Still reflecting on students' beliefs and attitudes about mathematics

During my teaching career I spent a great deal of time reflecting on the attitudes that students developed toward mathematics. That hasn't changed in retirement.

So now I am studying students' beliefs and attitudes and teachers' reflective practice in my Master's thesis with Dr. Christine Suurtamm as my academic supervisor.

During the Ontario-based Math 4 the Nines project, I, and two other teachers of grade 9 applied mathematics worked with a graduate student to developed and administered a before-and-after student survey on beliefs and attitudes about mathematics. A change-analysis of the survey revealed some interesting results that gave us pause to think and reflect.

In my research study, entitled Informing mathematics teachers’ reflectivity with student surveys on affective domain, I am inviting secondary math teachers to talk about their reflective practices, with a focus on the information gathered through a before-and-after student survey on beliefs and attitudes about mathematics (see below).

Participation in this study will involve a 30 to 40-minute Zoom interview at the beginning of the semester on reflective practices. Participants will then be invited to offer their students an online survey on belief and attitudes about mathematics. I will provide a Google Form for you to copy*. The survey will take about 10 minutes at the beginning of the semester and again at the end of the semester. A before-and-after change-analysis will be automatically generated. At the end of the semester, participating teachers will be invited to a second 30 to 40-minute Zoom interview and a 45-minute online Zoom focus group.

*NOTE: By making a copy of the Google Form, you and you alone will have access to the student data. Neither my supervisor nor I will have any access to student data or student information.

I appreciate that there are added time pressures on classroom teachers these days but if you are interested in participating or would like more information, please send me a DM @BDMcLaurin or an email to bmcla077@uottawa.ca or bruce.mclaurin@gmail.com

Thank you for your interest.

~Bruce

Sample items from the student survey on Beliefs and Attitudes about Mathematics

1.     The math that I learn in school is mostly facts and procedures that have to be memorized.

2.     In math you can discover things on your own.

3.     Making mistakes in math helps me learn.

Sample results from the Google Form survey

Sample results from the automated change-analysis

 

You can take the teacher out of the classroom but . . .

 

We all leave the classroom eventually. The question is when? How do we know when it's time? For some it is simply the 85-factor (age plus years of teaching). Or maybe a retiring spouse. Perhaps a change in curriculum which demands a reinvention of classroom materials or a mandated assessment practice that provides a push. Conflicts with colleagues or administration. Health issues. Maybe when you have been in automatic mode for a while and find the whole thing is getting tedious.

I was looking for a sign. I never really got one. My 85-factor came and went as did my 60th birthday. I was still in good health. I had only had a handful of sick days in 10 years. I loved the work and my colleagues. My career felt like it was still on an upward trajectory. I was excited by the changes I saw in mathematics education and was ready for new challenges. When I tried to explore the idea of retirement with my spouse, she would just look at me blankly. Later she explained her lack of support by telling me that the concept was totally foreign to her. Her parents were approaching 90 and still working. So retirement was a decision that I made on my own. It was not easy. It was probably the toughest decision that I ever made. I decided on a Saturday and broke out in shingles on Sunday. But many of my younger contemporaries had already retired. I was a grandfather. So I suppose it came down to how finite this life is. There is a big world out there and maybe there was another role for me to play, another adventure to explore and the only way I would know was to quit teaching.

My spouse once heard that it is important to retire not from something but to something. I didn’t really have anything to retire to. I had been so focused on teaching that my life outside of the classroom was limited. The only way that I could begin to visualize other life possibilities was to walk away, cold turkey. I quit. No supply teaching. No Twitter. No socializing with teachers. The first summer was like any other. A cycling trip in September was a distraction. October and November were tougher. I was embarrassed to be seen on the street. I had no job. I had lost my identity. An exchange at a dinner party reemphasized why I had set myself adrift. I was relaying the statistic that teachers in Ontario work an average of 25 years and collect pension for 30. Our guest contrasted that with the civil service where the average pension life is measured in months! Life is short.

I had made a vow to myself to stay completely away from teaching for one year. And I did. One year became two. I cycled in Quebec, California and Spain. I skied in BC and Austria. Did trail runs in Virginia, Washington State and Mexico, reconnecting with friends on my travels. I worked on rebuilding our cottage which was destroyed in a forest fire. We welcomed our second grandchild, born the day after we got back from a trip to Europe. 

Despite my best efforts I still wasn’t able to turn my mind off teaching. I continued to be drawn to news stories about education. It is where my mind drifted to on 20K runs. I had to admit that even in retirement, I am still a teacher. It didn’t take much to get me to apply for my Master’s. A colleague mentioned it and my spouse thought that it was an excellent idea if I did it. After all, it was something that I planned to do even as I was completing my B. Ed. so many years ago. When Chris Suurtamm agreed to be my academic supervisor, it tipped the scales. The chance to work with her was too good to miss. 

I enjoyed the graduate coursework and discussing education again. The work hasn’t been terribly onerous. I was able to squeeze in a trip to Baja, Mexico between classes and collaborate on a paper while poolside in Palm Springs. I just received the okay from the Ethics to go ahead with my thesis proposal. As a former department head, a workshop leader and a frequent participant in lesson studies, I once got paid to talk with teachers about their practice. Now I am paying for it. 

 

Never questioning the importance of questioning

The important thing is not to stop questioning. ~Albert Einstein

Last week Al Overwijk and I gave a one-day workshop entitled Everything we know . . . so far.  At one point Al was describing one of the lessons that came from our lesson study.  The main thrust of the lesson was to develop the criteria for asking good questions.  The impetus had come from an earlier lesson for my class where the students were given an assortment of materials and directed to ask a good question and then answer it.  The students floundered.  The questions were lousy.   As lousy as the disjointed irrelevant questions that we ask them too often.  So the lesson for Al's class was careful crafted to get at the heart of what makes a good question.  See Al's blog for more details.
 

After listening to Al's account of this lesson, a participant raised her hand and I smiled as she asked, "Why is it important for students to be able to ask good questions?"  Sure enough, Al responded with "That's a good question."  I jumped in.  I don't know exactly what I said. My words were sloppy but my passion was real.

The world is changing.  If this younger generation is to deal with problems that don't even exist yet in a rapidly changing world, then it is imperative that we help them develop the skills that they will need.  The first step in problem solving is problem-sensing and for that you need to ask questions - good questions - tough questions - good, tough questions.  Some days the toughest question I get is "Can I go to the washroom?"  We need to stop doing the heavy lifting for our students and let them do the bulk of the work and ask the questions.  We owe it to them.

 
A timely article concerning Jordan Ellenberg’s new book, How Not to Be Wrong: The Power of Mathematical Thinking appeared in Friday's Globe and Mail. To quote the article “Math is the science of not being wrong.” Ellenberg writes. In the real world, it doesn’t just find the right answers – it teaches us to ask the right question in the first place.  I couldn't agree more.

You Need to Change

Last week Al Overwijk and I had the opportunity to present at the Board's Leadership Conference.  To be fair, Al presented.  You see, our story is really Al's story.  There is no doubt that I changed my practice as a result of our collaboration but it was Al who really turned his practice on its ear.  In the presentation Al explained how after 3 semesters he was done teaching the grade 10 applied Math classes.  No matter what he tried it seemed that he couldn't engage the students and he was frustrated with the results.  He then said something in the presentation with which I took acceptation.  He reported that, as his department head, I told him he needed to change;  that he had nothing to lose because what he was doing wasn't working anyway.  He may have heard "You need to change" and maybe I did say it, but I like to think that what I did do was give him permission to take everything he knew about good teaching and put it aside.  To fill the void we went with two fundamental ideas.  First - let's just play - for the first month anyway, and teach everything through activities.  Second - we won't restrict and define our play within units of study, instead we will spiral back to all the big ideas again and again.  Al likes to say we had no idea what we were doing.  I like to believe that I knew exactly what we were doing.  We were casting off the moorings, pushing off from shore and slipping into uncharted waters.  We met almost every day to plan our next move.  The result of which fundamentally changed Al's approach.  He turned away from his rock solid chalk-and-talk and embraced an open-ended approach steeped in inquiry and observation.

We felt very good about our morning session and received great feedback.  With less time allotted for the afternoon session we knew that it was going to be a challenge.  With our principal and our superintendent and directors from other boards already in the room, our own director of education walked in with the keynote speakers Andy Hargreaves and Pasi Sahlberg.  The stakes increased along with Al's heart-rate.  It made me giggle to think that these accomplished educators whose work we have followed for years were now listening to us.  

In the end we both felt a little disappointed with the way the afternoon session rolled out because of time constraints.  As we were wrapping up we were offered a compensation prize of sorts.  A woman told the story of her friend's daughter who had been through three years of 'our program'.  At first I thought she said "You changed her life" but after consulting with Al and others I confirmed that she had indeed said "You saved her life." 

Slow Ed

Concerned with a population of increasing girth, Brazil recently came out with a new food guide.  Two of the basic principles are; cook from scratch and eat with others.  These are also basic principles in my classroom. 

Much of what we do in the classroom is the intellectual equivalent of fast food.  Relying on textbooks for example.  I would rather my students developed their understanding from scratch though activities.  Activities are the learning equivalent of slow food.  Research shows that eating alone can lead to unhealthy eating habits.  I believe that learning math in a group setting promotes healthier thinking.  The student desks in my class are arranged in pods of 4 to promote collaborative learning and encourage accountable talk.  (I do not want to discount the value of being alone with your thoughts when conceiving ideas but ultimately those ideas must be shared, tested and scrutinized by peers - hence this blog.)

Consider the number of questions in a textbook.  The 700 page Calculus and Vectors textbook that I have in front of me has over 2000 questions.  It is safe to say that most of these are not meaty problems; more like small bite-size snacks. Of course no one does every question in a textbook.  It is assumed that you will pick and choose or simply 'ace it'; "Do questions 1 to 25 odd only; a, c, e....".  Now, I know there are great and interesting problems in textbooks but most of the books tend to be filled with appetizers that are supposed to prepare your taste-buds for the main course.  Instead they tend to fill you up, leaving you bloated and sleepy.

I think that after reading The Man of Numbers by Keith Devlin, I have a better sense of our textbooks' pedigree.  It is an account of how Leonardo of Pisa, better known as Fibonacci,  published The Book of Calculation in 1202.  Early math textbooks were created to teach merchants commercial arithmetic skills. This pivotal textbook set the standard for years to come.  In fact, other than the addition of web links, textbooks haven't changed much since.   In an age where there are calculators in just about every kid's pocket, maybe it is time to start question what we value in Math class.

Historically the ability to do fast and accurate calculations has always been rewarded.  Students needed to know their "number  facts" to allow them to solve more complex questions without being  bogged down with basic calculations.  I get that.  But is it so bad to use a calculator for bigger numbers?  Unfortunately the value we place on snappy answers tends to permeate our work; question/answer wait times averaging less than a second; mad minutes; unconnected one-mark questions on evaluations; timed testing: rapid actions with limited capacity for nourishing deeper thought.

 

I know that calculation skills are important, but let's not be nostalgic for our glorious past of memorizing multiplication tables.  When someone in the back-to-basics movement complains that a young store clerk (a casualty of the 'new math') can't do simple arithmetic in an electrical blackout, I think of an old-school office administer who asked for my help with percentages.  How to best develop basic skills is not the main issue.  A bigger concern is our reluctance to even consider applying mathematics outside of the math classroom.  Even people with solid computation skills may never think of doing any math to see if an electronics store's extended service plan makes sense or if locking into a 5-year natural gas plan can actually save money.  Do our classrooms nurture or destroy math for most?

Another idea I have adopted from the slow food movement is 'eat local'.  If you close your textbook, you and your students may start to see mathematics as way to understand the world around us.  For example, when a local car drove into a sinkhole, I used the story in my grade 9 class to dig into concepts of area, volume, ratio and proportion.  One of the students knew the make and model of the car.  We found the car's dimensions online and calculated the volume of the hole.  We built scale models of the hole using linking cubes and built a life-size model with desks and plastic sheets. When the news reports stated that the hole had grown to the size of an Olympic-sized swimming pool, we used rope to outline a pool on the playing field.   What does two and half million litres look like?

But doesn't all that foolishness with ropes and cubes take time away from the hard math that these kids will need for the next grade?  There are so many curriculum expectations and so little time.  It seems we are always in such a rush to cover the curriculum.  To do that we need all the students to move along at the same pace.  It reminds me of the story about the man riding a bus in Moscow.  He watched as the driver failed to stop and pick up people waiting at each bus stop.  The man finally asked the driver why he didn't stop to pick up passengers.  The driver explained, "I am on a tight schedule and if I stopped I would no longer be on time".  We need to pick up all the passengers, even if it seems to slow us down.  But here is a trick.  We don't need to get every student to the same place at the same time.  Moving a group of toddlers down the street while they hold on to a rope is safe but it's not efficient. Some will be bored with the pace and others will be tripping over their shoe laces.  I say, let the faster students run ahead while we concentrate on helping the slower students. We will all be further along - in the long run.

Is getting everyone through the curriculum on schedule always effective?  I recall a colleague's account of an entire class of grade 11 academic students swearing that they had never seen the Cosine Law before.  They all were very adamant until one of the students found the Cosine Law on a test from the previous year; an evaluation on which she had done quite well.  We often introduce a concept with a concrete activity but quickly move to more abstract representations and concepts. Pushing into the abstract too quickly tends to lead to limited understanding and outright misconceptions.  Many students cope by desperately clinging to formulas.  My grade 9 students come to my class knowing that A = l x w but many need to count the squares one-by-one in a 5 by 4 grid.   They all know A = bh/2 but most never consider that a triangle has half the area of a rectangle.  Similarly, students can recite "a squared plus b squared equals c squared" and then tell me that 3 + 4 = 5, or 4² + 5² = 3².  I have had students tell me that they are quite positive that a positive times a positive is a negative.  And as every teacher knows that once something is 'learned' it is extremely difficult to 'unlearn' it.  Giving students a formula, rule or trick for the sake of expedience will ensure that we all get to 'teach' basic concepts over and over in grades 7, 8, 9,  10 and again in grade 11 and even 12.

Slow down.  Developing concepts takes time.  Making connections takes time.  Formulating good questions takes time. To solve complex problems without having the teacher slice everything into small chewable pieces takes time.  To reflect on the validity and reasonableness of a solution takes time.  To build an repertoire of complex skills takes time.  Slow down and in the end we will all be ahead.

The government of Brazil knows that fat-free, calorie-counting diets won't stem the rising tide of obesity.  Taking a page from the slow food movement, they are banking on a simple approach to save medical costs and ultimately people's lives. Fast food is not cheap.  It's nutritional value is minimal and the overall negative effects on our health and the environment is significant.  Thinking that you are saving yourself time and money with fast food is false economy.  Thinking that you are saving yourself time and effort in your classroom with snappy questions, tricks and shortcuts is also false economy.

 

I am not the first to use a food analogy for education.  Give a man a fish and he will eat for a day, teach him to fish and he will eat for a lifetime.  The word fish acts as a noun and a verb.  My attempt to rework it is not quite so nice;  give students some strategies and they will understand for test-time but teach students to strategize and they will create their own understandings for a lifetime. 

It's time to break from our diet of quick and easy questions, formulas and tricks, the cognitive equivalent of salt, sugar and fat. Time to construct mathematical concepts from scratch with the basic ingredients of investigation, questioning and accountable talk. 

Fast food is junk food.  Slow food is best.  Fast education is junk education.  Slow education is best.

Falling off the monkey bars

At EdCampOttawa today, the question came up; how does a teacher get to the point where they can let go of a prescriptive classroom approach that relies heavily on teacher-centred lessons? What does it take to allow students to have a bigger stake in driving the lesson? When are you ready to solicit questions from students and follow tangential lines of questioning while trying to stay within the boundaries of the curriculum?  Can a new teacher do that or do you need years of teaching experience to know what is important in the curriculum; to guide discussions and activities and have the confidence that experience brings to deal with issues, problems and questions as they arise?

How did I get to a point where it seems that I am more comfortable with ambiguity and uncertainty in my classroom than many teachers? Nurture or nature? My father was an accomplished engineer and from him, my 3 brothers and I learned not to be afraid of trying things that we didn't know how to do.

 

Today I thought of another training ground.  I was in a classroom just down the hall from the computer labs where I spent a number of years teaching computer science.  At that time I told my students that computer science provided more opportunity for problem solving than any other academic course in secondary school - including Math.  And the beauty was that I didn't have, or pretend to have all the answers.  I could offer approaches for finding bugs but I rarely knew what the problem was.  It is liberating to know that I could never compete with the skills of some of my better students who were spending 4 or 5 hours every night on their computer.  I wasn't the expert (thank god) and I was smart enough to get out of the way of learning. 

The other thought that I had today was how it seems that many of the youth of today are crippled with anxiety.  I have often blamed it on over-protective parents and policies that don't allow kids to fall off the monkey bars anymore.  Kids don't have the opportunity to learn that a broken nose won't kill you. 

I now believe that schools are just as culpable is this crime against children.  We packaged our lessons in bite-sized pieces so students don't choke and we keep safety latches on doors of ideas so students don't pinch their fingers.  We caudle our students with simple tasks so they won't fall and hurt their ego.  We spoon-feed our students in a misguided attempt to ensure their minds are nourished with everything that they will need.  No wonder intellectual engagement amongst students drops steadily  from the primary years to high school.  We are stifling curiosity and play.

We have removed the monkey bars from the playgrounds.  Our classroom must become the new realm of daring and sometimes scary lessons.

If we are afraid, cautious and timid then chances are that our students will be afraid, cautious and timid.

If we are bold, adventurous and risky then our students have a chance to be bold, adventurous and risky.  We want citizens who are comfortable living with ambiguity as they face an ever-changing world with an uncertain future.

Take chances, make mistakes, get messy.   ~Miss Frizzle

Teaching is Hard, Criticizing is Easy

A student left my class yesterday to go to the washroom.  A short while later, her friend who had no doubt received a text message, excused herself so she could go to the aid of said student who was having a panic attack.  What precipitated this attack?  Me.

On the fourth day of the semester I was introducing the concept of functions by having students walk the first initial of their name and record the walk as a distance-time graph.  When I called her to the front of the class, I knew she was uncomfortable but I went ahead.  I felt she handled it well and she received spontaneous applause for her accomplishments. 

She did not come back to class.  I followed up the incident with a phone call to her appreciative mother.  Apparently there are other factors but, by her choice, she is no longer in my class.  In fact she is no longer registered in any Math class this year.  Ouch.

This morning I watched a lecture on YouTube https://www.youtube.com/watch?v=3g2KjP2a20g by Amanda Ripley, the author of the recently published The Smartest Kids in the World: And How They Got That Way.  At the end of the video she emphasizes how hard it is to be a teacher.  She goes great lengths to explain that even though her mother was a teacher she really, really did not understand how hard it is to teach; 'fighter pilot hard, CEO hard'.  I concur.

I searched for the video after reading a review of Ripley's book in The Economist, August 17th-23rd 2013.  For her book she interviewed American exchange students in other countries and she examined test results from PISA (Program for International Student Assessment).  According to The Economist article, "Pupils in Finland, Korea, Japan, and Canada consistently score much higher than their peers in Germany, Britain, America and France."  This is not news to me: nor is the fact that the classrooms in the countries that scored the highest are devoid of hi-tech gadgets. 

Although I haven't read The Smartest Kids yet, I like what I see.  Like most people, I tend to absorb ideas that support my beliefs and ignore or dispel of ideas that don't fit with my paradigm. 

There have been a few articles lately of a different bent that I have been busy dispelling. An opinion piece in Mathematics Teacher, August 2013,  argues that differentiation for individual student needs in education is a myth and will continue to be a myth until there exists a sophisticated computer program that can deliver and test concepts for each student as they advance in the curriculum.  The author implies that a machine can do a superior job of assessing students' needs, better any teacher.  (Thankfully the periodical's editorial panel reminds the readers that mathematics is a basic human activity and teaching is a fundamental human activity).  It seems to me that that despite evidence to the contrary, too many teachers share the author's view; the way to catch up to the Asian countries is with more and better technology.  I am also disturbed by the implications that the goal of math class, from elementary school up, is to ensure that students can do calculus.  Mathematics professor Peter Taylor once estimated that 20 of the 1600 first-year calculus students at Queen's University will every use calculus. 

The next article is a little more challenging.  Margaret Wente of The Globe and Mail http://www.theglobeandmail.com/commentary/whos-failing-math-the-system/article14112165/ responds to the news that the latest test scores show that elementary school children's math scores are slipping.  She blames misguided educational trends that emphasis critical thinking and calls for 'back-to-basics'.  I quickly dismissed her premise but was curious to see how the public would react to the article.  Ouch.  

Ms. Wente struck a nerve, generating a flood of responses that are hard to ignore.  I read almost 200 of 275 responses, hoping that someone would challenge her ideas.  Almost all praised her views.  Blame was heaped on educational policy makers and teachers.  There were anecdotes  of how high school graduates who couldn't do simple arithmetic to complete a transaction when the electricity failed.  Sentiments expressed of how learning basic math facts like the multiplication table is boring but essential.  It seems many feel that that arithmetic is what mathematics is. 

Today's Margaret Wente column questions the fads and magic beans of educational pseudo-science. http://www.theglobeandmail.com/commentary/classroom-fads-and-magic-beans/article14168845/
The article quotes Tom Bennett, the author of Teacher Proof: Why Research in Education Doesn't Always Mean What it Claims; "I suspect that children learn when they are told stuff, and forced in some way to remember it, and practice it".  Ouch again.  Where's the research?  And we know this works because?  It worked in the past? I question how well it ever worked and I question how it could ever work with today's kids and I question how relevant any 'stuff' would be that I could tell students to prepare them for the future.  There are certain 'tried and true' approaches that I am glad we no longer subscribe to; like finding square roots with paper and pencil, corporal punishment, and blood-letting.  We don't need an industrial model of education that churns out human machines for our factories any more.  We need critical thinkers that can question and deal with all the challenges of an ever-changing world. 

I do wish that all my students could do basic math with speed and precision; it would make my job easier.  It bothers me that some can't and I am unsure how to change that.  But instead of trying to teach multiplication to 16 year-olds, I will spend classroom time doing more practical math like questioning the value of electronic stores' service plans and I will spend time on important mathematics such as examining the ramifications of exponential growth.

Parents sometimes bring up the same questions as Ms. Wente.  Are you experimenting on my child with new methods?   The gist of my response is "Every damn day".  I continue to search for more effective ways for my students to learn and I explore new methods to assess their progress.

My job is hard but I will also continue to listen to my critics.  It keeps me thinking critically about what I do and how I do it.