Friday, November 20, 2015

IBL Workshops in June 2016!

I am pleased to announce two IBL Workshops in June 2016.  The 4-day IBL Workshops will be located on the campus of Cal Poly San Luis Obispo.  These workshops are open to all college math instructors teaching undergraduate math courses at their institution.  Early-career faculty (grad students, postdocs, and assistant professors) are especially encouraged to apply!

For more details please visit the IBL Workshop page on the AIBL Website.

These workshops are funded by an National Science Foundation Improving Undergraduate STEM Education (IUSE) grant (NSF DUE - 1525058)

Thursday, November 5, 2015

Video by Jo Boaler, "Boosting Math"

Professor Jo Boaler, Stanford University recently posted a great video, called "Boosting Math."   It's intended for students (and parents, teachers, administrators) and highlights better mindsets and brain research that supports the notion that we can all learn Mathematics!

Friday, September 25, 2015

AMS Blog on Teaching and Learning Mathematics

Math instructors, check out a great blog on teaching and learning by Ben Braun, Priscilla Bremser, Art Duval, Elise Lockhart, and Diana White that is run out of the AMS.  Their latest series is on active learning and worth your time.

AMS Blog On Teaching and Learning Mathematics

Tuesday, September 22, 2015

Day 1 Activity: "What is one of your hobbies, and how did you get good at it?"

Day 1 is here at Cal Poly (Quarter System).  I know we are a month behind nearly everyone else, but day 1 is still day 1.  This quarter I am working with Professor Choboter on Calculus 1, and we implemented one of the strategies to build student buy-in our classes.

We asked students via email before the first day to think about one of their hobbies and how they became better at it.  In class, we have students talk to a neighbor to introduce themselves and share with one another their hobby and what they did to get better at that hobby.  We then asked each pair to talk to another pair and introduce their neighbors.  So each person interacts with 3 other people right away.

Next, students were asked to write on the board their hobby on the left side, and on the right side I asked them to write how they go better at the hobby.  Here are images of the boards.

How they go better at their hobby

No one wrote, "I sat back and watched someone, and that's how I got better at it."  I use their statements about how they get better by discussing with them that it's the same with math.  I mention that we get better when we practice regularly, both individually and collaboratively.  Therefore, we are running this class via IBL and we are going to work as a team on the math.  Students seemed ready to get on with it, so that was it.  With that done, it was time to dive into the first math activity of the term!

Ongoing work to continue building student buy-in is also important.  More on that in future posts...

Monday, September 21, 2015

Quick Post on Nuts and Bolts: Index Cards

I am teaching calculus 1 this fall quarter to mostly freshmen, and truly excited about being in the classroom again after a summer off!  Day 1 is tomorrow.

This short post is about the basics. I'm bringing index cards, my binder full of activities, and colored pens.     The index cards are useful for making name "tents" on day one.  I can start day one by learning names and calling on students to share their ideas and strategies.  One of my classrooms has stadium seating, which makes it difficult or impossible for me to get t the middle of the class. Using index cards (and maybe some creative seating strategies), I’ll still be able to engage with all students and learn their names.

Index cards can also be used to write student names to select students (or groups) randomly, for keeping track of student presentation, for mixing people into groups (by writing numbers or letters), and for writing question prompts to pass out to groups.  I am sure there are more uses.  I keep a couple of stacks handy at all times.

Edit:  I use some cards to help me learn names.  I use other cards to mix groups.  I used to mark presentations on a card, but now write notes and use a spreadsheet.  The point here is that cards can be used for lots of things, and having them around is really useful.

Monday, September 7, 2015

Quick Post: Positive Coaching

This is a quick post on a basic idea, called informally positive coaching.

One of the larger issues that comes up in discussions with new IBL instructors is student buy-in.  Students enter a class with default expectations about what a math class should be like.  Since teaching is a cultural activity, significant changes from the default requires the instructors to do some work to reset the norms.

Some of the usual expectations students come in the door with on day one are:
  • The instructor does all the work, and students are supposed to imitate the instructor to the letter.
  • Faster is better.  (Hence, slower is dumber.)
  • The teacher or the textbook is the external source that validates when an answer is correct.
  • Getting stuck is a sign of struggle and not getting it.
When students are in an IBL class setting, they are required to engage in mathematics differently, and they are assessed differently.  Some students may feel as if they don't know how well they are doing.  They are not getting graded the usual way, they are spending more time per problem, and they are appealing to logic and reason, instead of the instructor (or the back of the book) for knowing if they are correct.  It's understandable if students feel unsettled, especially in a first experience with IBL.  

One of the important roles for an IBL instructor is to continuously be a source of positive, constructive feedback (i.e. positive coaching) in ways that students know what they understand and what they need to work on to get better.

Positive coaching can come in many forms.  Here are some examples:
  • Letting students know that they are on track and succeeding and meeting your expectations.
  • Restating something that a student did (strategy, use a concept, etc.) that was useful or important for a solution.  Affirmation!
  • Solution recaps.  These are quick takes on a just presented solution or idea.  ("Let's look at what so-and-so just shared again... Here's what we discovered was needed in this problem...")
  • Emphasizing and praising productive failure.  ("Look what so-and-so discovered... This is great, because now we know what directions we can try next!")
  • Giving students feedback on how they doing a problem, when visiting groups.  ("Show me what you tried... That could work.  Keep up the good effort.")
  • Mini-talks or mini-lectures that set the stage.  ("This next set of problems is challenging, but we are going to work together to work through it...")
  • Once a solution has been presented by a student, to be willing to go over it with students again and again in office hours.
  • Giving students progress reports on their current grade.
All of the above should come with praise for students' efforts and creativity. In this way the instructor is giving affirmation and guidance on what students are doing that will help them succeed.

Last word.  Coaching and cheerleading are not the same thing.  See the Coaching vs. Cheerleading post.

Friday, August 21, 2015

Announcement: $2.8 Million National Science Foundation IUSE Grant (PRODUCT)

I am pleased to officially announce that the National Science Foundation has awarded a $2.8 million grant, called PRODUCT to (a) expand the IBL Workshop professional development network so that more participants can attend an IBL Workshop annually, and (b) offer IBL Workshops or the next five years for 300+ participants.  PRODUCT stands for PROfessional Development and Uptake through Collaborative Teams (PRODUCT): Supporting Inquiry Based Learning in Undergraduate Mathematics.

This is exciting news for us behind the scenes working in higher education reform, and much, much more it's provides further opportunities for faculty to attend IBL Workshops and provide research-validated, student-centered math courses at the undergraduate level.  What this means for the math profession is that we will be able to offer 300+ participants spots at 4-Day IBL Workshops over the next 5 years, and thousands more of students will have opportunities to experience IBL in the future.  Other main activities include developing short workshops for outreach purposes and hosting a professional developers meeting to share knowledge among those working in professional develop in higher education.

One facet of our work is based on ideas similar to those discussed in the book, "Moneyball."  That is, we use data-driven decision making as a core component of our work.  To cut through the fog of "conventional wisdom," "anecdata," and all that to find what is most likely true, our colleagues at CU Boulder's E&ER unit uses evaluation data and research to hone in the key variables to improve our programs.  Then we use a cycle of investigation to revisit ideas and further hone them and adjust.   There's no ideological affiliation.  We look at data, search for facts, create, and push the edges to improve Mathematics Education in the United States. 

I'd like to thank the National Science Foundation for their support and vision, and I would like to thank my collaborators, of which there are many, for their hard efforts over the years to improve mathematics education, who made this next project possible.  I look forward to working on PRODUCT with them, and will be posting information about future workshops on this blog.

Upward and onward!


Saturday, August 1, 2015

A Decade of IBL Workshops

This past month a team of 41 participants, professional developers, and evaluators converged at Cal Poly San Luis Obispo for another IBL Workshop, supported by the National Science Foundation.  This year marks the tenth year, since the first workshops were planned, and hundreds of participants have been implementing IBL, in a variety of forms, to thousands of students.

Attending a workshop is one of the best ways to get going with IBL for math instructors.  Math faculty can work intensely in the summer with experienced IBL Workshop staff,  build courses collaboratively with peers, and take advantage of follow-up support for the following year.  Thereafter, participants can engage ongoing activities and support via the IBL community.  Hence, attending a workshop has many benefits, including higher quality initial implementation of IBL, building connections with other IBLers, and access to the IBL community in the long run.

Interestingly, faculty have targeted a variety of courses.  In the past, the recommendation has been for new IBLers to start their IBL careers in upper-level math courses.   The reason for this is that it is easier to implement IBL in courses, where the students are more mature and there are fewer issues like the coverage issues.  The reasoning is purely based on pragmatic implementation issues, and is still relevant today.  It's relatively easier to implement IBL in upper-level courses, all else equal.  For many, this is not possible or desirable.  As IBL has spread, instructors have been interested in learning to implement IBL in a broader array of courses, including courses for non-math majors.  This makes sense, as some instructors teach at 2-year colleges or have specialized teaching areas. About half of the instructors indicated participants targeting courses for freshmen or sophomores.  Courses participants work on at IBL Workshops include:

  • Math for liberal arts
  • Calculus
  • Precalculus
  • Math for elementary teaching
  • Elementary statistics
  • Lower division Differential Equations
  • Linear Algebra (lower division)

IBL is a system for teaching that applies to all levels of Mathematics.  Participants are verifying this by voting with their feet.  Gone are the days of "IBL is for math track students."  Today, IBL has evolved into a broad, flexible framework.   If you are interested in attending an IBL Workshop in the future AND teach college-level Mathematics, more workshops are planned for summer 2016 (pending funding).  Details are forthcoming and will be posted here and sent out via AIBL listservs and the MAA.

Photos from the 2015 IBL Workshop

Monday, July 6, 2015

Checkout Westfield State's Blog: Steven Strogatz' Reflection (Part 1)

Westfield State is fortunate enough to have an amazing group of math faculty involved in IBL via their Discovering the Art of Mathematics project.  Check out the latest blog post (HERE), with guest blogger Steven Strogatz, commenting as a new IBLer, and having a great time in class.  Nice!

Kudos to Chrissi, Volker, Phil, Julian, Steven, and all the Cornell math faculty interested in IBL!

Thursday, July 2, 2015

IBL SIGMAA: Be a Charter Member

The Mathematical Association of America has special interest groups for members to organize around a topic or area of mathematics or mathematics education.  A special interest group of the MAA is called a SIGMAA.  The IBL community has decided to apply to form an IBL in Mathematics SIGMA, or IBL SIGMAA for short.  The purpose of the IBL SIGMAA is to have a group associated with the MAA to support the work of math instructors interested in developing, spreading, and supporting the use of IBL.

In order to create the IBL SIGMAA, charter members are needed.  Today, I am asking you to sign up!  Please use the link below to let MAA know that you are interested in being a part of the upcoming IBL SIGMAA.

Sign up to become a charter IBL SIGMAA member!

Many thanks to TJ Hitchman, Victor Piercey for spearheading this effort!

Tuesday, June 16, 2015

The Problems of "Good" Teaching and the Problem of "Excellent" Being the Enemy of Good

This post is about two dual problems.  A significant percentage of faculty or institutions are satisfied with "good" teaching, and a separate and overlapping group of faculty or institutions are paralyzed by perceptions that a high degree of "excellence" is needed to switch to active learning methods, such as IBL.  The purpose of this post is to offer a perspective on the dual problems in an attempt to minimize them and ultimately make change easier.

"Good" teaching comes in many forms, and I'll highlight just one prototypical case.  By "good" teaching I mean an instructor who gives clear lectures, gets good student evaluations, and students get the expected grades.  In 1988, Schoenfeld published an article "When Good Teaching Leads to Bad Results: The Disasters of `Well-Taught' Mathematics Courses," highlighting that beneath the surface things are quite the different than they appear.  Good teaching evals and within-spec grade distributions make things appear like their are going well.  Researchers have revealed since the 1980s that beneath the veneer of success are fundamental problems, such as strong negative beliefs associated to learning math.  These beliefs include statements like
  • Students perceive that the form of the solution is what counts (not the content)
  • All problems can be solved in just a few minutes
  • Students view themselves as passive consumers of mathematics
  • Students separate deductive and constructive geometry (as in they are unrelated)
Further studies show that the problem is in fact more widespread and deeper than what Schoenfeld revealed in his article.  There exists a long list of negative beliefs from the math education literature,  highlighted in a post I wrote previously.  Deep negative beliefs affect how students learn mathematics, and even if they do well on the usual timed-tests, they come away with crippled learning mindsets, which damages their future learning potential.  One can go on and on, looking at pass-fail rates in Calculus, lack of improvement in problem-solving ability, how US students lag international peers, and much of this is traced (at least in part) to how we teach (See Stigler and Hiebert).

One factor in perpetuating "good" teaching is over reliance on student evaluations as a measure for teaching effectiveness.  While student evaluations are somewhat useful, they are a highly limited and possibly misleading for assessing teaching effectiveness.  One reason is that a strategy to get high teaching evaluations is to (a) give easy tests and (b) tell jokes or be friendly and entertaining.  Studies also show there are gender effects (males get better ratings), and how the instructor looks (attractiveness) has an influence on student evaluations.  Students are also not able to measure items such as the precision and quality of using scientifically-validated pedagogies, and in fact students may be more interested in keeping things the way they are, even if they would learn more and better with an active-learning classroom.
“I do not think I would get on very well in my ideal school because I am too used to being told what to do.”  -- Frances, fifteen, (Claxton)
In addition to the problems of student evaluations as primary measure of teaching effectiveness, there are other facets of the math teaching culture that inhibit change.  Teaching is a cultural activity. Students, parents, teachers, and administrators have default expectations for what "good" teaching is. These expectations are not tied directly to deeper learning outcomes, and so there are forces that make it more difficult to change the status quo.

If all the signals are "good," then why change?  Consequently, societal, systemic, and cultural forces help to keep "good" teaching in place.  

Excellent Being the Enemy of Good
A dual issue is a particular notion of excellence.  Let me explain.  Even if one does the required homework and decides to try IBL, a factor that holds back instructors is the notion that one needs to be "excellent" at IBL before it can make a difference.  Another form of this is that a department chair may want an "excellent proof" that IBL.  I'm not saying we shouldn't strive for excellence.  Nothing could be further from the truth.  The point is that waiting until things are perfect and pristine is a mistake in terms of implementation timing.

Contributing to the problem of excellence being the enemy of good is the tacit assumption that there exists a safe neutral choice.  It goes like this.  If I am a "good" math teacher and my students like me, then I am taking a risk to try IBL.  Hence, I had better be excellent at IBL to make the switch.  

The reality is that there isn't really a safe, neutral choice in the current education climate.  The recent work by Freeman, et al published work in the Proceedings of the National Academy of Sciences, essentially states that if active learning vs. lecture was a medical study, then with the preponderance of evidence available today it would be unethical to continue using the lecture method.  

Further, studies like the Force Concept Inventory sheds light on the excellence issue.  In Physics education, we have learned the active learning group of instructors outperformed the traditional instructors, AND it did not truly matter if the active learning instructor was a novice and the traditional instructor was the award winning, inspirational figure.  In terms of learning outcomes on conceptual understanding teaching methodology won out as the most important factor.  You don't have to be an expert active-learning teacher to get solid outcomes.  All one needs is to be proficient enough at using specific active-learning strategies.  Some of the active learning strategies employed in Force Concept Inventory study are relatively easy to implement, such as peer instruction (Think-Pair-Share).  It's doable!

In summary, "good" teaching has an effect of lulling one into a false sense of success and security, which ultimately slows progress.  Further, waiting for things to be excellent also slows progress and makes change seem larger than it actually is.  Our teaching system and teaching culture unfortunately support these things (inadvertently), making it more difficult for change.  On a more basic level, just having productive discussions about these issues are difficult, because of the fact that many of the assumptions and cultural norms about teaching are embedded deep within us.  "It's the way it has been for long, long time. That's all I know..." is a commonly sung refrain.

Rather than basing our profession on labels, like "good" or "excellent," a more productive approach is to focus on implementing research-validated teaching practices (i.e. apply scientific knowledge).  Instructors can learn to use one or more specific active-learning methods (think-pair-share!) and start on the path towards deeper, more meaningful student engagement.  Workshops, IBL-specific conferences, mentoring programs, and regional or national math conferences offer opportunities for faculty to engage in discussions and find solutions for their particular situations.

Getting started with implementing active learning is a first step towards transformative experiences.  I encourage instructors to set their aim high in the long run, because full IBL courses have the potential to be transformative.  IBL instructors have repeatedly reported over decades stories of students, who initially did not see themselves as successful math students, go on to graduate school or careers that they did not think they could do.

Alfred discusses (~7:55 into the video) how he was about to drop out of college, but then buckled down and decided to go to graduate school in mathematics.

It's worth the effort!

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.

Friday, January 23, 2015

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

Attending a four-day IBL workshop is one of the best ways to get started with IBL or take the next big step in your teaching.  Long workshops are like retreats.  Away from the daily routine, faculty can focus on how to implement IBL methods, build up a target course, collaborate with colleagues, and build IBL-specific teaching skills.  

Full details are available at

The 2015 IBL Workshop is part of the Mathematical Association of America's PREP program.  More information about MAA PREP and registration for the IBL workshop can be found via this LINK.

*The IBL Workshop is funded by NSF DUE 1225833 (SPIGOT), and space is limited to faculty who teach at colleges and universities.  Early-career faculty (assistant professors, postdocs, and grad students) are especially encouraged to register!  A limited number of travel scholarships are available for early-career faculty.  

Saturday, January 3, 2015

Looking Back at the Previous Term: Reflective Practice with Efficiency

Just a quick post here as we start another year that applies to all teaching, not just math and IBL...

At the end of the term (or recently after a term ends), I like to look back at the courses I have taught and reflect on the successes and productive failures.  What I like to do is set aside 30 minutes or 1 hour, and write down my thoughts, go over my notes, look for patterns, and try to identify areas I can work on.  I look at the problems that were successful and the ones that were not so successful.

Then I put them into my to-do list manager (Omni Focus).  A variety of programs and workflows are out there related to to-do list managers, and the choice of software is not important.  It is important to use some system, however.   The upshot is that I like to put my ideas and thoughts into my to-do list app, and then set a date for to add more things in.  Repeat every week or so for just a few minutes.   Then when it's time to get moving with the next course prep, I have my thoughts about what I can improve ready for action, which leads to me adding in something new that is good or improving something much more frequently. 

Small bits of time used efficiently and computer-aided organization can help us learn faster and grow faster as teachers.  These small bits of time are like prepping the prep time.  This idea can be adapted to suit your needs, and I can see it as part of regular, ongoing reflective practice.

I don't think I'll ever be done learning about teaching, and truthfully that's a good thing.  It's akin to craftsmanship or being an artisan, where the work spans the arc of a career.  It's an enjoyable, fulfilling way to look at the profession, and it's a component of what makes teaching fun, positive, and optimistic (at least to me).

Warm wishes for a successful 2015!