Willpower Training

Who doesn’t want more willpower? Stanford lecturer Kelly McGonigal presents the latest research on the physiology behind success and failure of self-control in The Willpower Instinct. Both Kelly McGonigal and her twin sister Jane (who writes about self-improvement games in SuperBetter) have also given fascinating TED talks.

I was not surprised to hear that exercise, the universal panacea, has been found to have a strong effect on willpower. Even a 5-minute walk is restorative, particularly outside in nature. Similarly, sleep deprivation quickly depletes willpower, but naps replenish it. Slow breathing (four to six breaths per minute) can help when dealing with stress and temptation. Surprisingly, willpower reserve is closely correlated with heart rate variability (which is higher in fit people).

Willpower can be increased by small daily acts of self-control, even those seemingly unrelated to major goals. Hence perhaps, the Admiral William McRaven Make Your Bed phenomenon. Yet we need to watch out for subtle and unexpected pitfalls. Believing that we have done something virtuous may increase the likelihood of indulging (“moral licensing”). Congratulating ourselves on progress can reduce motivation. And who knew that frightening news stories increase our urge to spend on luxury goods?

I found McGonigal’s description of the neurology of the reward system and dopamine delivery disturbing, particularly as it relates to modern technology. Cell phones, email, and commercial web sites are all highly efficient dopamine delivery systems. She recommends finding positive ways to “dopaminize” our least favorite tasks by pairing them with something pleasant like music. The strategies she recommends as most effective in relieving stress are familiar, but worth repeating: exercise, time outdoors, reading, music, meditation, yoga, creative hobbies, religious practice, and time with family and friends.

A few other observations: willpower is contagious, just as rule-breaking is. Social support may be helpful in trying to reach a goal or form new habits. Self-compassion is more helpful than self-criticism. Think of people you respect who have had struggles and overcome setbacks. Keep in mind your long-term goals. And finally, picture how you will feel when you succeed in reaching your goals!

The Slow Professor: Ethics of Time in Corporate Academia

The first time I attempted to read The Slow Professor, by Canadian professors Maggie Berg and Barbara K. Seeber, I found it very tough going. As a mathematician, I felt myself adrift in unfamiliar and undefined terms such as neoliberalism, affective dimensions, instrumentalism, and agency. Nevertheless, I sensed that there was more to their message than doing less, slowly, so I persisted. What follows is my attempt to distill some of their ideas in the context of my own experiences and beliefs.

What is slow?

Slowness is interpreted as cultivating quality, understanding, and community.

Slow does not mean lazy. Slow refers to work that is “meaningful, sustainable, thoughtful, and pleasurable.” Slow means acting in ways “that promote both our own and others’ well-being” and resisting exploitive behavior.

Slow means breaking taboos about self-sufficient individualism.

Slow means creating the conditions required for scholarly and creative work.

Act with purpose. Reduce distractedness and fragmentation. Take time for reflection. Think long-term.

Why the corporatization of higher education can be harmful:

The corporatization of higher education prioritizes some research areas over others. It tends to favor research that is quantifiable, applied, and marketable. Emphasis is placed on productivity and efficiency. More nebulous values such as intellectual development, ethics, and citizenship may be neglected. Power is transferred from faculty to managers. Economic concerns dominate. Increasing workloads threaten the time needed to think, reflect, and discuss.

Why we should try to change:

Tenured faculty members have an obligation to improve working conditions for all. We should aim to reduce work stress, improve education, and cultivate deep thought.

Sources of stress and effects of stress:

The authors cite evidence for high levels of stress in academia in comparison to the general population, contrary to popular perceptions of professors as a leisured class. Many of the top stresses are related to time. Increasing workloads, short deadlines, a rapidly changing work environment, and constant pressure for efficiency all contribute to stress. Meetings, frequent interruptions, and the expectation of rapid response to emails fragment the working day. New technology, budgets, and assessments take away from the core work of teaching and research. Many faculty members feel it is impossible to do everything in their job description, no matter how hard they work.

Faculty stress in turn can affect student learning.

Life, liberty, and the pursuit of happiness:

The authors assert the right of faculty members to health, a private life, and personal activities. The slow movement supports the right to limit total working time to allow for these activities.

Emotional aspects of teaching and learning:

Slow means considering the emotional aspects of teaching and learning. Positive emotions improve attention and problem-solving. Devoting time and attention to the atmosphere of the classroom is as important as preparing content. The enjoyment of the teacher contributes to student enjoyment and learning. Student opinion forms show the influence of emotions in the classroom: students want classes that are inspiring, stimulating, and engaging, and teachers who are caring. It is important for teachers to find meaning in the ordinary events of every day and even in the difficulties. We should strive to give students the feeling of fulfillment of solving a difficult problem or understanding a difficult concept. Assignments, quizzes, and tests should not just be tools for evaluation but also useful and enjoyable for the students.

Too much stress contributes to cynicism. Overwork can cause teachers to dislike their students (and vice versa). Self-care is not selfish, but is vital to our ability to be kind to others.

Listen to student concerns, and notice students as human beings. Show positive regard, whether or not a student is successful.

Conditions for creativity and deep work:

In order to improve conditions for creativity and deep work, we need to change our relationship to time. Try to focus attention, resources, and energy on one activity at a time. Get offline, away from email, and instant messaging. Eliminate as much as possible from the schedule to make time for important high quality work. Fragmentation of time and energy reduces productivity. Regular blocks of time are needed to become immersed in a problem. The state of “flow” not only increases productivity, but also contributes to happiness. Stress is a major obstacle to flow.

Some specific additional suggestions are:

– Reserve your research day for research, not email, record-keeping, and administration.

– Use a transition or ritual to help focus, reduce anxiety, and create an atmosphere conducive to creativity.

– Silence the “inner bully.” Have the courage to explore new ideas.

– Time for rest is not wasted. We need pauses in the workday. Vacations are important.

Emotions, ethical issues and obligations to others:

We need to think about the consequences of our work for the world at large. We need to think about scholarship as community, not competition. The authors describe caring for oneself as an ethical issue. Being machinelike does not encourage compassion for others.

Time for self and time for others are related. Allowing time for others is an ethical choice. The distractedness of fast life can lead us to neglect others. Sometimes being ethical may mean being inefficient. We have an obligation to others. Psychological wellness is necessary for those in the helping professions. We need to structure our lives to prevent burnout.

Long-term view of development as a scholar:

We need to think of development as a scholar and human being, instead of just measuring ourselves by lines on a CV. Shifting the focus from product to development of understanding can help ease some of the pressure. We need to take time for unnecessary reading and following curiosity. It is important to think about our professions as a whole, including their implicit biases and assumptions.

The importance of language:

We are changed by the language we use. The language of corporatization can be demoralizing and contribute to unrealistic self-expectations. The authors suggest ways to alter the internal dialogue about academic work. They remind us that quality matters, thoughts take time, and more is not always better.

The role of community:

A sense of community is one of the most important antidotes to stress. Work pressures can make people feel that they are too busy for social interaction. Email and telework increase isolation. The corporate academic climate takes a toll on human relationships, even as work relationships become more important, due to increased working hours. A department can be a supportive community, not just a collection of people.

We need to be able to talk openly and honestly about our work, including professional challenges and difficulties. We should not discard face-to-face interactions as a defensive response to the culture of busyness. We need to help one another generously and avoid the “instrumental” view of colleagues. We should not let the corporate view of academia blind us to the role of emotions and human relationships. It is important to find pleasure in our scholarship and in our academic communities. We need to listen to understand, rather than to find weakness. The authors conclude on an optimistic note – despite the difficulties of academic life, we should not give up hope. We can create our own subcultures of mutual support and trust.

How Should Teaching Contributions be Evaluated?

This post is based on conversations with colleagues on how best to evaluate teaching. Student opinion forms do not necessarily provide a good measure of student learning or teaching effectiveness. Gender, race, and other irrelevant factors may affect student responses. Students cannot judge whether course content is at an appropriate level. Students can judge whether feedback is given regularly and in a timely manner, and whether an instructor is available for office hours. Student opinion forms are useful for identifying serious problems, especially in classroom atmosphere. Written student opinion forms give more useful feedback than numerical scores.

There is general agreement that we should not rely on a single metric. Some instructors suggest a checklist to reduce the effect of personal bias. A checklist would also help faculty members understand the process by which their teaching is being evaluated. Others note that a checklist might not support a diversity of teaching approaches. A checklist can help define what activities count as teaching. At some institutions, teaching is broadly interpreted to include academic advising and other forms of mentoring.

At a minimum, items to include in a checklist would be which courses were taught, contact hours, number of students, development of new courses, number of preparations, new preparations, project development, difficulty of courses, and supervision of student research. Other examples of items to document might be curriculum development, attending workshops, response to student feedback, and use of new techniques. Teaching courses outside an instructor’s usual area or courses which are hard to staff might be relevant. Course material such as tests, homework, and exams should be examined. Peer visitation can be useful in evaluating teaching, but can be subjective. There is disagreement about whether student scores on common final exams should be used to evaluate teaching. One instructor notes that any method of measuring good teaching can be gamed.

Some questions about teaching contributions are hard to answer. For example, how well do students perform on follow-up courses? Do students get regular feedback and sufficient opportunities to practice new skills? What is good feedback? Collecting homework and giving detailed corrections might be one answer. What is effective feedback? For example, how effective are computer grading systems such as WebAssign? What are good practices? How soon should graded work be returned? Some instructors put a lot of time into innovative grading techniques such as standards-based grading. Some instructors give many assignments and spend a lot of time grading. What are the departmental standards?

What are the expectations for availability for extra help outside of class? Some classes require more office hours than others. The amount may vary from semester to semester and is not always predictable. In our department, the amount can vary from 1 to 15 hours per week. It might be helpful to have a statement on limits of what students should expect from their instructors.

How can informal mentoring be measured? Are some groups called on to do more mentoring than others?

Some believe that teaching methods incorporating active learning should be encouraged. One checklist item could be whether students are actively involved in mathematical work during class time. Not all instructors agree on the effectiveness of active learning. Active learning may not be suitable for some courses, at least not if we want to cover a large amount of material. Furthermore, not all active learning activities are equally valuable. We need broad enough criteria to accommodate a variety of teaching styles.

A checklist can be useful to a certain extent, but is not enough to reach the top levels of achievement. These need to come from reflection and innovation. It may be easier to identify who needs improvement than who are the strongest performers.

A personal teaching statement summarizing the accomplishments of the year might be useful. But should teaching evaluation depend partly on how persuasively instructors can promote their contributions?

Another question is, in an institution in which faculty are expected to do teaching, administration, and research, how much does good teaching contribute to end-of-year evaluations and pay increases?

Finally, the evaluators themselves have only finite time and resources. How can we evaluate teaching fairly but without creating an unreasonable workload for the evaluators?

Zig Zag

For the last few weeks I’ve been meeting with a book group on creativity at my institution. We’ve been discussing Zig Zig, by Keith Sawyer, an associate professor of psychology, education, and business at Washington University in St. Louis.

The book is a compilation of ideas and exercises from many sources and at many levels. Sawyer’s goal is to capture the cognitive steps of the creative process. He emphasizes that the process is neither quick nor easy, but involves disciplines and habits of mind, applied over an extended period of time.

The following are some of my favorite ideas and suggestions from the book:

  1. Schedule idea time daily. Follow the principle of paying yourself first – try to save your peak time for creative activity. Guard this time from phone calls, email, and meetings. Some people prefer first thing in the morning, others the time right after lunch, and still others the end of the day.
  2. “There’s no creativity without some slack time.” Ideas often appear during undemanding activities such as walking or gardening. It is important to take breaks throughout the day, as well as some time off on weekends, and an annual vacation. Resist the cultural pressure to be constantly at your desk appearing busy.
  3. Keep an idea log or seed file, and review it every three to six months.
  4. Set a daily quota. Sawyer argues that many of the most creative people are also the most productive. He mentions well-known examples such as Albert Einstein, Johann Sebastian Bach, and Thomas Edison. A study of scientists found that their most influential papers often coincided with the most prolific periods of their careers. Keep more than one project going at once.
  5. Talk to many different types of people to expose yourself to ideas and outlooks very different from your own.
  6. Look for solutions that are simple, elegant, and robust.
  7. Once you have generated ideas and worked through the details, look for flaws in your work. Ask for the critical opinion of a friend or trusted colleague. Don’t get too attached to or fixated on a particular idea.

In the Arena

This weekend I attended a state robotics championship. The struggles the teams experienced with their robots reminded me of a favorite quotation from Teddy Roosevelt. This quote applies equally well to the challenges of doing mathematical research, or creative work in any field. This post is dedicated to all the robotics contestants who were brave enough to risk failing in a very public robotics arena.

It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasm and the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly …

What is Social Network Analysis?

This year I am supervising a student project on mathematical techniques of social network analysis applied to terrorism. We are using publically available data on the March 11, 2004 Madrid bombers, gathered by Jose A. Rodriguez from news reports. He represented 70 members of the terrorist group and the links between them as points (vertices) and links (edges) of a graph.

Social network analysis can help identify key members of a network, using various notions of centrality. The simplest is simply to count the degree, or number of links, of each point. Another measure of centrality is betweenness. A person can be important as an intermediary between two different parts of a network, even though the person may have few links to each subgroup. There are several more sophisticated measures of centrality which are already implemented into computer software such as Mathematica.

Graph theory algorithms related to the connectedness of networks can be used to estimate the fragility of a network and suggest strategies for disconnecting it into smaller and less dangerous parts.

Another approach to understanding terrorist networks is to try to identify possible models for the growth of a network based on the distribution of degrees. Statistical methods can be used to measure the fit of a data set to a probability distribution. Data can also be compared to simulations based on theoretical models of growth.

One obvious problem in studying terrorism is that data is incomplete and its accuracy is unknown. We are comparing the structure of the March 11 group with public data on other small social networks such as a group of jazz musicians and a collection of dolphins (both from the Wolfram Mathematica collection of social network examples). If we can find other social networks that have similar structures, this could help to understand how best to disrupt terrorist groups or inhibit their growth.

The Attention Merchants

How should we spend the limited resource of our attention? What is the value of our private lives? Where and when should we allow commercial intrusions? These are some of the questions asked by Tim Wu (http://www.timwu.org) in his new book, The Attention Merchants.

Wu relates the history of the advertising industry, from newspapers, to television, to computers (the third screen), and smartphones (the fourth). Harvesting attention has become an art and a science. Personal information gleaned from web searches is mined and sold for advertising. Wu dramatically describes the surveillance methods and online tracking of news web sites as a “more thoroughly invasive effort than anything NSA data collection ever disclosed …” The business of the attention merchants is to maximize quantities such as time on site, number of video views, and number of page views. Our attention is converted into revenue. Opting out requires an act of will.

Why should this matter? Wu answers with a quote from American philosopher William James,

Our life experience amounts to what we pay attention to.

Our attention is precious. Attention lost in bits and pieces throughout the day distracts us from focusing on personal goals and interacting with friends and family. The smartphone has become such a large part of our daily life, we should be aware of the costs that go along with the conveniences.

Wu recommends reclaiming time for serious concentration by periodically unplugging. Religion used to set limits on the reach of attention merchants. Wu suggests observing a digital Sabbath. He also urges us to be aware of how the culture of the commercial web is influencing us as individuals and as a society.

I think Wu has thrown a vivid spotlight on an aspect of daily life we rarely think about. He asks some very thought-provoking questions. Nevertheless, I am more optimistic than he is about the future of the web. I disagree with his conclusion that commercialism is crowding out individual creativity and innovation. Although our culture is changing rapidly, books like his help us to be more aware, and to make conscious choices about how to spend our attention.

Standards-Based Grading

One of my motivations for starting a blog was to document interesting teaching ideas I’ve learned about from colleagues. This post is based on a group discussion with several colleagues who have implemented standards-based grading (SBG) in their courses.

One instructor described standards-based grading as a way to get students minds right about what happens in a math class when they haven’t learned a topic. SBG gives them an opportunity to go back and master past material. This is especially important when it is needed for later topics. She gives grades in a simple manner, based on the question, do you know what is going on or are you lost? She doesn’t have to obsess over the exact amount of partial credit. Her three scores are: Wow!, OK, you got it, and No, you haven’t got it.

Students can ask for retests on any topic. A low score is replaced when a student does better, but a good score is never lowered. It may take several retests, but at some point they learn the material. A student with OK on every topic gets a B. If at least half the scores are Wow and everything else is passed, the grade is A. Students who don’t understand (pass) all the topics receive a C or lower, with a D if less than 75% was mastered and an F if less than 50%. Students respond pretty positively, except if they don’t understand that they can replace low scores.

An advantage of standards-based grading is that learning objectives are broken down systematically and clearly. The system forces you to decide what students need to walk away knowing – these become the standards.  There are typically thirty to forty standards for a course. Structuring a gradebook by standards can reveal what the class doesn’t know. It shows not just the average grade on a test, but what particular topics were not well understood.

Question: What happens if there are only two weeks left, and someone needs to master a lot of standards? Answer: Students are reminded throughout the course that they can come in and raise their grades.

Question: What is the role of the final exam?  Answer: The final exam is graded in the normal way and the mark is averaged with the term mark. For example the final exam might count for 30% and term work 70%.

Question: What if someone gets an A on the final exam and a C in the term work?  Answer: Maybe in this case the student could get an A. These are instructor choices. Some people allow students to miss up to 10% of the topics and still receive a B. Some instructors lower grades based on reassessments, e.g., on basic integration. This provides incentive to move away from cramming for an exam, and then forgetting the material.  If a grade goes down, the student needs to come back and be reassessed.

Question: Do you find yourself writing lots of tests?  Answer: Yes. That’s the investment.

Question: What do you do about homework?  Answer: One system is to treat homework as a way to learn but assign no grades for it. The grading burden may be pretty bad with other systems

One instructor assesses her class every day on some topic. She asks the students to read the question, then read their notes and talk to their neighbors, but not write. After a minute they stop, close their books, and write their answers. There is a lot of value in learning from each other. She allows her students to assess only once per day per topic. The number of times they have assessed on a topic is the number of practice problems they have to do before another assessment. She replaces the mark, whether better or worse, using a grading scale from 0 to 4. This instructor wrote 62 objectives for Calculus III. She gives fill-in-the-blank handouts in class. A box with a number on the handout indicates an objective. It is very time intensive at the front end to construct the test bank. She notes all objectives being used on each test question. Her gradebook is a huge color-coded spreadsheet with a row for each student and a column for each assessment of each objective. The system is labor intensive, but she finds value in tracking the history of assessments. It gives a picture of student progress and helps to reflect the level of effort. The burden of extra help in office hours is huge.

Question: Why not just average all the assessment scores? Answer: This instructor doesn’t want to give an A to a student who has no clue about 20% of the course.

Question: What if there are 10 topics in a course and a student masters 9? Answer: It depends on the instructor. The beauty of SBG, which you don’t see until you do it, is that this rarely happens. The motivated students are really motivated. One SBG instructor averaged 10 – 15 students every day in the week before final exams.

Question: Because of having to create so many reassessments, aren’t you limited to fairly simple and routine problems? Answer: This is an open issue and hasn’t been resolved yet. One instructor, who often teaches weak students, sees her role as giving her students the tools they need to pass physics and electrical engineering. She needs them to be able to do integrals. There is higher level thinking that she is not capturing. Another instructor gives both simple and complicated questions on her tests. Solving a complicated question earns a Wow, and a simple one an OK.

Question: Our students need to make sense of math, not just learn rote calculations. How does SBG address this? Answer: Yes, students are not always getting the big global picture. Sometimes students are asked to explain things in words, but we need to work at this.

Comment: Another problematic aspect of SBG is that there may be a strong tendency for students not to keep up with the course because they can always reassess.  There is a tendency to cram and dump one objective at a time. This will hurt them in the end.

Question: It seems that SBG involves a lot more work with office hours and reassessments. Is any aspect of teaching easier with SBG? Answer: Time can be saved by not assigning or grading homework.  However, some methods of implementing SBG are not feasible for an instructor who does research and administration.  It is possible to reduce the number of individual reassessments by doing reassessments in class if many students don’t pass a topic.

Question: Do students follow a trajectory, for example, are they all fairly close together in the list of assessment topics? Answer: There is some cramming just before grades are due. One advantage of the standard approach to grading is that students are all working on the same topics. An advantage of SBG is that it helps pick up the few students who have fallen behind.

Question: Is there a terminal time for reassessment? Do you ever clock out a topic? Answer: Some instructors limit the number of reassessments a student can do in a given week. However, even if a student is four weeks behind, the student still needs to know the course topics. No matter how late it is, reassessment is still valuable review for the final exam. Students do need to be reminded that there won’t be time to reassess many topics in the last week of classes.

Question: Do you still have time to talk about concepts in office hours?  Answer: One instructor uses two rooms, one for talking and one for quizzing.

Question: How do you encourage students to be considerate of your time? The most challenging thing about this system of grading is finding the time to allow students to come in for reassessment. It is already hard to find time for all the other things we need to do. Answer: One thing you can do is make rules – these are the hours you are available and students can only do a limited number of reassessments per week. With SBG, it is the students’ job to learn the material. When the material is broken into topics, there are fewer students saying, I don’t know anything – teach it to me. They know exactly what topics they need to master.

Comment: It would be interesting to compare the performance on final exams of students in SBG classes and students in more traditional classes.

Comment: It would be interesting to see how SBG plays out in the long-term development of the students. Are the reassessments adequate for what we are trying to teach?  We need to build an understanding of how to interpret calculus in applications. Response: We have to put things that are important on the list of standards. SBG forces us to define what the bigger things are, and we need to do this in a tangible way. Reassessments might look nothing like the original question. They are not necessarily the same question with another function plugged in. Reassessments are valuable because they focus student attention. Students don’t just ask for extra credit.

Question: How does SBG work with inquiry-based learning?  Answer: Great! It’s a really great match. But there are workload issues. One instructor did SBG in a fundamentals of math course. Each problem addresses multiple objectives. Consider the generic particular, which is philosophically hard for students to learn. To show that all elements of a set have a property, we show that an arbitrarily chosen particular element x has the property. This instructor required students to use the generic particular 10 times in proofs to get an A in the course. When students have to use this idea 10 times, they learn it!

Question: When students retake a standard, are they allowed to keep the question paper?  Answer: Some instructors keep the reassessments, so that they can be reused. Others are not concerned that students may possibly know the answers, because they have to show their solutions to receive credit. Some instructors pick reassessment problems at random, so that the students don’t know which one they’ll get.

Comment: Sal Khan automates this process. Have you thought about automating reassessments?  Answer:  There is value in looking someone in the eyeballs and asking if they have any questions. It provides a learning opportunity. There is value in spending time in person with our students. Also, we’ve seen with WebAssign that online assessment doesn’t always work well.

Question: Has there been any research on SBG? Answer: There are anecdotal accounts but there don’t seem to be any with numbers. There are differences in ideas about assessments. Some educators believe that assessing students in a standards-based model is fundamentally different than in a conventional testing model, and we are comparing apples to oranges. Opponents to standards-based grading say that this method is not assessing what we want to be assessing.

Question: As you build a question bank, how do you organize it? Answer: One instructor keeps a file for every objective and a page for every assessment (printed out). Other instructors keep a stack of reassessments in a folder, not organized. Some use even-numbered book problems. One instructor asks students to warn her in advance which topic they are reassessing.

With conventional grading, when students do poorly we may allow test corrections and retests and extra credit. We know they need to learn the material somehow. We create very convoluted systems to figure out what to do for extra credit. SBG solves this problem in a way that is respectful of the students. One instructor remembers how she hated busy work as a student. Extra credit assignments are busy work. SBG lets a student say, I know this topic and don’t need to do more practice problems.  SBG makes this instructor feel better about giving someone a low score.  The score says – you didn’t learn this topic yet. Trying is not enough. SBG teaches the student the lesson – if you failed a topic, it doesn’t mean you can’t learn it someday!

Digital Distractions

When I first started teaching in my current department, my biggest challenge in the classroom was to keep my students awake. Nowadays, thanks to smart phones and wi-fi in the classroom, I have few sleepers, but I fight a daily battle to compete with digital devices for my students’ attention. I looked up The Distracted Mind: Ancient Brains in a High-Tech World after hearing the authors interviewed on the Diane Rehm show on National Public Radio. I hoped to find arguments to convince my students that multitasking in class is an unproductive strategy. Neuroscientist Adam Gazzaley and psychologist Larry D. Rosen do indeed make a compelling case that overuse of modern technology interferes with our goals. They also throw cold water on my hopes by pointing out that even when students know this, they rarely change their behavior!

The statistics on smartphone use quoted by the authors are familiar and depressing, particularly as they apply to college students. For example, studies show that 90% of students text in class and use laptop computers for non-academic reasons. The authors describe the effect of distractions on cognitive control, including attention, working memory, and goal management. Suppressing irrelevant information is not a passive process and takes mental energy. Interruptions of work to check email or social media also take time and energy.

The authors describe how the expectations of society are changing. There is increasing pressure to answer cell phone calls, emails, and texts immediately, even outside work hours. Pressure is especially high for group emails.

Task-switching may fulfill emotional needs for social interaction. In experiments, heavy smartphone users felt great anxiety after only 10 minutes without being able to use their devices. Heavy smartphone use in class is linked not only to lower GPA’s but also to less life satisfaction. The authors briefly discuss the addictive nature of digital devices, noting that the shorter the time between rewards, the stronger the drive to repeat a behavior.

The authors suggest that a first step toward combating the effects of digital distractions is to be aware of our vulnerabilities and of how distractions affect our performance. Physical exercise, meditation, and being in nature may help combat the Distracted Mind. The following are some of the strategies recommended for focusing on important work:

– Partition the day into project periods during which email and social media are avoided.

– Let colleagues know when you will be available.

– Shut down all programs and apps not needed for the task at hand.

– Shut down email or at least turn off alerts.

– Silence smart phones and put them out of view.

– Limit oneself to a single screen and one website at a time.

– Take breaks from technology to reduce fatigue and stress. For example, walk around outside, read fiction or something funny, or talk to someone face to face.

As for my classroom, I am making my lectures briefer than ever, and my class problem sessions more challenging, so that students have greater incentive to stay on task. Assigning pairs of students to present particular problems on the blackboards is helpful when the class starts to lose focus. These problem sessions often end with a use of smartphones that even I can encourage – photographing the final problem solutions on the blackboards!

Recent Reading

Names for the Sea: Strangers in Iceland, by Sarah Moss. A university professor visits Iceland with her family and writes about volcanoes, redirecting molten lava, riots over the economic situation, smuggling in food, and beliefs in elves and hidden people. The writing is conversational, leisurely, and meandering, yet manages a “six-degrees-of-separation” feat of linking her experiences in Iceland to many disparate facets of modern life. As I immersed myself in this journal of Icelandic life, I was surprised how many times I found myself encountering Iceland in other contexts. For example, a mathematical article on detecting terrorist networks describes a model for disruption of terrorist cells based on Iceland’s response to the 1973 eruption of the Eldfell volcano. Icelanders attempted to protect their harbor by pouring water on the lava. The lava corresponds to terrorist plans, transmitted down a network from the top of a hierarchy, and mathematicians seek to find effective strategies for blocking the flow to the foot soldiers lower in the network.

Emotional First Aid, by Guy Winch. I looked up this book after hearing the author’s very personal TED talk, in which he described his close relationship to his twin brother, and his loneliness when they were studying in different countries. He makes the case that it is as important to learn to take care of our minds and emotional lives as it is to take care of our physical health. He talks about research-based strategies for dealing with pain that all of us feel at one time or other – loss, loneliness, and guilt. He asks us to consider how many problems of modern lives are related to emotional pain and what a high price our society pays as a result. He discusses the effects of loneliness on physical health and states that chronic loneliness is as large a risk to long-term health as cigarette smoking. I have seen evidence of the truth of this statement among my own friends and relatives. I think with sadness of a friend who was diagnosed last year with pancreatic cancer and our last conversation, about his regrets and loneliness. I think also of relatives with rich networks of friends, who lived long and social lives, without devoting themselves to any rigid regime of health and fitness. Although I picked up this book without any expectation of finding applications to teaching, I was surprised to find helpful insights into the psychology of struggling students. Winch has very interesting observations about self-esteem. He notes that a student with low self-esteem may resist all attempts to build his or her confidence, and may feel even worse than before after such attempts. Winch explains that “messages that fall within the boundaries of our established beliefs are persuasive to us, while those that differ too substantially from our beliefs are usually rejected altogether.”

Writing on Writing – A Personal Note

Today I’m celebrating submitting the semi-final draft of my first book! It is a mathematics book on topics in graph theory, co-written with a colleague, and is written at the advanced undergraduate or first-year graduate student level.

While the experience is fresh in my mind, I would like to record some of what I learned about the actual process of writing, particularly routines that were helpful in keeping me motivated and on schedule.

  • The actual writing went much faster than I expected. From our first bare bones proposal to the publisher, to the submission of this semi-final draft, it has taken about two years. Much of the work was done in the summers and in the spring of 2015, when I took a semester off from my teaching job to focus on writing. Of course, a great deal of credit is due to my prolific co-author!
  • It is useful to have a schedule and weekly quota. This statement may be obvious to experienced writers, but it was not obvious to me. It kept me from wandering off on interesting but nonessential tangents many times. Which leads me to my next observation …
  • It is invaluable to work with an experienced co-author. Our weekly meetings were always productive and motivating, even on weeks when I felt I had made little progress.
  • Banish perfectionism! The biggest lesson I had to learn was to operate in a fear-free zone and not to worry overly about mistakes. Perhaps some readers will argue that I embraced this precept a little too enthusiastically! I had to accept the inevitability that the final product would contain some errors, but done is better than perfect. The topics are in rapidly evolving fields of mathematics.
  • Expository writing can generate research problems. In academic mathematics, writing books is less prestigious than proving new theorems. Ironically, I have never had as many new research ideas as during the writing of this book.
  • Writing a book is a good way to learn a subject. I started the book with some uncertainty about my qualifications to write it. Indeed, at times I was daunted when trying to decipher the proofs in the research papers we used as references. My confidence grew as I resorted to trying to reprove the results by my own methods, and seeing result after result unfold before me like solutions of beautiful puzzles. My most exciting moment was finding a much more elegant proof of one of the results we wanted to include in our book!
  • Pleasant work conditions are important. Some people prefer quiet and solitude while working, but I like to listen to music and be around other people. Ottmar Liebert’s Nouveau Flamenco CD will always remind me of this book. I did everything I could to make the actual writing process as enjoyable as possible. I kept the phone nearby. I set a timer to remind myself to stop and take a break every 45 minutes. I started working as early in the day as possible. I rarely spent more than four 45-minute sessions a day at my desk, but I tried to work on the book almost every day. Some of my best ideas came to me while doing routine chores between and after writing sessions. I avoided listening to and reading the news.
  • Doing something creative is a wonderful source of happiness. When times were tough at my day job, working on my book helped me calm down and keep stressful events in perspective. It gave me a sense of professional identity independent of my school and math department.
  • When it comes to sheer exuberance and vivid use of vocabulary, I am still not as good a writer as my sons!

Leadership

Leadership is the art of influencing human behavior. It may be defined as “the art of imposing one’s will upon others in such a manner as to command their obedience, their confidence, their respect, and their loyal co-operation.” Put in everyday words, it is the ability to handle men. It is the chief task of the naval officer. The attributes of a good leader are the same throughout the world, regardless of his nationality or the type of organization in which he serves. The outstanding leader so infuses his followers with the desire to be led that they will do everything possible to comply with his wishes and support the policies of the organization whether the leader is present or not.

A leader cannot be made from a man who does not sincerely wish to become one, or from a man who is unwilling to make the sacrifices required of a good leader. The student who feels that leadership is all glory with little responsibility will be greatly surprised when he realizes that the glamour and prestige attached to the leader are outweighed by his worries, work, and responsibilities. To be an outstanding leader requires the hardest kind of work for which very little material credit will be forthcoming.

Naval Leadership, U. S. Naval Institute, 1949

This is my favorite leadership quotation. A friend of mine chose it for his retirement ceremony from the Navy. For many years I had it posted in my office. In difficult times, it has been an inspiration to help me put aside my ego, resist the temptation to give in to destructive emotions, and think about how to act as a leader, not only of my students, but also of my department and my institution. The first chairman I served under in my present institution embodied all the best characteristics of a leader. His question for us was always implicitly, “What can I do to help you do your job better?” He retired this year. He has been a role model for me, and this quotation reminds me of him.

Teaching Ideas from Colleagues

Some of the best teachers I have seen are members of my department. Every time I observe a colleague’s class, I learn something new about teaching. In this post, I discuss various ideas and techniques I have encountered recently and would like to try out.

Tegrity – One colleague has been recording her lectures for several years, using Tegrity. Her voice and everything projected onto a screen from her tablet computer are recorded, so that students can review the lesson later (or view it if they were absent). She is adept at setting up and putting away the equipment in the few minutes between classes, and she does minimal editing, so that the whole process of recording and posting online takes very little time. This system seems particularly well suited to upper level courses for which there is no good text.

Guided notes – Another colleague gets many student compliments for his skillful use of guided notes during lectures. Notes might include the statements of problems, but not their solutions, or outlines of results, with space for supplementary remarks. During lectures, he projects the notes onto a screen and adds handwritten solutions and comments by writing on a tablet. This technique could work for any class, but preparing the notes for the first time may be quite time-intensive.

Historical notes – I cannot remember any of my math professors mentioning the history of mathematics, with the notable exception of an undergraduate professor who promised us the story of Galois, and finished the course with a dramatic enactment of his final 24 hours and fatal duel. When a friend mentioned recently that he likes to put math in context by talking about the history of results and the mathematicians who discovered them, I thought it was a wonderful way to humanize math. I also realized that enriching courses this way is a long-term project. A few small comments, worked into appropriate parts of a course, represent hours of reading and planning, at least for me. My mind does not easily hold the necessary details or retrieve them quickly enough when needed. Then too, it takes some thought to pick anecdotes that will appeal to an undergraduate class.

Videos of applications – Technology has changed so much since I started teaching, that it took me by surprise to see a young colleague work YouTube videos of waves and other physical phenomena into an applied math lecture. I thought this was a wonderful idea and immediately started looking for appropriate physics videos for my differential equations course. I am making a list and collecting recommendations from other instructors.

Standards-based grading – A colleague talked recently in our teaching seminar about his experiments with standards-based grading. The idea is that students must achieve certain standards in all the main topics, but may attempt each assessment multiple times. Only the final (highest) grade is recorded. I like this idea in principle, but it sounds very time-consuming to administer. Also, there can be a certain amount of student resistance to new grading systems, and students may become discouraged and resentful if they feel that the bar has been raised too high. My department employs a smaller-scale version of this idea in the form of gateway quizzes on precalculus topics, basic derivatives, and basic integrals. Each quiz consists of 10 questions. There are multiple versions, with small changes in numbers and letters used, and with topics in different orders. A student must achieve a score of at least 9 out of 10 to pass.

Co-teaching – Finally, one of the most interesting teaching seminars I attended this year was a discussion of a team-taught course, involving two other departments, which combined lab measurements, computer modeling, and mathematics, to give students a taste of research. Several students from this course went on to do advanced full-year research projects in their senior year.

Changing the Culture of Sleep Deprivation

Arianna Huffington wants to change the cultural belief that work dedication requires sleep deprivation. Her new book, The Sleep Revolution, is at its strongest when it appeals to the values that drive high achievers to sacrifice sleep in order to excel. Her most compelling examples come from the world of sports. A series of experiments at Stanford University showed improvement in college athletes after just a few weeks of adequate sleep. Basketball players who increased their average nightly sleep from 6 ½ hours to 8 ½ hours improved their free throw and 3-point shooting percentages by 9%. Football players’ sprint times decreased. Swimmers set personal records. Professional football, basketball, soccer, rugby, and cycling teams all prioritize sleep as an important competitive tool and a critical element of successful training. Some teams even have sleep coaches who track athletes’ sleep and monitor sleeping accommodations while travelling. Huffington hopes that, with their impact on popular culture, athletes will help banish the “I’ll sleep when I’m dead” mindset.

Huffington does not minimize the difficulty of forming better sleep habits in today’s busy and complicated society. She emphasizes that each person’s challenges are personal and unique. She notes that it may take months to form new sleep habits and cautions that there will be setbacks. She wants to remove the stigma attached to napping: naps should be viewed as an aid to productivity, rather than a sign of laziness. Among the professional benefits of good sleep are improved focus, better decision-making ability, more patience, greater creativity, and increased memory capacity. At the institution at which I work, sleep deprivation among the students is routine and expected. Sleep deprivation has been a running theme in my own life too. I find Arianna Huffington’s story of the high price she paid for sleep deprivation and her struggle to reprioritize to be very personal, and I admire her courage for speaking up. By presenting herself as a high profile example that adequate sleep and success are mutually compatible, she has made an important contribution towards changing cultural attitudes toward sleep.

 

Optimization of Humans

Recently I’ve noticed a disturbing trend – the application of powerful mathematical optimization techniques to human behavior.

Retailers find ways to increase our purchases. The food industry researches ways to make us eat and drink more. Digital devices are becoming more addictive. Financial mathematics is used to acquire wealth without contributing value. Work schedules and assignments are designed to minimize costs and maximize profits. Humans have become the variables in mathematical problems, and decisions are made on the basis of mathematical models that may ultimately be optimizing the wrong quantities. How do we resist this trend, and is it possible to change what we quantify in our models, in order to move in a better direction?

In a calculus course, the study of optimization begins with simple problems such as finding the maximum height of an object that is tossed into the air. We then progress to geometric problems involving constraints, such as maximizing the volume of a rectangular box of fixed area, or minimizing the area of a rectangular box of fixed volume. We help imaginary farmers fence their fields at minimum cost, and imaginary oil companies locate oil refineries on the banks of straight rivers.

Linear programming uses numerical algorithms to solve large, complicated optimization problems. It is one of the main tools in the field of operations research, which applies mathematical techniques to decision-making problems. At my institution, enrollment in linear programming is growing so fast that we can hardly keep up with the demand for instructors. Linear programming is surprisingly accessible – many algorithms have been implemented into computer programs, so that it is relatively easy to get started formulating and solving fun problems such as scheduling sports tournaments, or designing a route for a triathlon, using free programs like gusek.

An introductory text in operations research, Network Flows, by Ahuja, Magnanti, and Orin, addresses optimization problems such as the following:

  • Schedule medical staff to hold staff levels as low as possible while maintaining satisfactory levels of health care and meeting demands at different hours of the day.
  • Schedule aircraft to routes to satisfy passenger demand at the minimum operating cost.
  • Find the most economical passenger routing strategy for overbooked airline flights.
  • Design minimum cost delivery routes for a fleet of vehicles to deliver goods to a collection of customer sites.
  • Schedule telephone operator shifts to employ the minimum number of operators to satisfy the requirements for each hour of the day.
  • Determine an employment policy that meets labor requirements and minimizes the costs of hiring and training new employees.
  • Allocate contractors to public works to meet job requirements and minimize costs.
  • Design a matching process to assign medical school graduates to hospitals based on the preferences of the graduates and hospitals.
  • Assign military personnel to job vacancies to maximize the utility of all assignments.

These situations are not merely theoretical. Operations research techniques are routinely used in business, in health care, and in the government. In a recent talk I attended, the speaker described strategies for emergency room staffing to prepare for a surge after a disaster. Surely this is a worthy goal, but I began to feel uncomfortable when he concluded that doctors should have less choice in setting their schedules, and that Friday golf games were expendable.

Human beings are not machines. Is a surgeon operating at 5 pm on a Friday the same as a surgeon operating at 9 am on a Tuesday morning? Do shift work schedules allow adequately for the reality of human biology? Should we remove yet more sense of control from the workplace in the name of efficient scheduling? Where are the variables in these models to measure motivation, job satisfaction, health, and happiness? Isn’t it time to move beyond optimizing only time and money?

These are questions I don’t have the answers to. As consumers, we can fight back against the push to buy more and more. Joshua Becker’s becomingminimalist.com site exposes the cultural pressures to spend and consume far more than we need. Sarah Wilson’s iquitsugar.com site urges us to avoid artificial products of the food industry and Just Eat Real Food. But what do you do when decisions about your job responsibilities and schedule are being made by a computer program? If you are acquainted with the computer programmer, you may be able employ diplomacy to obtain a more favorable outcome. You may find mathematical ways to outwit the program. You may even decide to write a better program yourself. But ultimately, these are judgments that should not be handed over entirely to a computer. We need to think more carefully about the application of optimization techniques to human beings.

Squaring the Curve: Exercise and Aging

Olga Kotelko took up track and field at the age of 77 and set 37 world age-group records before her death in 2014 at the age of 95. In his book, What Makes Olga Run?, Bruce Grierson tries to discover how Olga avoided much of the usual physical and cognitive decline of aging, even increasing her strength in her 80’s. Equally interesting to Grierson was what motivated Olga to continue to compete, setting new world records until weeks before her death. Grierson investigates Olga and other masters athletes who were successful in “squaring the curve,” replacing the usual long steady decline of health in later years with a short abrupt drop-off at the very end.

Grierson begins by discussing theories of aging and states that longevity is thought to be much more heavily influenced by lifestyle than by genetics. He examines the adversity hypothesis, that some acute stress, particularly during the period from the late teens to early thirties, is important in strengthening the immune system and developing resilience. Chronic stress, on the other hand, is known to shorten life.

Experiments with 60- to 80-year-olds show that exercise improves neuroplasticity, the ability of the brain to form new neural connections and strengthen existing ones. Exercise produces measurable brain growth, as well as improvement in many measures of cognitive function, such as reasoning, processing speed, memory, and decision-making. Grierson discusses the benefits of aerobic exercise versus weight-training, and concludes that both endurance and strength are important. He notes that exercise is both a beneficial acute stressor, and a reliever of chronic stress.

Grierson was surprised to find that it is possible to reach high levels of fitness, even after many years without training. An older person may be able to improve his or her VO2 max as much as a younger person. Weight-training may produce rapid and large responses. Many people can safely train intensely over the age of 60, 70, or even 80. In fact there seems to be an intensity threshold of about 80% of max for the most pronounced hippocampus growth and increase in production of growth hormones. Intensity of exercise seems to be correlated to feeling and looking younger. On the other hand, it is possible to overexercise. Too much long-duration and/or high intensity exercise over an extended period of time can cause heart and joint twenty years later.

The importance of sleep is mentioned, including its contribution to cell repair and insulin regulation. Exercising heavily increases the need for sleep. Shift work has been identified as a probably carcinogen. Stretching and massage can help muscles recover from workouts and improve immune function. Diet is important, not only what we eat, but consistency in when we eat.

Paradoxically, having routines and breaking routines are both important. Routines contribute to productivity. Breaking routines is good for growth. Yet too much disruption of routines can make people ill. New experiences such as travel and learning languages may help to protect and develop the brain.

Interoception, or attunement to subtle body signals, helps prevent injury and can boost athletic performance. One rough measure of interoception is the ability to tell time without a clock.

Personality can affect longevity. People who have a positive outlook and don’t allow other people to have a negative effect on them live longer. It is important to learn to shrug off stress and worry. Conscientiousness and sense of humor are other characteristics associated with longer life. Self-esteem helps protect the immune system. A common trait Grierson found in masters athletes was that they knew how to have fun – competitions were play and not work.

Grierson found that the desire to keep competing is related to finding ways to feel that one is still improving. More generally, a purpose-driven life contributes to both longevity and happiness.

Finally, there is evidence that social ties are even more important than exercise. In Olga’s case, the camaraderie of track meets was her greatest motivation. The shared competition promoted particularly rich friendships.

Grierson summarizes his findings in eight exercise and lifestyle rules to promote vitality, longevity, and happiness, concluding with a ninth rule: begin now!

Opinion: Research and Teaching

Why is research required for mathematics faculty at 4-year colleges? Why is “scholarly activity” such as giving talks and expository writing not enough?

I have asked myself these questions many times, and my answer has changed over the years. Recently, a new faculty member asked me the first question, and I remembered how I felt at that point in my career and tried to give the answer I would have liked to give my younger self.

First, I now feel fortunate that I get to do research, rather than dreading that I have to do research. Tenure is part of that, but I also feel the joy of taking more risks to explore on my own, rather than working on questions set by respected senior mathematicians.

I work at an institution that employs many instructors without Ph.D.s or even master’s degrees in math. Many of these instructors are wonderful teachers, with great patience and dedication. It is discouraging to feel that after years of graduate school, postdoctoral work, and college teaching, as an instructor I am so easily replaceable. It has taken time to recognize my own contributions.

Doing research has changed the way I approach my courses. I used to read the explanations in the text, and perhaps consult other texts before teaching a topic. Now, I try to arrive at my own explanations, and find new points of view. There is a difference between reading someone else’s proof and understanding a result to be true, and discovering one’s own route to that truth.

Doing research has developed my taste in mathematics. I have strong opinions on what types of problems belong in the syllabi of my courses. I no longer see the curriculum as a static collection of knowledge to pass on from one generation to the next. I want to communicate modern viewpoints and methods, and give my students a taste of the power of computers, even when they are required to do computations by hand.

When I struggle with problems at the limits of my ability, I experience the same type of confusion and frustration that undergraduate math students feel. When I talk to them of study methods, problem-solving strategies, and time-management, I am speaking from current personal experience. I want to help my students make a transition from passive learning to a more mature and independent working style. I want my students to understand that there is a transition to be made.

Finally, without attempting research, we do not even know what we do not know. Research gives us a glimpse of the vastness and complexity of the frontiers of mathematics.

V. I. Arnold

V. I. Arnold, 1937 – 2010, was the A in KAM theory in dynamical systems and ABC flows in fluid dynamics. He wrote hundreds of mathematical papers and many books, including well-known texts on ordinary differential equations and mathematical methods of classical mechanics. He completed the solution of Hilbert’s 13th problem at age 19. He has been described as the last universal mathematician. An asteroid, Vladarnolda, is named after him.

Arnold: Swimming Against the Tide, edited by Boris Khesin and Serge Tabachnikov and published by the American Mathematical Society, not only describes Arnold’s life and work, but also gives a glimpse into mathematical life in the USSR after World War II. After Perestroika, many of the leading Russian mathematicians emigrated to Europe, Israel, the United States, and Canada.

Arnold was born in the USSR into an intellectual family: he described himself as a fourth generation mathematician. At age 12, he started studying mathematics seriously, working through his father’s mathematics books. He participated in a math circle for 7th – 10th graders run by undergraduates at Moscow State University and was a contestant in the Mathematical Olympiads. After completing his graduate work under Kolmogorov (the K in KAM theory), he taught at Moscow State University for many years.

The Arnold seminar greatly influenced a generation of Moscow mathematicians. A participant recalls how Arnold would jump up in the middle of a speaker’s presentation to explain the origin of the problem, its connection to other problems, the significance of the main result, and the methods by which he would try to prove it. He would then debate with the speaker various implications and generalizations of the result.

In his lectures, he liked to start with the main and difficult ideas, explaining them in simple terms. Technical details came later or were left to the audience members to fill in on their own. One mathematician mentioned that Arnold liked to make small mistakes in his undergraduate lectures and expected the students to notice and correct him.

Arnold attributed the development of mathematics to discoveries of unexpected relations between different fields of inquiry. He believed, “… one needs mathematics to discover new laws of nature as opposed to ‘rigorously’ justify obvious things,” and refers to “the disastrous divorce of mathematics from physics in the middle of the 20th century.” He had particularly harsh words for the Bourbaki school of mathematics, blaming the decline in prestige of mathematics partly on those “who proclaimed that the goal … was investigation of all corollaries of arbitrary systems of axioms.” He agreed with Sylvester that “… a mathematical idea should not be petrified in a formalized axiomatic setting, but should be considered instead as flowing as a river.”

Arnold also liked to compare mathematics to mushrooms. The upper part of the mushroom corresponds to published theorems, but the enormous lower part in the earth corresponds to problems, conjectures, ideas, and mistakes.

In at lecture at the Fields Institute in Toronto, in 1997, in response to a question on how to locate something in the literature, Arnold listed some of his favorite older sources: the German Encyclopedia of Mathematical Sciences edited by Klein and published in approximately 1925, the Jahrbuch, the collected works of Klein and Poincaré, the Zentralblatt, the Russian Referativnyi Zhurnal Matematika, and the over one hundred volumes of the Russian Encyclopedia of the Mathematical Sciences, some of which have been translated into English. However, he warned, “Of course, in spite of all these precautions, you may discover too late that your result was known many years ago. It happened to me to rediscover the results of many mathematicians.”

In an interview for the Russian magazine Kvant, Arnold gave two conditions for becoming a research mathematician: a love of mathematics and good health. He thought vigorous physical exercise was as important as vigorous mental exercise. In the winter he skied 100 kilometers a week and in the summer he rode a bike and walked. Describing his work habits, he remarked, “When a problem resists a solution, I jump on my cross country skis. Forty kilometers later a solution (or at least on idea for a solution) always comes.” Arnold continued to be a prolific researcher to the end of his life and even changed his research direction radically a few years before his death, to study classical number theory and combinatorics.

Reflections on Fall 2015 Courses

Nowadays the syllabus of a course feels like the blueprint for a house, with the challenge to create something new and better each semester. There is more room for artistic license when I write my own syllabus, but there are still plenty of design decisions to be made when I receive a syllabus from a course coordinator. The budget consists of time and energy, the students’ and mine. This fall I taught two sections of a familiar multivariable calculus core course and a new course (for me): discrete math, an introduction to graph theory and combinatorics. Most of the comments below apply to the multivariable calculus course. I will discuss the discrete math course at the end of this post.

The multivariable calculus course introduces vectors and graphs in three-dimensional space. Students learn about partial derivatives, iterated integrals, and parametrization of curves and surfaces. The course concludes with Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. The Mathematica plot below shows a simple vector field and surface from the last test in the course.

hemisphere

I began the semester with several ideas. I wanted to reduce digital distractions, invite senior students to talk about their projects, and provide weekly handouts on applications of math or study tips. Ongoing were efforts to use as much class time as possible for problem-solving, to improve the homework I assigned, and to continue to use Wolfram demonstrations and Mathematica plots to introduce new concepts and help visualize examples. I had some additional ideas about interleaving material and giving cumulative quizzes and tests in calculus, but these turned out to be more difficult to implement than I had expected, due to the amount of material required to be covered.

Digital distractions: I asked students not to use computers and smart phones in class, and although there were inevitable lapses, and occasions when they had to be allowed for class purposes, on the whole I was happy with the results. Using an online text can be convenient, and I enjoy having students try out Mathematica, but the distractions of having on their desks the same devices they use for entertainment and social media can outweigh the benefits. I felt the classroom atmosphere and my relationship with the students was much better without digital distractions. I have used the WebAssign online homework system in the past, and found it useful for keeping weaker students on track, but did not use it this semester. I was surprised how happy my students were at this decision.

Homework: Rather than collecting weekly homework, I gave (long) weekly quizzes, and reminded the students frequently that it was up to them to determine how much of the syllabus homework they needed to do on each topic. A low quiz grade one week was usually a strong enough message that some recalibration of study strategies was necessary. Some of my students had covered most of the material in previous courses so I was faced with the dilemma of wanting to challenge them without forcing the rest of the class to cover additional material that was not part of the core. I decided to address it by giving students a choice of problems on written assignments they had to hand in. Some problems were fairly routine, and others were well beyond the usual level of the course. Surprisingly, many of the students deliberately chose the more challenging problems and showed great persistence in trying to solve them. One of their favorite problems was finding a parametrization for a torus and computing its area. Another was computing the two-dimensional Laplacian in terms of polar coordinates.

Wolfram Demonstrations: There are many beautiful Mathematica demonstrations of multivariable calculus concepts available at the Wolfram Demonstrations Project web site:

http://demonstrations.wolfram.com/

I keep a list of links for the course (updated every year), and use these demonstrations when introducing new topics.

Mathematica Plots: I enjoy creating Mathematica plots to illustrate examples in the course. I sometimes enjoy this aspect of the course a little too much, and can easily get distracted experimenting with ways to make more beautiful plots. Here is a basic plot of the “ice-cream cone” region I use to compare integration in Cartesian, cylindrical, and spherical coordinates.

ice cream cone

Here is a fun plot to illustrate an example in which we calculate flux of a vector field across the part of a plane that lies in the first octant (where x, y, and z are all non-negative).

triangle

Friday handouts: One innovation that was well-received and took relatively little time was the institution of “Friday handouts,” which were short articles on some application of math, usually from the American Mathematical Society Mathematical Moments series, which can be found online at

http://www.ams.org/samplings/mathmoments/mathmoments

Some of my favorites were Knowing Rogues, on rogue waves (large, unexpected, and dangerous); Making Votes Count, on voting procedures and how they can determine outcomes; Holding the Lead, on using random walks to model scores in competitive sports; Bending it like Bernoulli, on the mathematics of soccer kicks; Making an Attitude Adjustment, on repositioning spacecraft; and Tripping the Light-Fantastic, on the mathematics of invisibility. These handouts often stimulated discussions at the beginning of class, and cheered up the students by reminding them that the weekend was near. Near exam time, I also provided a short handout on some recent research on studying and learning.

Class visits and discussion of student projects: I invited two senior students majoring in math or operations research to visit each calculus class and talk about their senior projects. This gave my students a glimpse of what these majors were about, and allowed the senior students to present their work to someone other than their research advisers. I also invited a senior applied mathematician to talk about research and internship opportunities for undergraduates. All the visits were all well-received and I plan to make such visits a regular part of freshman courses in the future. My students also seemed quite interested when I described student projects I had supervised in my own capstone courses.

Class problem sessions: Most days I tried to allow at least 10 – 15 minutes at the end of class for students to start working on homework and for me to answer individual questions. Sometimes I had more formal problem-solving sessions, in which groups were assigned specific problems to work out on the board and present. I often gave special challenge problems for the more advanced students. Toward the end of the course, when the problems were longer and more complex, a full class period was needed to go over a set of problems and answer all the questions. I have always found problem sessions very useful in helping students get a quick start on homework and identifying difficulties. I also find that the more interaction I have with my students in problem sessions, the better the class atmosphere and my relationship with the students.

Discrete math: About a third of this course is elementary graph theory, and the rest is combinatorics, finishing up with generating functions. The students also learn to write proofs by induction. Some instructors introduce their own favorite topics, but I chose to stick with the basic content and have the students present about half of the material themselves. When students had difficulty with topics, I gave makeup quizzes on part or all of the material, to ensure that everyone mastered it. The class was small enough that I could give individual feedback on proofs in class, and ask the students to complete several drafts. I was very pleased with student performance on the final exam, and felt the slow pace and concentration on a few basic topics worked well. I would have preferred more serious applications in the text, and fewer problems counting doughnuts, candies, or colored balls. Finding more interesting applications will be my first goal when I teach the course again next semester.

Teaching and Desirable Difficulty

Make it Stick: The Science of Successful Learning, by Peter C. Brown, Henry L. Roediger III, and Mark A. McDaniel, covers some of the same ground as Benedict Carey’s How We Learn. Key strategies discussed by the authors include

  • Retrieval practice or self-quizzing. Although rereading and highlighting may give students an illusion of fluency, self-quizzing is much more effective for remembering and understanding material.
  • Spaced practice sessions. Waiting until forgetting has begun makes retrieval practice more difficult and more effective.
  • Interleaved practice rather than blocked practice. The authors point out that students may feel that better learning occurs with blocked practice, but research shows that mixing problems of different types produces better long-term results.
  • “Generation” or attempting to answer a question before seeing the solution.
  • “Elaboration” or making connections between new learning and prior knowledge.
  • Reflection on the learning process itself.

A theme that runs throughout the book is the counter-intuitive finding that when learning is more difficult, it is stronger and lasts longer. This raises the uncomfortable question: is striving to teach in a way that makes a subject seem as clear and easy as possible really beneficial to the students? How can I best introduce desirable difficulties into my courses?