student presentations in calculus A vehicle for getting students to talk about mathematics Increasing Student Engagement & Understanding through Active Learning Strategies in Calculus I Dana C. Ernst Northern Arizona University January 8, 2016

claims 1. An education must prepare a student to ask & explore questions in contexts that do not yet exist. That is, we need individuals capable of tackling problems they have never encountered & to ask questions no one has yet thought of. 2. If we really want students to be independent, inquisitive, & persistent, then we need to provide them with the means to acquire these skills. 1

what is ibl? ∙ Often involves very little lecturing & typically student presentations focused. ∙ Example: Moore Method, after R.L. Moore. ∙ Students should as much as possible be responsible for: 2. Guiding the acquisition of knowledge, 2. Validating the ideas presented (i.e., students should not be looking to the instructor as the sole authority). ∙ Two Typical Approaches/Modes to IBL 1. Student presentations. 2. Small group work. ∙ Most IBL instructors implement some combination. 2

my version of ibl in calculus ∙ Students responsible for digesting most new material out of class by working on a sequence of problems. ∙ Nearly all class time devoted to students presenting proposed solutions/proofs. ∙ Each batch of problems are meant to do some subset of the following: ∙ Introduce a new topic ∙ Develop intuition about a concept ∙ Synthesize ideas from a few concepts ∙ Prove a theorem ∙ Get practice doing routine or non-routine problems ∙ My students prove most of the theorems. ∙ Large-ish WeBWorK assignments due weekly. 3

my version of ibl in calculus ∙ Presentations typically take one of 3 forms. 1. An individual presenting their proposed solution to whole class. 2. An individual presenting their proposed solution to a small group. 3. An individual acts as a spokesperson for his/her small group & presents the group’s proposed solution to whole class. ∙ Instructor’s role: guide discussion & nudge students to ask the right questions. 4

my version of ibl in calculus 5

my version of ibl in calculus 6

my version of ibl in calculus 7

my version of ibl in calculus 8

my version of ibl in calculus 9

my version of ibl in calculus 10

more on student presentations ∙ Presentations are meant to drive classroom discussion, not to prove to you that Sally knows how to do Exercise 15. ∙ The perfect presentation is one that is interestingly wrong. “You will become clever through your mistakes.” — German proverb “You will become clever through your mistakes.” — German proverb ∙ One reason IBL works: Mode of engagement is different when listening to expert vs novice. “Student as skeptic.” ∙ Asking students to prove theorems, make conjectures, come up with examples/counterexamples, and come up with definitions generally make productive conversations. “You will become clever through your mistakes.” — German proverb 11

questions to ponder ∙ How much scaffolding will you provide on the problems? ∙ Mode of student presentations? ∙ How will you choose presenters? ∙ How will you assess presenters? ∙ Are students expected to generate proofs of theorems? Will proofs be assessed on exams? ∙ What constraints do you have on physical space? ∙ What’s your plan for obtaining student buy-in? 12

why ibl? The Colorado Study by Sandra Laursen et al. 300 hours of classroom observation, 1100 surveys, 110 interviews, 220 tests, & 3200 academic transcripts, gathered from > 100 course sections at 4 campuses over 2 years. IBL Interviews SALG Pre/Post Tests Transcripts Gender Observations Non-IBL 13

why ibl? Laursen et al. 2013 “Our study indicates that the benefits of active learning experiences may be lasting & significant for some student groups, with no harm done to others. Importantly, ‘covering’ less material in inquiry-based sections had no negative effect on students’ later performance in the major.” Laursen et al. 2014 “Despite variation in how IBL was implemented, student outcomes are improved in IBL courses relative to traditionally taught courses, as assessed by general measures that apply across course types. Particularly striking, the use of IBL eliminates a sizable gender gap that disfavors women students in lecture-based courses.” 14

personal reflections ∙ In an IBL class there are lots of issues that bubble to the surface that we blissfully ignore when lecturing. Feature not a bug! ∙ We are responding in real time to what the students are doing & thinking. ∙ Some IBL class sessions look rougher than others because students are in the process of learning difficult things. #PF ∙ In contrast, in a lecture class, we control everything that happens at every instant. This can look lovely to an observer but buries most of the messiness. IBL is jazz! 15

personal reflections ∙ With the right set of materials, content coverage is not really an issue. Pace accelerates. ∙ Keeping my mouth shut…and assessing ∙ If I spend 50 minutes talking, it’s unlikely I’ve done any assessment. ∙ In an IBL course, nearly whole class session is spent on assessment. ∙ Students presenting, discussing, & collaborating provides me & them with immediate feedback about how things are going. 16

examples Suppose a function f has the following graph. Assume that the graph is made up of parts of lines and parts of circles. −3 −2 −1 1 2 3 2 If F is an antiderivative of f such that F(−4) = 42, find F(−1). 17

examples As a preparation for the long bright summer days, Dr. Acula plans to store plasma in closed tin cans that have the shape of a cylinder. As an environmentally conscious member of the community, eh wants to use as little metal as possible. What should be the relationship between the radius and the height of his cans? 18

examples “Are we there yet?”, asked Kelly. This was not the first time this question came up and the odometer in the family Dodge was broken, so he did not know how far they had traveled. Luckily, Kelly’s brother Bud traveled with them and he diligently check the speedometer every 10 minutes. He had the following measurements in this notebook: 30 40, 35, 40, 20. What distance did they cover so far approximately? 19

ibl resources ∙ Academy of IBL (inquirybasedlearning.org) ∙ IBL mentoring ∙ Mini-grants ∙ IBL Workshops (iblworkshop.org) ∙ Discovering the Art of Mathematics (artofmathematics.org) ∙ Journal of Inquiry-Based Learning in Mathematics (jiblm.org) 20