Friday, March 17, 2017

Teaching with (Selective) Silence

"It's not the notes you play, but in the silence between them."  --  Miles Davis

By silence I mean selective silence at key moments. The technique described in this post is a variation on Think-Pair-Share or alternatively "Teaching with Your Mouth Shut" by Donald Finkel, and I think of it as an entry-level IBL technique. It can, however, be used in a broad range of IBL classes, and it's also a useful IBL starting point.

Suppose you are teaching a Calculus class, and you are at a point where an example would be useful. Instead of the instructor showing all the steps, the instructor can write the task on the board, and ask students to work on it and then discuss with a neighbor. Once students are talking to each other, the instructor can write the solution on the board.

The basic framework is presented here. You'll need to adjust the framework to fit your goals, the content, and the environment.

  1. Give Task  "Find the derivative of..." or "Here's the graph of the derivative, figure out if the function is concave up or concave down or neither of those."
  2. Ask students to try it and discuss with a partner.
  3. Silence Instructor waits in silence, and observes students working on the problem.  It helps to walk to the back corner, and then walk back up to the front. 
  4. Write When students start discussing, the instructor writes the steps on the board. Student can then compare.
Using this method at the beginning of the term, requires setup to encourage student buy-in. You should have some instructor guidance ready.  Examples of these are:

  • "I'd like you to try some problems sometimes before we look at solutions, so here's how it'll work when we do class activities..."
  • "It's important for you to practice and ask questions, so we can help each other..."
  • "This is like practice in sports or music lessons. It's time for you to try it, and for me to listen..."
This is Think-Pair-Share light. Students are not voting, suggesting, or sharing their solutions. But those options are on the table, and the instructor can opt-in to those. Instructors can elect to have students share their ideas when appropriate.

One advantage of this technique is that it does not require the same level of intensive preparation and class management as a more heavily student-centered IBL experience, where the sequence of problems and class discussions have to be organized carefully.  That is, you can throw this into your teaching toolbox and use it frequently.

One of the common instructor concerns is when students sit quietly and are not active.  I have talked to instructors who say that their students don't work in groups or don't want to work in groups. Getting students to try the problem and discuss is the instructor's responsibility. The main advice is to have students move their desks (if possible) or at the very least know who their partner is ("Point to your partner!").  The instructor should give clear instructions.  "Try this problem. Talk to your partner.  I want to hear you all talking." Then go visit quiet areas and gently ask them to talk to their neighbor.  Opting out should not be an option.  Moving to the side or to the back of the class momentarily helps visually cue that it's time for students to get to work. The act of walking off the stage sends a message that the instructor will be silent.

It's not a zero-sum game:  Frequent use pays off. What I mean by this is that as you get students more and more engaged, then it opens new possibilities to do even more inquiry. Students are more engaged.  They start asking more questions.  You get to know what their strengths and weaknesses are better, so you can make better informed teaching choices. And then this cycles in a positive feedback loop.

Interestingly, if you look at students' notebooks, you'll see nearly the same things as if you had done all the work on the board without asking students to take a turn. The difference is not usually evident on the pages of the notebooks. The difference is in the experiences getting those words and symbols onto those pages.

Lastly, one of the ways we learn about student thinking is to listen. It's much easier to listen when you are silent ;)

Miles Davis -- So What

Monday, March 6, 2017

IBL Mini-Conference at MathFest 2017

Here's a note from Alison Marr, Southwestern University and Patrick Rault, University of Arizona. Alison and Patrick are conference co-chairs of the IBL Conference at MathFest 2017, and are putting together the program for this year.

Interested in learning more about inquiry-based learning (IBL) and active 
learning? Have insights on how IBL works in your classroom? Then please consider attending the mini-conference, Constructing the Future of IBL: The Past 20 Years and the Next 20 Years, that will take place in Chicago as part of MathFest on Friday, July 28, 2017 from 5:30PM-9:30PM. The conference will include a panel discussion exploring various aspects of the diversity of IBL, a contributed poster session, roundtable discussions, and a plenary talk on the future of IBL by Stan Yoshinobu. More information about the conference and poster abstract submission can be found at

There is a $25 registration fee which includes heavy hors d'oeuvres, and you must register for the mini-conference when you complete your MathFest registration. We hope to see many of you there!

Alison and Patrick

Thursday, February 16, 2017

Part 2: Dr. Esselstein's Students

The following interviews are five of Dr. Esselstein's students mentioned in Part 1 of the interview. Five CSUMB students, five wonderful stories!

Transformative experiences are highlighted by the IBL community. Those of us who have used IBL methods in our classes have seen students flourish in ways that changes the trajectories of their lives. It sounds like we are exaggerating, as if we are making promises of rainbows and unicorns.  But there are many instances, when things go right, the students buy-in, and the entire class (including the instructor) comes together like a championship team.  It doesn't always happen, it's not easy to pull off, but it's why we work so hard to try and get to that special place.

In this particular instance, Dr. Esselstein taught a proof-based course for math majors, where the opportunities to build Math from first principles, parallel to how research mathematicians do Math.  Authentic success, the greatest confidence booster there is, did it's work.  This experience left a lasting impression on these students, who proceeded to earn their undergraduate degrees and then go to on to graduate programs.

Yes, you can make a real difference!

1. Diana and Alfred

2. Sandra

3. Helen

4. Daniel

Monday, February 13, 2017

Part 1: IBL Instructor Interview: Dr. Rachel Esselstein

This blog post is an interview in a Q&A style. In this edition, I interview Dr. Rachel Esselstein, University High School in San Francisco.  Her experiences teaching at both the university level and high school level, gives her interesting perspectives and insights into IBL teaching. Thank you, Rachel!  (See also Part 2: Dr. Esselstein's Students)

Hi Rachel! Thank you for agreeing to be interviewed. You’ve had an interesting and inspirational trajectory in your IBL teaching. Please tell us about it.  

My first attempts at IBL were as a professor at California State University, Monterey Bay where I worked for 7 years.  I decided to attend an IBL conference at University of Michigan after a discussion with a friend and collaborator, Fran├žois Dorais.  This led me to apply for a mini-grant which I used to create an IBL Intro to Proof course with Carol Schumacher as my grant mentor.  The course ran several times over two years and I found it to be a very challenging but rewarding experience.

In 2014, I accepted a position at the Bay School in San Francisco where I taught math for two years, learning the ins and outs of the independent school world and how to work with younger students.  Bay’s pedagogical philosophy is student-centered and so I had many opportunities to implement IBL in my classes.

This year, I started work at another independent school, University High School in San Francisco.  I am very happy in my work at UHS.  I am surrounded by other faculty who are incredibly talented both as teachers and as practitioners of their discipline.  UHS has a reputation for being an environment for students who appreciate a good challenge and it has been fun to work with amazing students who enjoy being pushed out of their comfort zone.

Please share with us how you use IBL methods in your classes.

It was a revelation for me to realize that I could use IBL when I found it applicable and use other pedagogies in other instances.  In other words, there is no need to commit to a full IBL course.  Now days, I pick and choose how and when to use IBL methods in my courses.  For example, in an introduction to linear programming for Precalculus, I gave my students an open ended problem with no prior instruction about optimizing the packing of supplies for a space mission.  Of course, they were not expected to discover linear programming for themselves in that one lesson but many of them created systems of linear inequalities, graphed them and then started hypothesizing about the optimal solution.  We then had a week-long discussion on why the optimal solutions would happen at the corners of the fundamental region and the students came up with two different ways to explain this phenomenon.  The unit wrapped up with a problem set in which the students worked collaboratively on very challenging problems that asked them to assimilate their understanding of the unit followed by a unit test that was fairly standard compared to what you might see at any other high school covering this topic.

On the other end of the spectrum, there are some topics that I have found are challenging enough and simply frustrate the students if they are asked to learn them via inquiry.  This is especially true for algebraic processes with younger students such as completing the square in an Algebra 1 course. In these cases, I still rely on group work and the unit will always end with a problem set of problems that ask the students to apply and assimilate their understanding but the material is introduced using other pedagogies.

How is teaching HS similar/different compared to teaching at the college level?

One of the biggest obstacles of using IBL at the high school level is working with the parents who may or may not understand and support their child learning in a style that does not reflect their own high school experience.  At independent schools, the parents are generally very supportive of the faculty but they also are very concerned with the success of their student.  I have found that it is always crucial to communicate the purpose and intention of IBL with my students but, at the HS level, it is also important to get the parents on board.  I try to make my grading scheme clear to both the parents and students and I make detailed notes and rubrics that help my students and their families track their progress.

At independent schools, our class sizes are much smaller than I had at CSUMB.  This means that working in groups is much easier to facilitate and I can take much more detailed notes about the work each student does when they are problem solving or presenting.

The resources for finding interesting problems to give my students are quite different.  While it is usually easier for me to construct challenging problems for high school level math than for college level math, my students are not as developmentally ready for things such as proofs or problems that require significant work.  They also seem less adept at working with open ended problems although I do believe that this is something they can overcome (as opposed to not being developmentally ready to handle).  I have found NRICH to be a wonderful resource for challenging problems as well as old Math League exams and occasionally old Math Olympiad tests.  The textbooks I am asked to use are good for practice problems for my students but they don’t provide the right problems for IBL so I mostly make my own materials.

Students are very self-conscious at the high school age.  This seems to be especially true for freshmen who are desperate to fit in and look smart or successful.  I have found it to be even more crucial to create a safe and welcoming environment in the high school classroom in order for students to truly engage in IBL.  Thankfully, my classes are smaller and the students are very well-intentioned and driven by curiosity.

I don’t emphasize presentations of solutions at the board as much at the high school level, especially with younger students.  I will scribe at the board (or get one of my more restless students to scribe at the board) while the presenter describes their work.  I have also found success with asking students to share back what they understood about their peer’s solution.  This helps them make sense of the work for themselves rather than just having them copy the work down on their own paper.

What have been some of your biggest challenges teaching Math?

The most challenging thing is that I spend copious time and effort creating a project or problem set that is successful for one group of students but falls flat with another.  Sometimes this is due to the abilities of the groups being different but it seems to mostly relate to the personalities in the class.  I find that IBL at the high school level (and somewhat at the college level) really relies on finding the right hooks; the problems must be engaging and tractable.  High school mathematics has many more venues for applications and so sometimes I can just tweak a problem to relate to a particular group of students.  Other times, I need to completely scrap an activity that I worked so hard to create the year before.

I also found collaborative teaching to be challenging when using IBL.  IBL is easiest to run if you have complete autonomy over the course and can adapt to the students’ interests and abilities.  I have found that my calibrations for my students are very different than other teachers’ for their own students and it means the courses might go at different paces and focus on different activities.

IBL at the high school level gets LOUD!  I have had to find ways to monitor the noise level when they are working in groups or debating a solution to a problem because their enthusiasm and energy cause the volume levels to get extremely loud.  High schoolers often have less social maturity than college students and so I have found it necessary to instruct them on how to give constructive feedback to their peers.

What have been some of the successes?

I don’t know that I have been teaching at the high school level long enough to see measurable successes in terms of students going on to find success in college level math.  I have had multiple students from my IBL courses at CSUMB go on to graduate school in math with great success and I have also had some students become high school teachers themselves who are using IBL in their courses.  I anticipate that we will see much more IBL in high school math as the Common Core State Standards and training around them continue to roll out.  Common Core has been a great touchstone for teachers to re-think their pedagogy and how it reinforces the mathematical practices and habits of mind.

I have absolutely seen an improvement in my students’ abilities to solve problems that are unfamiliar to them.  I also noticed the time my students spend working on a problem before asking for assistance has increased although I haven’t recorded or measured this formally.

Many of my students describe the IBL units as the “most fun” they have had in a math class.  Students like discovering the material and solving puzzles.  They love being challenged when they know that the stakes are non-threatening.  They find that IBL makes them feel successful in math because they aren’t just memorizing but understanding the material.

Here’s a video clip of two of your former students from CSUMB, Alfred and Diana. Something special happened in this class, where your students went were transformed learners. What happened in that class?

I don’t know how to describe what happened in that class but it really was magical for both the students and me.  I had Carol Schumacher as well as you, Stan, as resources and you both were very helpful in providing me with encouragement and advice when both were needed.  The stipend from the mini-grant allowed me to put much more time into preparing the course than I would have been able to spend otherwise.  Still, the money wasn’t commensurate with the workload.  It was a lot of work!  I had a great group of students who were open-minded enough to go through this experiment with me and I had very clear outcomes that I wanted to reach.  I think the success of this class was mostly due to the fact that IBL works when it is done well.

Any thoughts or advice for instructors thinking about using IBL but have not tried it yet?

IBL is a lot of work but it forces you to make your teaching student-centered. You will be surprised by the things your students discover and understand.  The best feeling is when a student or a group of students has a break-through and they want to celebrate it with you.  As mathematicians, we are used to the elation of a breakthrough on a tough problem but, for many of our students, this is the first time they have ever experienced this.  They cheer, give high-fives, post things on their refrigerators at home, and develop a more favorable opinion of math class.  It is a lot of work to run an IBL course but it is so much more fun for everyone.

Because it is a lot of work, find a teacher mentor who can help you troubleshoot any issues that come up.  It is best if they are at your school so that they know the school culture but even long-distance mentors are better than going at it alone.

Make the expectations clear to your students (and their parents) as well as your department chair and anyone supervising your work.  Students sometimes panic that you are “not teaching” and their complaints to supervisors can be detrimental to your career if the supervisors are not aware of your pedagogy.  Give the students ample positive feedback and opportunities to reflect on how far they have progressed.  Happy and well-supported students will put in significantly more effort than unhappy and frustrated students.

Observe a class that is using IBL and attend conferences such as the AIBL conference.  They are inspirational and motivational.

Continue reading Part 2: Dr. Esselstein's Students

Saturday, December 17, 2016

PRIMUS Issue on Teaching Inquiry and JMM Events

Hello IBL Community!  Brian Katz, Augustana College, has shares exiting news for IBL community. Please read below for special issues of PRIMUS and IBL-related events at JMM 2017 in Atlanta, GA.

PRIMUS special issue on Teaching Inquiry
Elizabeth Thoren and Brian Katz have organized the 19 papers and 2 editorials in this two-volume special issue around the ways that they contribute to the discussion of two questions: what is inquiry, and how do we support its development in students?

Part I, entitled Illuminating Inquiry,” focuses on the nature of inquiry, from discussions of its theoretical foundations and generalizations across disciplines to descriptions and analyses of the experience of inquiry from the inside. Part II, entitled “Implementing Inquiry,” focuses on approaches to offering inquiry experiences, from discussions of strategies to change student and instructor behaviors to descriptions and analyses of course design and project structures. Of course, a reader will find insight into both the nature of inquiry and approaches to achieving it in any paper in either part, and each part contains ideas for both instructors who have experience teaching with inquiry and those who are hoping to start. Each issue starts with an editorial that offers a more detailed overview and integration of the papers.

Illuminating Inquiry:
Implementing Inquiry:

  1. PRIMUS special issue on Inquiry-Based Learning in First-year and Second-year Courses (forthcoming). Guest edited by Dana Ernst, TJ Hitchman, and Angie Hodge
  2. MAA Instructional Practices Guide (forthcoming). This document is a companion to the MAA's Curriculum Guide that will focus on course design, instructional practices, and assessment. Doug Ensley and the project leadership are running focus groups related to this project at JMM 2017.

Joint Meetings of the American Mathematics Society and Mathematical Association of America, January 6-9, 2017, Atlanta, GA.

Here's a link to the full schedule:

The SIGMAA IBL is sponsoring a Contributed Paper Session with 55 talks on Inquiry-Based Teaching and Learning
  • Thursday 1/5 8:00am-12noon (Regency Ballroom, Ballroom Level, Hyatt Regency). This session is specifically designed to be accessible and useful for new or potential IBL practitioners.
  • Friday 1/6 8:00am-11:00am (International Room 6, International Level, Marriott Marquis). This session emphasizes lower-division and large course contexts.
  • This session emphasizes lower-division and large course contexts. Friday 1/6 1:00pm-5:00pm (International Room 6, International Level, Marriott Marquis).   This session first emphasizes secondary school contexts and then turns to equity in mathematics education. NOTE that this session flows immediately into the SIGMAA IBL Business Meeting.
  • Saturday 1/7 8:00am-12noon (International Room 7, International Level, Marriott Marquis).  This session emphasizes applied math and technology in inquiry.
  • Saturday 1/7 1:00pm-4:20pm (International Room 7, International Level, Marriott Marquis.  This session mostly emphasizes the Calculus context.
SIGMAA IBL is also sponsoring a panel featuring Susan Crook, TJ Hitchman, and Carol Schumacher Thursday 1/5 1:00pm-2:20pm (International Room 5, International Level, Marriott Marquis)

SIGMAA IBL Business Meeting (interactive discussion) Friday 1/6 5:00pm-6:00pm (International Room 6, International Level, Marriott Marquis)

There are many other presentations about IBL at the conference, and conference attendees are especially suggest you browse the RUME and SoTL sessions.

Friday, October 28, 2016

A Vision for the Future of Active Learning: Professional Development Centers

I've been participating in the Mathematical Association of America's Active Learning Day/Week (#activelearningday), I've shared some videos of students sharing their opinions and experiences in IBL classes. I've also truly enjoyed hearing from my colleagues about their experiences and what they do! What an inspiration!

I think it's appropriate at this point to share an idea I have been thinking about for several years. The idea is centered on the goal of really pushing active learning forward in Mathematics. The idea is fairly straightforward, and is to create several Professional Development Centers in Mathematics Teaching and Learning across the nation, loosely modeled after the Mathematical Science Research Institute. Because regional needs vary significantly, it is likely a better strategy to create more of these in smaller sizes, creating a more nimble and responsive network.

Some Issues and Gaps
Some gaps in our system exist for preparing college math faculty to teach using active-learning methods exists. I'll highlight some of the main ones I have noticed to shed some light on the gaps, and then we can see how a PD Center can offer significant help.

Education reform is hard, time-consuming work. It involves working with people in diverse settings and in the system is gigantic. Just in my state university system, the California State University, there are 440,000 students across more than 20 campuses living in learning in a vast range of environments. Impactful change in just one system is an immense challenge, much less the challenge presented by a nation with over 300 million people.

Teaching is a system and a cultural activity. Hence change is much, much more than merely changing textbooks or rearranging the schedule from MW to MWF. It involves shifting fundamental ways in which students interact with the subject, which involves a lot of inertia. Hence, this is yet another factor that implies that changes in teaching is a hard challenge for each individual instructor.

The Uptake Problem is a label I use to represent the fact that the percentage of faculty using active-learning remains low, despite knowledge of the existence of active learning. One study by Dancy and Henderson suggests that time and other complexities affecting the process of changing teaching methods are significant. It's not enough to know a solution exists. To find ones that work for individuals is a specialized problem confronting each instructor.

Grant work has been instrumental in accelerating the progress of active learning, and we have gained and learned a tremendous amount from this. We would not be where we are today, without grant-funded projects. The grant-funded projects, however, do not cover all the bases, so arguing that we have external grants isn't a full solution. For instance, when funding ends, projects can lose momentum or die off. We can say that we are at a point where we can see the finish line because of grant-funded work. We are much better off, but this line of work tends to be more focused and not the broad, long-term work we need to take things to the next level.

At present, there isn't a professional group that works solely on implementation of active learning on a full-time basis. This is in my opinion a weakness in the system, and slows progress. Grants allow some of us to do this part-time, but our time frames are short and our reach is limited compared to the size of the profession. At the moment, we are primarily a grass-roots effort (which is good and necessary, but not sufficient).

Centers for Teaching and Learning on college and university campuses are wonderful resources. I truly appreciate what they do and the services they offer. One area that CTLs can struggle with is in the area of the details of discipline specific active learning. Teaching Abstract Algebra or Real Analysis or first-year Calculus present different challenges to math instructors. Hence, mathematicians and mathematics educators are best positioned to be the ones to do the PD work. General frameworks for teaching are a good place to start, and can be learned from CTLs, but what happens when your students have trouble understanding nested quantifiers in the context of proving a function is not continuous? CTLs can be partners in this, but another force is needed to take things across the finish line.

Experienced faculty in active learning are generally diffuse and are (at least) somewhat isolated from one another. At the moment there are some efforts to coordinate, so it is understood by some that this issue needs to be addressed. Still much of the time the experience, skill, and knowledge of the profession is not being fully harnessed via collaboration, sharing, and consolidating knowledge, materials, and expertise. Consequently, a PD Center is needed to assist with coordinating efforts.

Hence, opportunities to learn about and get support using active learning are largely ephemeral, and all the while the challenges facing instructions remain significant.

What Professional Development Centers Could Do
I'll sketch the kinds of activities PD Centers could do. This isn't a comprehensive list, and is intended as a staring point to explore what might be possible.
  • Focused on professional development for all faculty interested in active learning and improving student learning of mathematics
  • Short workshops, during academic terms, where faculty could visit classes
  • Long workshops during academic breaks for intensive work
  • Traveling workshops to visit to departments
  • Create repositories for workshop materials, course materials, and relevant research studies
  • Postdoc and visiting scholar programs for developing young faculty and future leadership
  • Host small groups of visitors during the academic year to customize experiences around their specific needs
  • Create a video library of best practices for multiple types of classes, so visitors can sit in live classrooms and study video to ensure they have access to the range of skills and practices needed for successful implementation. The video library would also have value in formal workshop settings.
  • Outreach work to encourage an ever increasing number of instructors to lean in and try active learning
  • Develop workshop leaders so that workshops can be run across the nation.
  • Offer grants to support faculty and departments, who need time and materials to get up to speed
  • Full time faculty and staff, who focus on this line of work, perhaps including postdocs and visiting scholars
My sense is that a PD Center should be housed at a college or university, where teaching is valued and a variety of course offerings are available. In this case, it could offer many different options for professional development.

Our experiences running IBL Workshops inform us that changes in teaching requires significant time and effort. The typical new IBL instructor spends hundreds of hours to get going with IBL, even when starting with a week-long workshop.  The "activation energy" is high, and efforts are spent over many academic years. A PD Center could offer support that matches such longer-term time frames for faculty development and institutional changes, unlike grant-funded projects, which come and go and have more limited resources.

Let's look at the example of the state of California, which has about 1/8th of the nation's population. I'm choosing this example only because this is where I am. A PD center somewhere on one of the CSU or UC campuses, could be a place where math professors and instructors from colleges and universities across the state (and west coast) could visit within a half a day's drive or short flight. Annual workshops and conferences could be hosted, and specific regional issues could be addressed, such as the preparation of future secondary math teachers, how to teach Differential Equations to engineering majors, urban commuter issues in the SF and LA areas, and so on... Steady, long-term effort could be spent on a wide range of issues with the help and support of a PD Center that brings people together to work collaboratively on our big challenges.

Consequently, it can be argued that the existence of PD Centers would significantly impact the long-term successful implementation of active learning across the nation. These efforts could be expanded into other STEM fields, and in time (or in parallel) into K-12 Math and Science.  We have a robust group of mathematicians, mathematics education researchers, and math instructors across the nation, all doing great work. The ideas outlined above expose where our nation is lacking in infrastructure, and how some of these gaps could be addressed with PD Centers.

Great schools depend on great teachers. Focusing our efforts on teaching and learning is, in my opinion, the best bang for the buck. It would be wonderful if we could provide the support, leadership, and vision befitting the challenges, needs, and goals of our math teachers, our profession, and ultimately our students.

Monday, October 17, 2016

Does Class Size Matter?

The simple answer is yes.

Let's look at some data. An analysis released a couple of years ago by the National Education Policy Center uses an econometric take on the issue, where they try to disentangle data to find causal links. In their work, share evidence that class size increases harms students. This is from data that is "econometric" friendly, where the savings today in increasing class sizes (K-12) is offset in the future by far greater costs. (Penny wise, pound foolish?)

The point I want to get to is a very simple one. Another perspective of this issue is as a classroom teacher.  Let's look at the sequence of class sizes to see how teaching decisions are affected as class size increases. (The assumptions here include assuming we are talking about the college math setting.) Also I'm not going to list all possibilities. The goal here is to see a pattern as class size increases.

Class size of 10: Anything can be done at this class size. Projects, full IBL (where students present proofs or solutions), team/group work, seminar or whatever comes to mind.

Class size of 20: At 20, things are still manageable, and an instructor can get to know all of the students, customize materials and learning experiences, projects are still doable. IBL is doable in with small groups/pairs and students presenting individually.

Class size of 40: Things start to get to a point where there are too many students to let them present their work to the whole class.  Each student may only go to the board a couple of times a term. Somewhere between 30 and 40, instructors tend to switch away from some student-centered methods (such as students presenting their ideas at the board). Small groups can still be used, and the instructor may not be able to get to all groups on a particular task.

Class size of 80: Individual student presentation of mathematics is almost surely off the table. Small groups and peer instruction remain, so instructors can develop this area to support an active learning environment. Projects are highly difficult to implement at this class size, especially at institutions without teaching assistants. The instructor can still visit with some groups and make it around the room (depending on the space) say one trip per class period (approximately).

Class size of 160: At this point, you are well into large lecture territory.  Peer instruction (Think-Pair-Share) and similar methods (with or without clickers) remain in play, although the size or difficulty of the task may be on the less challenging side of the spectrum. Presentations by students, class discussions, projects, instructor visits to each group regularly are almost surely off the table.

As you go up in class size, you lose the implementability of teaching strategies that engage students and also anything that does get implemented needs to be done at a higher skill level and attendant time in preparing for class. Managing a discussion with 160 students takes skill, and most all instructors may be inclined to avoid it.

Consequently, what we are seeing is that there exists research evidence suggesting that a negative, material impact is a consequence of increasing class size AND from a practitioner standpoint instructors have fewer, high-impact strategies at their disposal as classes get larger. We have said nothing yet about how class dynamics can change as you get to larger class sizes. For example, it's easier to hide in a large class, and it's easier for students to checkout, be distracted, and show up late.

Another facet of class size issues is that class size is often a policy decision, not in direct control by an instructor. Instructors can have input on class size, but it's not up to instructors to set class size limits. This is done by policy makers or administrators, yet these decisions have significant impact on teaching decisions and hence learning outcomes. Faculty and administrators should take into account the big picture when it comes to class size. Focus on things like nominal efficiency, should be viewed while also heavily weighing learning outcomes, DFW rates and long-term impacts on student learning.