Oh the places you'll go...

Maybe its boredom or maybe inspiration but today I walked into my Grade 9 class and shared the screen below from my phone.

In particular I highlighted the number of steps: 2 408 434!!! 

I was curious to hear what they thought of this screen. It didn't take long to get the response I was hoping to hear...how far is that?!?!? I really didn't want an abstract answer of so many kilometres but preferred to know something we could all relate to. Could I have walked to downtown Toronto? How about Montreal?

Here is what one group determined.

I was curious to see what strategy the students would use to determine the number of steps per kilometre. I had brought in tape measures thinking they may want to measure a typical walking gate but no one ended up using them. This group was typical of the approach that each group used - they searched the number of steps online.

I spent some time after class mapping the different locations that each group derived. Here is the map. 

My personal favourite appealed to my love of the Red Sox - I can walk to and from Boston. They were even kind enough to tell me how long it would take me - 7 days 9 hours....one way. 

I'm not sure where this activity goes from here. I recorded the solutions for all of the groups and was thinking to share them with the class so we can discuss the strategies and whether all of the locations seem reasonable for the distance calculated. Maybe the experience of seeing so much math in a simple number that led to an engaging exploration that addressed the key expectations in 9 Applied of rate of change and proportional reasoning is good enough.

2016 Fields Medal Symposium Lecture

2016 Fields Medal Symposium

I finally had a chance to sit down and watch the Fields Medal Symposium opening lecture. The main speaker for the evening was 2014 Fields Medallist - Manjul Bhargava. Bhargava (born in Hamilton!) is an accomplished number theorist and professor at Princeton University. Initially I thought the lecture was going to be an exploration of obscure number theory (maybe this is why I delayed the viewing) and chuckles at inside jokes about rings and the Farnsworth Parabox. I was so pleasantly surprised by the accessibility of the content (even I could follow it!) and the wonderful connections that Professor Bhargava made to art and the natural world.

After viewing it I decided it would be a great item to share with my students in class and those who show a particular love for number theory. What I thought I would do with this post is breakdown the talk for those who haven't viewed it just in case you want to jump to the bits of particular interest in the event that you want share with your students. The lecture starts at about the 32 minute mark.

For the early going, Professor Bhargava shares his introduction to mathematics as nurtured by his mother and his grandfather, a Sanskrit scholar. In particular he talks about the role that number theory plays in mathematics. He introduces sequences of numbers. Some of these may be familiar to you and to your students. In particular he looks at the visual or geometric proofs for the sums of these sequences. This seems to be  very timely with the current focus on visuo-spatial reasoning. Jump to the 44 minute mark to see a great visual proof of the sum of the first n odd numbers. Here's some more highlights I noted in the talk.

47 min mark - Sum of ascending and descending whole numbers (e.g. 1 + 2 + 3 + 4 + 3 + 2 + 1)

48 min mark - Hex numbers - Connecting them to a visual arrangement. So cool!!!

52 min mark - Exporing the "atoms of our universe of whole numbers" - PRIMES!

57 min mark - Primes in Nature - Why do cicadas have a 17 year gestation period?

1 hour 1 min mark - The role primes play in encryption.

1 hour 6 min mark - What is special about honeycombs?

1 hour 11 min mark - Professor Bhargava shares his favourite problem as a child - Stacking Oranges

1 hour 16 min mark - The Fibonacci Sequence - Professor Bhargava illustrates an amazing connection to Sanskrit poetry and lots of connections to the natural world.

1 hour 40 min mark - What's so special about the number 142 857?

1 hour 44 min mark - Fractals!

As you can see there is so much here to explore and share with your students. Each and every topic is explained in very accessible language and presented with such clarity it is hard not to be excited by the connections that number theory has to the world around us.

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.