Teaching Problem Solving in Large Introductory Classes: The View from Physics

1. Teaching Problem Solving in Large Introductory Classes: The View from Physics Ken Heller School of Physics and Astronomy University of Minnesota 20 year continuing project to improve undergraduate education with contributions by: Many faculty and graduate students of U of M Physics Department and the U of M Physics Education Group Details at http://groups.physics.umn.edu/physed/ Supported in part by Department of Education (FIPSE), NSF, and the University of Minnesota

2. A Guide for Discussion • Problem Goals • Why Solve Problems? • What are Problems? • Experts and Novices • Teaching Problem Solving? • Modeling a Framework • Coaching • Supporting Real Problem Solving • Designing Problems • What is Context-Rich? • Why? • How Well Does It Work

3. Private Sector Gov’t Labs High Schools Employment Problem Solving Interpersonal Skills Technical Writing Management Skills Adv. Computer Skills Spec. Equip. & Proc. Business Principles Statistical Concepts Knowledge of Physics Advanced Mathematics 0 50 0 50 0 50 Percent Reporting Frequent Use Survey of Physics Batchelors, 1994-AIP

4. Survey of Faculty in Majors Requiring Introductory Physics Highest Rated Goalsscale 1 - 5 • Algebra-based Course (24 different majors) 1987 • 4.7Basic principles behind all physics • 4.2General qualitative problem solving skills • 4.2Overcome misconceptions about physical world • 4.0General quantitative problem solving skills • 4.0Apply physics topics covered to new situations Calculus-based Course (88% engineering majors) 1993 4.5 Basic principles behind all physics 4.5 General qualitative problem solving skills 4.4 General quantitative problem solving skills 4.2 Apply physics topics covered to new situations 4.2 Use with confidence • Biology Majors Course 2003 • 4.9Basic principles behind all physics • 4.4General qualitative problem solving skills • 4.3Use biological examples of physical principles • 4.2Overcome misconceptions about physical world • 4.1General quantitative problem solving skills (*3) • 4.0Real world application of mathematical concepts and techniques • 4.0 Know the range of applicability of the principles of physics

5. Lowest Rated (Biology Faculty)

6. Free Faculty Responses - Goals (Biology Faculty) 1. In your opinion, what is the primary reason your department requires students to take this physics course? Problem solving/math Underlying Principles Application • To understand the basic laws of physics; to be able to apply physical principles to other problems; to overcome fear of math, quantitative approach to science. • General understanding of how 1st & 2nd order linear differential equations explain behavior of various physical systems(mechanics, thermodynamics, electricity). • Living things rely on a number of physical principles. Concepts we cover in lecture & techniques/equipment used in the laboratory require an understanding of physics. Physics is fundamental to many biological processes, & develop skills in problem-solving & modeling. • Provide basic concepts in physics as applied to biological functions; learn how to think quantitatively about these applied physics concepts.

7. “Process of Moving Toward a Goal When Path is Uncertain” “Problem Solving Involves Error and Uncertainty” Exercise vs Problem What Is Problem Solving? • If you know how to do it, its not a problem. Problems are solved using general purpose tools Not specific algorithms A problem for your student is not a problem for you M. Martinez, PhiDeltaKappan, April, 1998

8. Means - Ends Analysis identifying goals and subgoals Working Backwards step by step planning from desired result Successive Approximations range of applicability and evaluation External Representations pictures, diagrams, mathematics Some General Purpose Tools General Principles of Physics

9. Students’ Misconceptions About Problem Solving You need to know the right formula to solve a problem: Memorize formulas Memorize solution patterns Actions that reinforce the misconception Test requires students to remember important equations Allow students to bring in "crib" sheets It's all in the mathematics: Manipulate the equations as quickly as possible Plug-and-chug Numbers are easier to deal with Plug in numbers as soon as possible Actions that reinforce the misconception Single step problems. Multi-part problems.

11. Remembering 311941483526616430678538799514282739 random 106614921620177618121860194120002006 pattern 1066 1492 1620 1776 1812 1860 1941 2000 2006 246810121416182022242628303234363840 One step rule based 011235813213455891442333776109871597 Two step rule based

12. Student Difficulties Solving Problems • Lack of an Organizational Framework that links previous experiences, knowledge, and procedures • Physics Misknowledge • Incomplete (lack of a concept) • Misunderstanding (weak misknowledge) • Misconceptions (strong misknowledge) • No Understanding of Range of Applicability • Always True • True under a broad range of well-defined circumstances • True in very special cases • True in this situation • Lack of internal monitoring skills (reflection on what they did and why, asking skeptical questions about their actions)

13. Students need instructional support to solve problems Cowboy Bob is camped on the top of Table Rock. Table Rock has a flat horizontal top, vertical sides, and is 500 meters high. A band of outlaws is at the base of Table Rock 100 meters from the side wall. Cowboy Bob decides to roll a large boulder over the edge and onto the outlaws. Determine how fast Bob will have to roll the boulder to reach the outlaws. Algebra-based Physics (second of four tests - 1989) Circled statements from evaluator

14. Components of Course • Teach Students an Organizational Framework • Emphasize decisions using physics • Rule-based mathematics • Use Problems that Require • An organized framework • Physics conceptual knowledge • Connection to existing knowledge • Use Existing Course Structure • Lectures (given by Professors) MODELING • Discussion Sections (run by TAs) COACHING • Labs (run by TAs) COACHING

15. General Problem Solving Skills (i.e. Polya 1957) 0 Problem-solving FrameworkUsed by experts in all fields How to solve a problem when you don’t know how Recognize the Problem What's going on? Describe the problem in terms of physics What does this have to do with physics ? Chi, M., Glaser, R., & Rees, E. (1982) Plan a solution Can I use what I know to get an answer? Execute the plan Get an answer Evaluate the solution Can this be true?

16. Problem Solving Worksheet used at the beginning of the course Plan the Solution Construct Specific Equations Execute the Plan Solve the Equations Focus the Problem Picture and Given Information Question: Approach: Describe the Physics Diagram and Define Quantities Describe the Answer Does it answer the question? Target Quantity(s): Does it have correct units? Quantitative Relationships: Is it unreasonable? Page 1 Page 2

17. Teaching Students to Solve Physics Problems • Solving Problems Requires Conceptual Knowledge: • From Situations to Decisions using internal knowledge • • Visualize situation • • Determine goal • • Choose applicable principles • • Choose relevant information • • Construct a plan • • Arrive at an answer • • Evaluate the solution Students must be taught a problem solving framework that does this explicitly

18. The Dilemma Start with simple problems to learn expert-like framework. Success using novice strategies. Why change? Start with complex problems so novice strategy fails Difficulty using new framework. Why change?

19. What Using Cooperative Groups Does for Teaching Problem Solving 1. Following a logical problem solving framework seems too long and complex for most students. Cooperative-group problem solving allows practice until the framework becomes more natural. 2. Complex problems that need a strategy are initially difficult. Groups can successfully solve them so students see the advantage of a logical problem-solving framework early in the course.

20. What Using Cooperative Groups Does for Teaching Problem Solving 3. The group interactions externalize the planning, connection, and monitoring skills needed to solve problems allowing students to observe them in others. 6. Coaching by instructors is more effective 4. Students practice using the language of the field, "talking physics“, and explicitly connecting it to their existing knowledge base. 5. Students must deal with and resolve their misconceptions. External clues of group difficulties Group processing of instructor input

21. Email 8/24/05 I was reading through your 'typical objections'. Another good reason for cooperative group methods: this is how we solve all kinds of problems in the real world - the real academic world and the real business world. I wish they'd had this when I was in school. Keep up the great work. Rick Roesler Vice President, Handhelds Hewlett Packard Cooperative Groups Having Students Work Together in Structured Groups • Positive Interdependence • Face-to-Face Interaction • Individual Accountability • Explicit Collaborative Skills • Group Functioning Assessment

22. Student Reaction to Learning Problem Solving Changing a deep held way of thinking is traumatic Death of your beloved ideas and way of doing something. Death of a loved-one (Elisabeth Kubler-Ross) • denial • anger • bargaining • depression • acceptance

23. 5 stages to a common traumatic event : Problem Solving! DENIAL --- I don’t really have to do all that? Try it again my own way! And again. Read the book or ask someone and then..., try again. ANGER --- "%$@^##& professor!", "I shouldn’t have to take this course. I should wait until someone else teaches it. This has nothing to do with what I need." Crumple up the paper and throw it away! “These problems are tricky, unclear, and just weird." BARGAINING --- "Oh please help me pass. Can I do extra work for extra credit. Just for once give us enough time to solve the problems.” DEPRESSION --- “What am I going to do. I'm going to fail. I give up. I’ll never be able to pass the course with this rotten professor. What's the use". ACCEPTANCE --- "Ok. I really need to have a logical and organized process to solve these problems. These problems really are the kind of thing I need to be able to solve. I can actually use this technique in my other classes."

24. 0 Why Group Problem Solving MayNot Work 1. Inappropriate Tasks 2. Inappropriate Grading 3. Poor structure and management of Groups

25. The Monotillation of Traxoline (attributed to Judy Lanier) It is very important that you learn about traxoline. Traxoline is a new form of zionter. It is montilled in Ceristanna. The Ceristannians gristerlate large amounts of fevon and then brachter it to quasel traxoline. Traxoline may well be one of our most lukized snezlaus in the future because of our zionter lescelidge. Answer the following questions. 1. What is traxoline? 2. Where is traxoline montilled? 3. How is traxoline quasselled? 4. Why is it important to know about traxoline?

26. 0 A Complex Process The procedure is quite simple but you may have to go somewhere else if the facilities are not adequate. Before the process begins, you form different groups. Of course, one group may be sufficient depending on how much there is to do. Next you get started. Be careful, a mistake can be costly.It is important not to overdo things. It is usually better to do too few things than too many. This is especially important when issues of compatibility arise. At first, the whole procedure might seem complicated since timing can be crucial. With practice, it can all become routine. After the procedure is completed, form groups again to complete the process. This whole cycle will need to be repeated often. Answer the following questions. 1. What is the process being discussed? 2. What facilities are needed? 3. What are some compatibility issues? 4. Why is it important to form groups? Household task Laundry

27. Appropriate Problems for Problem Solving The problems must be challenging enough so there is arealadvantage to using a problem solving framework. 1. The problem must be complex enough so the best student in the class is not certain how to solve it. The problem must be simple enough so that the solution, once arrived at, can be understood and appreciated.

28. 2. The task must be designed so that • the major problem solving heuristics are required (e.g. physics understood, a situation requiring an external representation); • there are several decisions to make in order to do the task (e.g. several different quantities that could be calculated to answer the question; several ways to approach the problem); • the task cannot be resolved in a few steps by copying a pattern.

29. 3. The task problem must connect to each student’s mental processes • the situation is real to the student so other information is connected; • there is a reasonable goal on which to base decision making.

30. The Form of the Question Your task is to design an artificial joint to replace arthritic elbow joints in patients. After healing, the patient should be able to hold at least a gallon of milk (3.76 liters) while lower arm is horizontal. The biceps muscle is attached to the bone at the distance 1/6 of the bone length from the elbow joint, and makes an angle of 80o with the horizontal bone. For how strong a force should you design the artificial joint? (The weight of the bone is negligible.) • Gives a motivation – allows some students to access their mental connections. • Gives a realistic situation – allows some students to visualize the situation. • Does not give a picture – students must practice visualization. • Uses the character “you” – allows some students to visualize the situation. • Cannot be solved in one step by plugging numbers into an equation – students must practice organized quantitative decision making and use mathematical skills. Called Context-Rich Problem

31. Context-rich Problems • Each problem is a short story in which the major character is the student. That is, each problem statement uses the personal pronoun "you." • The problem statement includes a plausible motivation or reason for "you" to calculate something. • The objects in the problems arereal(or can be imagined) -- the idealization process occurs explicitly. • No picturesor diagrams are given with the problems. Students must visualize the situation by using their own experiences. • The problem requires the student to make decisions. It can not be solved in one stepby plugging numbers into a formula.

32. ABSOLUTE SCALE “If you win, I do NOT lose.” Grading EVERYTHING WE WANT STUDENTS TO DO IS GRADED “If you don’t grade it, they don’t learn it!” • Always write physics principles and a logical, organized problem solving procedure. • Only basic equations given on test are allowed . • Small, but significant part of grades is for group problem solving. • During lecture, answers to questions are occasionally collected and graded (can be done electronically). • Predictions for lab problems are graded. X

33. Grading Rubric for Students Scaffolding 4 PHYSICS 1201.200 Final Exam December 19, 2005 This is a closed book, closed notes quiz. Calculators are permitted. The ONLY formulas that may be used are those given below. Define all symbols and justify all mathematical expressions used. Make sure to state all of the assumptions used to solve a problem. Credit will be given only for a logical and complete solution that is clearly communicated with correct units. Partial credit will be given for a well communicated problem solving strategy based on correct physics. MAKE SURE YOUR NAME, ID #, SECTION #, and TAs NAME ARE ON EACH PAGE!! START EACH PROBLEM ON A NEW PAGE. Each problem is worth 25 points: In the context of a unified solution, partial credit will be awarded as follows: a useful picture, defining the question, and giving your approach is worth 6 points; a complete physics diagram defining the relevant quantities, identifying the target quantity, and specifying the relevant equations with reasons is worth 6 points; planning the solution by constructing the mathematics leading to an algebraic answer and checking the units of that answer is worth 7 points; calculating a numerical value with correct units is worth 3 points; and evaluating the validity of the answer is worth 3 points. The 30 multiple choice questions are each worth 1.5 points.

34. Scaffolding 5 Control of Equations that are Allowed Equation sheet on the final exam

35. 1. What is the "optimal" group size? three (or occasionally four) 2. What should be the gender and performance composition of cooperative groups? heterogeneous groups based on past test performance. : - one from top third - one from middle third - one from bottom third two women with one man, or same-gender groups Structure and Management of Groups

36. 3. How often should the groups be changed? For most groups: stay together long enough to be successful enough change so students know that success is due to them, not to a "magic" group. about four times first semester, twice second semester Structure and Management of Groups

37. do not use groups of two. assign and rotate roles: - Manager - Skeptic - Checker/Recorder - Summarizer Structure and Management of Groups 4. How can problems of dominance by one student and conflict avoidance within a group be addressed?

38. 5. How can individual accountability be addressed? • assign and rotate roles, group functioning; • seat arrangement -- eye-to-eye, knee-to-knee; • individual students randomly called on to present group results; • group work is practice for individual tests; • each student submits an individual lab report. Structure and Management of Groups

39. Course Structure Four hourseach week, sometimes with informal cooperative groups. Model constructing knowledge, model problem solving framework. LECTURES One houreach Thursday – cooperative groups practice using problem-solving framework to solve context-rich problems. Peer coaching, TA coaching. RECITATIONSECTION Two hourseach week -- same groups practice using framework to solve concrete experimental problems. SameTA. Peer coaching, TA coaching. LABORATORY Friday lecture -- problem-solving quiz & conceptual questions (2 problems, 10 multiple choice) (1 group problem in previous discussion section) every 3 weeks. TESTS

40. model coach fade Cognitive Apprenticeship Instruction influence development of existing ideas INSTRUCTION filter give meaning to new ideas Learning in the environment of expert practice • Why it is important • How it is used • How is it related a person’s existing knowledge sights and sounds Collins, Brown, & Newman (1990)

41. Scaffolding Additional structure used to support the construction of a complex structure. Removed as the structure is built Examples of Scaffolding in teaching Introductory Physics • An explicit problem solving framework • A worksheet that structures the framework • Cooperative group structure that encourages productive group interactions • Grouping rules • Group roles • Group reflection • Limit use of formulas by giving an equation sheet (only allowed equations) • Explicit grading rubric for problem solutions to encourage expert-like behavior • Problems that discourage novice problem solving • Explicit grading rubric for lab problems to encourage expert-like behavior

42. Math pre vs FCI pre – Biology Students R2 = 0.14 Correlation between a math skills test and a physics concept test General test taking (preparation, IQ, …)

43. FCI pre vs Math pre – Engineering students R2 = 0.20 Correlation between a math skills test and a physics concept test General test taking (preparation, IQ, …)

44. Problem-Solving vs. Math pre – Engineering Students

45. Problem-Solving vs. Math pre – Biology Students

46. PS vs FCI pre – Engineering students

47. PS vs FCI pre – Biology Students

48. The End Please visit our website for more information: http://groups.physics.umn.edu/physed/