Interview: Professor Matthew Boelkins, Grand Valley State University and Active Calculus

Hi everyone! A massive thanks to Professor Matthew Boelkins, Grand Valley State University, for taking the time to share some info about his project, Active Calculus.  -SY

1. Please tell us about yourself.
I've had the good fortune to serve on the math faculty at Grand Valley State University in Allendale, MI, for more than 20 years.  I'm also one of the two editors-in-chief of PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies), and have held that position for 5 years, following 5 years as associate editor.  My interests span a wide range of undergraduate teaching and learning of mathematics, but much of my recent creative energy has been focused on developing textbooks that encourage active learning.

2. Briefly describe “Active Calculus”? How much does it cost to the student?
Active Calculus (there are both single and multivariable texts; I'm the lead author of the single variable one, and will focus on that in my responses) is a free, open-source textbook designed for a standard calculus sequence taught from an active learning perspective.  Rather than lots of worked examples, most of the text is structured around activities for students that are designed to be completed before or during class, the latter with encouragement and feedback from peers and instructor.  Interested people can learn more at https://activecalculus.org/.

3. What are some of the reasons why you decided to write “Active Calculus?”
There were two main reasons.  First, I read an article in MAA FOCUS about the \$21M home built by the author of a popular calculus textbook.  I figured that if the author had earned that much in royalties, the publisher had likely made \$210M.  It was no longer tenable for me to ask my students to pay \$150 for a textbook that had ideas in it that had been well understood by humankind for decades, even centuries.  I wanted to write something that would be free for them.

In addition, while at the time I began writing (around 2010) I didn't fully know the scholarship that tells us why active learning is better,  I used a lot of active learning in my own teaching and had found my students to be more successful than when I used to lecture.  Having developed a collection of activities for calculus to use in my own teaching, I had the idea to use those as the foundation for the textbook, and that's what led to Active Calculus.  When the Freeman report came out in 2014 (https://www.pnas.org/content/111/23/8410), that provided added motivation for making the text a good one.

4. What is a typical day like in a course using “Active Calculus” when you teach using it? Can you use it with flipped learning or with IBL methods?
For each class meeting, I prepare a written script with estimated times.  In advance of most classes, students complete a "daily prep assignment" that normally consists of a short reading, a preview activity from the text, and 1-2 additional questions; students spend 30-45 minutes completing these, and their work is graded on effort and completeness for 5\% of their semester mark.  Most classes then begin with a "daily prep debrief & discuss" (6-8 minutes) where students check in with one another and see what questions they might want to discuss as a class.

From there, we usually engage in some brief (5-7 minutes) lecture & discussion to build on daily prep and set stage for an activity.  Students then work in groups of 3-4 on an activity from the text for 15-20 minutes, followed by or including some discussion for closure, transition, and new ideas (adding another 5-10 minutes), and then we are on to the next activity for 15-20 minutes.  Often I teach our 4-credit calculus class on a schedule with two 2-hour meetings a week, so we basically rinse and repeat this schedule for a second hour, but without a daily prep assignment to start the 2nd hour.

I've written a blog post that has some more information and detail, including references to key preface sections of the text for students and instructors, as well as a video for students on how to use the text: https://opencalculus.wordpress.com/2019/08/05/how-to-use-active-calculus/

I think the text is particularly well-suited to a flipped learning setup:  when I have students complete daily prep assignments, they are doing some key basic learning on their own outside of class.  The in-class activity-driven style also fits with engaging students during class in the some of the most important and demanding work of the course.  Some of my GVSU colleagues have created screencasts that accompany the text, and these would work especially well for a flipped experience:

• Chapters 1-4
• Chapters 5-8

For IBL practitioners who have worked with that approach using more traditional texts like Stewart or Hughes-Hallett, I think you'd find Active Calculus to be a suitable companion for such a course.

5. What are some of the responses by faculty and students after using “Active Calculus”?
For views of some faculty who have reviewed or used the text, see https://open.umn.edu/opentextbooks/textbooks/active-calculus-2-0.  I get a lot of email traffic about the text, and many of those responses express gratitude for the text.  Recently, an instructor who is a first-time user this fall sent me a very kind thank-you note: "I wanted to write to say how much I've appreciated Active Calculus.  I was a bit suspect (I've been using Stewart), but I am so impressed with it.  I like the problems, the students working in class, the organization ... it's really quite good.  I'm excited to be teaching this class again.  And thanks for saving our students some money as well."

This fall, I also surveyed Active Calculus users via my email list and Twitter, and in an open comment part of the survey, I got additional feedback such as:

"I appreciate you and your work on this so much. It's really let me teach calculus in the way I want to without having to create the material from scratch. A million thanks."

"The book is well thought out with great examples. When I teach the class, I really feel like I am actively working through the book with the students. One comment our Quantitative & Symbolic Reasoning Center director told me was that students in Active Calculus sections are asking why a concept is true and being stuck on the theory, while students not using AC are asking questions on the algebra and not thinking about the concepts of the course. "

"I really appreciate the structure and the way that the text works. It has been a game changer for my teaching of Single Variable Calculus! Our pre-calc teacher is now using Active Prelude as one of her core texts and we are excited to see how that influences students being prepared for my class."

"I've been using Active Calculus since my first year of teaching AP Calculus AB and I can't imagine using any other curriculum.  I cannot thank you enough for the time and energy that you have put into your work and my hundreds of students over the years thank you as well."

As to what students say, I think that many students find the book different at first, and thus at times frustrating.  They have been trained to expect a book that has lots of fully-worked examples, followed by exercises that are similar to the examples.  Active Calculus is not that way.

Some of the instructors who responded to my survey commented on what students say:

"It's an excellent book. Students and I both really appreciate its clarity and accessibility. Just yesterday I was recommending to a student the very nice "Summary" bits at the end of each section, and they said, "oh, that's exactly what I was looking for.""

"I really like the text and got mostly positive comments from the students on the course evaluations last year.  In the past, the comments were almost all negative or neutral."

"I asked my students and they expressed appreciation for how easy it is to read. Some students also said they wished there were more examples, which gave me an opportunity to remind them why there aren't more examples. :)"


6. How widespread is “Active Calculus”?
One of the challenges of having a text that's free and available online is knowing exactly where it is being used.

I'm aware of at least 17 4-year universities, 2 2-year colleges, and 5 high schools that have formally adopted Active Calculus as their required textbook.  People from at least another 30 institutions have responded to my recent survey to say they are using it fully in their own course, even though the text hasn't been adopted by all of their peers.

Here's the list of adopters I have at present:
California State University, Monterey Bay
Carroll College
Doane University
Dordt University
Lane Community College
Laurentian University (@ St. Lawrence College)
Lebanon Valley College
Lenoir-Rhyne University
Nevada State College
Scottsdale Community College
Sonoma State University
St. Mary's College of Maryland
Texas Lutheran University
The College of Idaho
University of Northern Colorado
Vermont Commons School
Vernon Hills High School
Westfield State University
Westminster College (SLC)
Westmont College
Wyoming Seminary

7. Could this book be used in high school?
Absolutely, and several of them have publicly shared their experiences.  For instance, Dave Sabol of St. Ignatius High School has used AC for several years and writes about corresponding activities he has developed on his blog, https://therationalradical.wordpress.com/calculus-resources/.  Jim Pardun and Steve Korney of Vernon Hills High School recently presented on their work teaching AP Calulus using AC at the regional NCTM conference in Nashville.

8. What are your future plans for your texts?
The overall goal is to keep making both Active Calculus and Active Prelude to Calculus better.

For Active Calculus, the original text was written in LaTeX and later converted to PreTeXt, which allows the HTML output.  I'm realizing that I haven't yet taken advantage of many of the features PreTeXt offers, such as having better cross-referencing, a better index, and more interactive features.  A first goal is to make revisions to take advantage of those features.  I am also considering adding Sage cells to the text to offer some embedded computation and experimentation for students; having some interactive computational opportunities is a second thing I hope to incorporate in the not distant future.  And a third significant goal is to have some additional exercises, ideally with many of them focused on modeling and applications.

For Active Prelude, this is the first year the text has been public, so I'm waiting on some user feedback to see where to focus energies next.

9. Anything else?
People can learn more at https://activecalculus.org and https://opencalculus.wordpress.com/, and I always appreciate hearing directly from users or people interested in the text by email at boelkinm at gvsu dot edu.

Even Mr. Miyagi Had Student Buy-In Issues Initially (Humor)

Even Mr. Miyagi had student buy-in issues. If you've seen the movie, The Karate Kid, you know these "wax on, wax off" training scenes, which is clearly active, Daniel-san-centered teaching. You wouldn't just want to only watch videos or watch demos, if you need to face real humans in a karate tournament. It seems like avoiding a kick to the head might require more than factual knowledge of what a kick is and knowing about the existence of a block or an avoidance move.

Let's take a look at a key segment, where Daniel doesn't see the point of all his practice, and has a blow up.

When viewing this scene from a teaching point of view, it comes down to Mr. Miyagi not framing and signposting the activities.  "You're doing this to get stronger at ..."  Instead, he assigned Daniel exercises, and did not tell him what these exercises were for. So naturally Daniel was upset, because he could not see the connection to learning karate. Finally, Mr. Miyagi does some student buy-in work, and has Daniel demonstrate that he actually has strengthened his defense abilities. At that point, Daniel made a big step forward in student buy-in, after getting some positive feedback from this teacher.

(Now of course this a movie, and drama is needed. I wouldn't change the movie, nor is this a film critique.)

If we boil things down to their essence, the situation is (a) instructors know what the connections are and the purpose of the work, however (b) students don't always see (or can't see) these connections, since they lack expert insight.  Hence, it's vital for teachers to signpost, "We are learning in this way, because..."

Returning to The Karate Kid, at some point, there has to be authentic meaning to the work. Daniel fully buys in after the "anniversary scene", where it is revealed that Mr. Miyagi had suffered tragedy in his family life, while earning a congressional medal of honor for his military service to the United States. This experience provides much needed perspective for the young man. The next scene is Daniel practicing with intent on his own - a sign of full buy-in.  Game on!

This suggests that buy-in has multiple levels. It's one thing to see why you are doing an exercise. It's another to be fully committed to the path forward. This is where assignments and activities focused on the meaning and value of the work can be useful. This can include but is not limited to the value of problem solving, growth mindset, and inspirational content, where students have an opportunity to learn universal lessons about what their education is for.

Wax on, wax off,
Wax on, wax off...

Cold Calling Pitfalls

This blog post is about cold calling. Randomly calling on students can go well, but it's not so easy as pulling a name and asking, "What's the answer!?"

The pitfalls of basic random cold calling is that it's not an equitable practice in its barebones form. Some issues include (but are not limited to) the following.

1. Making a mistake in front of the whole class can be stressful. This is a high risk situation.
2. Not everyone thinks fast, so they may be in the process of figuring something out, and then called on the spot to have the goods, right then and there. 
3. Thinking fast is not necessarily smart, and thinking slowly is not necessarily not-smart. But having to be quick on your feet biases the definition of smart as having the answer quickly in class on the spot.
4. Stereotype threat can be triggered. Some students are the only one of their identity in your class. If this student makes a mistake, then the stereotypes could be "validated." 

What can we do?

Randomizing who contributes does have a positive facet. It spreads the chances around. That's good. The key is to do it in an inclusive way.

Here's one example:

Instead of going with something like "What's the answer!?"you can add think-pair-share into the mix.  First, state the question or task, and ask students to think first and the talk to their partner. Then when you select a person at random, the prompt could be, "Share with us what you discussed with your partner." What students share could be a partial answer, a question, a comment, the whole answer, or perhaps asking for another minute to finish their conversation. If framed as a way to generate discussion and engagement, rather than having to be right, this can reduce anxiety and be a more inviting experience.

Riffs: We can think of think-pair-share as the base layer, and add on or adapt layers above it. Instead of randomly calling on a pair, you could visit a few groups and see what they tried. You can usually very quickly find a pair willing to share. Quiet students who have a useful idea can be encouraged to share, and this specific teacher move is where you enact inclusion! The quiet student is invited in by the instructor and is validated for having something worthy of being shared.  

Keeping track of who has shared will then help you spread chances equitably.  This riff works well earlier in the term, as you are developing student buy-in and comfort with talking about and discussing math. It's lower stakes, and doesn't put people on the spot.

Another riff is for classes that include student presentations. Assigning randomly the first chimers is a way to spread who makes the comments. Otherwise, it's the usual people raising their hands again and again. This can be done by selecting two students, then asking everyone to talk to their partner about the presented work. The selected students chime in first. They can ask a question, comment, or give a compliment. Again the contributions are done in a way, where students have time to think, discuss, and prepare remarks. Once the first chimers have made their comments, you can open the floor for further comments.

Cold calling can be warmed up with some additions and tweak. Using a small amount of time, allowing for thinking and talking, and taking perceived judgment off the table, creates an environment where students are less concerned about how they might appear and more focused on the Math. 

Student Buy-In Strikes Back

An Arstechnica article is making the rounds essentially about student buy-in. There are several layers to unpack here.

First, here's a link to the article: College students think they learn less with an effective teaching method They don't even realize they've learned more.

The key point of the article is that students liked lectures more, but did statistically better when taught via active learning. The article ends with suggesting that instructors give a short lecture on the benefits of active learning to deal with the issue.  Giving a pep talk is a good starting point, but not enough.

Why do we even need to work on student buy-in? The overarching reason is because teaching is a cultural activity. When people walk into a classroom, they have default, often unacknowledged assumptions about what is "supposed to happen." Deviations from the norms can create tension.

We need to unpack this further. In math, we have Math anxiety that messes up a lot of students. In general, we could describe this as having fixed mindsets about intelligence. "I'm not a math person" or "I learn best when shown all the steps that I can memorize..." come up as signals of this. This matters when we get to a point where students get stuck. Getting stuck is exactly the point where we have to confront our images of ourselves.  Getting stuck has been implicitly learned as equivalent to being stupid. The "smart" ones get it fast, and if you're not fast you're not smart. This is actually something that comes in education research (under the heading "Nonavailing beliefs", which are beliefs that inhibit or do not support learning).

When an instructor uses active learning that sets students up to have to make sense of something actively, then it's natural for students to get stuck sometimes. And when students get stuck, all those issues mentioned above get activated.

Another layer is the "answer getting" culture we've created. Much of school success has been about getting the answer. The most common questions that are asked in class are "What's the answer?" and "Is this right?" Rarely is it about, "Why is this true?" or "How else could we approach this idea?" So when we ask students to process ideas at a deep level, rather than crank out answers, then it creates yet another tension -- "Is this going to be on the test?"

Smart has been co-opted. Let's re-co-opt smart. Smart is working on ideas and problems, getting stuck, trying new ideas, collaborating, and so on... Smart is thinking of education as a journey. We need to educate students (and parents) what being smart is. With growth mindset research, we have a framework to have productive discussions about this.

Teaching is also a system. So if you teach X, but test Y, there's a problem obviously. But life is more subtle. If we focus on process in active learning, but test the easy-to-test things for whatever reason, then our assessments are saying we value Y, but our activities are saying we value X. Actions speak louder than words, and assessment is where you put your money where your mouth is as a teacher. The point here is the conditioning students go through is not just about what happens in class, but about the whole experience. Assessment is one of the big pieces of a class, and affects how students view learning. It's not just what activities we use in class. We also need to align our assessments (both summative and formative).

Perhaps one of the more troubling ideas from the article is that students can't identify they learned more. This made me pause.  This isn't new news, but it's a reminder.  Let's think about this.  In almost any other context this is truly odd. If you're learning to play the trumpet and are learning to hit high notes, you know when you've learned it. Of course, there's nuance in learning music, so I'm not trying to make it a binary learning outcome. But if students don't recognize they have learned more, it's a sign that there's more than just the specific teaching that's not right. One thing that jumps out is feedback and coaching. Students need regular feedback that they are learning and making good progress. Pointing out successes regularly and equitably is essential and goes a long way. "We learned this... Way to go, and we learned this because we worked on it, got stuck, and figured it out. That's smart!"

Now all this sounds like I might be blaming students to some extent. I'm not. Not in the slightest. This is about unpacking the layers of our system. Circling back and putting the layers together, we get the outcomes our teaching culture is designed to achieve, whether we realize it or not. We still have holdovers from the roots of the industrial revolution, where our model for education was created using a factory model. You know, bolt on the knowledge and you're good to go. But our goals are different today, and we are shifting towards humanistic education. That is education for developing people as human intellectuals.

Student buy-in is generally about this broader cultural shift. When students walk in the door it's our job as teachers to help them make this transition in mindset and purpose. If we just change the way we teach, and don't inform students, it's on us if they walk away with a bitter aftertaste.

How to get started? Let's get practical. After all that blabbing above, we need things we can do in class that work.  Linked below is a post from earlier this summer with a collection of links from what to do on Day 1 to ongoing strategies to digging deeper into Math Anxiety.

Student Buy-In In Practice Overview

And just yesterday I wrote a letter to students that can be used as a starting point to get students on board.

Letter: Dear Student

Letter: Dear Student

This letter is to students in college math classes, but might apply in other settings such as secondary math or other subjects.

Dear Student,
I am writing this letter to you, because your instructor, other instructors, all of us care. We care deeply about your success. We care about your future, and the future of others. That's why we went into teaching in the first place, a profession notorious for long hours, high commitment, and not the highest wages. Teaching is a calling, and our calling specifically is to help young people today to prepare them to solve tomorrow's problems. Teaching is a social responsibility to young people, to prepare young people with the knowledge, creative thinking, and values needed to live healthy, successful, impactful and meaningful lives.

Consider this next idea for a few minutes about time... Children who are in kindergarten today will retire in about  60 years. I write this in 2019, and that means a student in kindergarten will retire around the year 2079.  2079!  Think of the difference between what your life is like today, compared to 1959. The world has changed drastically in ways that people in 1959 could not predict. No way they could foresee smart phones, google, climate change, automation, globalization, etc.   The main point I want to get to is that there are questions and problems that young people (you) will have to solve that have not even been thought of yet by anyone on the planet. Your education today must prepare you to solve these unstated, problems far out into future.

What this means is that getting the answer in the back of the book isn't nearly enough.  Yes, it's good to check your answers sometimes, but that's just a small part of your education. Yes, it might have worked to get you into college or through your last math class. I totally get it. But learning isn't a set of check boxes or getting the answer that is in the back of the book. The people who invented the iPhone or the people who put the first human on the moon or your favorite band or your pediatrician or whomever else you admire or appreciate, they solved real-world problems and looked far beyond the back of the book. At some point, logic, reason, and creativity is what you need. Answers might be good for checking things off, but really the learning is the process to getting to the answer and being stuck on good questions that make you think.

Let's unpack some of the details. It's okay to ask, "Is this right?"  Please ask for help, and ask for help every time you need it, even for small questions. But I hope you also go farther, and ask questions that expand your thinking. "Why does this work?" or "Why doesn't this work?" or "Is there another way to look at this?" Memorizing isn't thinking, by the way. We can teach computers to memorize better than humans, and thus memorizing isn't as important as it might seem. Sure it might help you get back a multiple-choice test, but really in your future life multiple-choice tests won't be how we tackle something big like climate change.  The big goals of your education include deep understanding, being able to explain complex ideas with nuance, being able to learn from others, and being able to use ideas creatively in new ways often collaboratively.  Hence asking for help should be part of a larger process to make sense and expand your understanding and thinking. Learning is fun when it makes sense! And if it doesn't make sense, then keep on trying to understand and get help.

You can get help from various resources. Resource number 1 is your instructor. That's the person who is responsible to your learning. Next there are your classmates, your textbook, the tutoring center, and perhaps the internet. Try and talk to as many humans beings as you can first. Math is learned better with more human interaction.

Office Hours: Office hours are for you, and if you are stuck on something, even a small thing, go to office hours. It's not an imposition when you show up, and your instructors want to help you. Even if you instructor seems completely different than you, you can and should ask for help. While it might seem a bit scary, it's ok. I know a ton of math instructors, and so far all of them are human beings, and are really nice in office hours. Some even have a sense of humor! I know that might be shocking, but it's true. And when they go home, they go home to things like cats and children, and watch TV or text their friends about Friday plans, just like you.

Yes, there exist legends of math geniuses, who work in their attics for years to invent math. That works for them in certain contexts, but none of them worked alone or got to that  point all by themselves. They have collaborators, consultants, books,... they actually went to school with other human beings at some point in their lives (and interacted with them). They read journal articles, they attend seminars, they go to conferences. Some people gave them a job, so opportunities were given to them. No one is 100% self made. Therefore, work with other people regularly, even if you don't view yourself as "social" in the everyday sense of the word or are introverted. In this letter, I mean social in a school or workplace sense. We all have to communicate with others to give and receive feedback, as well as brainstorm new ideas.

And really I hope you get stuck a bunch of times in your learning process (in a safe learning environment, not on tests).  Get stuck??? Yes, get stuck. Because you need to push your personal abilities. Each time you get stuck and unstuck, you learn what works and what doesn't work and you get smarter. Through working on problem solving in math (or any field), you are doing something like going to the gym for your brain. Your brain will get stronger, and you'll learn new ways to think, see, and feel.

Math anxiety is a real thing. I've written about it many times on this blog. I've talked to hundreds of students about it. I'm sorry about this. Math anxiety should not exist. Not everything in the world is right, and math anxiety is one of those wrongs that we are trying to fix.  In the meantime, if you had experiences that led you to math anxiety, what you need to know is that it's not your fault! You're not dumb, you're capable, and there is a way out.

The way out is doable. It's shifting from a fixed mindset (where one views their math abilities as fixed at birth), to a growth mindset, where one views effort and practice as the ingredients for getting smarter.  Think about one of your hobbies or interests. How did you get better at it? You practiced. You might have had a teacher or coach or watched videos, but at the end of the day you put in the focused, dedicated hours, and did the work. That's being smart, and from now on, we are co-opting the word smart.  Getting smarter at math is exactly same. A good teacher will provide you with a positive class environment and support you through your specific learning challenges, and when you practice, think, ask questions, collaborate, and do all those things people do in every profession and hobby, you'll make real progress.

Only watching videos of other people doing the math isn't going to cut it. Look I get it. Khan Academy is one click away. It's a useful resource, and I even watch KA sometimes to see which ones might help my students. Whenever I need to fix something in my house, I find a video on how to fix it. That's a good way to get information that you lack. However, learning math is like learning to be a musician or athlete. It's not just information and facts, but also about developing thinking and problem-solving skills. Doing better at math requires thinking mathematically, which is analogous to learning to ride a bicycle. You can't be taught to ride a bicycle beyond the basics by watching a video, because there are things your brain and body have to construct by actually doing it in order to build that skill.  Mathematical thinking is the same in that you can't just get info uploaded into your brain like a firmware update. You also need to construct understanding and meaning for yourself, just like your body and brain have to construct things in order to ride a bicycle safely.

Another example is learning how to hit a baseball/softball. We could watch tutorials all day and understand what we need to do. Basically it's swing a bat and hit a ball. But only watching videos is obviously not enough.  We need to actually swing a real bat and hit a real ball and get ongoing feedback from coaches. And then practice, play games, strike out, reflect, rest, repeat.  It takes time to get good at it. That's the perspective you should have about Math and watching videos. Sure watch videos sometimes to get some info, but don't stop there. Start there, and do the work. Do your own reps on real problems. Otherwise, you'd just watching, and that is just sitting in the stands. You need to be on the field, because this is your life and your future. Get in the game!

Hopefully, your instructor will ask you to work with your classmates sometimes on a question or task. In education this is called active learning and is part of inquiry-based learning (IBL). These methods are designed specifically for you to engage and think for yourself. Listening to someone isn't enough. Sometimes we need to hear ideas from the instructor that we can't easily build ourselves, but like sports or music or any hobby, you ultimately need to be the one engaged in the process, asking questions, and taking ownership of your development.

Learning to work with others is critically important. Working in groups is not only about helping one another, although that's a good aspect of group work. One of the main benefits of group work is learning through discourse. Sometimes we need to talk things out in order to make sense of what is going on, and hearing other people's ideas can also benefit all of us, and helps us engage in the process of trying and refining new ideas. Another benefit of group work is learning to communicate. In an era when more and more repetitive tasks are being automated, the ability to do humanistic work, such as communication and problem solving, is much more important and valuable.

Try to contribute to group discussions and regularly invite your group mates to share... "So what do you think? What did you get? [smile]"  It's not about one person getting the answer for the group, and everyone else copies. It's about giving everyone a chance to think, try, share, refine, and see ideas from multiple perspectives. That's good for you!

In summary, focus on problem solving as a process, embrace and be patient with being stuck and not having answers right away, think about the long game of your personal intellectual development, develop a growth mindset, and work on learning with your classmates. These are things you need to prepare for your future. All people, especially young people, have immense capacity to learn, grow and get a lot smarter! Believe in yourself by actively investing in how you learn.

Best wishes on a successful new school year!

Sincerely,
Professor Stan Yoshinobu

Pronouns, Early Term Simple Survey

Many thanks to the NSF PRODUCT team for bringing this up in an email discussion! This short post is on pronouns, and a result of that discussion. These are not my ideas, and I'm sharing the main ideas from that discussion.

Pronouns matter, and it makes a difference if we ask students to tell us their pronouns without assuming what they are. Why does this matter? Here's a quote from www.mypronouns.org

"Often, people make assumptions about the gender of another person based on the person’s appearance or name. These assumptions aren’t always correct, and the act of making an assumption (even if correct) sends a potentially harmful message -- that people have to look a certain way to demonstrate the gender that they are or are not."

One way to get this information is to ask. And while you're asking, you might as well find out what they preferred to be called and what might help them succeed.

1. Your name____ (and email optional)
2. Please call me by this name (please add a phonetic pronunciation, if people often pronounce your name incorrectly) _____ 
3. Please use these pronouns for me (don't want to share this? no problem; not sure what this means? read about pronouns here: https://www.mypronouns.org/) ____
4. For me to be successful in this class, I need you to know that ____

Working towards an equitable classroom is an important consideration today (as the 4th pillar of IBL), and we should be more conscious of the issues and be action oriented. The last question on what would help students succeed can help you understand specific needs right at the start of the term, which can make a significant difference for student success.

One more thing. I put "he/him/his" in my email signature. An easy and impactful thing to do.

In short, I learned not to assume. I learned that we should simply ask and share with our students why we are asking, since they may need to learn as well.

Student Buy-In In Practice Overview

Student buy-in is one of the issues that comes up frequently at workshops and in hallway conversations. Student buy-in isn't a simple thing. I've written about it previously on this blog, and I think this topic needs to be visited regularly.

Getting stuck is hard. Fruitful struggle and productive failure aren't usually taught and learned. Making mistakes has often been equated with failure (in the negative sense of the term). Students aren't usually encouraged to explore, experiment, and tinker.  Thus, the conundrum is that in order for learners to grow, they need to be challenged appropriately, which means being stuck on some ideas, yet being stuck is equated to being dumb.

Luckily today we have the advantages that can help change learning experiences into authentically positive ones.  Growth mindset work has zeroed in on beliefs that lead to becoming smarter. We know more about how to use active learning to open up learning spaces, and we have a growing collection of videos on productive failure that can direct students toward successful mathematical practices. Instructors can assign videos as homework with reflective writing prompts every week or so for the first part of the term.

Day 1 of a course is important. Linked below is one way to open a course, by starting with students' hobbies and how they got better at the hobby.
https://theiblblog.blogspot.com/2019/01/opening-course-and-launching-winter.html

See also Dana Ernst's "Setting the Stage" opening.
http://danaernst.com/setting-the-stage/

Ongoing strategies for student buy-in are posted here. It's not enough to only do something on day 1, because it's a journey.
http://theiblblog.blogspot.com/2019/01/ongoing-student-buy-in-strategies.html

Nudging students to engage more is one way to address student buy-in. We all need a break sometimes, and we can all use a bit of support. One way to keep students going is to nudge them.
http://theiblblog.blogspot.com/2018/11/nudges-as-teaching-technique.html

Attend to Math Anxiety, because knowing where students are coming from can help us be better teachers. Math anxiety is a thing, and most students have some level of anxiety. Ignoring it only limits student learning, so we might as well deal with it. Math anxiety is linked to (lack of) productive failure, and fixed mindsets. Here's a post on the iceberg diagram and math anxiety and how instructors can detect math anxiety and fixed mindsets from statements like, "I don't learn this way..."
http://theiblblog.blogspot.com/2018/01/iceberg-diagram-fixed-mindset-math.html

Digging deeper, math anxiety is something you can read about from students directly. Here's a collection of math anxiety quotes to give you a sense what lies underneath. If you've never asked, try adding a math autobiography assignment at the start of the term. Let students share their experiences.
https://theiblblog.blogspot.com/2015/03/math-anxiety-realities-student-voices.html

Sharpening your IBL skills is also important, because a well-taught class is part of the equation. Problems that are too hard or leaving students struggling for too long works against student buy-in.  Also making things too easy is also. The IBL Blog Playlist is collection of posts organize by topic. If you are new to IBL, we also have a video series to get you going.

Standards-Based Grading Example in an IBL Course

I'm sharing an outline of standards-based grading I've recently used in a course for future elementary school teachers, although much of this is generally applicable to other courses. I'll list the main features and then get into some of the details below.  Also this is just one example, so do not assume that what I am sharing is representative. It's really a form that works for the specific course.

Here are the main features:

1. Gateway exams
2. Reading assignments
3. Homework assignments
4. Productive failure
5. Class contributions and participation
6. Final project

How these fit together is that if a student earns a passing grade on all items, they earn a B in the course. Students can raise their grade to an A/A-/B+ with an excellent grade on the final project. Students earning one or more non-passing scores in any of the categories will earn a grade lower than a B, with specific grades reductions based on the nature and quantity of the unsatisfactory grades.

1. Gateway exams: These exams are based on the IBL units we work on regularly in class, and are based on the math being learned in the course. Students are required to pass all of the problems on the gateway exams (i.e. get the correct). For any problem that was not successfully passed, a student must retake that problem on the retake exam.  The retake exam is given about two weeks after the initial exam. Problem done correctly do not have to be retaken.  The first retake is done in class. Subsequent retakes are completed in office hours or alternatively completed in writing and submitted for review.  This past term, I gave 2 exams, and was limited by the quarter system (10 week terms) in how many retakes can be given in class.

Retakes can become a logistical challenge for large classes or in courses where there is a significant amount of material to cover. One has to weigh the costs and benefits of this and plan accordingly. The strategy I've taken is to start with a course that I thought would be relatively easier to manage, and then work my way to other courses where I feel I would be better off with more experience.

2 and 3. Reading assignments and homework assignments are graded for process and completeness. Accuracy feedback is given, however, the goal of these assignments are for students to think and reflect on math and math knowledge for teaching. Points are not taken off for mistakes or incorrect answers, and instead feedback is given when necessary and points are awarded for good process. For example, if a student gets a problem wrong, but writes questions or explains what they did and what they still need to work on, then they earn full credit for the problem. 

4. Each student is required to present one productive failure (i.e. #PF) per term (in a 10-week quarter) about a mistake or something the student was stuck on. The format is to discuss (1) the mistake or issue, and (2) to share what they learned from the process.  (In some courses the number of #PF presentations is 2.)

5. Student contributions to the class discourse is another component. Students work in groups and are expected to show up to every class, contribute to discussions, be effective group mates (i.e. be good at listening, supporting, and sharing), and present math ideas sometimes. More or less this is participation grade, but with stipulations about expected behavior. 

6. In lieu of a final exam, students must submit a final report. The report is based on 4 tracks related to mathematics teaching in the elementary school and the course content (in this case fractions for teachers).  Each track has a lead source (article or book). Students are required to do library research, branching out from the lead source, to find learning challenges (for children) established in the Math Ed literature. Lastly, students are required to create rich mathematical tasks that address the identified challenges that build from starter problems to middle problems to goal problems. 

I get asked if creating math tasks is pedagogy.  The answer is no. Creating math tasks to address specific math learning goals is a teaching specific math activity. Identifying the main math ideas, ordering and sequencing math problems, and building up from first principals is doing a math (applied to teaching children). 

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Some general comments.

Gateway exams require students to learn all the standards of the course. There's no partial credit for problems, and students are required to demonstrate they know the math they need as teachers. Students have multiple chances to make sure they get problem completely correct. The retakes can be logistically challenging, if you are not organized. Overall, the workload is about the same, because retakes eliminates the time needed to determine partial credit, and there's a tradeoff that more or less washes out (for me).

Further, the overall assessment structure aligns the class to the mathematical work of teachers and the philosophy of IBL. The focus is on learning, and guiding students to what they know well and what they need to work on further. What students need to work on is clear, and this I find one of the main benefits of standards-based grading.

The reading assignments, homework, productive failure, and class contributions, as an ensemble focuses on process and prospective teacher beliefs. The shift is away from "answer-getting" without deep understanding.

Final projects or final exams can be implemented in ways that work with standards-based grading. In this specific case, I decided on final projects, since it gives future teachers the opportunity to connect they math they are learning, the research literature, and connect that to the classroom. In other courses, I have used standards-based final exams. 

If you're thinking about trying standards-based grading, I highly recommend giving it a go.  If you have been using standards-based grading, please share what you do! 

Edit: Dr. Kate Owens published A Beginner's Guide to Standards-Based Grading the AMS Blog.

IBL Blog Q&A: The TIMES Project, Karen Keene, Justin Dunmyre

This blog post is an Q&A session conducted via email with Dr. Karen Keene and Dr. Justin Dunmyre. They are sharing information about the TIMES project. Thank you Karen and Justin!


0. Please tell us about yourselves.

Karen Keene has her Ph.D. in Mathematics Education from Purdue University.  Karen was introduced to active learning in undergraduate mathematics education while she was a graduate student involved in the creation of the Inquiry-Oriented Differential Equations materials. She has been serving as a project leader on the TIMES project where inquiry-oriented instruction, one form of active learning since 2013. She is currently an Associate Professor of Math Education at North Carolina State University and is currently serving as a rotating Program Officer for the National Science Foundation.

Justin Dunmyre has his Ph.D. in Mathematics from the University of Pittsburgh, and is a Brown ’13 Project NExT fellow.  He is currently an Associate Professor and Chair of Mathematics at Frostburg State University. Through Project NExT, Justin got interested in active learning, and subsequently participated in the IBL Workshop.  This transformative experience led him to wonder what IBL would look like in his discipline (differential equations) and almost as soon as he had that thought he got an email through the IBL mailing list about this exciting TIMES project!

1. We'd like to learn about the TIMES project. What is the main idea behind this effort?

The TIMES project began as a collaboration of second-generation authors of varied inquiry-oriented (IO) classroom materials.  By second-generation we mean that Michelle Zandieh, Sean Larsen, and Chris Rasmussen wrote the original IO materials for Linear Algebra, Abstract Algebra, and Ordinary Differential Equations, respectively.  The TIMES Principal Investigators, Christy Andrews-Larsons, Estrella Johnson, and Karen Keene were each graduate students of these original authors and launched the TIMES project to study how they might support other faculty in using these materials.  That’s why TIMES actually stands for Teaching Inquiry Mathematics: Establishing Supports. These supports were three fold: providing the curricular material, a 3-day summer workshop on using the materials, and weekly online working groups. The three supported curricula were IOLA (Inquiry-Oriented Linear Algebra) and IOAA (Inquiry-Oriented Abstract Algebra formerly known as TAAFU - Teaching Abstract Algebra For Understanding) and IODE (Inquiry-Oriented Differential Equations).

Justin was in one of the first cohorts of TIMES fellows for IODE, and became involved in running the online working groups and became a coauthor on the materials, along with Nick Fortune whose dissertation research was supported by TIMES project.

2. What classes do you have materials for?

A first course in Linear Algebra, an abstract-algebra course focused on groups, and a first course in differential equations.

3. What is a typical day like in an IO class?

A typical day is centered around the guided reinvention of particular mathematical concept(s).  The tasks are based on the principles of the instructional design theory of Realistic Mathematics Education (RME), of which one of the tenets is that the material must be experientially real for the students.  The Inquiry Oriented materials are grounded in contexts that the students can initially understand and reason about, maybe from a less sophisticated viewpoint, even if they’ve never had that specific experience before.  For example, in IODE, one tasks early in the materials is focused on population growth of owls in a forest, which students may not know a lot about, but can understand. As in any IBL classroom, students are constructing the mathematics for themselves, and taking ownership of that mathematics. The students work through a series of tasks, often encountering the the tasks for the first time in class. Therefore, we rely on small group work for an initial translation from the context to concepts relating to the learning outcomes of the class.

The instructor facilitates this guided reinvention by primarily using four instructional components (Kuster, Johnson, Keene, Andrews-Larson, 2018) , not all of which may happen on the same day.  These four instructional components are general instructional goals: eliciting student thinking, building on student thinking, building a shared understanding of the mathematics reinvented in the classroom, and connecting the students’ mathematics to formal mathematics.

In any given day in our classes, one would see the students sitting in small groups, working on tasks. One would also hear whole class discussions where the instructor is soliciting student input, re-voicing, and, reshaping it, innocuously,  to guide the conversation with an eye on the instructor's mathematical agenda. Occasionally, the students would make presentations of their ideas, but this would not always happen. The whole class discussion and small group discussions do happen every day.  Finally, when the class has finished an idea, or perhaps developed a need for notation to express their ideas, the instructor connects their work with formal language and notation.

4. What are some of your best moments as a teacher in an IO class?

Justin:  There are so many!  One of my favorites came from the first time I taught our optional unit on bifurcation theory.  This unit starts with a task where students are challenged to model the introduction of a parameter that represents harvesting of fish from an otherwise logistic model.  After settling on a simple shift of the form dP/dt = 0.2P(1-P/25) - k, where k is the harvesting parameter, students are asked to come up with a one page report to explain to the owners of the fish hatchery the ramifications of varied choices in k.  The one page report is the trick! By requiring students to use space efficiently, they can actually invent the bifurcation diagram for themselves. What really surprised me was how many different forms this bifurcation diagram can take. I’ve seen students use spreadsheets that show if dP/dt is positive in green or negative in red (the bifurcation diagram then emerges as the change from green to red), carefully stacked phase lines, analytically drawn bifurcation curves using the quadratic formulas, and more.  When the students present these ideas to one another, they realize they’re all saying the same thing, and absorbing insights from other groups result in very deep understanding of this sophisticated concept. The first time I taught this unit I was giddy, I couldn’t believe that these bifurcation diagrams were emerging before my eyes, completely invented by my students!  We wrote about this task sequence in a PRIMUS paper here (Rasmussen, Dunmyre, Fortune & Keene, 2019). 

The “salty tank” problem is practically a rite of passage for students in my differential equations classes.  It’s developed its own legend here on campus, because the discussions are so robust, and it is the first time that students are really asked to develop their own equation.  The students really marvel at how they can have an 50 minute long debate over a prompt as simple as “A very large tank initially contains 15 gallons of saltwater containing 6 pounds of salt. Saltwater containing 1 pound of salt per gallon is pumped into the top of the tank at a rate of 2 gallons per minute, while a well-mixed solution leaves the bottom of the tank at a rate of 1 gallon per minute.” This problem is the seat of another favorite moment of mine. Ideas were flying all around the room, what does it mean to be well-mixed, what should the input term look like, what should the output term look like?  A student said something to her group, but she didn’t want to cut into the whole class discussion. Her group thought it was important though, so one of her more outgoing group members interrupted the discussion and he said “I think we should all hear what Sarah has to say.”  Of course, it was a critical insight that helped reframe the conversation in a productive direction! But the act of one student elevating the status of another student, that was a powerful moment that will always stick with me.

Karen:  When I taught this course to math and physics majors together, my favorite times were when we talked about the difference between instantaneous rate of change and rate of change over a "very very small time interval".  The conversations were always spirited and deep, with the Physics Majors declaring it doesn't matter if there is a difference and the Math majors wanted it to matter and try to understand what a "limit" really is. Of course, I was always rooting for the Math Majors, but it didn't really matter, as it was a situation where the students were engaged in thinking deeply about the math and taking the authority of learning on themselves.  Of course, ultimately, we had to agree to disagree and the physics majors usually could go along with the idea of instantaneous rate of change as that being the foundation of a differential equation.

I can think of other times that after we had small group discussions, two groups would present their ideas about a particular task- and they were not the same.  When that happened, I would encourage each side to state their case. Then I would send the students into small groups to continue the discussion and decide what they thought.  This might go on for much of a class. I know it took up a lot of time, but it was worth it--- they were not waiting on me to tell them- but making their own mathematical judgments.  Most of the time, it all seemed to move the agenda forward. I do remember one time that the whole class agreed on something that I knew was mathematically wrong. I made the decision (there was no test or assignment the next day) to let it stand.  On the next day, I brought of the decision with a question that led them to believe they were wrong-- and all was forgiven!

5. If you could give some advice to math instructors thinking about using active learning, who have not tried yet, what would you say to them?

Justin: There is evidence in the research that supports your decision to try active learning, so you should proceed with confidence.  For the IO curriculum, it is extremely exciting to be a sort of curator of the conversation. You don’t know what students are going to say, and you get the exhilaration of thinking on your feet to fit their ideas into your agenda.  So, my advice is: this is hard work! Be kind to yourself when you just can’t reshape their ideas the first time. We have found that, although active learning is our main mode of instruction, there is a “time to tell.”


Karen: Take it a step at a time— some instructors might go all out first time around, but trying one or two days or tasks and seeing how it works in your classroom is just fine. Ultimately, students will be more engaged and take ownership of the mathematics they learn in your new active learning classroom.

6. How can readers learn more about the TIMES project and get involved?

The TIMES NSF grant has essentially run its course, so we are no longer running workshops.  You can find the course materials, including instructor’s notes, at these websites:

IODE:  iode.wordpress.ncsu.edu
IOAA:  www.web.pdx.edu/~slarsen/TAAFU/home.php
IOLA:  iola.math.vt.edu

When you begin to investigate these materials, please don’t hesitate to contact us; we are more than happy to help!  With sufficient interest, we may even run informal online working groups.

References:
George Kuster, Estrella Johnson, Karen Keene & Christine Andrews-Larson
(2018) Inquiry-Oriented Instruction: A Conceptualization of the Instructional Principles, PRIMUS,
28:1, 13-30, DOI: 10.1080/10511970.2017.1338807

Chris Rasmussen, Justin Dunmyre, Nicholas Fortune & Karen Keene (2019) Modeling as a Means to Develop New Ideas: The Case of Reinventing a Bifurcation Diagram, PRIMUS, DOI: 10.1080/10511970.2018.1472160

Guest Post: Ed Parker on Graduate-Level Math Teaching and IBL

SY :  I’ve had several people sympathetic to IBL methods suggest that as IBL methods become more widespread at the undergraduate level, they will be unnecessary in graduate mathematics programs.  Do you have any thoughts on this?

Ed Parker:  Since I’ve never taught in a department that offered the PhD in mathematics, I’m probably not the right person to be asking.  But I’m certainly willing to respond.  You just need to understand that any expertise I may have is either historical or based on my limited teaching experience with master’s level students or my experiences as a graduate student.  It is somewhat ironic that the issue has arisen.  When I wrote Getting More from Moore back in 1988, IBL was, at least grudgingly, accepted as a viable option for graduate mathematics education while undergraduate IBL was pretty rare.  Several programs such as Auburn and North Texas had committed graduate programs and others such as Emory and the University of Texas had a visible presence.  When discussing IBL’s possibilities for undergraduates, I remember well being told by multiple persons from multiple places that IBL might work with well-prepared upper level majors, but that the students needed to be “ready for rigor” before it could have a chance to work.  And demonstrating and having the students reproduce what they had seen, then apply the theory to examples was apparently the way to make the students “ready”.  Readiness seems to be a given at the graduate level due to admission requirements.

First, I would suggest that anyone interested in the issue watch the video of John Neuberger’s talk on graduate mathematics education, delivered at the 2007 Legacy Conference. Rather than focusing on technical math education issues, he began with the challenge of turning curiosity into passion for the hunt and then proceeded to relate some of his experiences relative to benefitting from, and implementing, IBL.  Even an impartial judge (which I am not!) would likely judge his career as a teacher a successful one.  In 1977, I became his 13^th PhD student and he continued to produce productive PhD’s after his move to the University of North Texas the following year.  A notable aspect of his talk is his view that graduate teaching is a natural adjunct to a mathematician’s research.

I entered Emory University’s graduate program in 1973 after a four-year hiatus following my graduation from Guilford College, during which time I had taught secondary mathematics at Bayside High School in Virginia Beach, Virginia.  My draft board had decided public school mathematics teaching was in the public interest and granted me an occupational deferment in lieu of processing my application for conscientious objector status.  (That draft board had not dealt with a CO hearing since my twin uncles during World War II.)  My undergraduate education at Guilford was decidedly IBL.  In the core courses, we mostly read stuff, talked in class about what we read, and then wrote about what we had read and talked about.  With the exception of the calculus sequence and differential equations, none of my mathematics courses had a textbook. Although Mr. Boyd had us buy Heider and Simpson’s Theoretical Analysis for analysis and Greever’s Theory and Examples of Point-Set Topology for topology and Mr. Walker had us buy Birkhoff and MacLane’s A Survey of Modern Algebra for algebra; all were used as reference points and problem sources.  We did not reproduce textbook proofs in any of these courses. 

At Emory, I took the algebra sequence and the analysis sequence my first year, taking only two courses since I had only a 2/3 assistantship.  David Ford’s introductory analysis course in Lebesgue measure and topological vector spaces was totally IBL.  Trevor Evans’s algebraic structures course was given from course notes.  He had us buy Herstein’s Topics in Algebra and the Schaum outline on group theory as resources.  Dr. Evans talked through the course notes three days a week, stating additional problems as he went, and the fourth day was student presentation day.  A typical presentation by me went like this: I presented.  Dr. Evans would stare at the board, stroke his chin, then put down his pipe and say,  “I suppose you are correct, Mr. Parker, but WHY DIDN’T YOU THINK OF THIS?”  Then he would take the chalk from me and show the class a “good” proof.

As a second-year student, I took the third first-year course, a topology course in Moore spaces given by William Mahavier through IBL.  I had come to Emory with the idea of studying foundations and Dick Sanerib was offering a course on Model Theory that year. However, Emory would not give me credit for the fall quarter due to a course in symbolic logic that I had done at Guilford, so I audited model theory which was done by straight lecture following Bell and Slomson’s text and took John Neuberger’s Functional Analysis and Differential Equations, which was given by IBL, for credit.  I solved three problems that quarter:

There is a single function, call it $f$, so that if $x$ is a number, then $f’(x) = f(x)$ and $f(0) =1$.

There is a single function, call it $f$, and a largest non-degenerate connected set containing $0$ that is the domain of $f$, so that if $x$ is a number in the domain of $f$, then $f'(x)=-f(x)^2$ and $f(0)=1$.

Suppose that $x$ is a number.  Then  $\Sigma_{n\in\mathbb{N}}\frac{1}{n!}*x^n=\Pi_{n \in \mathbb{N}}(1-\frac{x}{n})^{-n}$

Imagine, if you will, knowing that  exp_e was the answer to the first question but having no idea how to make it appear, or that you could solve the second differential equation by “separation of variables”, but realizing that assuming a solution existed begged the question.  Needless to say, I didn’t think I was doing very well.  Looking back on it, I’m kind of glad I didn’t think of producing a power series from thin air, then proving that it worked since the path I took led through the Fundamental Existence and Uniqueness Theorem.

Near the end of the fall quarter, Dr. Neuberger stated a list of eleven problems that I later found to be, if one took the collective hypotheses and conclusions, the Hille-Yosida Theorem.

At the end of the quarter, I had to decide whether to continue Model Theory or Functional Analysis and Differential Equations.  Neither professor recruited me and I still don’t know why I chose to continue FA&DE.  Did I mention that I didn’t think I was doing very well?

I finished Hille-Yosida in early March.  (It took my classmate only three weeks once he went to work on it!)  Within a calendar year of when I finished Hille-Yosida, I had the theorems that formed the core of my thesis although I had still not passed my algebra qualifying exam.

Heading into my third year, having passed my analysis and topology qualifiers and failing my algebra qualifier and having taken complex analysis in summer school, I was scheduled to take the second level topology course and Dr. Neuberger’s research seminar.   A note of comparison is in order here.  A student of Dr. Evans pursuing an algebra thesis was expected to spend his “year of preparation” reading the pertinent literature.  On the other hand, I didn’t even know I was beginning work on a thesis.  Dr. Neuberger gave me a paper of his on Lie Semigroups and a short paper of Tosio Kato’s that had distilled (brilliantly!) a very long paper of Miyadera’s which had originally proven the dense differentiability of non-expansive semigroups on Hilbert spaces to work through.  I was given no guidance of which I was aware about why or how.

Before continuing on this line, I should mention that I seriously considered dropping out after getting the news that I had failed the algebra preliminary exam.  The birth of our second child the day after news of having failed the algebra prelim rescued me psychologically, but it also added yet another level of family responsibility to my table.  I talked with Dr. Neuberger, who was teaching complex analysis and he said that it was fine for me to use the course time to write a master’s thesis and that Dr. Mahavier had described to him an example I had made in spring quarter of the first-year topology course that would likely provide the substance for the master’s thesis.  I talked to Dr. Mahavier and he agreed to supervise the thesis.  Ironically, Dr. Evans, with whom I had taken the first-year algebra sequence, whose second-level seminars I had attended, and who never seemed to like my proofs, suggested that I should continue.  That, together with my wife’s encouragement, won the day.  My assistantship was renewed and I embarked on my third year.

In the research seminar, I tried to work my way through the two papers.  I had never been good (as in quick) in following other persons’ arguments, but I dutifully slogged my way through, with a cognizance of the structures Dr. Neuberger had appropriated from Hille-Yosida.  The elegance of Kato’s argument made it easy (even for me) to follow, but I realized that I was just verifying details.  This caused me to set out on my own, mimicking Dr. Neuberger by thinking about Hille-Yosida structures in non-linear contexts.  The Cesaro mean (I later found out that was what it was called) was the vehicle to a theorem on non-linear semigroups that I formulated and proved.  In seminar, Dr. Neuberger listened to my argument without changing his expression.  When I finished, he gave me a copy of Glenn Webb’s landmark example of a non-expansive semigroup on a Banach space that contained an open set in its domain where it was nowhere differentiable and asked me to see if his semigroup satisfied the premise to my theorem.  That night my euphoria turned to despair; I could prove that the premise was satisfied, and once I understood Glenn’s example, the verification of the application was dirt simple.  Thus, because it was easy even for me, I was sure my theorem must be no good.  In seminar the next day, Dr. Neuberger asked if I had been able to do what he asked and I mumbled something like “It can’t be any good; it’s too easy,” and showed him my argument.  He became instantly animated and told me, “This is the sort of theorem that theses are built around.”

At this point, Dr. Neuberger gave me some entries into the literature through which I learned about the work of the Japanese school that Kato had consolidated and gained access to the (then) current work of Brezis, Pazy, Crandall, Martin, and Liggett, and Neuberger’s seminal paper which had ignited the Japanese school’s initial successes.  An ancillary aspect of learning to use the library was to make sure that my theorem and its application were original.

There was still the issue of passing my algebra qualifier.  At Emory, the rule was two strikes and you’re out.   I was auditing Mary Frances Neff’s first year algebra sequence which I continued for the year and passed the algebra qualifier on my second try.  The department was kind enough to expand the two 4-hour qualifiers from the year before to two one-day tests.  They kicked me out after 9 hours the first day and 8 hours the second.  Both days, there were still problems I thought I could do.

I finished the following year.

What inferences can I make if I add to the mix what I have learned from talking to colleagues about their graduate school experiences?

Broadly, IBL at the graduate level, if the goal of an advanced degree is to certify readiness for original problem solving and ability to pass the mathematical canon of one generation to the next, is a super-charged version of what happens at the undergraduate level.  In contrast to the undergraduate entry, at the graduate level a four-year baccalaureate mathematics degree is in place as well as an entry test result that gives comparisons with other such students and may show some level of breadth and recall of some curriculum.  The main question probably should be, “How can this base best be nurtured?”.  The traditional response has been to give a “graduate level” broadening and strengthening by “mastering” carefully selected texts and/or the arguments of the professors’ lectures to create a base, then to certify the students’ readiness with a battery of barrier examinations.  Those deemed worthy are then given a second dose of the program in more concentrated contexts, usually the advisors’ research specialties, and embark on their own research missions, often as colleagues of their advisors.   In IBL, the students recreate the canon by solving the problems fundamental to it as seen by their professors and realize some breadth as they are held accountable for the work of their peers.  The battery of barrier exams appears as an institutional commitment, but, according to what the students have demonstrated in their individual trips through the canons, the turn toward research is not much more than a continuation of what they were already doing, the major difference being that the questions are chosen closer to the frontiers of the subjects of the courses and the classes are smaller. [WARNING: The above characterization is the author’s and may not represent any consensus opinion!]   As results are achieved, the students are directed to the literature with the goal of broadening their knowledge of others’ efforts and increasing the effectiveness of the students’ abilities to find their own problems.

My prejudices in favor of the IBL model are likely too strongly held to give the traditional model a fair hearing.  However, a late ‘80’s/early 90’s tome out of our professional societies exhorted us to make mathematics education “a pump, not a filter”.  The traditional model, in its insistence on early graduate education being a preparation for its barrier examinations certainly looks like a filter to me.  A colleague (whom I greatly respect) who spent the bulk of his career at an urban state university in the same city as an elite private university once told me, “If we could just get the students that Elite U blows away, we would have a better graduate program than they do.”  Perhaps this was an idle boast; perhaps it was not.  The pump effect stands out in Lee May’s recounting of the “sheep and goats” parable in his book on IBL methods.  Lecture and test does not provide for the opportunity William Mahavier seized to split his topology class and recombine it two quarters later as a class of peers.  At a Legacy Conference a year or two after Robert Kauffman of University of Alabama-Birmingham had died, a former colleague spoke to the gathering.  Robert had fought what IBL practitioners might call “the good fight” for many years, standing on the principle of academic freedom to teach in the way he considered most effective.  The colleague, who admitted he was reluctant to become an ally of Robert at first, recounted how he and many of his peers, often after decrying the lack of preparation of their graduate students, would remark how lucky Robert was to get so many good students in his classes.

Stan has written thoughtfully and insightfully on the coverage issue, which is often used to justify criticism of IBL instruction at the graduate level.  Udayan Darji, at a Legacy Conference in the early 2000’s, used his time at the podium to remind the audience that, if there were gaps in what they “should” know, part of their research time should be spent in filling them.  If one looks at my experience in Neuberger’s Functional Analysis and Differential Equations course, it should be clear that anyone capable of doing graduate mathematics could “master” proofs of the three theorems I proved by reading them with the investment of less than a week’s work time rather than the two months it took me.   Similarly, one could likely slog through the Hille-Yosida treatment of the Hille-Yosida Theorem in less than a month rather than the three-plus months it took me.  But would the appropriation of other people’s ideas, at the expense of nurturing your own, get you to a thesis the following year?

Where then, might an IBL student get her/his breadth?  I would first point out that the library will always be there.  But budding mathematicians need not master its entirety before beginning to think on their own.  Considerable breadth is achieved in being handed the responsibility of verifying the veracity of classmates’ presentations.  I still remember Tom Pate’s proof of a theorem in Fourier Analysis, for which I had a “brute force” argument, using soft analysis.  I have not viewed linear algebra the same since that day.  I owe similar debts to Margaret Francel, Everette Mobley, and Terry McCabe, to name just three.  Lessons I learned from them gave me alternative outlooks when I would work through textbook proofs as I put together my own courses.   And, as one teaches with IBL, the students will direct you to “natural” lines of reasoning.  Accumulated experience as well as preparatory learning can also build a mathematician’s repertoire.

In conclusion, I return to Neuberger’s talk:  Your continued commitment to research will fire your teaching and your teaching will abet your research.  So let your students in on the hunt from the get-go.

No tome of mine is complete without a baseball analogy.  Cy Slapnicka became a legendary scout for discovering Bob Feller.  My question is, “Who could have seen Feller throw and not realize that he would become a star?”  In the modern game, the same could be said for Bryce Harper.  But they are the baseball equivalents of the students that Harvard, Duke, or Chicago recruits for its Putnam team and it is doubtful that any form of instruction in graduate school will keep them from succeeding.  There is another group of students that clearly has big-league possibilities and the minor league experience is expected to build into a body of players producing major league level play.  Certain organizations are known for “growing” these players while others let the cauldron of competition weed out the “weak”.  I would suggest that there is a strong analogy here with schools that admit only the testibly top students and then still blow many of them away.  But, in baseball, these two categories of players are not enough to fill all of the rosters.  Finding latent talent and nurturing it is responsible for developing the rest of the big-leaguers.  There are lots of mathematics majors out there with highly developable tools.  I suggest that IBL  does not inhibit the development of super-stars and is likely superior in the development of a far larger number of students.