1. Invitation: Visit my classes any time (with advance notice)
Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, about how much longer will it take for the snowball to melt completely?
1 hour, 30 minutes
3 hours
3 hours, 35 minutes
4 hours, 20 minutes
11 hours, 30 minutes
To vote via phone:
Room COX6694
Invitation: Visit my classes any time (with advance notice)

2. The Joys and Challenges of Implementing IBL in Calculus
Jonathan A. Cox
Fredonia Sigma Xi
November 10, 2017

3. The problem
Julia and Nancy – analysis of the academic behavior of Fredonia students after receiving a D, F, or W in calculus (December 1)
The DFW rate in calculus classes at Fredonia is sometimes 40% or higher.
Fredonia recognized in 2012 as having a “Particularly Successful Calculus Program”!
I’ll mention a couple broader and older examples of the type of data that’s concerning.

4. Data cited in Talking About Leaving by Elaine Seymour and Nancy M
Data cited in Talking About Leaving by Elaine Seymour and Nancy M. Hewitt, 1997.
Between 1966 and 1988, freshman interest in
mathematics majors declined from 4.6% to 0.6%
physical science majors from 3.3% to 1.5%.
A 1993 report indicated a relative 40% loss in SME (i.e., STEM) majors between freshman and senior years.
Another series of studies estimated between 34% and 40% of high school graduates intending to major in STEM abandoned the idea at or before enrollment in college.
35% of STEM majors switched to other majors in the transition from freshman to sophomore year.

5. Data cited in Talking About Leaving
Another project focused on asking women about their experiences in freshman STEM classes. 71% reported a variety of negative experiences, including, in order of importance,
poor teaching or organization of material
hard or confusing material, combined with loss of confidence in their ability to do science
cut-throat competition in assessment systems
grading systems that did not reflect what students felt they had accomplished.

6. These trends have NOT reversed.
Yousuf George’s Randolph Lecture at last month’s Seaway Section meeting.
A 2016 statement from the Conference Board of the Mathematical Sciences (CBMS) reinforced a central theme from Talking About Leaving: “even well-supported and well-prepared students who intend to enter STEM field face inherent barriers to success in our current mathematics education system.”
Familiar conversation with an advisee

7. Foundation for solutions: Active Learning
The CBMS is a consortium of fifteen American mathematics organizations.
It released a statement Active Learning in Post-Secondary Mathematics Education in Some quotes from the statement:
“[W]e use the phrase active learning to refer to classroom practices that engage students in activities, such as reading, writing, discussion, or problem solving, that promote higher-order thinking.”

8. Quotes from CBMS statement
“A wealth of research has provided clear evidence that active learning results in better student performance and retention than more traditional, passive forms of instruction alone.”
“These methods have been shown to strengthen student learning and achievement in mathematics, to foster students’ confidence in their ability to do mathematics, and to increase the diversity of the mathematical community.”

9. Quotes from CBMS statement
“[W]e call on institutions of higher education, mathematics departments and the mathematics faculty, public policy-makers, and funding agencies to invest time and resources to ensure that effective active learning is incorporated into post-secondary mathematics classrooms.”
An appended document with background on the statement elaborates “The MAA National Studies of College Calculus projects found that, when combined with a foundation of good teaching practices, AL [active learning] techniques have a positive impact on student confidence in Calculus (Bressoud, 2015, 2016).”

10. Inquiry-based Learning (IBL)
Definition. Inquiry-based learning is a pedagogical framework which engages students in answering questions or solving problems without giving them answers or solutions. It aims to develop in students the ability to investigate problems independently. Typical (but not universal) features follow.

11. Inquiry-based Learning (IBL)
The course is based on a carefully constructed sequence of tasks (problems to be solved, propositions to be proved or disproved, etc.).
The tasks are provided in the form of course notes that may also contain assumptions, definitions, a few motivating examples, and some background explanation, but not much else.
There is typically either no textbook or a textbook designed for an IBL course.

12. Students presenting solutions is the primary classroom activity
Students presenting solutions is the primary classroom activity. Students actively participate in critiquing the solutions and contributing ideas; these ideas and explanations drive progress in the course.

13. Students are engaged in apprenticeship into the practice of mathematics.

14. The learning curve
The background to the CBMS statement further advises that “effective implementation of AL techniques requires a substantial learning curve for both faculty and students” – Committed to transform all classes to IBL 2017 – All classes all IBL throughout the year Possible due to sacrifice and favorable circumstances: Sabbatical in Fall 2016, teaching only two different courses in Spring 2017 and two different courses in Fall 2017.

15. Change is difficult: Institutional obstacles
Three major institutional obstacles contribute to the entrenchment of the currently dominant model of teaching calculus. 1. Textbooks (There will be more on this later.) 2. Online homework systems which frequently accompany textbooks (and create similar issues) 3. Midterm exams – “Teaching to the test”, not best match with course goals (More on this)

16. Progress on overcoming institutional obstacles
The Fredonia Calculus Committee approved phasing out our current traditional textbook (James Stewart) and replacing it with an open source textbook (initially trying Active Calculus by Matt Boelkins, et al.).
The committee also approved eliminating required use of the online homework system (WebAssign) for Calculus I as of Spring 2018.
I am not giving any midterm exams in calculus this semester. (But there are daily assessments.)

17. The learning curve Steep for full implementation
The slope starts small for partial implementation
It was for me. Project NExT: how I got on the curve.
Since about 2005: 2-3 presentations/semester
CBMS statement: “Developing expertise with unfamiliar teaching techniques is an incremental process, one that is best conducted in partnership with a community of colleagues, and with supportive resources from professional societies.”

18. What is involved in converting a course?
Turning lecture examples into problems that they would solve and present.
Course notes need to fill in background details and be neat and precise.
One lecture example sometimes needs to be split into multiple, more manageable problems for students.
I occasionally include diagrams.
Solutions from the lecture notes are removed.

19. Comparison Lecture notes IBL course notes

20. Comparison
Lecture notes
IBL course notes

21. Joys: Problem 44 Most difficult problem up to that point
Presented incorrect solution
Sat down to work on it more
Came back with a correct solution
Applause broke out.

22. Joys From “Google told me to do this.”
To “I spent HOURS figuring this out!”
Class on the day I’m not there
The suspense during presentations: I walk into class never knowing what will happen that day.
When a student explains or solves something in a way that I never thought of

23. Snowball Problem – summary of pieces
Problem 69 Describe the behavior of the radius 𝑟 as a function of time 𝑡 and express the rate of change in derivative notation. Sign of derivative? Problem 70 Express as an equation the information that it takes 3 hours for the snowball to decrease to half its original volume. Problem 72 Give a formula for surface area of a sphere, and express as an equation that the rate of change of the snowball’s volume is proportional to its surface area.

24. Snowball Problem – summary of pieces
By now a differential equation has been obtained. Problem 90 Solve it for 𝑑𝑟 𝑑𝑡 and simplify. Problem 91 What type of function is 𝑟 as a function of 𝑡? Problem 92 Solve for 𝑟(3) in terms of 𝑟(0).

25. Snowball Problem – summary of pieces
Problem 93 Find a formula for 𝑟(𝑡) that includes 𝑟(0) as a parameter. Problem 94 Find the 𝑡-intercept of 𝑟. Problem 95 Answer the original question: How much longer will it take the snowball to melt? Express it in hours, minutes, and seconds, rounding to the nearest second. Answer: About 11 hours, 32 minutes, 31 seconds Goal: Make this sort of sequence standard for exploration of content.

26. When were these problems presented?
Section 1
Section 2 69
2-08
2-15 70
3-24
2-13 72 90
3-22
2-22 91 92
3-27
4-03 93
3-31
4-07 94
4-19 95
4-05
4-21
Given 2-06
Given 2-15

27. Topics – first part of the course
Precalculus review Velocity and Tangent Problems Infinitesimals and differentials Instantaneous rate of change Basic derivative rules Snowball problem Marginal cost Chain Rule

28. Topics – remainder of the course
Properties of functions Special types of functions Transformations of functions Combinations of functions Limits Limit laws Continuity Algebraic techniques for evaluating limits Limit definition of derivative Exponential functions and their derivatives
Product and Quotient Rules
Review of trigonometry
Limits and derivatives of trigonometric functions
Inverse functions and their derivatives
Logarithmic functions and their derivatives
Related rates problems
Maxima and minima

29. Standard topics NOT covered
Infinite arithmetic Intermediate Value Theorem Limits at infinity Higher order derivatives Implicit differentiation Linear approximation Logarithmic differentiation The Mean Value Theorem Derivatives and the shape of graphs (e.g. concavity, curve sketching) L’Hospital’s Rule Optimization Antiderivatives

30. Challenge 1
How can a calculus instructor cover all the traditional and expected topics while conducting an IBL course in which student progress drives the pace?

31. Clarence Stephens “Go fast slowly.”
Provide a smaller number of problems on each topic, enabling deep contemplation of each.
Also from Ted Mahavier
I am aspiring toward this standard.

32. Challenge 2
How can a calculus instructor keep students from passing on the opportunity to present solutions, and the keep the class as a whole from skipping over difficult problems?
Students called on may choose an available problem or pass.

33. List of available problems
Sample day in Calculus I 70, 82-87, 89-95, 102, 130, 131, 137, 141, , , , 161, 164, 166, 167; Revisit: 122; Reserved: 145, 163

34. Strategy 1 Deduct presentation point(s) for passing.
From the syllabus:
“Each student will receive two free passes on presenting for the semester. Additional passes will cost one point from the student’s presentation total. Furthermore, passing two classes in a row will cost two points, and so forth, up to a maximum deduction of four points per day.”

35. Strategy 2: High priority problems
Oldest available problems are so designated
Any student could volunteer to present one of them and immediately get to do so.
Calculus I: Some problems still didn’t get presented and “expired”.
Calculus II: High priority problems which still aren’t presented become eligible to be placed on a quiz or assigned for out-of-class write-up.

36. List of available problems
Sample day in Calculus I 70, 82-87, 89-95, 102, 130, 131, 137, 141, , , , 161, 164, 166, 167; Revisit: 122; Reserved: 145, 163 Sample day in Calculus II 124, , and ; Problems 107 and 123 will be revisited. Problem 138 is reserved. (Problems through 130 were high priority.) Conclusion: Reasonable solution to Challenge 2

37. Challenge 3
Def. A batting order is a predetermined list of the students in a class, generated for each class meeting, that indicates the order in which students will be given the opportunity to present a solution. Challenge 3. How can an algorithm for setting the batting order be designed so that all students regularly have the opportunity to present, while at the same time discouraging passing and giving students with fewer presentations the chance to catch up?

38. Batting order for Calculus I
Prez. batting order
Number
Last
Just in action?
Ethan 6
3-Mar N
Tara
Robert
21-Apr
Elan
24-Apr
Calston 7
1-May
Jordan 8
19-Apr
James
5-May Y
Sarah 9
28-Apr
Elizabeth 11
Danli 12
Kermit
Andie 13
14-Apr
Mackenzie 14
Carly
3-May
Hanna 15
Lauren 17
Skye
Samuel 18
Aurora
Say Ra 20
Austin 24
Batting order for Calculus I

39. Batting order for Calculus II
Prez. batting order
Number
Last
Just in action?
Cesar 3
25-Sep N
Jessie 7
11-Oct
Tyler 8
Ian
Steven
13-Oct
Jessica
Crystal Be
Jamie 9
Cordelia
Alex
16-Oct
Josh 10
Coco
Rory 12
Eythan 4
18-Oct Y
Jake 6
Zach
Dave
Jason
Batting order for Calculus II

40. Solution to Problem 3 The primary sort is on “Just in Action?”.
Priority goes to those who were not active last class.
On average, those who did not present last class are the ones that have the opportunity in this class.
Those who passed last class remain at the top of the list, so I continue to hold their feet to the fire to get up there and present!

41. Questionnaire responses this week
“The teamwork of the class fostered by this teaching style is really incredible. If I ever become a professor I will definitely consider this approach.” “I do like that we need to present in front of the class because it helps to put words to a problem and explain it, I am not nearly as nervous as I was in the beginning of the semester.” “Overall the course … is teaching me important skills about problem solving and thinking for myself.”

42. Questionnaire response this week
“Because a significant number of students in the class can’t handle the class structure of essentially teaching themselves, I often end up spending several hours of my time each week helping a number of these students so they don’t fail…. This has caused a significant amount of stress for me, and has even triggered an anxiety attack…. It’s not uncommon for me to spend 3-4 hours helping 3-6 people 2-3 times a week. I feel like this wouldn’t happen if we discussed the topics in class before we get the course notes on them. I love calculus but I hate this class because of the situation that arose due to the class structure.”

43. A New Challenge
How do I prevent a situation where a student experiences excessive stress due to feeling compelled to help other students learn the material? I don’t have solution to propose yet. Ideas on this? Discussion on anything else? If there’s still time, I can describe my grading scheme for daily assessments.

44. EMRN rubric (Robert Talbert)

45. More reasons I’m not giving exams
No one tells me I must give midterm exams. Nonetheless, there is a long and well-established tradition or convention of doing so that makes eliminating them hard to even imagine.
Generally, taking a test is not a real world skill.
Exams take student focus off the mathematics and place it on grades. They also disrupt the flow of the course.

46. More reasons I’m not giving exams
Ultimately, I am interested in what a student can do at the end of the course.
The results of a midterm exam commonly have two detrimental effects on students. a) Students who do poorly on an exam become demoralized and in effect give up, concluding they cannot do well in the course. b) Students who do well on an exam may slack off, figuring they’ve got it made.

47. Then why give a final exam?
Students have an expectation of exams because they are ingrained in the system. It provides a level of familiarity. A final exam does measure some aspects of what students know at the end of the course, and I am interested in measuring this. It provides a potential point of comparison between an IBL course and a non-IBL course. Back to the main story