Lessons Learned from Back in the Day

One of the neatest experiences of being a high school teacher is those too infrequent visits from recent graduates. I love catching up with them and finding about what they are up to and how they are adjusting to life post-high school. A bit of honesty here - I tell all of my current Grade 12s that my memory is dreadful and please do not be upset if you return and I don't immediately recall your name. Actually...even after some time I probably won't recall your name. Now bizarrely, after a brief awkward moment where I admit to forgetting their name, I will invariably blurt out their last name! Not sure what that says about my brain but I can dig out an obscure surname as soon as I hear a student tell me their given name.

On Friday, I was graced with a visit from two students who graduated in June of 2014 - Jaydev and Nuan. As is my habit, I didn't recall their names but no need to worry. I was about to walk into my Gr 12 Data Management class and asked if they wouldn't mind speaking to these prospective grads about their experience post-high school. I was able to skirt the name issue by asking them to introduce themselves to my class and to talk a bit about their university experience. Aha - got the names! 

I have told all prospective grads about my experience in university but I have been honest about what translates from the time I was in school to now. Some of my experiences are of course DATED! My first year tuition was a grand total of about $1500! I had to attend lectures or ask a friend who did attend for their notes if I missed one - no podcasts or online modules to access. The only technology you saw in a lecture was maybe a personal tape recorder. 

But there are experiences that have transcended the passage of one century into the next - yes it was that long ago. I worked doggedly to understand what I didn't. I left no resources or stone untouched. I accessed office hours, teaching assistants and any other resource that may help me in my pursuit of understanding what I didn't. I tell the prospective grads that when they are paying for the privilege of education that they should get their money's worth! It was so good to hear the recent grads echo this advice in the time they spent with my class.

A common refrain on the first day of my classes is that the best resource that a learner has is probably sitting right near them. We are all on this journey to learning and understanding and the experiences of classmates is most like our own. They are most close to our level of understanding and can help when there are any misconceptions. Jaydev and Nuan related similar experiences to mine. Often we would be sitting in libraries, cafeterias and even in the pub occasionally as we worked through a linear algebra assignment or discussed the reading from philosophical theology. We laughed together, sometimes cried together but most often LEARNED together.

The structure of the day in college or university can sometimes lead to a lot of down time between classes. When you aren't working on the latest assignment or reading it is critical to find an outlet that engages you in another way. I'm not talking about a pub crawl. I'm talking about being involved in a club, activity or something that can be a distraction from all of the pressures of student life. It is not critical that this be connected to your school community but it is helpful to form a connection to your school that is not purely academic. I was a commuter student which presented its own set of challenges but I still tried to be involved. I wrote for the school paper. Jaydev especially stressed this point. He didn't get involved in first year and he felt that disconnect. So he made a concerted effort to get involved. It gives you chances to connect with students that aren't necessarily in your academic program and year. 

Lastly (and predictably I think most important), be open to the path of your passion. I went to school enrolled in a business program. Applied to law school and was admitted to attend Western after two years. Declined that offer and found my passion in a history of math course. Jaydev was initially in health sciences but switched to urban planning when someone recognized something in him that told them that he may like the program. He loves it! Always be open to your bliss as Joseph Campbell would insist. The path may be winding but be open to it and be okay with the occasional detour. 

I wonder if I was to travel back in time and ask a student at the University of Padua  in Renaissance Italy (I have no idea if there is or was a university in Padua) and observed the life of a student if many of these same eternal truths would be evident. The life of being a student - actually of being a lifelong learner may change at the periphery but it doesn't change at its core. With a passion for learning and surrounding yourself with like-minded hungry learners - it is a good life. 

A Lamentable Lament

I have been very fortunate to be working with a teacher candidate from a local faculty. She is a very conscientious teacher and we got into a good discussion on the nature of teaching math during her last classroom visit. She made the comment that sometimes what her and her teacher candidate friends see at the faculty sounds very good in theory but when the rubber hits the road and they are standing in front of a room of students that they need to present the content and make sure it sticks. And so the model of teaching that emerges is very much the same method that they may have been exposed to when they were students. Stand and Deliver.

I really appreciated her candor and I know that this isn't unique. I've been fortunate enough to visit faculty candidates in another school over the last few years and I usually hear this comment made by a few of the teacher candidates. I know I felt the same way when I was in the faculty at Mount Allison in New Brunswick. It was survival mode and so with a very limited repertoire of strategies I fell back on what was familiar. Although there was the one moment at Amherst High School in Nova Scotia where I strode into my Grade 11  class with a powdered wig trying to do my best Isaac Newton impression in an attempt to connect Newton's fluxions and Leibniz's development of the calculus. It morphed into an study of the history of math and that was a glimpse into what was possible outside the box. Its a message that I have often repeated to teacher candidates. The placments they get are opportunities to try the strategies they are learning. They shouldn't feel so afraid to make mistakes in their teaching...isn't that how learning (whether its math or teaching math) occurs?

At one point I asked if she had read Paul Lockhart's "A Mathematician's Lament". She promptly pulled it up on the computer and started reading. I recall the impact when I read it. I read it at a critical moment in my own teaching career. It is an indictment of the way math is taught in too many classrooms. I was ready for the message at the moment I read it and it confirmed to me my desire to change the way I taught math. Coincidentally, Lockhart's Lament came up again over the weekend. I have been very lucky to meet some amazing educators and quite honestly thinkers (you know who you are) in my teaching career and I would include Sunil Singh in this group. Sunil was a guest on the ZPC podcast (Zone of Potential Construction) hosted by Chris Brownell. The podcast is Episode 5: Mathematics, Happiness and the Joy of Discovery. Sunil was talking about his soon to be released book "The Pi of Life" (Nice title Sunil...any tigers in this one?). During the podcast, Sunil talked about the inspiration that Lockhart's Lament provided him in his career and the impact it had on the way he views the teaching of mathematics.

And so I wondered what is keeping us from addressing what are systemic issues in the teaching of math. I am not sure I have resolved the issue...not sure anyone ever has or will. But one thing that keeps nagging at me is this perception that the constucts within which we teach math are its greatest enemy. By the constructs I am referring to the insitutionality of our system of education. It may be time to consider that we are to trying so hard to fix something within a system that may make it impossible. There are some amazing educators doing some creative and innovative things in the classroom but, in the end, even they are limited by the parameters within which they work. Our boards and schools do their best to support teachers with grand schemes of support  that usually happen in some banquet hall to little or no affect. It may be time to think bigger. It may be time to significantly shift what we consider to be the way students are taught math.

All this from a single conversation with a very honest teacher candidate. And perhaps that is a reason for optimism. The next generation of teachers will take up the torch and at least raise the questions again and again. But I hope it isn't an infinite number of times.

The Evolving Thinking Classrom - OCMA Keynote

OCMA Conference Keynote

I had the pleasure of providing the keynote at the Ontario College Math Association conference on Thursday, May 26. It was great day at the Fern Resort in Orillia. We even ended up playing the Game of Frogs with some in attendance standing in for our frogs. Thank you to everyone in attendance.

The title of the talk was "The Evolving Thinking Problem-Solving Classroom". A mouthful and this was an omen of what was to be included in the talk. I tried to provide a perspective on my own journey in teaching math from a problem-based perspective and stuff as much content into 90 minutes as could conceivably could be done. I think the theme for me is comprised in one comment I made near the start - They <<our students>> are inflexible in their thinking because we made them rigid in their learning. I have become more and more convinced that we are at a critical moment in the teaching of math. The confluence of so much we have learned about teaching and the technology we have available will change what the math classroom will look like in the near future. It is great to be a witness and sometime participant in this evolution and revolution.

The keynote also provided me a platform to thank some educators that have added to my professional journey and made the journey richer. Some are mentioned in the slides and I know I missed many more. My apologies. I have been more a learner than teacher in the last few years and that has made all the difference. 

Here are all of the slides from the talk with comments to provide perspective on what is displayed on each slide. Hopefully it provides a bit of a narrative and makes sense.

https://drive.google.com/file/d/0BywXsUXbYYuXNTBRRU9ST1I5TzQ/view?usp=sharing

https://drive.google.com/file/d/0BywXsUXbYYuXNTBRRU9ST1I5TzQ/view?usp=sharing

Something for the Data Teacher

The most recent More or Less Podcast (Friday, Feb 5) contained a fair bit of fodder for the math classroom as usual...in particular for teachers of Data Management.

Student Absence and Student Achievement

At about the 16:10 mark, the story that is investigated is the impact of student absences (especially parents who take their kids out of school for vacation outside of the regular school holiday time). Even though this is UK data, the story resonates with teachers anywhere. I know I cringe a bit when a student presents me with the form that indicates a prolonged period of time will be missed due to a family vacation. But how much of an impact does that absence have? A great discussion ensues on the other factors that actually impact achievement beyond attendance. If you are teaching about the different types of causation that links variables, you can find a great example of correlation but necessarily causation here.

Probability and Birthdays

At the 22:40 mark, the podcast looks at the probability that a parent shares a birthday with two of their children. In particular, it mentions a number that had been reported in the past by Dr. David Spiegelhalter that there are 8 families in the UK where three children share a birthday. So that got me thinking. How did he come up with that number? I would let your students struggle a bit with this. What do they need to know to derive this number?

Hopefully after some good discussion, they may raise the following points:

• the number of families in the UK with 3 children
• the chances of having 3 people share the same birthday (a great tie-in to the traditional birthday problem)

Try it out. I did a quick search and found the family size data in 2012 for the UK from the Office for National Statistics but you may find more recent data. The great thing is if they can work backwards and determine the probability correctly and then using the number - 8 families with three children sharing a birthday - the number of families with three children determined from this calculation is very close to the statistic found in the data (off by a margin of 3.1%).

Now, the actual question of the probability of a parent sharing a birthday with two of their children is surprisingly similar (or maybe not much of a surprise). The probability is determined in the same way except now the two children can match EITHER parent which reduces the probability. The numbers that Dr. David Spiefelhalter mentions are rounded but you can confirm them easily. 

Here's an example for two children families (roughly 3 000 000 in UK in 2012):

Probability of two children sharing a parent:

(Note Spiegelhalter uses the number 1 in 67000)

Then multiplying by the number of families, you get:

So about 40 families have a parent sharing their birthday with their two children in two children families (as is mentioned). It turns out that the caller who asked the question actually has three children which raises the chances of him sharing his birthday with two of his three children. A great lesson in counting - how many ways can you select two children from the three? 

Here's my calculations showing the number of families where one parent shares their birthday with two of their children in families with 3 children. Note that the approximate values used by Spiegelhalter are also shown (he mentions there are 1 000 000 families when in fact there are 1 100 000 families).

Now, how do those numbers compare to Canada? You can find the data from the 2011 Census and get your students to work on it. My quick calculations show that there should be 33 families with two children that share a birthday with a parent. COOL! 

This assumes that birthdays are randomly selected. We of course know that they aren't. If you want to tie in looking at the distribution of this data, here's the distribution of birthdays by month from 2008-2012. The end of the podcast discusses this distribution but its interesting to note that the distribution in Canada doesn't mirror that in the UK exactly but it is pretty close. I love finding real world ways to look at probability. Hopefully this is something you can tie into your probability lesson as well. 

OAME 2015 Leadership - A Recap

Presidential Address from Tim

So OAME Leadership is over for another year and I was lucky enough to get funding to attend this year. I need to say a big thank you to OAME President Tim Sibbald (@TimSibbald) for his vision in organizing the conference. The theme of his term is connectivity and this was the common thread that tied the excellent sessions offered throughout the weekend.








Thursday kicked off with Ralph Connelly. Ralph provided a historical perspective to the leadership conference and how it has been on the forefront of math education changes since the first conference in 1974. The evening finished off with some much needed mathematical thinking courtesy of Ian VanderBurgh. Here are the two problems that Ian shared with the group. The first is a great one to do with your class.

Friday was jam packed and started off with University of Windsor professor Dragana Martinovic. Dragana challenged us to consider how we model leadership by reviewing some of the work she has done in collaborative inquiry models. The afternoon consisted of a presentation from Queens University math educator Lynda Colgan. Lynda provided a great overview of the impact parents have on the learning of their children. Check out this parent involvement quiz that served as a minds on for her session. Lynda also shared a recently released resource produced by the Council of Ontario Directors of Education. The resource is a parent tool kit that can serve as a plan for hosting a parent math night. It includes research-based information that can help parents support their child's math education. The toolkit includes resources, support materials and videos in five modules for Grades K-8. A secondary toolkit is in the works. I am hoping to use the Grade 8 module to host a parent math night at my school for the incoming Gr. 9 class.

The fun continued on Friday evening with an author night. Participants were treated to presentations from authors that have written books with a mathematical slant. The authors were:

• Richard Hoshino "The Math Olympian" (Check out Richard's website for lots of great stuff related to the book.) I first met Richard at a University of Waterloo summer workshop. I still use some of his activities in my teaching including one of my favourites "The Game of Frogs"
•  Siobhan Roberts "Genius at Play - The Curious Mind of John Horton Conway". If you haven't heard of Conway, he is a bit of a character. I would highly recommend some of the Numberphile videos that feature Conway. In particular the one on his inventing of the game of life. Siobhan has also written a book on Donald Coxeter called the "King of Infinite Space". Check out her website for more info.

• Ed Barbeau also presented some of his books of problems. If you want a great supply of problems from Ed, check out his webpage. 

Unfortunately I was unable to attend the end of the conference on Saturday. Particularly disappointing since Ian VanderBurgh was closing the conference with some more mathematical mayhem courtesy of his many mathematical conundrums. 

Thanks to Tim again for his vision in putting together a great program. I think the measure of a conference is always how quickly I take back what I saw in a workshop or session. And it didn't take long for me to share the many problems I enjoyed tackling and the resources that I collected over the three days. Now I have some books to read.

My Comfort Quote

I'm pretty sure I'm not alone in that I have a quote that I come back to once in a while that helps keep me going and reminds me of why I am doing what I am doing in my classroom.

My quote came from an article written by Ann Kajander in the September 2005 OAME Gazette. Ann is an associate professor in the department of undergraduate studies in education and the recipient of 2014 Margaret Sinclair Memorial Award. I highly recommend her regular feature in the Gazette: Mb4T: Mathematics by and for Teachers.

The article was entitled "Supports for the Effective Implementation of The Revised Grade 9/10 Curriculum: The Role of Manipulatives in Making Algebra Meaningful".

September 2005 OAME Gazette

The sentences that hit me squarely between the eyes were:

"Put bluntly, learning based purely on procedural methods has little meaning or long term retention for most students, and does not facilitate the depth of understanding needed to use math for any worthwhile endeavor outside (or inside) of school. Even more bluntly, its boring, demeaning, and encourages the all-too-prevalent perception that mathematics is difficult, useless and unintelligible."

Sometimes it isn't a quote itself that makes the impact although it plays a big part. Sometimes you have to be ready for the message. I guess the quote hit me at the right time and the right place. I had been struggling for a while in my teaching and recall relating to a colleague that was teaching Grade 10 Applied math with me that I was bored out of my mind. And if I was bored out of my mind I couldn't imagine how the students felt! That feeling coupled with what I read in the article was the catalyst that changed everything for me in my teaching. We redesigned the course with the idea that a good task was key to bringing the math out and would lead to that deep understanding. 

I can clearly remember playing the game of frogs to introduce quadratic relations and having the vice-principal walk in while students were jumping around the room. Or building tetrahedral pyramids that outgrew what our classroom could contain. All of the work we did to redesign the grade 10 course made us want to share the learning and so we presented at OAME in 2006. Our workshop title was Grade 10 Applied: The New Frontier. And on the last slide of our slide deck was what all good workshops should end with...a quote. Guess which one appeared on our last slide?

I would love to hear what quotes have inspired you in your own teaching. Feel free to leave them in the comments section. Thanks!

My First Blog Post

Its the THINKING stuped!

I tell this story often to anyone that will listen about my experience teaching math since the curriculum revisions in Ontario that occured in 1999. So my apologies to my colleagues who have heard me drone on about this before.

But first a look back...lets go back to a time long ago in a classroom that may look suspiciously like the ones we see now. Pre-1999, the curriculum in Ontario was an anomaly in North America...well specifically at the high school level. Students completed a five year program with the university bound students taking OAC (Ontario Academic Credit) in their final year. College bound students could choose from technical math credits or business math credits in their final year. Students needed 30 credits to earn their Ontario Secondary School Diploma (OSSD) and two of those had to be math credits.

The assessment and evaluation experience was also significantly different. A typical course outline might include the following in the determination of a student's mark:

Tests ____%
Quizzes ____%
Homework _____%
Participation ______%
Attendance ______%
Final Exam _____%

I taught in a Catholic school when I first got into teaching and can recall including a student's adherence to the uniform policy being included in some final evaluations.

How did things change when the new curriculum was introduced and the 5th year of high school was phased out?

The curriculum changed! What was taught in Gr. 9  through 12 changed to reflect the reduction from 5 to 4 years. The names of the courses were intended to now reflect the pathways that students were intending to pursue once they left high school (Academic -- University, Applied -- College, Locally Developed -- Workplace). The OAC courses that university bound students had taken in the past were replaced with Grade 12 University level courses that reflected a similar content to those in the OAC courses.

But the changes went a lot deeper than just labels. The spirit of the curriculum emphasized exploration and investigation over the rote memorization of skills. A value was placed on reasoning and justification that was not as evident in the past. But math was still math. Students still needed to solve equations, factor, derive and all of the things they were asked to in the past.

One of the most significant changes occurred in the assessment and evaluation practices that were outlined in the curriculum and supporting documents. Most of the items that had been included in the past such as participation or homework were no longer to be included in a student's final mark. They were now commented on as learning skills that were the same from Grade 1 to 12 (responsibility, organization, independent work, collaboration, initiative and self-regulation). Student marks we now to be determined by their performance in four achievement categories: Knowledge & Understanding, Thinking, Communication and Application (p28-29 of the Grade 11-12 curriculum documents).

What is fascinating in looking back at the changes is how the community of math educators reacted to these changes especially around assessment and evaluation. Speaking from my own experience at the time, what most of us did in the classroom was take our old tests and started placing Ks, As, Cs and Ts next to questions as we started the exercise of categorizing the questions we were asking. Easy peasy. New curriculum and new assessment and evaluation practices - no problem. If you were dropped into a math classroom in 1995 and then in 2005 I would challenge you to try to tell the difference. In fact, I would challenge you to tell the difference in 2015 for some math classrooms.

You can probably guess at what types of questions were labeled with each category label. And so we boldly stepped into a new day of curriculum with the assessments we had always used in the past. What were the consequences of this approach? It didn't take long for the alarm bells to go off. We could now see the areas where students struggled. It was like lifting up the rug after sweeping a lot of dirt under it for years or not realizing there was dirt there all along. Which category gaped like a black hole showing us what we had neglected for so long? THINKING! Students struggled in solving what were designated as thinking questions. The questions they were asked were those that we usually didn't assign from the homework - those questions from section C in each textbook lesson.

The focus of this post isn't the questions themselves - another time another post for that discussion but rather the reaction many of us in the math community had once we saw the results. The majority of schools that I encountered just DECREASED the amount that thinking factored into the determination of a student's mark. I know that's what we did at the school I was in at the time. Reflecting on this and recognizing what is taught it still boggles my mind that the achievement chart category that emphasizes planning, executing and reflecting on solving a problem is worth the least in a mathematics category.

My challenge as a department head and my challenge maybe to you is think of strategies as to how you can improve student performance in this category. It isn't solely for the benefit of seeing a bump in marks but for what it can indirectly provide. It can provide you the opportunity to dig deeper into the curriculum and assessment documents (e.g. Growing Success) but more importantly in my mind is the opportunity to value what we should value in teaching and doing mathematics --- SOLVING PROBLEMS!  We didn't get into teaching math to do 30 factoring questions. Another byproduct could be the fostering of persistence in problem solving that doesn't come from 30 factoring questions. The answers to the problems that our students will be solving will not be found in the back of the book. We need to provide them with opportunities for solving rich problems that demonstrate WHY we teach the skills we teach. Just like a carpenter isn't regularly tested on how to use a hammer but rather on how well they can put together a sturdy enough house. Let's value what our subject is about....its the thinking.