Skip to content
ADVERTISEMENT
Sign In
  • Sections
    • News
    • Advice
    • The Review
  • Topics
    • Data
    • Diversity, Equity, & Inclusion
    • Finance & Operations
    • International
    • Leadership & Governance
    • Teaching & Learning
    • Scholarship & Research
    • Student Success
    • Technology
    • Transitions
    • The Workplace
  • Magazine
    • Current Issue
    • Special Issues
    • Podcast: College Matters from The Chronicle
  • Newsletters
  • Events
    • Virtual Events
    • Chronicle On-The-Road
    • Professional Development
  • Ask Chron
  • Store
    • Featured Products
    • Reports
    • Data
    • Collections
    • Back Issues
  • Jobs
    • Find a Job
    • Post a Job
    • Professional Development
    • Career Resources
    • Virtual Career Fair
  • More
  • Sections
    • News
    • Advice
    • The Review
  • Topics
    • Data
    • Diversity, Equity, & Inclusion
    • Finance & Operations
    • International
    • Leadership & Governance
    • Teaching & Learning
    • Scholarship & Research
    • Student Success
    • Technology
    • Transitions
    • The Workplace
  • Magazine
    • Current Issue
    • Special Issues
    • Podcast: College Matters from The Chronicle
  • Newsletters
  • Events
    • Virtual Events
    • Chronicle On-The-Road
    • Professional Development
  • Ask Chron
  • Store
    • Featured Products
    • Reports
    • Data
    • Collections
    • Back Issues
  • Jobs
    • Find a Job
    • Post a Job
    • Professional Development
    • Career Resources
    • Virtual Career Fair
    Upcoming Events:
    College Advising
    Serving Higher Ed
    Chronicle Festival 2025
Sign In
casting out 100

Casting Out Nines

Where math, technology, and education cross.

The Chronicle Blog Network, a digital salon sponsored by The Chronicle of Higher Education, features leading bloggers from all corners of academe. Content is not edited, solicited, or necessarily endorsed by The Chronicle.

Better examples through peer instruction

By Robert Talbert October 25, 2011
3088780713_f631c9e901_z

I just gave midterm evaluations in my classes, and for the item about “What could we be doing differently to make the class better?”, many students put down:

To continue reading for FREE, please sign in.

Sign In

Or subscribe now to read with unlimited access for as low as $10/month.

Don’t have an account? Sign up now.

A free account provides you access to a limited number of free articles each month, plus newsletters, job postings, salary data, and exclusive store discounts.

Sign Up

3088780713_f631c9e901_z

I just gave midterm evaluations in my classes, and for the item about “What could we be doing differently to make the class better?”, many students put down: Do more examples at the board. I think I’ve seen that request more often than any other in my classes at midterm. This is a legitimate request (it’s not like they’re asking for free points or an extra day in the weekend), but honestly, I’m hesitant to give in to it. Why? Two reasons.

First, doing more examples at the board means more lecturing, therefore less active learning, and therefore more passivity and dependence by students on authority. That’s bad. Second, we can’t add more time to the meetings, so doing more examples means either going through them in less detail or else using examples that are overly simple. In the first case, we have less time for questions and deep thought, and therefore more passivity and dependence. In the latter, we have examples that lack the conceptual clarity and generalizability to be of any use when solving a real problem. That’s also bad.

I think what students really want when they ask for “more” examples is to be more involved with the examples they get. They want more than a false sense of competency that often follows a well-constructed example that’s performed by an expert -- they want first-hand experience with constructing solutions. That’s great! But this can be done poorly too, for instance through group work that throws students into the deep end of a difficult problem and leaves them to fend for themselves. What’s needed is a middle ground between pure lecture and pure group work.

ADVERTISEMENT

For me, that middle ground is peer instruction. I’ve been using PI with all my classes this semester, but especially in Calculus 2 it seems to provide a happy medium for students and for me.

Here’s an example PI question from a unit on sequences:

pi calc 2 1

Students were asked to think quietly by themselves for one minute, then vote using clickers. This slide followed a 10-minute minilecture that included a graphical interpretation of what “convergence” and “divergence” of sequences means, the technical definition of convergence, and five small examples of sequences and their convergence and divergence behaviors. When I say “small” I mean all five could be done in less than three minutes. So this wasn’t a lot of practice.

The correct response here is (d), Diverges, because the sequence does not get arbitrarily close to a single numerical value as \(n \to \infty\). On the first vote, the votes were 8% for response (a), 20% for response (c), and 72% for response (d). That 28% off the right answer was a little bothersome, so I just showed the histogram of clicker responses and had students get into groups of 2 or 3, with the instructions that they were to convince the other members of their group that they are correct. (Or if they are all in agreement, come up with a solid verbal explanation.) On the second vote, 100% of the class voted (d). All we needed was a quick debrief of why that answer was right, and we’re done. (That was for one section of Calculus 2. My other section had very similar numbers on that question.)

This PI question was better than an example worked at the board because the students were directly involved with it and because it highlights the concept at hand without getting bogged down in calculations. In subsequent problems, we could refer back to this clicker question and get a lot of use out of it. For example, students often mistake being divergent for being unbounded, but whenever that came up, we’d just say, “Remember that clicker question where the sequence was just \(-1, 1, -1, 1, \dots\)?” Yes, it does take longer than an example at the board -- a PI question will take a minimum of 3 minutes for thinking, voting, and peer instruction without factoring in any debriefing. But the investment in time pays off in terms of having a longer shelf-life. Students tend to remember PI questions that generate a lot of discussion, as opposed to lecture items in which they have no vested interest.

Here’s another PI question:

pi calc 2 2

This is a “loaded question” item because there’s no definition of “best” in this context. On the first vote, about 59% of the students voted (c) for the Integral Test, and about 38% voted for either the Comparison test or the Limit Comparison Test. Interestingly, this was at the end of a class session on the two comparison tests, but students didn’t default to those two methods for their answers just because that’s what we were covering. I put students into groups to discuss/convince, and at the second vote, the numbers were 82% for the Integral Test and 18% for the Limit Comparison Test. At the end, I asked the Integral Test voters why they thought that was best -- and they told me how to set up the integral and calculate it. Then I asked the Limit Comparison Test voters the same thing, and they told me how to set up and calculate the limit. I added the Comparison Test approach at the end because I happen to think that’s the easiest way to do this. So out of this one PI question actually came three examples.

ADVERTISEMENT

I’m telling students that peer instruction is better than examples because, although we don’t do as many of them as we might do lectured examples, we go into them more deeply and they get their hands on them first.

What about you? Are you using peer instruction in your classes, or maybe some other way to leverage better examples into your classes with active learning? Let’s hear it in the comments.

Image: http://www.flickr.com/photos/attercop311/

We welcome your thoughts and questions about this article. Please email the editors or submit a letter for publication.
Share
  • Twitter
  • LinkedIn
  • Facebook
  • Email
About the Author
Robert Talbert
Robert Talbert is a mathematician and educator with interests in cryptology, computer science, and STEM education. He is affiliated with the mathematics department at Grand Valley State University.
ADVERTISEMENT
ADVERTISEMENT

More News

Vector illustration of large open scissors  with several workers in seats dangling by white lines
Iced Out
The Death of Shared Governance
Illustration showing money being funnelled into the top of a microscope.
'A New Era'
Higher-Ed Associations Pitch an Alternative to Trump’s Cap on Research Funding
Illustration showing classical columns of various heights, each turning into a stack of coins
Endowment funds
The Nation’s Wealthiest Small Colleges Just Won a Big Tax Exemption
WASHINGTON, DISTICT OF COLUMBIA, UNITED STATES - 2025/04/14: A Pro-Palestinian demonstrator holding a sign with Release Mahmud Khalil written on it, stands in front of the ICE building while joining in a protest. Pro-Palestinian demonstrators rally in front of the ICE building, demanding freedom for Mahmoud Khalil and all those targeted for speaking out against genocide in Palestine. Protesters demand an end to U.S. complicity and solidarity with the resistance in Gaza. (Photo by Probal Rashid/LightRocket via Getty Images)
Campus Activism
An Anonymous Group’s List of Purported Critics of Israel Helped Steer a U.S. Crackdown on Student Activists

From The Review

Illustration of an ocean tide shaped like Donald Trump about to wash away sandcastles shaped like a college campus.
The Review | Essay
Why Universities Are So Powerless in Their Fight Against Trump
By Jason Owen-Smith
Photo-based illustration of a closeup of a pencil meshed with a circuit bosrd
The Review | Essay
How Are Students Really Using AI?
By Derek O'Connell
John T. Scopes as he stood before the judges stand and was sentenced, July 2025.
The Review | Essay
100 Years Ago, the Scopes Monkey Trial Discovered Academic Freedom
By John K. Wilson

Upcoming Events

07-31-Turbulent-Workday_assets v2_Plain.png
Keeping Your Institution Moving Forward in Turbulent Times
Ascendium_Housing_Plain.png
What It Really Takes to Serve Students’ Basic Needs: Housing
Lead With Insight
  • Explore Content
    • Latest News
    • Newsletters
    • Letters
    • Free Reports and Guides
    • Professional Development
    • Events
    • Chronicle Store
    • Chronicle Intelligence
    • Jobs in Higher Education
    • Post a Job
  • Know The Chronicle
    • About Us
    • Vision, Mission, Values
    • DEI at The Chronicle
    • Write for Us
    • Work at The Chronicle
    • Our Reporting Process
    • Advertise With Us
    • Brand Studio
    • Accessibility Statement
  • Account and Access
    • Manage Your Account
    • Manage Newsletters
    • Individual Subscriptions
    • Group and Institutional Access
    • Subscription & Account FAQ
  • Get Support
    • Contact Us
    • Reprints & Permissions
    • User Agreement
    • Terms and Conditions
    • Privacy Policy
    • California Privacy Policy
    • Do Not Sell My Personal Information
1255 23rd Street, N.W. Washington, D.C. 20037
© 2025 The Chronicle of Higher Education
The Chronicle of Higher Education is academe’s most trusted resource for independent journalism, career development, and forward-looking intelligence. Our readers lead, teach, learn, and innovate with insights from The Chronicle.
Follow Us
  • twitter
  • instagram
  • youtube
  • facebook
  • linkedin