Saturday, May 16, 2015

David Bressoud Discusses Calculus At Crisis

David Bressoud is weighing in on the Calculus crisis, which I think many may not see as a crisis.  This series will be interesting, and I'm looking forward to Bressoud has to say.  This month Bressoud started a several part series on his blog Calculus At Crisis: The Pressures, with the first post focusing on the larger, societal forces affecting enrollments at the "macro" level.  Bressoud has written often about Calculus, and it's worthwhile reading for math educators.

The quick version of Bressoud's latest post is that more students (often inadequately prepared) are taking calculus, while math departments are seeing resources and support dwindle. Simultaneously, a call has been made to increase the number of STEM graduates, yet resources, human capital, pedagogy, professional development infrastructure, and general education infrastructure upgrades have not been made.

Calculus has long been a hotly debated issue in undergraduate education.  Much has been said, and can still be said.  My hope is that discourse will pivot towards using data and science.  One major positive aspect of the MAA Calculus Study is that we can now discuss issues with better data and deeper insights.  MAA plans to produce an MAA Notes guide to share what successful programs do, and how other institutions can make changes that make material impact on student success in Calculus.  Hence, there exists the opportunity to make data-driven or data-influenced decisions and upgrades, as opposed to merely relying on intuition and anecdotes (AKA anecdata or conventional wisdom).

I am reminded of a mental experiment regarding Calculus that expresses the issues on a human level.  How many of our students say, "That was the best idea ever!" after finishing a Calculus course?  The vast majority of us would say zero.  This is an utter shame.  Let's think about this.   Calculus is one of the greatest human inventions of all time.  In the absence of Calculus entire fields in science and technology cease to exist, and modern life as we know it would also not exist.  Yet, few students walk away from Calculus with a deep appreciation and rich understanding of it.  It's as if Calculus has become a standardized test.

It doesn't have to be this way, and there's reason to be optimistic.  The MAA Calculus study provides a framework for how to think about the issues and how to make improvements to our classes or programs from a "system" perspective.  Teaching is a system (and a cultural activity), and we have knowledge and insights, based on the hard work of many creative individuals, pointing the way towards solutions and specific areas where efforts will be productive.

Link:  MAA Calculus Study

Tuesday, April 14, 2015

"Activation Energy" and IBL Uptake

Here's a basic question.  What does it take to switch from traditional to IBL teaching?   This question can be answered in many ways.  I'm going to come at this from an education system reform angle.  Essentially the general problem is the implementation challenge in education.   Perhaps the biggest challenge for the education community now is implementing what works in the classrooms on a broad scale, and open questions remain about what it takes in time and investments to make the necessary changes.   In this post the notion of activation energy is used to shed light on what it might take to make reform stick.

In Chemistry, activation energy is a term that means the minimum energy that must be input to a chemical system with potential reactants to cause a chemical reaction.  Implementing IBL is analogous in that there is a substantial initial investment by an instructor to learn the necessary skills and practices to be successful in the classroom.  What is the activation energy required for an instructor to switch to IBL teaching?

Instructors of course vary greatly, due to experiences and teaching environments.  For our purposes a "back of the envelope" computation is all that's necessary to illustrate the main points.  Let's take as an example an assistant professor who attends an IBL workshop.  In this case, we have an instructor who elects to attend a workshop, is motivated to learn to use IBL, is invested in her job and institution, and wants students to learn authentic mathematics.

So here it is.  The activation energy required is 100+ hours, plus resources to travel to a workshop,  prepare for a course, and engage with the IBL community.   The breakdown is as follows.
  • 10 hours pre-workshop preparation
  • 40 hours of workshop time
  • 50+ weeks to prepare for a target course (begin course materials adaptation, plan activities for the first few weeks, syllabus, other course management.)
  • Plus hundreds of hours more through the first few terms of using IBL.
The aggregate investment per faculty is roughly between $5,000-10,000 to attend a workshop, materials, mentoring, and so forth.   Most of this cost is imbedded in the professional development infrastructure.  Experts need to be hired to develop the materials and run the workshops.  My initial estimate of the the IBL activation energy is approximately 100hrs + $10K (or 100+10 for short).   This is the base investment to get started.  Becoming an expert IBLer is a much, much longer effort that takes years.

It's noted that this investment level can go down with economies of scale, but we are nowhere near that level of uptake or infrastructure today. The infrastructure across the nation is not yet established to offer lower-cost options.  In fact, at this point in history a focus on efficiency is likely a major strategic mistake.  The infrastructure needs to be built up first, and then in the future as uptake increases, one can economize.  It's called economies of scale for a reason.

The situation quickly gets more complicated.  Colleges, CTLs, professional development groups likely do not have good estimates of the activation energy required for uptake of student-centered pedagogies (or are even aware to think about it this way).  Further, as we develop more sophisticated teaching frameworks, the complexity of professional development and the specific supports needed by faculty become more technical and discipline specific.  The activation energy is dependent on the PD available, the subject matter, and the varies by instructor, course, and institutional environment.  Teaching Calculus isn't the same as teaching Math for Elementary Teaching or Topology.

Further, an echo from the past that continues to be felt today that inhibits progress on implementation is the factory model mindset.   Lingering to this day is the sense that instructors are delivery devices for information.  With the factory mindset is the belief that fixes to the system are in the form of tweaking courses, chipping away at the margins, and changing textbooks.  This is one reason why I am presenting the activation energy concept for education reform.  When we view instructors as delivery devices and/or underestimate for whatever reason the real effort necessary to become an effective IBL instructor, then the level of support and resources allocated to the problem is too low.   Invariably some new IBLers will struggle (due to inadequate preparation), and then the next, linked fallacy that results is something like, "I can't teach via IBL" or "IBL doesn't work at my institution."

The Pendulum.  There exists a defeatist belief among some in education that there is an education pendulum, and that's just the way it is.  Things repeat like a sinusoidal function, over and over again.   I've written about this before here.  Those weary of repeated efforts to make changes have a reason to be this way, and I am sympathetic to a degree.  They've seen things swing one way and then another, and those with hopes for a brighter future have seen their efforts crash and and burn, which is highly demoralizing.  The activation energy idea sheds some light on the matter.   Let's think of it as an absurdly flawed road trip implementation.  If we put in only half the gasoline needed for the trip, and each time we get towed back home, then one interpretation of these events is that this is what vacations are about. We go, don't make it to your destination, because we run out of gas. Then get towed home.  Hence the pendulum.

But it doesn't have to be this way. Education is a human construction.  It's not like the stars and the moon, where we have no way to affect the universe outside our planet in significant ways (as of today).  We built it.  We can change it.   If we follow our own teaching philosophy, then we should ask good questions.  Why is there a pendulum?  What are the causes?  Is there a better way?

Let's the put 100+10 estimate into context relative to current institutional practices.  Faculty training programs often have 1-2 days of "new faculty orientation" or a weekly seminar that meets for 1 hours.  More or less this is an order of magnitude below what I am seeing as necessary, without considering the nature and quality of the programs.  It is fairly well known that current practices of TA training or new instructor training are not sufficient to address the broader uptake problem, where low percentages of faculty in STEM actually use proven, student-centered teaching methods.  With the activation energy perspective, we can start to quantify how much more effort is needed and what it would cost.  While we are below the mark that I estimate is necessary with current practices, on the positive side we know ways to get people past the activation energy.  Solutions exist!

On a personal teacher level, getting started with IBL is hard.  IBL, however is a "sticky" idea in the sense that once instructors use it well, they stick with it.  Indeed, once you see your students think creatively and transform how they think and think of themselves as learners, there's no going back.  And that's easily a worthwhile 100+10 investment!

Friday, April 3, 2015

IBL Workshop 2015, July 7-10, San Luis Obispo

One of the best ways to learn how to successfully implement IBL is to attend a weeklong workshop.  The IBL Workshops hosted by AIBL are specifically designed for college math faculty.  These hands-on workshops address the practical obstacles that faculty face in the transition from tradition to IBL teaching.  Participants of the workshop adapt or develop IBL course materials, work collaboratively on IBL specific teaching skills, engage in discussions about IBL video lesson study, learn specific nuts and bolts issues in smoothly running an IBL course, and participate in a yearlong mentoring program.

A handful of spots remain for the NSF funded workshop this July.  If you are thinking about making the transition to IBL teaching, please go to to learn more.

Tuesday, March 31, 2015

Quiet Classes: Using Pairs to Generate More Discussion

Here's post on a common theme, but it's worth revisiting this topic.  Quiet classes (or students) are ones we need to work harder at, and efforts can make a significant difference.

Quiet, low-energy classes are the ones that make me the most nervous as a teacher.  As a long-time IBL instructor I have become accustomed to loud, boisterous classes, where students are engaged, communicate, work together, and ask questions to me and to each other.

At the start of the term some of my classes have a quiet or passive personality.  Students may not have had an IBL class yet, and hence are likely to expect to sit, take notes, and generally not interact with others during class time.  A potential problem with quiet students is that they give you little to no information about what they are thinking (and not thinking).  Student thinking is critical in day-to-day IBL teaching.  An IBL instructor doesn't know exactly the lesson plan for tomorrow, until today's class is over.  Using what is known about what students are doing well and what they need to work on, is bread and butter in IBL teaching.

What can an instructor do to liven things up, when students are (initially) reticent?

There are several options.  My main goal is to reset classroom norms so that students understand that making their ideas and questions known to others is the default.  Reseting norms can be accomplished via tasks that require students to engage verbally.  My most preferred setup is to use pairs.  When you are talking to one person, it's awfully difficult to hide in the conversation.  In contrast, within a group of 4 students, one or two students could sit back and let the others talk.  Hence, I like pairs (while simultaneously admitting that personal preference is a part of the decision).  I also use other sizes, but pairs are the default, especially for quiet classes.

In a full IBL course, pairs can do a range of activities, from working on problems, reviewing a presented solution, getting started on a new problem set,...  Asking students to share what they did in pairs is a safe, easy way to open discussions.  The stress of having to be right isn't a factor, and students are more likely to offer thoughts, questions, or ideas that can generate a productive discussion.

Making a pairs seating chart and ensuring all pairs are called on regularly ensures that all students are regularly involved.  In a full IBL course, the requirements to present and comment are built into the course.  Getting quieter students to offer comments, can be done via the pairs structure.  For example, ask students to review and discuss a solution or proof in pairs (after a proof is presented) will generate more and better comments and questions.  I normally phrase the task as, "Please review the proof and come up with 2 questions or comments."  Then I call on pairs rather than ask for volunteers, and spread the work around to ensure all pairs get called on regularly.

In "hybrid" IBL courses, where the course has a larger percentage of instructor talk time, then the implementation of pairs becomes more specific.  What I'm envisioning in this situation is an instructor just getting started with IBL, or an instructor in a situation where significant IBL time is not feasible.  In this case, the instructor may be using some Think-Pair-Share or group work as components to their teaching.  Here are some examples:
  1. Let pairs discuss a concept or strategy:  "In pairs, discuss for a minute a strategy you might use for this problem."  Listen in on a few conversations, and choose one or two to share to the whole class.
  2. Check for understanding even when building skills:  "Please work in pairs to apply the techniques discussed to the following cases."  As you walk around, you can ask students how they are doing with the exercise. 
  3. Pause for moment to let students try something on their own first.  This could be the next step of a problem or proof.  Or it could be that you ask students to justify a statement or review what just happened.  After the pause, ask students to check in with one neighbor.   As students talk, you can visit a few pairs and ask a pair to chime in.  (Be sure to visit different people each time you do this to spread the work around.)
The above are just examples.  An infinite variety of ways to get students to talk more exist, and the main point I want to get across is the framework.  Use clear, specific mathematical tasks and pairs to get students to think and then talk.  Once they start talking, they can open up.

Tuesday, March 10, 2015

Call for Abstracts, RLM and IBL Conference

College math faculty -- please consider submitting an abstract to present at the 18th Annual Legacy of R. L. Moore and IBL Conference, June 25-27, Austin, TX.  We hope to see you in Austin!


  • My favorite IBL activity
  • Teaching Inquiry and promoting questioning
  • Student success outside of academe: IBL fostering success in business and industry
  • Flipped course environments and IBL: Blending ideas and methods effectively.
  • IBL Innovations: new happenings
  • Poster Session 

Link to Conference Web Page

Tuesday, March 3, 2015

Math Anxiety Realities: Student Voices

In the interest of trying to ensure things are interpreted appropriately, I need to mention some very important caveats.  I deeply respect the people who work in education at all levels.  No one I know, who works in education, intentionally creates or supports building anxiety.   This post isn't about pointing fingers at specific groups of people.  This is why we do scientific research.  We seek to find out whether what we are doing is working or not.  This post is intended to be a call to action, and an invitation to open, intellectual dialogue about a critically important issue.

Math anxiety is real.  In our discussions about education reform, an overlooked piece is what students think and feel about Mathematics.  Learning skills, concepts, and habits of mind are part of the core of education.  Associated to this is the enjoyment of learning or lack thereof, in the case of math anxiety.  If a student learns how to do math and hates it, it's clearly not the outcome we desire.  Standardized testing does not measure attitudes about math, and because of this our debates are skewed in that we are ignoring very real problems.

My motivation for sharing student quotes comes from a couple different places.  One is that instructors rarely ask students to write about their own personal experiences with Math.  Getting to know your students and what they think and feel about math is valuable, so I encourage math instructors at all levels to include an assignment that asks students to write a math autobiography (or something equivalent) to get a sense of what your students are coming into class with.  The more you understand your students, the easier it is to find a place to meet them and build something positive.

Another source comes interestingly enough from talking to parents about CCSS.   As a mathematician I get asked about my opinion about CCSS Math.  (My short answer is that science is there, and that we need to work out the significant implementation challenges.)  The discussions often end up in a category I call the "back in the day" category.  It goes something like this.  We talk about some of the evidence I have seen in my work in IBL, how it's important to learn skills on a foundation of understanding, and so on.   But then the conversations ends up with, "Well, when I was a kid..."  The usual thing I learn about that person is the assumption that what we did as kids, back in the day, was good, effective math education.  I mean, look at us.  We turned out okay, right?

One common misunderstanding is that Math is equated with doing basic calculations.  It's about getting answers fast to computations and doing algorithms.  Math is not viewed as the subject mathematicians know it as.  This topic has made the rounds again and again, and it's likely to be here so long as we continue traditional teaching practices.

Another more subtle misunderstanding is that the reform efforts have to prove themselves better than the tried and true traditional form of math instruction.  There was never any base of high quality.  See John Dewey's writing at the turn of the 20th century or Warren Colburn (1830) The reality is that education systems need to evolve like healthcare systems.  New knowledge leads to new practices.  Teaching is not a fixed, static entity, and education systems that adapt and incorporate scientifically-validated methods are the ones that are going to set their societies up for success.  Further, decades of education research studies show that we (in the U.S.) need to make significant changes to curricula, methodology, assessment, teacher professional development, and more.   Traditional, memorize-and-regurgitate curricula and instruction has some unintentional and catastrophic consequences.  Many students hate math.  These students think it's only for geniuses to understand.  Problem solving, independent thinking, creativity, and curiosity are not associated with Mathematics.

The tacit assumption, however, is that we don't need to move forward and evolve.  It's understandable for people, who are not in education, to feel this way. They don't have access to data (at least easily), and can only base their opinions on the information they get from mass media, word of mouth, and school report cards.

Consequently, I'd like to bring to life math anxiety through the words of real college students.  Math anxieties are carried into adulthood, by highly intelligent people, and I believe it affects how they learn, interact with new ideas, and perceive themselves.  The quotes were collected from just two sections of a course for future elementary teachers and two sections of a G.E. math course at Cal Poly.  I could share similar quotes from my previous job (Cal State Dominguez Hills), despite being from a demographically different group of students, suggesting this is a widespread phenomenon.  In courses for future elementary school teachers, it is typical for 50-75% of students to have negative or somewhat negative associations with Math.

Student Voices
"The first time I remember doing anything with numbers was in kindergarten... I wrote all the numbers from 1 to over 500.  I remember feeling very proud of myself.  And that is the last time I felt brilliant at math.  I do well in the classes but I don’t feel as though I understand it." 
"There is a very popular American phrase, 'love at first sight,' which describes an intense overwhelming sensation felt by a person who has experienced something or someone that they can not stop thinking about for the first time.  This phrase would be completely inappropriate to describe my relationship with Lady Math.  In fact, quite the opposite.  My first memory of math dates back to 2nd grade.  I remember taking the 'Times Table Test' every friday, and wondering to myself, 'why on earth is this relevant to anything I will ever need to do for the rest of my entire life...' As I made my way through elementary school, I would find myself wondering the same thing more and more each year." 
"My math experience overall has not been a positive one... I remember always feeling extremely embarrassed because I didn’t know the answer as fast as the other students.  Since then I had a fear of math."
"I always dread math classes because of the many negative experiences I have had with them."
"Math has become one of my personal enemies..."
"I believe there are two types of people in the world: math people or people who despise math.  I fall into the latter category."  
"My first memory of math is the times tables in third grade.  I liked the competition, and the racing to the end, but I wasn't so hot on the accuracy.  And then when we got to fractions... my struggles really began.  In fifth grade, I effectually stopped doing all math homework and doodled my way through class.  This lasted until my first parent teacher conference, at which point, my parents became aware of my math maladies."
"Math and I are old adversaries.  It all began in first grade, when we were learning simple math.  We used to get these little math worksheets for practice... I was one of the slowest kids to move through my worksheets and advance on to the more complicated puzzles the entire year... I would plod along miserably through my homework and worksheets and pray every class period that I had somehow suddenly developed the ability to become invisible.  Sadly, that was never to be the case. Every time I was called upon to give an answer became an embarrassing spectacle of epic proportions complete with cherry-tomato-red faced stuttering and incorrect answers." 
"In this math tutoring program [in elementary school] I was drilled over and over again on simple mathematical concepts.  Doing pages of multiplication and division on a time limit, and doing pages of simple problems for prolonged amounts of time.  Naturally, I grew to despise math even more than I had before, and it hardly improved my bad habit of solving problems too quickly and making small mistakes... I was placed into a Trig class that renewed my dislike for math.  Once again my ability to correctly answer was impeded by rushing too much through the problems, and I struggled once again to solve problems correctly." 
"I attended math tutoring at the tender age of six, where I was rewarded with graham cracker and stickers for the correct answers, and I felt that this was something to be ashamed of.  In seventh grade I began pre-algebra, and I had a teacher who was just entering her second year in the field... I repeated the course in eighth grade, with better success.  I felt a lack of self efficacy at that point, highlighted by the notion that I was not meant to excel in math, and that I should accept that and work towards other goals." 
"From a very young age I always had several words I would attach to my response to anything math related.  These lovely describing words included anguish, struggle, frustration, and confusion.  Needless to say, I do not have a good relationship with math... In high school my math grades began to reflect an amount of incompetence in my math classes, which irked me to no end, as I was doing quite well in my other courses.  Earning high grades was something that was expected of me, and I felt like I was floundering...  I had to hire a tutor to pull me through trigonometry and calculous and I still have many foggy areas in both subjects (as well as a deep seeded hatred of the courses)... A key reason I decided upon my major and career path was to avoid math at all costs..." 
"To say the least I was scared of math.  It has always been the subject that no matter how hard I worked, or how many problems I did, the test always was a struggle... I think I became ingrained and almost have an irrational feat of it.  I became so used to saying 'I'm not good at math' 'I don't like math' that I became used to not being successful at it." 
"My experiences in math through my much of my life have been horrendously painful; I would begin every summer in middle school and high school crossing my fingers that my final would raise my math grade to a C-... It really wasn't always like that, some of my earliest memories involving math were those speed tests where you got a sheet filled with simple addition and you had to see how fast you could solve them.  I always remember being one of the fastest in my class.  But I guess that was more memorization then problem solving at that point, because when algebra began my grades declined drastically." 
"Throughout my education, I have earned A's and B's in average level math classes, but those grades were not earned easily.  As an underclassman in high school, I worked my way through Geometry and Algebra II without too much strenuous work, but as I got to Pre-Calc/Trig in my junior year the difficulty level rose while my success slowly declined.  That year I needed all the extra tutoring hours I could get, and I still felt unconnected, uninterested and as confused as ever with all of the equations and formulas that were drilled into my brain.  My problem with math is not that it is hard -- I realize that most things in life will be difficult and I always enjoyed the work that goes into achieving greatness.  My problem with math is that I often feel like I just don't get it no matter how hard I study, which only adds to my stress levels." 
"... I can remember that third grade was the year where we were forced to learn and memorize by heart our times tables, up to the twelves... I didn't realize it at the time, but looking back on that year it is obvious that my anxiety and doubts about my own math skills started then." 
"Tenth grade was a horrible year for my appreciation in math... I took the final and passed with a C- after that year I never wanted to do math again."
"Math and I have not always been the best of friends.  Since the beginning, it's never been my best subject.  I struggle with remembering all the different formulas required for getting problems done correctly;  I even struggle with the fact that everything has to have one certain answer and no free thinking is allowed.  Your own thoughts and ideas are not involved whatsoever in math, and I have always had annoyed feelings toward that." 
"I honestly cannot remember my first negative emotion towards the subject of math, but I can say that it developed early on.  Math is the only subject that I try to avoid at all cost.... Thinking back now, I can recall the first time I realize that I would struggle to achieve a high understanding of math.  In the second grade, I remember the timed tests we would be given weekly to test our addition and subtraction skills... I hardly ever finished the entire sheet before time was up.  Every week, I felt defeated.  If anything, these weekly quizzes degraded my spirit.  I wasn't learning how to practice my math skills or challenge my intelligence;  I learned to despise the subject that had me seemingly beat." 
"Math became painfully difficult for me around the fourth grade.  I could not for the life of me learn my times tables beyond 6.  Week after week I took the test for 7 and week after week I was unable to pass it.  Everyone in the class had a magnet that moved along the wall from number to number as we passed each test. One by one my classmates moved along the wall all the way to 12 until I was one of just a few children left.  Of course, I was humiliated... High school was much of the same.  I worked long nights with a tutor for C's."   
"My experience in math has been very negative, long, and discouraging." 
"Saying I do not like math is an understatement."

And the good ones leave Math too...
"Math became my favorite subject from then on, I just never admitted that to anyone because I didn't want to be a nerd.  The simplicity of math is what I continued to love:  there was almost always one correct answer.  No more and no less.  I received A's in every semester...  Moving along the math train pretty quickly, I found myself in algebra 2 my freshman year.  This was advanced for my high school, at least.  After A's in both semesters, I moved onto trigonometry/pre-calc.  Again, I received A's as the norm.  This was the point that I realized it.  Math bored me.  I knew I would be one of the most unhappy people in the world if I was doing nothing but math problems for the rest of my life..."  
"The reason I was not a fan of math was the lack of discussion involved.  I loved learning about the stories involved in history, and reading about experiences in English as well, but when it came to math it was just cut and dry; formulas and data compilations.  I also associate with math the feeling of frustration and disappointment."   
"Since elementary school I have always done pretty well in my math courses, but I never seemed to enjoy the subject quite as much as other subjects... it was never as appealing or enjoyable to me because it seemed to be more rote based and did not capture my attention because it did not allow for creativity...  there was always one right answer..."

Links and Further Reading

  • Jo Boaler on Math Anxiety
  • Post on Students' Attitudes and Beliefs in Mathematics
  • "Talking About Leaving: Why Undergraduates Leave the Sciences" by Seymour and Hewitt is a careful study about why students leave the Sciences, which is a related topic. While math anxiety isn't a identified to have a role with undergraduates leading the sciences, the broader issue that is connected is that education systems can unintentionally push away students from subjects.

Friday, January 30, 2015

Creating Time for More Engagement

Sometimes we teach in situations where we cannot use a full IBL approach.  Perhaps the class is too large or there are "coverage constraints" that are outside of your control.  Despite such restrictions due to environmental factors outside of the instructor's control, there exists strategies that be used to create time to engage students.

Think-Pair-Share (TPS) is a core piece.  What's nice about TPS is that one can expand or contract the size of the TPS task so that it fits into your existing system and/or the specific need at the moment.  Simple, quick concept questions to longer problem-solving tasks span the typical range of TPS uses in Math.  So the goal is to get time to insert TPS, and engage students through specific tasks where they get to think for themselves.

The main topic of this post is creating time.  Below are some strategies to buy a little time.
  1. Instead of coping the statement of problems, exercises, definitions on the board only for students to copy them into their notes, a handout can be used where all of this is printed or available on electronically.  The time used for all this writing can now be used to process, think, and discuss.  Handouts potentially save quite a bit of time.  In traditional courses, a chunk of each class period is spent on writing on the board and writing in notes, often of things that are already printed in the textbook.  Further, the time it takes to write your lecture notes, can be used to type up a handout.  It really doesn't take that much more prep time.
  2. An example of the above is the typical "working an example" for the class.  Normally a teacher works through each step and shows what to do to the students in class.  If this example was printed on a handout, then those 5-6 minutes of instructor show-and-tell could be made much more active.  It can be from a quick, "Think of how you'd start this example... then check in with a neighbor" to a more involved "Try it, and we'll have someone share it on the board in a few minutes."
  3. On a related note, having students read or work on basic material outside of class can create time to do something deeper in class.  This is based on the "flipped class" idea.  Flipped classes of course can create much more time, but not all faculty can invest this much time for every class to do this.  An intermediate step between zero and fully flipped classes is to use reading assignments or similarly constructed tasks done outside of class to get students up to speed.
  4. If you're going to use group work for part of the time, ask students to move into groups at the beginning of the class.  Then they are ready to go, and discussions ramp up faster.  Less time is spent on the students getting up to the point where they are talking.   It's noted that when students are in rows and have been passive for long stretches, there's more inertia.  This inertia can be a time sink.  Setting up the class "Feng Shui" can move things along faster.
  5. Use more efficient prompts and questions. Several times during a lecture instructors often ask "Do you understand?" or "Are there any questions?"  And wait a minute... and no one has anything to say... You might as well swap out those few empty minutes in class for a more useful bit.
  6. Cull optional material from the course.  Some topics are more valuable than others.  Learning how to minimize some topics, gives you extra time to spend on more important material in a more engaged way.  The uniform distribution of time across topics isn't likely to be optimal, since some topics are more important than others.  
One or two of these strategies gets you a fair amount of time to get things going.  In aggregate, you actually get a significant chunk of time.  It's similar to Moneyball in major league baseball.  Small advantages are pushed in aggregate over large sample sizes.  An extra 15-20 minutes of quality time, every day over a term or year can lead to significant learning results.