Math for America San Diego

1. Math for America San Diego Holistic Problems and the Common Core Standards Susie Amoroso, Yekaterina Milvidskaia, Ovie Soto

2. Outline Background and Purpose A Student’s Perspective A 2nd Year Teacher’s Perspective A 15th year Teacher/Support Provider Perspective Q&A – All

3. Purpose • To introduce attendees to holistic problems (HP’s). • To discuss the implementation of HP’s. • To demonstrate the ability of HP’s to support the Common Core Standards’ eight mathematical practices.

4. BACKGROUND: Common Core’s 8 Mathematical Practices • 1. Make sense of 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.

5. BACKGROUND: Connecting Common Core Math Practices with Proof Schemes Taxonomy • Common Core: • Reason abstractly and quantitatively • Use appropriate tools strategically • Look for and make use of structure • Look for and express regularity in repeated reasoning • Deductive (Proof Scheme): Harel and Sowder (1998) • Ability to pause and probe into the meaning of the symbols • Make changes to expressions/equations/geometric objects, anticipating their effects and compensating for them in a goal-oriented fashion • Attend to the generality aspect of conjectures • Attend to regularity in a process using it as a reason results generalize

6. BACKGROUND: Synthesis of Common Core’s 8 Mathematical Practices • ARE YOU NUTS??!! • Does counting on fingers qualify as “Using appropriate tools strategically”?

7. BACKGROUND: Synthesis of Common Core’s Mathematical Practices • Students come to see mathematics is a humanactivity. • Make sense of problems and persevere in solving them. • Teachers extend the locus of authority. • Construct viable arguments and critique the reasoning of others. • Teachers help students learn to reason deductively. • Reason abstractly and quantitatively. • Look for and make use of structure. • Look for and express regularity in repeated reasoning. • Attention to precision.

8. BACKGROUND: A Teaching Dilema • Algebra A polygon has n sides. An interior angle of the polygon and an exterior angle form a straight angle. • What is the sum of the measures of the n straight angles? • What is the sum of the measures of the n interior angles? • Using your answers above, what is the sum of the measures of the n exterior angles? • What theorem do the steps above lead to? • California Geometry (Prentice Hall): #46, p. 162

9. BACKGROUND: A Teaching Dilemma • If the problem were changed (see below), what potential benefits and obstacles might a high school Geometry teacher encounter? What might students try? • A polygon has n sides. An interior angle of the polygon and an exterior angle form a straight angle. What is the sum of the measures of the n exterior angles? • Adapted from California Geometry (Prentice Hall): #46, p. 162

10. BACKGROUND: A Teaching Dilemma Formal Mathematics Students’ Mathematics Teacher’s Knowledge Enacted Curriculum

11. BACKGROUND: Teaching Practices Problems constrain teachers and teachers constrain problems. A teaching actionrefers to what teachers in a particular community or culture typically do in the classroom. A teaching behavioris a typical characteristic of a teaching action. Teaching Practices Teaching Behaviors Teaching Actions

12. BACKGROUND: Holistic vs. Non-Holistic Problems A holistic problem refers to a problem where one must figure out from the problem statement the elements needed for its solution—it does not include hints or cues as to what is needed to solve it. A non-holistic problem, on the other hand, is one which is broken down into small parts, each of which attends to one or two isolated elements. Often each of such parts is a one-step problem.

13. BACKGROUND: Purpose • Questionswe hope to address: • What are some examples of holistic problems (HP’s)? • What are some potential benefits and obstacles in using HP’s for teachers? • How do HP’s connect to the common core mathematical practices? • What teaching practices do we see as closely connected with the successful implementation of a holistic problem?

14. A Student’s Perspective BA in Mathematics (UCSD). Math 121A and B Finishing a Masters in Education (UCSD). Completed 1 year of student teaching.

15. A Student’s Perspective Connections between different areas of Mathematics Algebra, Geometry, and Calculus Multiple solutions to a single problem Multiple approaches and ways of thinking Problems Necessitate Conjectures and Theorems

16. Find the Center of a given Circle

17. Step 1: Pick any 3 points on a circle and connect them to make a triangle. A2 A1 A3 Geometric Approach #1 Step 2: Construct Perpendicular Bisectors of the Triangle. This can be done using a Compass

18. Step 3: The perpendicular bisectors intersect in a point, call this point O. Continued Claim: Point O is equidistant from the vertices, and hence is the center of the Circle. Step 4: The Claim is proved using ThePerpendicular Bisector Theorem:Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment

22. Algebraic Approach Step 4: Set the two expressions equal to each other to find the point of intersection. This point will be the center of the Circle.

23. Geometric Approach # 2 Step 1: Construct a chord that is a diameter (proved by Thale’s Theorem). Step 2: Repeat. Claim: The point of intersection of two diameters is the Center. This can be proved by definition of a diameter and intersection of lines.

24. Step 1: Fold paper to create two semicircles Step 2: Fold paper again to create another two semicircles The Tactile Approach

25. No hints or cues as to how to approach the problem No “Apply this Theorem, or Use ____ Formula” What Makes the Problem Holistic? • Multiple ways of solving the problem • Geometric, Algebraic, & Logical • Multiple ways of thinking • Perceptual, Empirical, & Deductive • Necessitates a Particular Mathematical Concept • Why must the bisectors of the chords intersect at the center of a circle?

26. Promotes academic debate among students. CC: Construct viable arguments and critique the reasoning of others. A Teacher’s Perspective Real-life thinking - Students need to figure out an approach before applying tools they have. CC: Use appropriate tools strategically. Holistic Problems afford different learning modalities. This give students better opportunity to find an approach that works for them.

27. Who am I? • Susan Amoroso • In 2008, earned B.S. in Mathematics and was selected as a Math for America Fellow • Just completed 2nd year of teaching • Low-income, high-minority high school in San Diego County, CA • Attended first MfA San Diego Summer Institute in July 2009

28. My Intro to Holistic Problems At MfA San Diego Summer Institute in July 2009. Knew right away this was different: • No lecture • No explicit discussion of teaching strategies • Given holistic problems to solve • Worked in small groups • Every group shared its solution

29. #1: At what time after 4:00 will the minute hand of a clock overtake the hour hand? Step 1: Started with a picture of such a clock.

30. Step 2: Divided the clock face into 360° Thus, the starting locations are 0° for the minute hand and 120° for the hour hand. In one minute, the minute hand moves 6° and the hour hand moves 0.5°.

31. Step 3: Create a Table

32. Step 4: Write and Solve an Equation 6t = 120 + 0.5t 5.5t = 120 t ≈ 21.82 ≈ 21 minutes 48 seconds The minute hand will overtake the hour hand at approximately 21.82 minutes, or 21 minutes and 48 seconds.

33. Follow Up Problems #2 – At what time after 7:30 will the hands of a clock be perpendicular? #3 – Between 3:00 and 4:00 Noreen looked at her watch and noticed that the minute hand was between 5 and 6. Later, Noreen looked again and noticed that the hour hand and the minute hand had exchanged places. What time was it in the second case?

34. What I Learned About Learning Mathematics Power of images (and alternatives) • e.g., Degrees vs. proportions Okay for students to see that math is messy Mathematical procedures are tools; quantitative reasoning is the goal

35. What I Learned About Teaching Mathematics Intellectually intriguing Multiple points of entry • Some students solved proportionally • Some solved algebraically Lend themselves well to differentiation

36. Connection to Common Core Standards Reflect real-life nature of problem-solving Require students to make sense of the problem Often provide multiple points of entry – students develop their own problem-solving strategies based on their strengths

37. Connection to Common Core Standards (Cont’d) Require students to reason quantitatively (in context) and attend to the meaning of quantities Students must communicate effectively by making clear mathematical arguments, explaining their reasoning, justifying conclusions and critiquing others’ solutions

38. Fast Forward Two Years Knew this was the type of learning environment I wanted to create in my classroom Currently, I teach very traditionally • Homework review • Toolbox Notes • Independent Practice Effective use of holistic problems is a complex process

39. Systems of Equations #1 – Tree A is 2 feet tall and growing at a rate of 1 foot each year. Tree B is 6 feet tall and growing at a rate of 0.5 feet each year. In how many years will the trees be the same height and what will that height be? Students: Represented their work in a table… My Goal: Solving by Graphing

40. Systems of Equations #2 – Tree C is 4 feet tall and grows 0.5 feet each year. Tree D is 7.3 feet tall and grows 0.25 feet each year. In how many years will the trees be the same height and what will that height be? Students: Represented their work in a table… Goal: Solving by Substitution or Elimination

41. Challenges for Students Not being able to get started Poor organization Addition errors, particularly on #2 Lack of perseverance Calculating growth for fractions of a year

42. Partial Table of Values for #2

43. Challenges for Teacher How to help students get started Getting students to share solutions with the class Time issues Moving from contextualized to de-contextualized

44. The Big Picture The value and benefits of using holistic problems Understanding that change takes time

45. A Dual Perspective: Teacher/Support Provider • Personal Background: • 15 years teaching high school mathematics. • BA – Math, MA – Math, PhD – Math Ed (May 2010) • Continue to teach 2 classes at Patrick Henry High School • MfA Summer Institute Facilitator • MfA Program Associate Master Teaching Fellows and Field Support • Graduate of a previous summer institute

46. Prepare a poster to be presented to your classmates answering the following question. Will these two lines meet on the left side of the page, the right side of the page, both sides of the page, or never? Explain your reasoning with the following rules: You must find something to measure and say something about how to measure it. You must use your measurements in some way to defend your answer. Where do they meet?

47. Interesting student thinking that surfaced: Parallel did not mean the same thing to all students even though it had been defined. Alternative conception: (Jasmine) near and far perceptual intersection. Where do they meet?

48. Jasmine’s distinction helped authenticate the task. Students came up with several productive ways to tell if lines are parallel: If the “distance” between them ever shrinks, the lines are not parallel. “Distance” can be measured with a ruler. If you have the Geometer’s sketchpad, you can measure the slopes of the lines. Equal slopes  parallelism. If the lines head in the same direction, then the lines are parallel. Draw a perpendicular line to one. If it is perpendicular to the other, then the lines are parallel. Draw a transversal. Check for congruent angles in the “same position of both groups of angles formed”. Where do they meet?

49. Given the information below, will lines AB and DF meet on: The half-plane containing point C. The half-plane containing point D. Neither A nor B. Both A and B. Where do they meet?

50. Given the information below, will lines AB and DF meet on: The half-plane containing point C. The half-plane containing point D. Neither A nor B. Both A and B. Where do they meet?