Friday, July 27, 2018

The Bumps in the Road

I just finished reading Mary Bourassa's latest blog post. Mary has courageously and honestly shared her experience this year teaching and the challenges that it presented. I have known Mary for some time and I can honestly say that every time I see her I walk away from our encounters taking away something that I can use in my teaching. Mary exemplifies what is best in teaching. She is constantly examining her practice and pushing herself to make changes that are in the best interests of her students. Mary, whether you read this or not (and quite honestly my expression of gratitude may be more for my benefit), on behalf of everyone (students, teachers, parents...everyone)...THANK YOU!

Her post reminded me of a file that had sat abandoned on my computer. The name of the file is "The Bumps in the Road" and the date it was last opened was February 23 of this year. Here is what was in the file:

The Bumps in the Road

My journey of going grade less is now more than a year old. Wait that’s not entirely accurate. It isn’t just my journey. I have some companions that are on this journey with me. At our department meeting to end the first semester we took a moment to reflect on the journey and in many cases to nurse some wounds. It hasn’t been easy. It has been hard. But my belief in the value and integrity of removing the focus from grades and marks to reflecting on the mastery of learning goals remains steadfast.

I am on the downward half, home stretch, nearing the finish line – pick your crummy metaphor - of my teacher career. There are less school years ahead of me than behind me and I was thinking that if the manner in which we teach/assess/evaluate math didn’t look too different from the time I entered teaching to the time I left teaching than I would be completely disheartened.

I want to take this moment to finish what I started. That start of a post was in response to what seemed like a wave of criticism and complaint against the changes we had implemented in our department. It seemed like everyday at the start of semester two brought another crisis (cue the Supertramp...Crisis? What Crisis?). We had introduced gradeless in Grade 9 but had also made an effort to spiral our content and I can't remember how many years since we abandoned textbooks. Why would we take on the work of using feedback and learning goals to drive student learning? And it was a lot of work! Why would we redesign our courses with a focus on exploring content with more detail as we progressed through the course with special attention to making connections and using rich tasks in our instruction? Every change we have made in our department was made in what we felt was the best interests of the students.

But I know that the start of that post was in response to what seemed like a siege on all of the work we had done. I was definitely feeling a moment of doubt. I am grateful that I have an administration and department that has remained united in our mission to make these changes and has ridden out those bumps in the road. Right now, thinking about the year ahead, I know that there will be challenges again. But I am committed to what we started. I am not committed to it out of a stubborn desire to make sure that its my way or the highway (please don't cue Sinatra's...My Way. Besides I like the Sex Pistols version better).

I am completely willing to change if someone shows me that students would be better off with textbooks, with teaching that presents content in disconnected chunks, with a system that places more value on the answer than the process. But we all know that EDUCATION is so much more than just textbooks, unit tests, recommended calculators and memorization of facts. We are dedicated to the process of giving our students time to explore with rich tasks that challenge their thinking and allow them to truly experience what we love about math - the discovery of something we didn't know before because we DISCOVERED it rather than were TOLD it. And we are dedicated to the fact that we don't have to place a number/letter on that process of learning in order to give it value. The process of LEARNING is abundantly valuable and doesn't need that percentage to give it any more credibility.

So...I do have a few years left. And it would be easier to just keep doing it the way I have always done it. Dust off the unit tests, worksheets and markbook. Definitely would be a smoother ride.  But what's a few bumps in the road when the journey is so much more enjoyable in the end.

Tuesday, January 2, 2018

My V

Five Books That Have Shaped My Mathematical Worldview

Thanks to Matt Oldridge (@MatthewOldridge) for providing the impetus for this post. Thanks Matt for including me in the tweet and for nudging me to give this some thought. It was enjoyable to look at the book shelf and identify those personal seminal works but also a little depressing as I saw all of those that remain untapped.

I haven't looked at Matt's post at the time of writing as I didn't want to be influenced in any way by seeing his top five. I will as soon as this post is complete...I promise Matt.

A Concise History of Mathematics (Dirk J. Struik)
There are many other history of math tomes and there are probably many that are better written than Struik's history but timing is everything. This is the text that accompanied Israel Kleiner's third year History of Mathematics course at York University and that is the course that sealed my passion for the subject. I still have my notes for that course and that short 256 page book crammed in all of the wonder and joy of mathematics.

Good Questions: Great Ways to Differentiate Mathematics Instruction (Marian Small) and More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction (Marian Small and Amy Lin)
Hard to think of a list without Marian Small (@marian_small) on it at least once! Cheating a bit here as I am thinking of these as one when they are in fact two separate books. It is also a good way to give a shout out to notorious cat-lover Amy Lin (@amylin1962). I tell the story frequently to those who haven't tired of hearing it that these two books remain the most popular resources in my department. Everyone in the department asked for their own personal copies of these books when they were published. These books changed the way we teach and assess. Thank you Marian!!!

Elementary and Middle School Mathematics Teaching Developmentally (John A. Van de Walle)
A truly seminal work in the teaching of mathematics and one that I would argue belongs on all math teacher book shelves regardless of division. I really delved into this work during my time as a resource teacher and it cemented in me my belief that the foundation of learning mathematics must first come from a visual approach and then proceed to the abstract. Some call it concreteness fading. I used it to introduce concepts such as integers and fractions when my own kids were going through the intermediate grades and have incorporated many of its lessons in workshops to parents and my own teaching.

Damned Lies and Statistics (Joel Best)
This is a super short book (about 200 pages) but I was so happy after reading it that I made it required reading one year when I taught data management. I have read many books about the importance of being data literate and a few of them contain more recent examples but this was the first and as I read it, I kept thinking that students could easily digest the message of the book. The book hits many of the historical blunders of statistical incompetence but also introduces some subtle concepts that aren't covered in any data management texts!

Dataclysm (Christian Rudder)
A book recommended to me by Judy Mendaglio (@judy11235813) knowing my enjoyment of books that explore our relationship to data. Rudder is the co-founder of OkCupid and his book is by far my favourite (especially since it was more recent) book about data. Not a day or week would pass without me sharing a nugget I read with my (exasperated) department and classes. This book is jammed with content that will amaze, amuse and anger.

Two Honourable Mentions...

The Incident of the Dog in the Night-time (Mark Haddon)
A work of fiction. A soon as I saw that the chapters were numbered using primes, I was hooked. Lots of math is embedded in a story that will keep you turning the pages.

Proofiness (Charles Seife)
Another book with data and statistics at its heart but this one serves as a cautionary tale. It harkens to that old saying: with great power comes great responsibility...and some are not being very responsible with numbers.

Well that's it but there are plenty that I left off that deserve mention (e.g. Boaler, Humphries, Burns and on and on and on) but only so much time to write and read. And this doesn't include the articles and shorter pieces that I return to from time to time to inspire and revive that passion like Apostolos Doxiadis's Embedding mathematics in the soul: narrative as a force inmathematics education. I could go on but I need to get to the stack left to read.

Saturday, October 28, 2017

The Parent Conference - A Conversation Without Grades

This past Wednesday was parent-teacher conferences at our school and I was booked solid from 5:00-8:00 with appointments for parents and guardians of primarily students in the Fusion program. At one point before the interviews, I looked at my colleagues in the Fusion program and asked them what they were making available to parents to use as evidence of student progress. I have to admit there was this momentary panic as I thought of those times where I would print out mark summaries for parents during interviews and for the first time ever, I had NOTHING of that sort prior to the interviews. I had not produced a single mark in either my Fusion classes or in my Grade 11 class but I think I had assessed more student work and provided more feedback than any time in the past up to this point in a semester. There was a sense of security that I think that mark printout provided. The blunt numbers were there as a stark summary of what had been learned and what hadn't been learned - at least that's what I thought and so did everyone else in the interview. They were not to questioned. THEY WERE THE MARKS!

So my colleagues and I wondered aloud - what should be in an parent-teacher conference that doesn't focus on the grade up to this point in a semester? Here is a summary of some of that thinking and some of the things that also came up organically during the course of the interviews - persistent patterns that made the conversations meaningful for the parents, students and the teacher!

Start with them!
I like to always start the interview by asking about their own questions or concerns regarding the student's progress. This sets a tone that I think is important. This interview is for them. It is not for me to lay out a laundry list of concerns or plaudits.

Evidence? What evidence? Oh!!! That EVIDENCE!
Your better have something! We use an online platform for the collection of student evidence related to each learning goal. And I do have summaries of the feedback provided to students throughout the semester. On top of that, we scan student work that is done in class using the copier so that we have an archive of those products. Also include evidence of the learning skills which are probably more informative at this early stage in the semester than anything else. I like to review the learning skills evaluation and the rationale behind each with the parent and student. Speaking of the student....

Who is this about?
I noticed this year more than any other that the conversation was continually steered back to the student. But before that can happen a key ingredient must be present - THE STUDENT! I insist that they be present for the conference. They have the most skin in the game and so they need to be present. As the interview progresses - once the parent had a chance to raise any questions or concerns and I had given my quick summary - I want the student to take over and lead the remainder of the interview. Here are some of the questions I asked:

  • What learning goals have you struggled with and what learning goals have you felt you mastered?
  • Give me an example of feedback that you received and how you acted on that feedback?
  • What are you doing to ensure that you master the learning goals you may be struggling with?
  • What learning skills do you think need more attention and what can you do to improve upon those learning skills?
  • What is your overall assessment of your achievement and what evidence supports that overall assessment?
This part of the interview is critical. It reinforces to me the most important part of trying to teach without grades - the responsibility for learning lays squarely with the student. They need to be continually reflecting on where they are in mastering the learning goals set out for each cycle of learning.

The most common parallel I draw with my students is my flailing on the guitar. I take lessons and I practice (admittedly not enough). But having my guitar mentor tell me that my playing is at 42% is about as informative as Donald Trump reminiscing about global warming patterns.  The 42% tells me something - it tells me I stink. But having him tell me, as I try to master "Highway to Hell", that I need to work on my fingering for the transition from D to F# is something that DOES HELP! In much the same way, that 42%, 52%, 65%, or 91% relates something to the student and parent but beyond just comparing your position to what seems like an arbitrary benchmark - very little more.

Some of the features outlined above exist in any interview - grades or no-grades. They reflect the communication that evolves when there is a sense of trust between the student and teacher and the parent and teacher. They reflect the belief that the parent recognizes that the instruction and the assessment of what a student has done up to this point in the semester is grounded upon what is best for that student. We had a parent information night about our Fusion program and that did a lot in the way of helping parents see that the enormous amount of effort we were putting into the program was grounded upon OUR belief that what we were trying to do was best for student learning.

If you have any FEEDBACK for ME on other features of the parent conference that can make it a meaningful meeting for everyone involved, please leave a comment. Thanks!

Thursday, October 5, 2017

On the Road Toward Gradeless Teaching/Assessment

On the Road Toward Gradeless Teaching/Assessment

I just read a great piece by Jo Boaler entitled "Math Class Doesn't Work. Here's the Solution" published in Time. Boaler points to the primary culprit in classrooms across the US (and I would contend also in Canada) - an emphasis on a performance culture in school. The article resonated with me because it called to mind conversations I have had with students, parents, administrators and other educators about what we have been trying to do in the math department at Fletcher's Meadow.

During the 2016-17 school year, we piloted a program in Grade 9 called Fusion. We ran Fusion in first semester with three sections of Grade 9. Those three sections took place in the same period and allowed us to expand our collaborative tasks on a larger scale. In addition to this unique feature of the Fusion program, the team decided to also try to spiral through the content in the Grade 9 program and embed gradeless teaching/assessment.

By the end of the semester, we had gathered data from a variety of sources including:

  • an online portfolio for each student
  • formative in-class assessments
  • observations gathered during collaborative tasks
  • traditional summative assessments
Reflecting on the process of assessing and evaluating student achievement, we thought we had gathered more data than we had ever used to consider how well a student had done in achieving the learning goals for the course. And I think that was the revelation for me. The focus had shifted from a pressure to achieve an artificial benchmark to an emphasis on learning the content. Students were continually reminded that the absence of grades was not meant to hide a truth that only their teachers were privy but rather to shift their focus to reflecting on what they knew and how well they knew it. The feedback provided on all of the sources mentioned above was like the diagnostic test that my mechanic performs on my overworked VW Wagon that is pushing 240, 000 km - it gave them an idea of things that needed their attention and things that they had mastered. The only difference is that my VW will eventually succumb to the laws of nature whereas our hope is that the students focus on learning will lead to deep recall and retention of that learning. 

Note that the relationship as well between teacher and student also changed from a hierarchical relationship inherent in a process of awarding grades to a partnership of learning. The teacher is tasked with providing the feedback that will assist the student in their mastery of each learning goal. But critical to this relationship is the acceptance of the responsibility on the part of the student for their learning which is something that needed to be EXPLICITLY told to them since this is so different from their experience in the past.

And this is the point that Boaler makes so well in the Time piece. In particular she points out that "Our grading and testing practices are largely responsible." for the pressure that many students feel to perform rather than to LEARN! We expanded our Fusion pilot to 9 of our 11 sections of Grade 9 math this year and I have been blessed with a risk-taking department that has adopted gradeless teaching/assessment in other courses. And this idea that the focus should be on learning is the point that I stress each and every time I talk about the rationale for our move to gradeless teaching/assessment. We had a packed parent information night this year and the response although initially unsure was overwhelmingly positive once they were given the explanation for the move in our assessment and teaching practices. 

This is a journey and we have a long way to go in our own learning. We are still wrestling with providing feedback in a timely manner and making it manageable as we expand to multiple courses. But as I reflect on where we are and where we have been, the tough part to imagine is retracing our path in the journey and returning to grade based performance feedback. We have seen the benefit to student learning and a reduction in anxiety around performance. I know that there will be bumps and perhaps we will venture down some dead ends in our journey but I'm glad we have ventured down this road. It has made all the difference - so far.

Monday, March 13, 2017

7 4 7 - Teaching Through "EDOC" - The Director's Cut

At the T3 International Conference in Chicago, I was asked to be a part of session titled "Seven for Seven" - seven speakers speaking for seven minutes on a topic that they are passionate about. It was a great experience and I want to thank Kevin Spry (@kspry) for the chance to be a part of an amazing lineup of speakers. For those not able to be at the session or conference, the session was on Facebook Live. Along with my talk on coding in the classroom (Teaching Through "EDOC"), attendees got a chance to listen to:

  • Sherri Abel (@sherriabel1014) - Student-centered Teacher-facilitated Engaging-minds Math-science (aka STEM)
  • Todd Morstein (@tmorstein) - Demo Friday: Challenging Students to Question and Problem Solve
  • Valerie Hudson (@vhudson_math) - Helping Student Conceptual Understanding Soar to New Heights
  • Stephanie Ogden (@SoSogden) - A World Without Teachers
  • T3 Leadership Award Winner Marc Garneau (@314Piman...not the astronaut) - Who's Doing the Math
  • Michelle Rinehart (@HowWeTeach) - Transforming Into Our Teacher Leader Selves
The process of creating one these talks is about as enjoyable as a tax audit but as I reflected afterward I thought about how it forced me to distill into a short time frame what I really wanted to relate on the topic of coding in the classroom. In particular, the process of writing out what I wanted to say really helped me organize it into a coherent narrative. And so I thought I would share the text of what I wanted to say here...all of it including what I FORGOT to say.

Hi! My name is Paul and I teach math. And I code. And I make my students code. There used to be a time when computers seemed intimidating but we would be hard pressed to go without them for a day now and we probably figured out what the steering wheel is for. My goal in just 420 seconds is to relate to you that coding can and should be as familiar to us as our use of any technology.

To arrive at that end we need to agree on what we are talking about. When we think of coding we thing of computer programming - the set of instructions a machine follows to complete a task. But my hope is to also show you that it is much more than that especially in a math class.

I listen to a podcast called All Songs Considered and I am struck once in awhile by the sheer volume of music released. It seems like so many more people are making music available. And so I wonder is it because the way we perceive and therefore make music has changed. What used to seem like something that could only be created between five lines can now be created with anything...including technology.

And so does coding suffer from the same preconceived notions. When our students think of coding, how many of them think of lines of code or the programmer as geek or nerd? Or just the magic that happens behind the screen? By coding in our classrooms we make it accessible to them and maybe get them to think it's okay to be a geek.

Changing perceptions is just one outcome of coding in the classroom. We want to also create an environment that invites collaboration, encourages perseverance, mandates that students challenge themselves and their peers to make their thinking clear and of course celebrate their triumphs.

These traits of a thinking classroom create the climate for what I think is an even greater payoff for the math teacher. There are two broad benefits that I've reflected on as critical to the math classroom once you are coding.

First, the computational proficiency that is inherently required to code is intrinsic to coding. For the program to work, the math has to be right! Writing a simple code to output the area and perimeter of a rectangle or the hypotenuse of a right triangle can only help reinforce the mathematics we are teaching.

Secondly, and in my opinion, an even larger impact is the insistence that coding should be the way we think in a math classroom. We all have seen or use different problem solving models like Polya's classic and we all break down problem solving in the same way needed to be a good coder.

I know the challenges of fitting all that we are told to do into the already crammed minutes of a period. I don't do this everyday in class but once I've covered a skill....I usually say, "I bet a machine could do that." and once the groans subside (they do die away after a time...remember we are changing perceptions), we code that skill! I don't see it as an add-on but just the right spice to make the dish taste better.

So how to start? Well how about coding something that isn't mathematical - get at the computational thinking by coding something that is familiar to them. I'm not too sure how many students know the chicken dance but here's the code! Or even better, get them to code a skill they are good at like flipping plastic bottles to land upright. That's big at my school for some reason.

And be hard on them when they start writing their code because you know the machine will be..."What do you mean syntax error?!?!?". You can use out of order cartoons as a starting point. I like coming back to this idea by giving them a program with lines of code mixed up and they need to fix it. Or here's another example I use every year in Grade 9. It only works once. If you want to try it, follow along. Here's my code: Draw a square. Draw a trapezoid. Draw a circle. Draw a triangle. Alright...everyone done?

Does your picture look like mine? No? That's because my code stunk! There was an error that I'll say was planned especially to my students but repeat to you because I missed it when I made the slide! But also my code needed so much more to make my thinking evident. This is going to speed up now.

Ken Ken. The size of the puzzle tells you the numbers you will use. This is a 3 by 3 Ken Ken so I will use the number 1, 2, and 3. Number 2 will go in the top left because I need to obey the rules of the bold boxes called cages. For the 6x, the numbers are 2 and 3 but like Sudoku I can't repeat a number in a row or column so 3 goes in the top left and 2 right below it. I only have one number left in that column which is 1. What divides to 2 and involves 1? It must be 2 to the right of the 1 which leaves 3 as the last number in the bottom row. Then 1 goes above that since that is the only number left and 3 goes right in the middle for the same reason. The last number is 1 to complete the puzzle. I love the deductive logic needed to solve these puzzles and as the puzzles get bigger and involve all 4 operations they reinforce basic number sense.

Once you've tilled the soil and planted the seeds with puzzles and coding skills, the ground is fertile for coding. This is what one of my coding challenges looks like. A simple program that requires the output to be the savings on a sale item. I want to give a shout out to fellow crazy person doing one of these talks, Michelle Rinehart for showing me this model called pseudo-code. It is the thinking needed before jumping into the programming. The parallel to Polya's problem solving model is obvious. And the reflect step is when they test their program.

And so they code. Once they feel comfortable with their pseudo-code, the journey begin and they have to bring along passengers. But it doesn't take long before they hit a pothole - syntax errors, logic errors, structural problems like nothing is being displayed. This is not the house I had pictured in my mind! But I am here to tell you...celebrate the bug. Embrace it!

I didn't anticipate it but coding perfectly aligned with some of the work my department had done around growth mindset and the work of Carol Dweck and Jo Boaler. I'm going to break one of the rules of presenting and read this quote because it was exactly what I needed to reinforce why coding was important and it will eat up some time!

Once coding became a part of my classroom it helped with my vision for what I wanted my classroom to be a thinking space. A place where they could think and wonder and make mistakes. Once of the physical changes I made to also help with that was using vertical spaces rather than horizontal desks as the place for collaborative thinking. A shout out to my colleague (and roommate) Tom Steinke for introducing me to the work of Peter Liljedahl and his research around vertical non-permanent spaces.

Here's an example from my class in Vectors. Students asked, "Is it possible to just input two algebraic vectors and have the calculator determine the resultant geometrically?". And of course I said, "I wonder." There was coding as we defined the inputs, figured out what to do with those inputs and generate the output. And you know along the way, it helped them remember the math as well!

So to close, I was trying to think of a flashy last slide that would leave you remembering something like "Everyone can code"...but this just didn't seem right. It felt like, implied in that was "Why aren't you?"

And so I changed it. But this still didn't capture it. You can code...yes but meh. And so I settled on this.

Thank you!

Monday, December 19, 2016

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 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 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.

Monday, December 5, 2016

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.