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

Quick Start Guide to IBL Teaching

Disclaimer:  This is just a quick start guide, not a manual, and getting started in IBL is hard work. Please visit the IBL community to learn more at www.inquirybasedlearning.org

Step 1: Determine the overarching goals of the course, and use these big goals to help with decisions you have to make.  For example, when teaching Real Analysis 1, one of the big goals of the course is to understand deeply the fundamental concept of convergence. 

Step 2: Find/Adapt a sequence of problems.  Some course materials are available from www.jiblm.org.  Another option is to build your own set of problems, using a textbook as a guide for how to sequence the problems. 

Step 3: Understand your role as an instructor.  IBL instructors select the math problems, make assignments, select presenters, moderate and facilitate discussions, use group work appropriately, mentor, pose sub problems or special cases when the class is stuck, ensure all students are deeply engaged.

Step 4: Marketing what and why IBL to students and colleagues.  Marketing is used in the best sense of the word here.  Students and colleagues need expectations reset, especially if IBL is not commonly used at your institution.  Student buy-in is critical, especially at the beginning of the term.  Because teaching and learning are cultural activities, there exists a set of default, often unconscious assumptions of what math is, what teaching math is, and what students are “supposed” to do in a math class.  Based on the “distance from IBL” your students are, use the appropriate amount of regular, ongoing marketing and sign posting of tasks.

Step 5: Pick a rubric for grading presentations and homework.  Here’s one example.

            4 points = correct

            3 points = mostly correct, except for technical or clarity issues

            2 points = there exists a logical flaw

            1 points = went to the board

Allowing students to pause and return later once is a regularly practiced by IBLers.   Bonus points for productive failure are awarded by some IBL instructors.  A score equal to a 1 is rare.  Other rubrics exist. This is shared as a common choice.

Step 6: Keep a course diary with notes from class and thoughts as you prepare, grade, and reflect on your teaching.

Step 7: Use a spreadsheet with names in the rows and problem numbers across the top.  Use this spreadsheet to keep track of who has presented.   Separately, keep another spreadsheet with the names in the rows and days of the term across the top. This second sheet is used to keep track of participation, and helps you make data-driven decisions regarding who to call on and how to build groups that work effectively.

An Insight from Burger and Starbird

I'm in the process of reading "The 5 Elements of Effective Thinking" by Ed Burger and Mike Starbird. I'll start with a quick insight that can help your teaching right now.

Consider the two ways to approach the usual situation where a teacher tries to see if the class understands what is going on.

"Are there any questions?"

vs. 

"Talk to your neighbor for sixty seconds and come up with two questions."

Asking "Are there any questions?" is no longer as useful a line of approach or teaching technique as we'd like it to be.  How often have we been met with silence?  The problem with silence is the lack of data about student understanding.  If you want to know you need them to produce something (an answer, an idea, an example, a question,...).  And you should want to know what your students are thinking or not thinking.  Knowing exactly where your students are at is vital to teaching, just as making a proper diagnosis using evidence is necessary to medical doctors.

Another way to look at this is the following.  If a certain teaching strategy doesn't elicit the response you need, then find another approach.  If you do not give your students a chance to opt out of thinking of a question, then they are going to be more engaged.  Moreover, students will learn how to ask questions and also to seek to find new questions.  Thus, setting the stage for questions has benefits far beyond checking for understanding.  Questioning becomes part of the intellectual life.

Marketing IBL to Your Students

One of the common issues new IBL instructors face is student buy-in.  This is a real concern for all instructors.  In this post, I outline the issue broadly, and then provide some tips for how to ensure that your students get with the program in their minds and in their hearts.

First, let's talk about the issue broadly.  One important thing to remember is that Math is culture.  There exists default expectations about what a math class is and the roles for students and instructors.  These default expectations are often unconscious -- we don't think about them.  When you meet someone new or talk with your boss, you probably do not realize all of unconscious things you do (or don't do) in these interactions.  Likewise, in a math class students have certain expectations that are almost always aligned to traditional instruction.  Students expect instructors to show, and their job is to follow dutifully and write down notes and perform these tasks on exams.

IBL classes are aligned differently, of course.  Students are asked to solve problems they do not know the answers to, to take risks, to make mistakes, and to engage in "fruitful struggle."  These are all very different from normal expectations (as of today -- hopefully that will change).

Tools for making sure your students are on board are

1. clearly defining students' role in the class
2. providing a clear rationale for IBL (regularly)
3. creating a safe and successful classroom environment.

Students need to know what their job is in an IBL class, and it is the instructor's job to make this clear.  Students must know what they are supposed to do (solve problems, write math proofs/solutions, communicate effectively,...).  Students need to know the instructor's role (provide appropriate tasks, coaching, mentoring, adjusting the challenges as needed, moderating discussions,...)

Why IBL?  Well there are lots of reasons.  Research shows it's better for students.  We are now in the era where information about anything is available on your cell phone, and one can run Wolfram Alpha on a cell phone, too!  In other words, all lower-order thinking levels (as per Bloom's Taxonomy) are now nearly worthless due to advances in technology.  Effective thinking is now where it's at.  Thus, IBL is the way forward for students.  This line of reasoning addresses items 1 and 2 above.  What about 3?


The heart is the heart of the matter.

Telling people, "Medicine is good for you!" isn't sufficient.  Students need to know that the instructor is their advocate for learning.  Students need to see themselves as successful mathematicians (where they may never have thought this before in their lives).  Thus it is important for students to struggle, but struggle within reason.  It is suggested that IBL units start off at a basic level, where all students in the class can achieve some success.  Then the problems should ramp up in difficulty as appropriate for your students.  When in doubt, include more easy problems than less, especially at the beginning of the course and at the beginning of new material.  The worst case scenario is that you spend a few extra minutes on them or just skip them entirely in class, and assign them as homework.  There is no cost to including more problems.

In this sense, establishing a safe and successful classroom environment is asymmetrical.  Erring on the side of being "tougher" is fraught with perils.  First, you are going again previous experiences and the traditional classroom culture.  Second, many students have negative attitudes about math.  Third, telling students that are stuck to "just keep going" can lead to the perception that the instructor is not helpful, and thus not teaching.  Struggle is good if the students feel that the struggle leads somewhere.  This type of scenario is often the case for students writing negative comments on course evaluations.  Students may in their minds understand that IBL is good for them, but they experienced too much frustration to truly enjoy the experience in their hearts.  In other words, the experience was not an aesthetic experience.

I also note that saying that you told them is not enough.  You need to know if the students feel it in their hearts.  Look at them and see if they are enjoying the math and interacting positively.

Of course, there is good reason for students to struggle and perhaps not solve a problem.  Such experiences are fruitful on many, many levels of learning.  BUT this is something that should happen down the road, once students are off and running, enjoying math and doing math successfully in a positive and supportive learning environment.  Training for a marathon has similarities to teaching an IBL course.  You don't coach a new runner with hard interval training on day one followed by long 20-mile tempo runs.   Athletes train by exerting an appropriate training load and recovering.  Then they repeat and then move on to new things gradually.  Math is no different.  Tasks should match students' experiences and abilities and grow with them.

If you are not positive, how can your students be positive?  If you never smile, why would your students smile back at you? Be positive!  It's important to let your students know that they are working hard and progressing.  I thank my students for their participation, and I try as best I can to make classes a supportive environment.  Pointing out the good parts of solutions, ideas, and efforts should be a part of daily practice.

Start easy. Establish the learning culture from day 1.  Build on positive class experiences to challenge students to do more and more.


General IBL Points
Some points you can use as a base for discussing IBL classes with your students.  This is a list of talking points to help you find your own way of conveying the message that IBL is good for the mind.

1. IBL is a student-centered method of teaching similar to the Socratic method. It requires more work for me (the instructor), but it's better for you.  Research shows that students who are actively engaged learn better.  While you may not be used to it, I'll do my best to make sure you are comfortable with it and will be successful in this class.
2. One goal of IBL is to help students learn to think independently, and become a successful problem solver.  In other words, a goal of IBL is to help you get better at thinking effectively.  That's really what we will work on.  And you can't learn to think effectively, if someone does all the thinking for you...
3. IBL emphasizes the process of problem solving and theorem proving rather than the memorization of facts.
4. IBL is not experimental.  It has been employed successfully since the days of Socrates.
5. The reason why books and other outside resources are not allowed is because we will discover the ideas ourselves.  We will collectively work on the tasks and come up with our own ideas.
6. It’s OK to be stuck.  Being stuck is a noble state of mind.  It means you are just about to learn something new!
7. It’s OK to be frustrated.  You’re doing fine -- try to slow down and enjoy the process.  We'll get it eventually.
8. Being stuck is natural.  Whatever you do, don’t give up.  If you're stuck, there has to be question in there that you can ask.  
9. What's the best way to learn to play the piano?  Should you just watch videos of pianists?  Do you need to do something else besides watch someone else play?

Teaching is more than content delivery.  Addressing the learning challenges as part of the course is a good thing to do, and acknowledging where students are coming from and building a bridge for them to cross is a core component of effective IBL teaching.

Reflective Writing at the End of Year

The end of the academic year is a busy, busy, busy time of year.  Commencement, grading, fatigue, summer plans, submitting finals grades,... so much going on.

My recommendation is to stop for 30 minutes.  Get a cup of tea or coffee.  Grab a sheet of paper and a pen, and write about your year in teaching.  Write down what your teaching successes were and what you want to improve on the year.  The hard won knowledge and experience from the academic year, from all the grading, and class time can slip away with time.  But if you just spend a few minutes to write your thoughts, it makes it much easier to recall what happened.  This makes it much more likely that you will see big gains the next time.

As they say luck is when preparation meets opportunity.  In teaching, part of preparation is having a framework that effectively guides the next round of preparation.

Take notes, then take a break.

Congrats to all new graduates this month and next!

Architecture and Education Part 2

In part 1 I described an ideal situation where the physical space of a building can encourage learning.  Now I turn to a pragmatic issue -- how to make your classroom physically and mentally a good learning space. Small groups are useful in a wide variety of situations.  Although some IBL instructors forbid collaboration of the group work kind, many instructors use group work highly successfully.  Both style have merit, and what I highlight here are ways to setup a classroom, should you choose to use group work (which I recommend as a good way to get started using IBL methods).

Physically most of our college classrooms are setup with the factory model.  Students are sitting in rows.

In a room like this, the desks can be moved.  In this case you can easily ask students to move their desks into small groups of size two to four.  One suggestion, if possible, is to arrange it so that it is easy for you to walk through the classroom. Creating a boulevard in the middle allows you to walk down the boulevard and get to all the groups (e.g. ask the first two rows, and last two rows to move together) . 

If your room has fixed desks, all is not lost.  You have to ask students to work with neighboring students and get them to turn their bodies at face on another.  It's not ideal, but it works.

Don't be shy about moving people. This is part of IBL instruction -- move students to where they will be successful!  It is better to be mildly intrusive than to allow other factors to inhibit learning.  

Non-physical caveat:  The buildings, setting up groups, desks, these are all physical and can assist with developing a healthy, productive learning environment.  By themselves they are not sufficient of course.  Instructor skill, leadership, facilitation, and coaching are major drivers and can overcome almost any physical boundaries.

Ideally we would have small tables in one part of the room for small group work, and chairs in another part of the room for whole-group discussions and presentations.  I hope that future buildings are designed for class discussions, collaboration in small groups, as well as some private space for those times when students need to think by themselves.  This is doable and low cost.

Teaching Tips: Chaos

Another title for this post is "Chaos" vs Chaos.  True chaos is not appropriate in a classroom for obvious reason.  As an instructor transitions from lecture to IBL methods one has to recalibrate what and orderly classroom looks and sounds like.

Talking -- there will be more talking, especially if one uses collaborative groups.  If the tasks are implemented appropriately, then there will be noise.  This is a good thing.  Listen to the discussions about math and get insights into how students think.  

Side note: If students are talking about other stuff, redirect.  Ask them them what they tried, if they can work on the next problem, give them another task,...

Non-textbook route -- students who are learning math should not be expected to produce clean proofs like a seasoned, professional mathematician.  It's hard for instructors to remember the challenges we faced when we were learning ideas for the first time.  To instructors the work looks chaotic, disorganized, messy.  But that is a fact of life.  Learning is messy and nonlinear.  That's a good sign.  Students are trying and working and building and exploring.   If there's nothing on the page, that's when you need to step in and provide guidance and mentoring.  When there's stuff happening, keep an eye on them, but don't mess it up by intervening.

For pure Modified Moore Method classes, such interactions in classrooms as described above may not happen.  That's fine.  You know it is happening outside of class, but it is important to know and be conscience of the fact that students are struggling.  If they come in for help, then it is our job as instructors to listen compassionately and understand that the students asking for help are stuck and are developing.  We can shape the path for them without giving away answers, and we can be supportive and point out all of the positive steps they have taken.

The purpose of class discussions, presentations, homework, portfolios, etc. is to put all this chaos into a final form that makes sense.  Our goal is to produce mathematics, and the way mathematics is produced is by proving or justifying why statements are true.  Class discussions, presentations, homework, portfolios are some ways to channel the chaos into a product.

Short story:  Embrace good chaos.  Channel good chaos into a final product of some sort.  Don't expect students to take your path to a proof or solution (i.e. what you think is chaos might just be another way).  Help and support those who need it without diminishing the intellectual value of the task.

Note:  If the homework collected is not high quality, use homework templates to encourage your students to improve their process.

Teaching a Course for the Second Time (or Third...)

Winter term has started here at Cal Poly.  In fact we are in week 3.  This winter quarter I am teaching two sections of Math 330 Algebraic Thinking with Technology for future elementary school teachers.  The course title isn't the point of this post, though I may talk about this course in the future.  What I want to get to is that I taught this course last term for the first time.  This quarter I am at an advantage as an IBL instructor, since I have laid out the course materials once.  For the second time around I can now review my notes, look for problems that did well and try to fine tune problems to get students into the learning zone, where the problems are not obvious and not out of reach.

A hidden "side" advantage of IBL is that we become much more efficient as we repeatedly teach a course.  So prep time declines with each iteration until it reaches a steady state.  Moreover, we gain valuable insights into the learning issues.  This is part of Math Knowledge for Teaching that helps instructors help students.

Example 1:  Let's say students really struggled with Problem N last term.  They tried it, but no one got it and things stalled big time.   Some extra time and special cases were needed to get students going.  This time around I can put into the problem sequence an exploration, where students investigate some given special cases and are asked to build several more examples.  Then using what they learned from this experience, they hopefully are better prepared for Problem N.  I don't want to make Problem N too easy or give away too much, but want to get the core ideas out into play through investigating related examples.

Example 2:  Problem K was way easy for nearly all of the students.  This time I'll leave Problem K as is, but add in a few more challenging problems after Problem K to ramp up the level.  (Note: In math finding really hard problems isn't the issue.  The issue is finding appropriate challenges for the developmental level of your students.  This changes with each class, so one is always getting things "in the ballpark" and adapting to each situations.)

It's Jazz.  You have Rhythm Changes in mind, but it's not the same every time.

Nuts and Bolts: Assessment of Presentations

IBL courses deserve assessments that match what students are asked to do in class.  In most full IBL courses, presentations form a significant portion of the grade.  Grading breakdowns could be something like this:

30% Presentations
30% Final Exam
20% Midterm
10% Written work
10% Portfolio

In this post I'll focus on presentations, and discuss the other items in future posts.  Student presentations are one of the key components of an IBL course.  Students should present their work to their classmates to be vetted for correctness and clarity.  This process by itself is immensely valuable.  In fact, it lies at the heart of IBL.  Students work on hard problems at home.  They bring back their findings, get feedback and repeat, until a problem is solved.

Assessing presentations can take on many forms.  One way is to use a point scale.  Here's a rubric which can be adjusted to suit the style of an instructor.
10 Completely correct and clear
8 or 9 minor technical issues, but the proof is correct
5, 6 or 7 Proof is incorrect, has a significant gap(s)

Points don't tell the whole story.  Instructors should have in mind the overall qualities they want from students.  Below is a general guideline that can be adapted to match your own criteria and your institution's.  Students should be encouraged and rewarded for their intellectual contributions to the class.  Generally students should present regularly and also participate meaningfully in discussions in groups and in class.  Students who are able to prove major theorems and present regularly in class should earn an A for presentations.  They are capable of doing original (to them) mathematics.   Students who present regularly, but none of the harder problems typically earn a B for presentations.  Students who show up to class regularly and present only occasionally earn a C for presentations.  Students who miss a significant number of classes and/or do not present more than once typically would earn a D or F for their presentation grade.

*Some instructors also include some bonuses for creativity and ingenuity.  When a student does something that you have not seen before that shows real creative thinking, it should be rewarded in some way.  I take notes in class and write comments and jot down the creative idea.

The main message is that students have an incentive to

• Show up to class having worked on problems
• Participate and discuss mathematical ideas
• Prove theorems on their own
• Contribute to their own and their classmate's intellectual development

Are these not the qualities we want from our students?  Aligning incentives in positive ways to what society values in people is a something we should strive for.  Assessment is not merely a way to determine grades -- it's also a tool to encourage students to be young mathematicians!

Nuts and Bolts: Getting Students to Ask Good Questions

By Matthew G. Jones, Cal State Dominguez Hills <mjones@csudh.edu>

One of the keys to making IBL work is to make sure that students really engage with the topic. There are lots of ways to do this, but I will focus on two ways to get a whole class discussion going. The first of these ways is to use question starters. I have done this in two ways. In the first case, I simply write a question on each of three or four index cards, and distribute them around the classroom while another student is presenting a solution. The index cards have a single question type, such as, "How did you know where to start?" "Could you explain how you went from ... to ...?" "Can you explain what the question is asking?" "Was this the first thing you tried?" The students are told that if they are given a card, they must pose a question, and they can use the one on the card if they wish. This way, those few students are thinking of a question to ask while the presenter is working. The presenter also knows to expect questions, and the class understands that discussion of the solution is the norm, rather than passive silence. The other way I have seeded questions is to hand out a sheet to the entire classroom, and to tell them that you will call on a few students to pose a question, and that they can use the handout to help them formulate a question.

In either scenario, if I call on a student who claims to have no questions, then I will ask the student to paraphrase a specific part of the presentation, such as, "Could you explain what you think the presenter means by this line?" or "What is the goal of this part of the solution?"

The second way to get a whole class discussion going is to let a presenter complete his/her solution, and then to give the student observers 2 minutes to discuss the solution with a neighboring student in the room. Sometimes I will ask a pointed question to prompt the discussion, such as, "What kind of proof did the presenter use, and why do you think that was his/her choice?" and sometimes it is left open. Then, I open the whole class discussion by calling on students and asking, "What did you and your partner discuss?" This kind of question diffuses the pressure for students to report themselves as confused, because they often give replies like, "We were trying to figure out..." or "We weren't sure about..." The main idea is that students are more comfortable reporting on their actions in a partner discussion than identifying their personal confusion or misunderstanding.

See What They Can Do

My good friend and colleague, G. Edgar Parker, Professor of Mathematics, James Madison University, has written a wonderful story for The IBL Blog.  Thank you, Ed!


See What They Can Do

The bedrock for what became the basis for my teaching philosophy was forged when I was an undergraduate at Guilford College.  I was in college for all of the wrong reasons; I didn’t like school, but I knew my parents expected me to go to college and I was in no hurry to find gainful employment unless I could make a living playing baseball.  I was at Guilford for all the wrong reasons; I didn’t want to follow my older brother Elwood for another four years (the expectations associated with coming behind a straight-A student who was also a two-sport varsity athlete, all-conference in one sport and team captain in the other, even if they were imaginary, seemed real to me), but I went anyway because I had seen the baseball teams that my two older brothers played on there and knew that I could pitch better than anybody I had seen pitch in those six years (and besides, it had snowed on my campus visit to the school I visited that I liked the most and it was the week before the regular baseball season’s schedule back home was to start).

During the advising session before my first semester, fall of 1965, I jettisoned any ideas of following the subject that intrigued me most, Biology, because I found out that, at Guilford, you went to labs as a priority over practice and made up labs if you missed them for games.  My advisor, J. R. Boyd, placed me in two mathematics courses, Calculus I and Mr. Boyd’s Linear Point-Set Theory.  Interestingly enough, it was calculus that scared me.  I had actually heard of that and didn’t think I was prepared.  I should have guessed that Mr. Boyd was cut from different cloth; when I protested that I had only had four weeks of trigonometry in high school, he had “reassured” me by telling me, “Don’t worry, if you need to know more, you’ll learn it.”.

I was so naïve that I just assumed that the way point-set was being taught might be something that people did in college, so I just tried to solve the problems and hung on.  Since Mr. Boyd never said anything negative, for all I knew, I was doing okay.  I was getting some problems, or at least Mr. Boyd seemed satisfied with what he wrote on the board when I described my arguments.  Since I didn’t know the difference between a hard problem and an easy problem, I was trying them in the sequence they came in the notebook.  And then I ran into the problem that, looking back, could have been stated as “the continuum is not denumerable”, but was stated in a way that tempted a student to find a map from the natural numbers onto the numbers.  I thought I had an argument, but Hal Phillips, whom I considered to be the best student in the class, chose the problem when his turn came.  As Hal presented, I saw his ideas (which I had discovered myself) crash one by one and finally doom the argument on which I had worked so long and hard.  I was, at that time, still pretty meek in public and very reticent to call attention to myself, but the ardor of the moment overwhelmed me and I burst out in class in frustration (Mr. Boyd had made no suggestion as to how to remedy the now-evident flaws in the argument), “Mr. Boyd, why don’t you just show us how to do it?”.    Without missing a beat, he turned to me and said, “Mr. Parker, why should I limit you to what I know?”.  It was an opportunity to learn a lesson about teaching, but at the time, I was just relieved at the way Mr.
Boyd handled the situation and didn’t add to the embarrassment I had created by calling attention to myself.  Besides, I still wanted to pitch in the Big Leagues and education was just a diversion to enable me to continue to play the game I loved.  Education was not my intended vocation.

Fast forward to 1977.  I had finished my PhD in mathematics at Emory University, had courses given by Moore method from David Ford, Bill Mahavier, John Neuberger, Phil Tonne, and Mary Frances Neff, and chosen academe as my vocational home.  I recognized by now that Moore method is what allowed me to blossom creatively and provided me with the tools to learn other persons’ mathematics as well.  Still possessing, at that time, some sense of humility, I concluded that Moore method should provide the same growth potential for others it gave to me and decided to use it as the core for teaching the mathematics courses in the major at Pan American University that I gave.  Over the next seven years, for each such course, I painstakingly constructed a problem sequence with the self-assurance that, anybody who could do the problems in sequence would have to be able to see how to do the next one in the sequence because the connections were so obvious to me.  And each time, as often as not, the students leaped right past my lemmas to the important problems and found ways I hadn’t thought of to do the problems or ways that I considered “less natural”, or patiently proved my lemmas, but then showed me what the lemmas should have been by proving the theorem without using the lemmas.

Fast forward to the summer of 1988.  I had been at James Madison University since 1984 and I now understood how to follow the students’ ideas in the major courses, but still thought, for some reason, that non-majors had to be led to the fountain before they could drink.  That summer I was assigned yet another section of Mathematics 103: The Nature of Mathematics, the lowest numbered mathematics course in the catalogue and a course that I had taught every semester since it was introduced in 1985 (I had developed such a course for a less well-prepared clientele at Pan American, so, naturally, even though I had voted against our department offering such a course for General Studies at James Madison since I thought a school of our pretension should make calculus the core requirement, I was picked to give one of the initial offerings and was blessed with it on a continuing basis.) .  On a lark, I had the following conversation with myself: “The students are in this course because all they want is a mathematics credit.  So no one will fail the course.  But I will not tell them that and I will teach them as if they were majors.”  So I designed a course in which I “taught” them rigor by having them justify, on the basis of the field axioms applied to the words number, +, and $\times$, some of the fundamental techniques from high school algebra, and then put them to work on proving, as theorems, the field axioms stated for the ordered pairs of numbers with addition and multiplication defined so as to make the structure the complex numbers (without having i a part of the notation, I hoped nobody would recognize that they may have studied this algebra already and I was correct in this guess).  The only modification of Moore method I made was that the class was split into eleven groups, each with two or three students, and each group was responsible for proving a single theorem (chosen by lot), with all students responsible for certifying the correctness of the arguments and for reproducing and using what they certified to be correct.  The class got them all!  It was not that the students populating this class could not do the mathematics, it was my not believing they could do it that never gave them a chance to do it.  The summer of 1988 may have been an act of God.  I have used this problem set many times since, and no other class has gotten them all.  But every class has gotten some of them, and most classes have gotten the existence of reciprocals, the problem I consider the most difficult in the problem set.

Dr. Parker, why should you limit your students to the way you think about things, and why should you impose limitations on them?  


The longer I teach, the more I wonder why I didn’t ask myself this question earlier.

G. Edgar Parker, James Madison University

Nuts and Bolts: Small Groups

The topic of this post is group size in collaborative learning environments.  There are differing opinions about this, and I will not go into any data regarding small group sizes.  Rather I'll talk about what I do, and you can use this as your starting point.

Size:  I like groups of size 2 or 3.  Clearly 2 is the smallest group size.  Once groups get to size 4 or greater, students can hide.  This is something I want to avoid.

Mix 'em up -- generally it is a good idea to move students around.  One can use the Random() command in excel to create a random list of numbers, and then sort students on that column.  Then take bunches of 2 or 3 students to form the groups.

Random shuffles aren't always the best, however.  For instance, there may be certain personalities you want to keep away from each other.  In this case, one has to do some group engineering.  Experience tells me that as you get closer to the boundary of a classroom (i.e. the walls), there exists a greater probability of personalities, especially if groups were allowed to form naturally.  I also look for students who are quiet.   Then I make a list of personalities and quiet students.  Seed each group with exactly one from this list, and then fill the groups with the other students.

The number of people in the group is but one aspect of heathy groups.  Students can get noisy, chatty, or a person can start dominating discussions.  It is important to be an active coach and mentor.  An instructor might be called upon to manage personalities and guide through gentle questions/directions like, "Can I count on you guys to work as a team today?" or "Let's stay in mathland (to the student texting)."

It can also be useful to talk about what it means to be a good group member.  A good group member shows up to class ready and on time, listens carefully to their partners, is supportive, and offers appropriate levels of help at the right time (i.e. no blurting out answers and having a genuine respect for learning).

Short version:

• Find a good size (2 or 3, in my opinion) and mix of students.  
• Coach and mentor group dynamics until they work as teams.

Nuts and Bolts: Homework Templates

One big idea I learned in college English and Literature courses was the notion of writing and rewriting drafts.  Constructing an essay is a process -- you read, you think of some idea you want to argue for, you construct an outline, you write a draft, and then you rewrite and reprocess until a finished product (or the deadline) arrives.

Of course you learn this in any subject.  To master any discipline require a long, long process of thinking, working, reworking,... Discipline and focus lead over time to an increase in skill and understanding.

All too often in math courses, the content is broken down into bite size pieces.  That is, the material is overly preprocessed for "easy" consumption.  Students follow.  Teachers get high marks.  Life is good.  But we know that's not good enough.  Certainly this doesn't develop the kinds of work ethic, habits of mind, and problem-solving ability we value.

One issue that needs to be dealt with is the specific process students use in their daily practice.  Do they just try something and either (a) get it or (b) give up after getting stuck?  This is probably the case for far too many students.  Poor process and practice leads to poor results, and diminished intellectual develop in the long run.   To shift the nature of the practice is not an easy task, and what I propose is not *the* solution -- it's just a start to one.

How does it work?
In my classes, I require students to use a homework template.  It's just a word document with the name of the course at the top and two little prompts.  The first is the statement of the problem or task.  The second is the solution/work/whatever the student has.   Students are instructed to do scratch work on separate paper, and most importantly analyze their work and transfer it to a final version onto the template.  One problem per page, unless the problems are really short.  Students are given the option of including their scratch work to the problem.

This system forces the drafting process that is much more explicit in the humanities and is easy to pull off in math classes.

A sample Template

Does it work? Yes.  The quality of student work improves considerably.  Instead of getting scratch-work quality written work, students are now required to take their homework more seriously.  Students also comment that they feel proud of their work, and why not?  If you think of an original idea, and present it well, that's something you should be proud of.  The homework has good content, it looks good, feel professional,... a job well done!

Does it solve all problems in the math process "pipeline"?  No. Templates and requirements of drafting do not address all issues.  Teaching is a complex system, and addressing students' process of doing homework is but one part of the picture (though a very big one).  It is, however, a very welcome step in the right direction.

Mistakes Are Good!

One of the messages in an IBL that goes along with "Being stuck is okay!" is "Mistakes are Good!"

Mistakes are generally stigmatized in U.S. Math Education.  When a students does something wrong, it is unusual if the student thinks of the mistake or error as an opportunity for greater, perhaps even profound insight.

There exists many reasons why we build prototypes or practice in a batting cage or simply use scratch paper.  We need to see how things work.  We need to practice and fail, so that we can learn to do what is right.  In short, practice and experimentation.

One cannot grow without experimenting or trying things.  It would be nice if our students could all have the disposition to say things like "Let's see if the idea works for a special case..." or "Let's see if we can check our thinking..."

If students fundamentally believe that mistakes are bad, then the very nature of their interaction with mathematics is limited.  Over time this leads to poor self image and then ultimately poor habits of mind and work ethic.  That would be the nail in the coffin.

One of the ways to get students over the negative image of making mistakes is to provide opportunities for students to experiment, and to allow for mistakes to play a central role in the learning process.  In fact, in an IBL class students make *great* mistakes.  They say or do things in ways that maximize their learning.  As an instructor I no longer make these mistakes, because I already know the material.  First time learners of a subject reveal, through their mistakes, what they know and what they don't know.  This is where the learning zone is, and this is where one can create magical learning experience!

Student: <Writes or says something that is incorrect>
Teacher: "Oh, did you just say/write... Well I'm really glad you brought this up.  How many of your were thinking about this the same way?  Good!  Let's rewrite this as a question, and then investigate it further to get to the bottom of this."

Respect the Struggle

In IBL classes, being stuck is a critically important part of the experience for students.  One of the greatest lessons a student can learn in school is how to manage being stuck.

One of the issues we face, particularly in the U.S., is that mistakes are stigmatized, and this context makes teaching problem solving more difficult.  When a student gets a problem wrong it is far too often interpreted as a criticism of their mathematical ability.  Yet, in contrast to this problem-solving ability and creative thinking are highly regarded attributes in all areas of life.

What can we do as teachers?  Students must know that

It's okay to be stuck!

In fact, being stuck is a noble state.  It's when we are stuck that we learn to learn. It's when we are stuck that we construct new ideas, and discard or improve upon ones that are not good enough for our current situation.  Being stuck is good!

One facet of effective teaching is respecting the struggle.  What this means is to allow students to struggle and think, in such a way that they are not stressed out, rushed, or feel that making mistakes is "bad."  Ensuring that students feel that they are allowed to explore, think, experiment, and build ideas that may not work is critical to building a positive learning environment.

Giving away answers or letting students flounder excessively are two ways we can get off track.  As a teacher one should monitor students and manage the struggle so that students are challenged, making progress (over time), and not overly frustrated.

How do you know if you have a positive learning environment in your classroom?  Ask yourself if your students feel it's okay to be stuck.

Classroom Strategy: Think Pair Share

One of the most effective and easiest ways to get students involved in your classroom is to use Think Pair Share.  Harvard Professor and Physicist, Eric Mazur, has been one of the strongest proponents of using peer instruction.

How does Think Pair Share work in a mathematics classroom?  It goes like this...

• Pose a question or task to the class, such as "Give an example of..." or "Which ones of these, if any, is an example of...?"
• (Think) Let students think about the question/task individually for about a minute (or whatever appropriate time) 
• (Pair) Ask students to explain their solution/idea/thoughts to one person sitting next to them.
• (Share) Involve the entire class in a discussion of the question/task.  A good way to start off is to walk around the class while the pairs are discussing and ask one or two pairs to share their ideas.
• (Recycle) If necessary, a class may not arrive at a consensus.  In such cases the class can re-enter the pair phase and work with their partner to sort out the details.

Below are some examples chosen from elementary Number Theory.  But these ideas can be easily adapted to any math course from Calculus to Math for Elementary Teaching to Real Analysis.

Example1:  (Starter question)

• State the definition of $n|k$, where $n,k$ are integers.
• Question: In your own words, interpret what $n|k$ means.  Write a few sentences.
• Share with your neighbor your sentences and revise if necessary.
• Pick a few groups to share their work.

Example 2:

• Task: Determine which of the following statements is true.  
1. If $n$ is even, then $2|n$.
2. If $n$ is even, then $n|2$.
• Think for yourself which one is correct.
• Convince your neighbor of your answer.
• Class discussion. Recycle if there is confusion or lack of consensus.  

Example 3:

• Question: If $n|a$ and $n|b$, then $n|(a+b)$.
• Think of some strategies you could use to prove this theorem
• Discuss your strategies with your neighbor.  Write questions, if you have any.
• Pick a few groups to share their strategies and/or questions
• Make a list of the ideas, and let students continue to ask questions.  Then one can move on to the next task, leaving the proof as a homework problem that will be shared later.  Another option is to let students come up with a sketch of a proof in class and clean it up at home to be turned in/presented the next time.