A Mathematician’s Lament

1. A Mathematician’s Lament Inquiry Based Learning Pete Anderson pganders@henrico.k12.va.us Teacher of Mathematics Glen Allen High School HCPS

2. This workshop is entirely inspired by this text by Paul Lockhart.Free: http://www.maa.org/devlin/lockhartslament.pdfPurchase:http://www.amazon.com

3. Workshop Topics Web Site: http://gahs-math.wikispaces.com • What does it mean to “teach”? • Modeling • Problem Solving Techniques • How to Break the Cycle • Redefine the meaning of “Word Problem” • What have you done lately?

4. Who was your favorite teacher? Why?

5. bottom of page 1 “I had this one kid, though, man was she sensational! Her sheets were impeccable – every note in the right place, perfect calligraphy, sharps, flats, just beautiful. She’s going to make one hell of a musician one day.”

6. Elmer “Sonny” Sampson

7. Why go to music school?

8. Bottom of page 2 “I just use the Paint-by-Numbers kits supplied by the school board. Sadly, our present system of mathematics education is precisely this kind of nightmare.” Read on

9. Top of page 3 “The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, ‘math class is stupid and boring,’ and they are right.”

10. Spring 2010 – my Algebra 1 class

11. Bottom of page 3 “Simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.”

12. Modeling with Technology • Triangle in a Box • http://teachers.henrico.k12.va.us/glenallenhs/anderson_p/Lament/applets/Triangle_in_Box.html • Two Segments – Four Sided Shape • http://teachers.henrico.k12.va.us/glenallenhs/anderson_p/Lament/applets/TwoSegments_FourSidedShape.html • Point on a Grid • http://teachers.henrico.k12.va.us/glenallenhs/anderson_p/Lament/applets/Level1.html • Line on a Grid • http://teachers.henrico.k12.va.us/glenallenhs/anderson_p/Lament/applets/Level2.html • Parabola • http://teachers.henrico.k12.va.us/glenallenhs/anderson_p/Lament/applets/Level3.html • Angle of Incidence & Reflection • Sketchpad file

13. Bottom of page 4 “Now where did this idea of mine come from? How did I know to draw that line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck.”

14. Problem Solving Techniques • Solve an Easier Related Problem • Physical Representations • Work Backwards • Venn Diagrams • Algebra • Finite Differences • Other Ways to Organize Information • Other Ways to Change Focus • Other Forms of Spatial Organization • Draw a Diagram • Systematic Lists • Eliminate Possibilities • Matrix Logic • Look for a Pattern • Guess and Check • Sub-problems • Unit Analysis

15. A text by Ted Herr and Ken Johnson • Problem Solving Strategies: Crossing the River With Dogs and Other Mathematical Adventures(ISBN: 1559530685 / 1-55953-068-5 ) • Available at AbeBooks.com for as little as $1 (used). • GAHS Math text samples

16. Let’s Solve a Problem Crossing the River ... • Level 1 – with a wolf, goat, cat, and cabbage • Level 2 – with Missionaries and Cannibals • Level 3 – with Dogs In groups of ___, you have __ minutes to solve the problem.

17. Level 1: Wolf, Goat, Cat and Cabbage • Sailor Cat needs to bring a wolf, a goat, and a cabbage across the river. The boat is tiny and can only carry one passenger at a time. If he leaves the wolf and the goat alone together, the wolf will eat the goat. If he leaves the goat and the cabbage alone together, the goat will eat the cabbage. How can he bring all three safely across the river? • Online version • Flipchart

18. Level 2: Missionaries and Cannibals • Three missionaries and three cannibals must cross a river using a boat which can carry at most two people, under the constraint that, for both banks, if there are missionaries present on the bank, they cannot be outnumbered by cannibals (if they were, the cannibals would eat the missionaries.) The boat cannot cross the river by itself with no people on board. How can they all cross the river alive? • Online version • Flipchart

19. Level 3: Crossing the River with Dogs • The situation: The five members of a family and their five dogs (each family member owned one of the dogs) were hiking when they encountered a river to cross. They rented a boat that could hold three living things: people or dogs. Unfortunately, the dogs were temperamental. Each was comfortable only with its owner and could not be near another person, not even momentarily, unless its owner was present. The dogs could be with other dogs, however, without their owners. The crossing would have been impossible except that Lisa’s dog had attended a first-rate obedience school and knew how to operate the boat. No other dogs were that well educated. • The task: Explain in detail how do the five family members and their dogs cross the river?

20. How did you solve the problem? • What skills were required? • What previous knowledge was required? • Is there more than one solution? Is one solution better than another?

21. Rubrics Drive Instruction Original | Mine | HCPS – draft w/ template

22. Shall we do another? • Anno’s Hat Tricks • Oh Crap. It’s Trash Day! • The Locker Problem • anything from “The Canterbury Puzzles” – 1908 Rubrics: Original | Mine | HCPS – draft w/ template

23. middle of page 5 “By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject.”

24. Break

25. Turn Evil into Good

26. Turn Evil into Good Completing Exercises vsSolving Problems

27. Top of page 6 “… a self-perpetuation monster: students learn about math [from a traditional text and methodology] from their teachers, and teachers learn about it from their teachers.”

28. Middle of page 6 “… Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions.”

29. How to Break the Cycle • Problem Level • Unit Level • Nine Weeks Level • PBL • Flipped Classroom • Backwards Design

30. Problem Based Learning

31. Original Lesson Plan Area & Circumference lesson plan

32. The 4 Ds to Problem Solving Define: Write a list of questions but do not answer them yet. “What do you need to do or know in order to solve this problem?” Design:       Create a working sketch and equation that will solve the problem. Do:             Gather information needed to substitute into the equation and solve. Debrief:      Turn in your work with the grade you believe you have earned. Discuss with the teacher why you deserve this grade.

34. PBL Lesson Plan Area & Circumference Sample lesson plan with and with PBL approach http://geometryonline.pbworks.com/PBL

35. Always with a Rubric

37. Teaching for Tomorrow Teaching Content and Problem-Solving Skills by Ted McCain Amazon.com

38. How To Solve It – G. Polys A Four Step Process • Understanding the Problem • Devising a Plan • Carrying Out the Plan • Looking Back • Intro to Text • Full Text • Amazon.com

39. Flipped Classroom

40. The Flipped Classroom • What do you have to teach? • What do you want to teach? • Why not do both? The flipped classroom is a model of teaching in which a student’s homework is the traditional lecture viewed outside of class on a vodcast. Then class time is spent on inquiry-based learning which would include what would traditionally be viewed as a student’s homework assignment.

41. Traditional Pacing Guide This is what I haveto teach.

42. Flipped – Pacing Guide

43. The Flipped Classroom • My Lesson Plan • My Vodcast– Geometry • My Youtube Channel

44. Backwards Design a.k.a. Understanding by Design

45. What’s your best “big” project?

46. Backwards Design Stage 1 starts with educators identifying the desired results of their students by establishing the overall goal of the lessons by using content standards, common core or state standards. In addition, UbD's stage 1 defines "Students will understand that..." and lists essential questions that will guide the learner to understanding. Stage 1 also focuses on identifying "what students will know" and most importantly "what students will be able to do".

47. Backwards Design Stage 2 focuses on evidence of learning by assessment. Teachers plan performance tasks and evidence of understanding. Performance tasks determine what the students will demonstrate in the unit and what evidence will prove their understanding. This can include self-reflections and self-assessments on learning.

48. Backwards Design Lastly, stage 3 lists the learning activities that will lead students to your desired results.

49. Example of the UbD Approach Geometry: Unit One • What are the Standards of Learning • Unpack the Standards using Bloom's  • Essential Questions • How and why do we communicate through measure? • How and why do we mathematically describe naturally occurring phenomenon? • When, if ever, is a mathematical solution practical?  • Performance Based Assessment - "It's in the Hole!" • Directions • Sample • Rubric • Photos • Activity/Lesson Plan #4 • Activity/Lesson Plan #3  • Activity/Lesson Plan #2  |  Henrico 21 Entry • Activity/Lesson Plan #1 • Plan a schedule for the marking period and fill in with needed traditional material 

50. End Result