Developing Secondary Math Curriculum

1. Developing Secondary Math Curriculum Thomas Rye Joel E. Ferris High School Spokane, WA For a preview, feel free to read the FAQ on pp 25-26 of your workbook.

2. Essential Questions • What is understanding? • How does starting with the end in mind impact our teaching, as well as student learning? • How can Understanding by Design be aligned with varying teaching styles and student populations? • What would we expect to see in classrooms where teaching for understanding is consistent?

3. Essential Questions • To what extent is there a connection between state standards (national standards) and what is truly mathematically important?

4. Essential Questions • In what ways are learning mathematics and doing mathematics the same? • How much time do we spend answering the questions? • How much time do we spend questioning the answers?

5. Essential Questions • How can we get students to think and communicate mathematically? • What does it mean to think mathematically? • What does the communication of mathematics look like?

6. Setting the Stage • Where do you teach? • Whom do you teach? • Why are you here? • What is the extent of your knowledge regarding Understanding by Design? • What is the extent of your experience regarding Understanding by Design?

7. Setting the Stage • What is the role of mathematics in our society? • Arithmetic • Algebra • Geometry • Statistics • Calculus • Advanced Mathematics

8. Everyday Math? The algebra is fun, but was it necessary?

9. Setting the Stage • What is mathematics? • What does it mean to do mathematics? • Can students do mathematics? • What does it mean to think mathematically?

10. Setting the Stage • So why teach mathematics? • What about mathematics is worth teaching? • Mathematics is done by a few yet benefits all.

11. Setting the Stage • What mathematics did you or your colleagues teach this last year? • What mathematics did your students learn this last year? • From your list, mark with a #1, the things they will find beneficial to them over and over again next year. • From your list, mark with a #2, the things they will find beneficial to them over and over again through the next five to seven years. • From your list, mark with a #3, the things they will find beneficial to them over and over again through the next twenty to forty years.

12. What about the book? • To what extent is there a connection between your textbook(s) and what is mathematically important? • Short term • Long term

13. Understanding by Design • Curriculum framework designed by Jay McTighe and Grant Wiggins • A means of developing units of instruction and curriculum on a larger scale

14. Understanding by Design • Stage 1 – What do I want students to understand? • Stage 2 – How will I know when they understand? • Stage 3 – How will I get them to understand?

15. The 6 Facets of Understanding(p. 23) • What does it mean to understand mathematics? • Explanation • Interpretation • Application • Perspective • Empathy • Self-knowledge

16. Understanding Mathematics • Marble Madness • The continuum of understanding

17. 3 Stages of Backward Design 1. Identify Results 2. Determine Evidence 3. Plan Learning Experiences

18. Stage One – Identify Results • During this session, you will design the following for the unit of your choice: • Enduring Understandings • What “big ideas” do I want kids to remember 40 years from now? • Essential Questions • What essential question(s) will guide and focus the teaching and learning? • Knowledge and skills • What skills, concepts, and procedures will be needed in order for students to understand?

19. Stage One – Identify Results • Examining Content • What is important for students to know / be able to do / understand for 40 days. • What is important for students to know / be able to do / understand for 40 months. • What is important for students to know / be able to do / understand for 40 years.

20. 40 days (the test) 40 years (life) Enduring Understandings 40 months (college)

21. Facts Worth Familiarity Concepts/Strategies Important to Know/Use “Big Ideas” Worth Understanding Stage One – Identify Results

22. Stage One – Identify Results • Mathematical Enduring Understandings • 40-year learnings in mathematics • So, of what I’ve been teaching the last 15 years, what has been of long term use to my students?

23. Stage One – Identify Results • What is one of the most difficult state (or national) standards to teach? • In what ways is it difficult? • What misunderstandings are common? • In what context do students need to know this?

24. Mathematical Understandings • Consider… • what you want your students to take away from your class. • what you want your students to remember about your class 40 years from now. • what you want your students to remember about mathematics 40 years from now.

25. Overarching and Topical Understandings • As we consider framing “Enduring Understandings” for our classrooms, it helps to keep in mind that there are two types: • Overarching: Broad ideas, transferable to other topics/content areas. • Topical: Specific to the unit topic.

26. Overarching Understandings • At the heart of the discipline • Run across grade levels • Many times run across curricular areas • These may be understandings for your entire course • Try sketching out a few ideas…

27. Mathematical Understandings • Students should… • see the mathematics in their world. • think. • communicate effectively.

28. Mathematical Understandings • Mathematics is a useful language for symbolically modeling and thus simplifying and analyzing our world. • Mathematics is a logical and objective means of analyzing and solving problems. • The effective communication of mathematics is essential to its application.

29. Mathematical Understandings • What do students think? • Math is a method of describing relationships between numbers and other measurable quantities. • Math can give visualization to what cannot be seen.

30. Mathematical Understandings • What do students think? • Mathematics is something that existed that we had to find. • Math simplifies our world through determining information hypothetically.

31. Mathematical Understandings • What do students think? • Mathematics opens up universal potential for each human to contribute to our understanding of our world.

32. Topical Understandings • Also big ideas at the heart of the discipline • Specific to the unit • Broad enough to transfer beyond the unit • Narrow enough to guide the teaching

33. Connecting to Standards • Which of the following are enduring understandings? • Understand measurable attributes of objects and the units, systems, and processes of measurement. • Understand patterns, relations, and functions. • Compare and order fractions, decimals, percents, and numbers written in scientific notation. • Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

34. Understanding by Design Templates • A tool for organizing curricular, assessment, and learning plans effectively and efficiently. • The templates embody the first stage of “Backward Design”. • Workbook p31, 36-37, 46-51 • 1-page template • 3-page template

35. Misunderstandings • Misunderstandings related to the unit (or course) may help identify enduring understandings • Consider… • What is behind the arithmetic mistakes students make? • Why is algebraic manipulation difficult for some (many) students? • What causes students to mix up slope?

36. Write Understandings • Write overarching and topical understandings for your unit (or course) • Filters to consider… • Is the understanding a big idea at the heart of the discipline? • Does the understanding allow for transfer to other topics or fields or to life? (the “so what” test) • To what extent does the understanding uncover common misunderstandings and/or demand “uncoverage”? • A few Examples on page 109

37. Share Understandings • Share your understandings with the people at your table • Do they pass the filters? • P242 (first 2 bullets on stage 1) • Sharing with the group

38. The Road to Understanding… • How do we get students to engage in the process of understanding? • Through the process of inquiry, or, as Understanding by Design puts it, through “Essential Questions.”

39. Types of Essential Questions • Overarching • Topical

40. Essential Questions • Have no one obvious right answer • Framed to provoke/sustain student interest • Raise important questions across content areas • Reflect conceptual priorities • Recur naturally

41. Writing Essential Questions • Derived from understandings • Written to engage students in discussion • Sometimes essential questions lead to additional enduring understandings

42. Essential Questions • Write some essential questions of your own • Keep in mind the filters • No obvious right answer • Engaging • Raise important other questions • Examples on page 98

43. Knowledge and Skills • It is important to consider what key knowledge and skills will be developed during the unit. • Consider two areas as you develop these: • Knowledge/skills called for by content standards • Knowledge/skills needed by students to be successful in reaching desired understandings.

44. Declarative Knowledge • Facts (dates, names, statistics) • Identify the base, exponent, and coefficient in a monomial expression. • Abstract concepts • See how the base and exponent impact the magnitude of the expression. • Rules and principles • Know the rules for exponents.

45. Procedural Knowledge • Skills • Identify “like” monomials. • Step-by-step procedures • Express numbers using scientific notation. • Complex processes • Simplify monomial expressions and ratios of monomial expressions in which the exponents are integers.

46. Time to Produce • Finish up your Stage 1 draft • Self-Assess your work against the standards on page 242.

47. 1. Identify Results 2. Determine Evidence 3. Plan Learning Experiences 3 Stages of Backward Design

48. Assessment Needs • Consider the most appropriate and efficient way to determine if a student “gets”… • Enduring Understandings • Knowledge • Skills

49. Stage 2 Determine Evidence • Consider how students might be able to demonstrate their understanding. • What things will they do? • In what formats or settings?

50. Stage 2 Determine Evidence • During this stage… • You will outline a performance task for your unit. • Sketch out a rubric to assess the performance task.