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.

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!

Sessions:

• 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

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.