Wednesday, May 10, 2017

Finding a Balance in Math

For the past month or so, I’ve been participating in the Not A Book Study look at Cathy Fosnot’s work on constructing multiplication and division.

This has been a tremendous opportunity to make great connections with other math educators across Northern Ontario, and really deepen my knowledge of how students construct mathematical concepts. I’ve been searching for something deep to sink my teeth into for a while, and I find this book - and all the discussion surrounding it in the #notabookstudy - has been rich and extremely thought-provoking.

As a secondary math teacher, I learned a lot in teachers’ college about tips and tricks to teach math through the intermediate and senior years. But we didn’t learn very much about how students learn math. How do they make the jump from thinking additively (counting 3 + 3 + 3 + 3 + 3 + 3 + 3) to thinking multiplicatively (7 x 3)? How do they approach the concept of division? How do they go from being able to problem solve in a specific context, to a generalized one, eventually using variables and equations?

Needless to say, I’m learning a lot about learning by reading this book, and this will certainly enrich how I approach teaching mathematical concepts in the future.

But I find I’m struggling with another aspect of teaching math as I continue learning about how we learn - finding the balance between discovering the concepts and practicing the skills.

In secondary, there is very little discovery (I’m generalizing here, based on my experience teaching). Most of what the students learn is taught to them in a very procedural way, with perhaps a bit of inquiry or hands-on activities, but almost never where the students are discovering the math themselves. And the students certainly aren’t owning the math.

On page 48, Fosnot says “Mathematics cannot be learned through transmission…” and I agree - students need to understand what the mathematics is telling them in order for them to remember what they learn. And yet we, as teachers, are still merely transmitting information.

However, in elementary (again, a generalization from what I’ve seen, mostly in primary and junior), there is a lot more discovery of the math, with lots of emphasis on concept construction, but not as much procedural learning. While this can promote a better understanding of what the math means, and a better growth mindset and approach to problem solving, the drilling of facts or repetition of procedure doesn’t seem to be, pardon the pun, part of the equation.

Partially because of this, in secondary we are seeing students who struggle more and more with basic math facts. That struggle leads to frustration, cancelling out any gains that may have been made from understanding the math initially discovered in earlier years.

Can students understand where the math comes from AND become procedurally efficient?

I’d like to think so, but we as teachers need to work on that balance. With respect to automaticity, Fosnot says “The issue here is not whether facts should eventually be memorized, but how this memorization is achieved: by rote drill and practice, or by fostering on relationships?” (page 86).

The former is shallow but “quick;” the latter is deep but takes time. And if “It is not up to us to decide which pathways out students will use [as they move toward constructing understanding]” (page 18), how can we ensure each student has the time to properly develop their thinking AND their automaticity?

Wherein lies the balance? My journey continues...

Wednesday, April 26, 2017

Snapchat Stories

I'm at my first CONNECT conference this week, in beautiful Niagara Falls!

Typically at conferences, I track some of my learning through Twitter (follow #CanConnectEd this week for some AWESOME learning!). But for this conference, I wanted to try something different.

So I'm trying out Snapchat stories as a way of capturing what I'm seeing and learning. 

Follow along!


I like that in the end, you get a running chronological story of what you see throughout the day. I like that you can caption images as you go, and though I haven't drawn on the images (or added stickers!), I like the idea of being able to do that quickly, too.

I appreciate the speed at which I can snap a photo, caption it, and get it published. And even though a normal snapchat will disappear soon after it is viewed, posts in Snapchat stories stick around for 24 hours before self-deleting.


But I don't like that it's not widely interactive - no one can see my stories unless they follow me, and you can't connect with others as easily as you can with a hashtag. And you can only have one story on the go at a time... I went for a hike yesterday and wanted to share some pictures, but didn't want to tack them into the conference story. It would have been nice to be able to make a separate story.

For the long term...

At the end of the day, I can save the entire story to my phone and then export the video. Here are my stories so far!

Note: upon uploading and viewing the videos here on the blog, I realized they are all blurry! Is there a way to fix this? They're not this blurry in the original story... :)

Day 1

Day 2

Day 3

Day 4

End Result?

In the end, I'm not sure that I would use Snapchat stories again to track conference learning. I really liked the speed of snapping and captioning, but I missed the networking and connecting that often comes along with sharing on Twitter or Instagram through a hashtag.

With only a few edu-peeps following me on Snapchat, too, I'm not sure it was worth "sharing" this way, either. It might have been better to just snap pictures right into an app like Evernote.

Why did I try this? We often talk about bringing the learning to where the learners are. With so many students on Snapchat, is it worth considering delivering content to them through this media? Is it feasible to have students document their learning through snaps?

Have you tried Snapchat stories before? Do you have any advice? :) What way of keeping track of what you see at conferences do you prefer? Have you tried using Snapchat with your students?

Saturday, April 22, 2017

Tracking Observations & Conversations

In many of the collaborative inquiry projects I'm working on this year, the teams are choosing to focus on student communication. In some cases, we are looking at how a student can best communicate what they have learned. 

Traditionally, this takes place on a unit test (or other culminating activity), supported by a number of smaller assessments leading up to the test, such as quizzes, worksheets, projects, etc.

However, these are all product-based. How can we, in math, move away from assessing primarily through products, and more through conversation and observation?

This is a scary concept for many high school math teachers, who are so used to assessing products. It's easy enough for teachers to observe, or to have a conversation, but whereas you track a level/mark on a product (16/23 on a test, say) that you see directly on that assessment paper, how do you track what you see off-paper?

As I think about being back in the classroom, this is something I want to improve upon. Instead of chalking it all up to my "professional judgement," I'd like to be able to track what I see and hear, and offer that as a record of student learning.

But I have no idea how to start that tracking. The newness of it (to me), and the openness of it, makes the process feel overwhelming. A lot of math teachers are in the same boat.

So who better to ask about tracking observations and conversations than the experts - Kindergarten and Grade 1 teachers! Where so much of the learning is done without textbooks, worksheets, and tests, these teachers are the pros in recording what they see.

I did a quick poll of K/1 teachers on Twitter of how they track the learning that happens in their classrooms (click here to see the Storify archive of the conversation), and here are some of their responses:


Some teachers are using paper checklists, created in advance, highlighting the look-fors and the learning goals the students will be working to complete. They can be stored in a binder, always within reach in the classroom for when learning is observed or when the teacher and student engage in conversation.

Amy (@Teach_Laidlaw) shared an amazing post detailing her checklist process. You can find it here:

Photos & Video

Aviva (@avivaloca) is the QUEEN of documenting her students' learning through captioned photos and video. She and her teaching partner collect them and review them throughout the day, using them to not only assess the students, but also to plan future activities.

Brigitte (@BrigitteDupont0) has a busy French classroom with lots always on the go - she uses pictures and video too. She also keeps everything in Google Drive for easy access.

Stickies & Colour-Coding

I think Marcie (@MarcieLew) is one of the most organized people I know. Her tracking method of choice is stickies and colour-coding for at-a-glance overviews of how her students are doing. She can then walk around the classroom with a clipboard and stickies, observing and conferencing throughout the lesson.

Google Forms

Amanda (@amandakmalo) creates online versions of checklists using Google Forms. There would be a drop down list for the student's name, another list for the subject, and then an open answer for comments and quotes. 

Not only could Amanda fill in the forms on a tablet as she was going around the classroom, but the ECE and prep teachers also had access to the form to record what they observed. With a running record of comments for each student, it made report card writing much easier, as well as learning stories.


Geeta (@geetaranikumar) also recorded her observations digitally, but through Evernote, which includes the capability of attaching images and audio files to a "note." Geeta also mentioned that she could add tags to the pages/notes for easy reference and sorting later. 

I'm so grateful to my PLN for sharing their practices with me. I'm a big fan of the paper checklists, but I'm leaning toward the convenience of the digital record keeping. How do you keep track of observations and conversations in your classroom?

Saturday, March 18, 2017

When do they Learn That?

With the introduction of collaborative inquiry (CI) projects in our secondary schools this year, it's been very interesting to see what schools identify as their students' greatest learning needs, and choose how to address those needs.

For our Numeracy CI projects, many schools identified various aspects of communication as a significant need. It was interesting (but, some would argue, not surprising) to see evidence of this need from a number of different data sources, including past individual EQAO results, overall EQAO school trend results, student IEPs/psycho-educational assessments, and/or anecdotal information.

One school decided to look at vocabulary acquisition. How could explicitly teaching and placing emphasis on math vocabulary help students when it comes to problem solving?

Throughout the project, we explored different interventions for students, tried different strategies in class, examined student data, and listened to student voice. In the end, we were able to help a lot of students find success where they didn't experience success before.

But it's one of the spin-offs of this project that I wanted to mention and share.

When do they learn that?

During one of our team meetings, we were discussing which vocabulary words we expected students to know coming in to grade 9 math (and know well), and which words might be relatively new to students.

For instance, we figured students should know the word "area," since they start learning about the area of basic shapes in an early grade. However, the word "radius" is only introduced in grade 8 - having only been exposed to the word for one year, students might not yet have a solid grasp of its meaning.

(And, as an aside, a student who was on a modified curriculum before high school may not have seen the word "radius" at all. Important to note if we are now teaching these students in grade 9.)

But what about the word "angle?" 
Or "factor?" 
Or "ratio?" 
How well can we expect students to know these words?

A Vocabulary Continuum

We took a look at the glossary at the end of the Ontario Grade 1-8 Mathematics Curriculum, and the glossary at the end of the Ontario Grade 9-10 Mathematics Curriculum, and placed vocabulary terms in a continuum based on when they were first introduced. Check it out:

(scroll down to see the five strands, and across to see the different grades)

Being not familiar with the elementary curriculum at all until this year, I found this layout fascinating. I had no idea students learned fractions as early as grade 4/5 (many grade 9 students still struggle greatly with fractions), or that scatter plots - which I've always considered a "basic" skill in grade 9 - is a concept students would have only just learned in grade 8.

Some notes:

  • We chose to only look at the grade 9 & 10 applied stream (not the academic stream) to both keep the chart simple, and keep it aligned with our focus courses in the project.
  • In grades 9 & 10, the names of the strands change. We kept the same strands as in the elementary curriculum and tried to place the grade 9 & 10 words appropriately.
  • The placement of the words in the various grade levels was based on when they first appeared in the curriculum - this might actually be different in real life (and indeed, different from teacher to teacher, or from school to school).
  • As a spin-off of this spin-off, we also made a list of "tricky words" that have more than one definition - you can see those on the second tab at the bottom of the sheet.

What do you notice? 
What surprises you? 
How can this help when it's time to introduce new concepts?

Thank you so much to Kris Oliverio, Megan Parry-Jamieson, Iain Brodie, Randy Porter, and Richard Duffy for double-checking the placement of these words for me!

Friday, February 10, 2017

Finding Validation

I've been home, sick, for the past two days. It's hard for me to come full stop, but I've learned that the best thing to do when you're sick is to heal. And if that means a lot of sleeping and tea and soup and video games and episodes of Star Trek: TNG (all interspersed with coughing fits), so be it.

One day of resting to heal is fine. But on the second day, I'm restless. I even tried to go to work today, but got turned around on my commute. When I walked back in the door, my husband looked at me and said, "I told you so." :) I then fell back asleep for another 4 hours.

But on this second day of rest and healing, I started to think about everything I was missing while I was home, sick - an inaugural event at local schools on science and tech, a separate regional event that all the other board co-ordinators attended, the interactions with my colleagues, and coaching opportunities with my gymnastics team. 

In a profession where I thrive on all those interactions with others, I was feeling left out, and unvalidated. So many people are accomplishing great things today! What am I doing?? Healing. As productive and contributive as my white blood cells are being, I certainly don't feel productive or contributive.

And that got me thinking about my students. When they are in school, what makes them feel valuable and worthwhile? Do they drift through their days feeling as though they haven't made an impact or a difference? Do they feel as though they contribute to their learning, or the learning of those around them? How can we make sure they feel productive and contributive?

Allow students to contribute

What do our students already know? What could they quickly find out? Instead of listing examples of amphibians in a note, could students seek out and find their own examples (and non-examples)? Can they come up with a way to design a lab, instead of always following instructions on how to set one up? How are they creating content for the course's curriculum?

Allow students to share

What can students teach others in a small group setting? How can they best share within the class? What can they communicate with a wider audience? How can students share what they're learning with other classes within the school? How can they connect with community members? Who looks to the students to learn?

Allow students to drive their learning

What interests students outside of school? How can their passions be used to further what they're learning in class? How can we provide choice in what or how the students learn? How can we make sure they are making gains in their learning every day? 

Allow students to provide input and feedback

How do the students feel about how they are learning in our classes? Are students included in the planning and assessment of the presented content? Are their actual learning needs the same as what we perceive them to be? Is their voice heard? How do their opinions on learning shape what happens in class?

How do you make sure your students feel valued, productive, and worthwhile in their pursuit of knowledge?

Monday, January 30, 2017

From Grade 8 to Grade 9

This year, I'm learning a lot about how we can help students (particularly math students) transition from grade 8 to grade 9 successfully. 

There are a lot of reasons why this transition is not successful for all grade 9 students in many boards. Off the top of my head, this is because differences include:

New school:
  • Students go from knowing their way around their school very well to not knowing where to find anything or anyone.
  • Students go from having two nutrition breaks in the day to one lunch break (that's gotta make the period before lunch soooo hard for hungry students!). 
  • Students go from having opportunities to play and get outside (recess is a given) to having nothing like this scheduled (students can go out at lunch time, but don't have to).
  • Students spend more time on the bus. Bus rides, in a rural area like ours, get longer as there is only one high school servicing a large geographical area.

New teachers:
  • Students go from having one teacher who knows them (and the 30 other students in her/his class) really well to having four teachers who know them (and the 70 other students they teach that semester) very little.
  • Students go from knowing all the teachers in their school (having been taught by most of them) to knowing NONE of the teachers in the high school.
  • Students go from knowing what one teacher expects from them to having four potentially very different teachers, and having to juggle a myriad of expectations.

New peers:
  • Students go from having the same peer group to support them all day to different peer groups that change throughout the day.
  • Students may not even be with their usual peer group if their timetable is significantly different than their friends'.

New courses:
  • Students go from unstreamed classes to streamed classes (that carry their own stigmas).
  • Students go from a modified curriculum (such as working at a grade 6 level in math) right into the grade 9 curriculum.
  • Students move into an environment where grades matter, exams are written, and credits need to be achieved. There are consequences for not passing a course.

New Freedoms:
  • Students can leave the school at lunch to walk up to a local restaurant, or just hang around outside the whole time. Some students aren't very good at disciplining themselves to come back to school in time for third period after this new-found freedom.

In short, there's a LOT that students need to adjust to when moving from grade 8 to grade 9. Part of my job this year is to examine how that transition is happening and what we can do to help students better bridge the gap.

So I'm looking for suggestions. What can we, as grade 8 and grade 9 teachers, do to help our students be as successful as possible as they manage these changes? Which of the above factors can we control and exert influence over? If we could make a TOP TEN list of ways to help students make the transition, what would that list include?

Sunday, January 22, 2017

Humans vs. Computers

We hear a lot these days about how computers (and robots) are taking over all the mundane tasks we humans do on a regular basis; how we have to change the way we're teaching in order to prepare students for a very different-looking workforce.

Robots/computers can vacuum our floors (hello, little Roomba!), mix fancy drinks for us, and can drive our cars. On assembly lines, robots complete tasks more efficiently and more consistently than we ever could as a species. They can even create abstract works of art and write newspaper articles.

I've often wondered what the limit is: 
Where is the threshold for what humans can do pretty well, but computers still can't do at all?

Last month, I discovered one possible answer: composing music.

Just before Christmas, a sound clip of the first computer-composed Christmas carol was released. This computer was fed hundreds of hours of traditional Christmas carols which it analyzed, decomposed, pulled the most common elements from, and then used to synthesize something completely original. 

Here it is - take a listen:

It's AWFUL. I'm pretty sure some of the youngest students I teach, who have very little experience in music, could come up with something better. Especially those lyrics. Yikes.

A computer can try and combine the most popular elements of existing songs - basically pulling from a huge resource bank, larger than any human would have access to - but can it really push the boundaries of music? Can a computer be daring? Can a computer take creative risks? At this point, I would venture that they cannot.

Humans, however, can. My husband (@christheij) is a music teacher. Recently, while looking for new choir music for the spring season, he found this gem by Katerina Gimon. Take a listen - it's well worth it.

Perhaps my favourite part of this piece is the score, which contains, among other unique things, the following as notation (seriously, this is actually written into the music) (yes, that's forte fire):

Humans are able to take knowledge of music and not just synthesize from it, but expand on it, creating completely unique music that sounds GOOD, even from seemingly random noises. Computers? Not yet.

So what are we preparing our students for? If the mundane and routine jobs will become automated, but computers still can't be THAT creative, then it's probably a good thing to focus on those 6 C's of 21st century learning: Critical thinking, Collaboration, Creativity, Communication, Citizenship and Character.

But that's not to say we should give up rigour. We humans are still pushing: deepening our knowledge of how our brain works, and translating this into robots and artificial intelligence. The "hard" skills of the scientific method, understanding mathematical processes, and logic sequences that come with activities such as coding need to still be at the forefront. 

With these as skills, our students may one day be able to write a program that allows a computer to compose music that actually makes sense to our ears.