works fine with stamina

41. Time • Slow down for learning, thinking, and language • The press of time against the scope and depth of curriculum • The press of time against the engagement, language processing, and cognition of ELLs • The press of time against instruction in two languages • Time for teachers to learn, to think, and to give feedback to students

42. Participants: where to find the time • Some students need more time, more feedback, and more encouragement than others to learn. Where can the more time come from? The additional feedback? Encouragement?

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

44. Why standards? Social justice • Main motive for standards • Offer 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

45. 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

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

47. 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

48. 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

49. When to use Direct Instruction Every day Usually at the end of the lesson Once a week at the beginning of the lesson Students have to be prepared for direct instruction

50. Direct Instruction • When there is significant variation in prior knowledge • Students must be prepared for direct instruction • Lesson starts with variation in prior knowledge and ends with direct instruction