A guide-on-the-side approach to calculus

Rewind a few years. Glowing student evaluations, as well as recurring teaching awards, indicated that I was effectively doing my job. However, two observations made me reconsider how well I was really doing. Namely, many of my students seemed to heavily depend on me to be successful and retain only some of what I had taught them. Inspired by a desire to address these concerns I began transitioning away from “sage on the stage” towards “guide on the side.” In particular, I began adopting an inquiry-based learning (IBL) approach in my proof-based courses. Yet, due to larger class sizes, significant content expectations, and a desire to maintain some level of sanity while I retooled many of my courses, I continued to teach calculus via direct instruction. Fast forward to the present. Consistent with a growing body of evidence, I have witnessed improved student outcomes in my IBL courses, as well as in subsequent courses. Compelled by my experiences, together with an increasing number of students in my first-semester calculus courses that have previously taken calculus in high school, I decided it was time to chuck my lecture notes and embrace an IBL paradigm in calculus. In this talk, I will relay my experience teaching calculus in the fall of 2014 using a modified-Moore method.

This talk was presented in the MAA Contributed Paper Session on First-Year Calculus: Fresh Approaches for Jaded Students on January 12, 2015 at the 2015 Joint Mathematics Meetings in San Antonio, TX.

Transcript

1. a guide-on-the-side approach to calculus
   First-Year Calculus: Fresh Approaches for Jaded Students
   Dana C. Ernst
   Northern Arizona University
   January 12, 2015
   

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2. about me
   ∙ Assistant professor at Northern Arizona University
   ∙ Spent 4 years at Plymouth State University prior to NAU
   ∙ When I started teaching, I mimicked experiences I had as a student
   ∙ By most metrics, I was an excellent instructor. But:
   “Things my students claim that I taught them masterfully, they
   don’t know.” — Dylan Retsek
   “Things my students claim that I taught them masterfully, they
   don’t know.” — Dylan Retsek
   ∙ Inspired by a desire to address these concerns, I began
   transitioning away from direct-instruction towards inquiry-based
   learning (IBL).
   “Things my students claim that I taught them masterfully, they
   don’t know.” — Dylan Retsek
   1
   

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3. about me (continued)
   ∙ 1st exposed to IBL during a Project NExT workshop run by Carol
   Schumacher in 2008
   ∙ Taught 1st full-blown IBL class in Fall 2009
   ∙ Currently Special Projects Coordinator for AIBL & mentor for new
   IBL practitioners
   ∙ But my calculus courses have been IBL-lite at best…
   2
   

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4. what is ibl?
   ∙ Students given tasks requiring them to solve problems, conjecture,
   experiment, explore, create, & communicate.
   ∙ Rather than showing facts and/or algorithms, the instructor guides
   students via well-crafted problems.
   ∙ Example: Moore Method, after R.L. Moore.
   ∙ Students should as much as possible be responsible for:
   1. Guiding the acquisition of knowledge,
   2. Validating the ideas presented (i.e., students should not be
   looking to the instructor as the sole authority).
   3
   

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5. why have i been stalling on ibl in calculus?
   ∙ Larger class sizes
   ∙ Significant content expectations
   ∙ Coordinated sections at NAU
   ∙ Desire to maintain some level of sanity while I retooled many of
   my courses
   4
   

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6. why is now different?
   ∙ Peer pressure
   ∙ Most of my excuses were gone & I was ready
   ∙ I was assigned the Honors section!!!
   ∙ Smaller class size
   ∙ I became a co-coordinator for Calculus 1 & Calculus 2
   5
   

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7. class demographics
   ∙ 24 honors students (10 female, 14 male)
   ∙ Majors: engineering, chemistry, biology, physics, CS, forestry,
   business, anthropology, comparative cultural studies (no math)
   ∙ All but one student was a first semester freshman
   ∙ All but 5 had taken calculus in high school
   ∙ Scored a mean of 14.1/20 = 70.5% on our calculus readiness test
   compared to 13.6/20 = 68% for all sections (range: [10, 20])
   ∙ Math biographies:
   ∙ “Mathematics is awesome. I’m really looking forward to this
   class.”
   ∙ “I always get an A in my math courses, but I don’t enjoy it.”
   ∙ “I hate math.”
   6
   

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8. classroom
   ∙ Four 50-minute class sessions
   ∙ Undergraduate peer TA for 3 days per week
   ∙ Large white boards on 3 walls
   ∙ Rows of large tables (not easy to move)
   ∙ Hard for me to move around and hide
   7
   

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9. course structure
   ∙ Used Brian Loft’s Differential Calculus IBL problem sequence (free
   at Journal of Inquiry-Based Learning in Mathematics)
   ∙ Course ran as a modified-Moore method:
   ∙ 5-10 problems assigned each class meeting. Due next class
   ∙ Students responsible for digesting new material out of class
   ∙ Nearly all class time devoted to students presenting proposed
   solutions/proofs (10% of course grade)
   ∙ Two modes of presentations:
   1. Volunteers presenting live, one problem at a time
   2. Number off, groups simultaneously write up solutions, select
   spokesperson
   8
   

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10. course structure (continued)
    ∙ Huge WeBWorK assignments due each week on a non-class day
    ∙ Exams:
    ∙ 4 exams & cumulative final exam
    ∙ Given on same days as other sections
    ∙ Covered same material as other sections with some common
    questions
    ∙ Additional conceptual questions (proofs, examples,
    counterexamples)
    ∙ I lectured roughly a total of 3 hours all semester
    9
    

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11. course structure (continued)
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12. course structure (continued)
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13. course structure (continued)
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14. course structure (continued)
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15. course structure (continued)
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16. course structure (continued)
    15
    

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17. reflections
    ∙ I had so much fun & was continuously impressed
    ∙ Getting buy-in took serious effort
    ∙ Having such a large percentage of students that have had calculus
    before was a blessing and a curse
    ∙ The students proved nearly all the theorems on the board with
    occasional gentle nudging
    16
    

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18. reflections (continued)
    ∙ Had trouble keeping up with other sections for first half of
    semester
    ∙ I was stalling & adding new material by end of semester
    ∙ My students increasingly out performed other sections
    ∙ All but one student was sold on the approach by the end of the
    semester
    ∙ A couple students had transformative experiences
    17
    

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19. student quotes
    ∙ “Even after having calculus previously, I understand the material
    so much better now.”
    ∙ “I think the environment created was really conducive to learning
    and trying out different tactics. Even though I hated it at times, I
    think the homework presentations really help and force you to
    know what you’re talking about; when you have to present it really
    made me want to have a solid understanding of the material so
    that I would present correctly.”
    18
    

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20. student quotes
    ∙ “I really liked how you explained where many of the formulas or
    theorems came from, rather than just having us rely on simply
    believing you.”
    ∙ “…class time was both productive and fun.”
    ∙ “The style of teaching itself is difficult for people who haven’t
    taken calc before. The majority of the class has, making it much
    easier. Whereas learning ideas myself with no experience takes
    time and effort that is difficult to have in a college setting. Also,
    expecting to memorize some formulas can be annoying.”
    19
    

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21. questions
    ∙ What if I don’t have the honors section?
    ∙ What if the class size increases?
    ∙ What would I do differently to accommodate the students I often
    have in calculus that are unprepared?
    20
    

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