Mathematics: It’s not about doing one thing or the other thing. It’s about the kids.



So, the past few days, as test results have been released, have brought with it the annual debate about mathematics instruction and it’s current state in Ontario. While I am not an expert by any means, I am someone on the ground, in the classroom, on the battlefield, if you will. And I’m someone who is exceptionally passionate about mathematics instruction and assessment.

I’m going to say what I think, straight out. So, here goes.

  1. We need to make our mathematics classrooms places of student voice, engagement and wonder.  This doesn’t mean that we don’t engage is rigorous mathematics that involves computation,  the development of fluency, and a strong foundation of mathematics principles.
  2. Mathematics instruction needs to be grounded in understanding WHY mathematics works the way it does. This means that students must be active participants in the learning though asking questions, debating procedures and discovering some of the beauty on their own. But it doesn’t mean that we don’t also learn and practice standard algorithms, formulae and procedural aspects of mathematics.
  3. Mathematics assessments need to be authentic — they need to give students a chance and opportunity to share their thinking and their understanding. This doesn’t mean that we don’t give tests, but perhaps it means that our questions give students opportunities to show their thinking in  a different way than we are anticipating in our answer key.
  4. Mathematics classrooms need to be safe places, just like all our other classroom spaces. For me, this means we work on our memorization and our mental math abilities, our fluency skills, our number skills in a way that is not hampered by competition, by time or by penalization for not knowing the answer right away.

In the end, I believe that mathematics instruction needs to be responsive to our students. We need to know who they are as learners and build our program around them. As one of my most respected mentors said this summer, we need to make sure we are doing enough mathematics to be good at mathematics. And so, for me, in my classroom, this means that we engage in problem solving on a regular basis, that we do number talks, and practice our mental math skills. That we debate answers, and that we tackle big concepts. But it also means that I tailor direct instruction (which is not the same as ‘surface’ instruction) to what my students know and where they need to go next. It means we practice standard algorithms and formulae when we are ready to. After all, those were developed over hundreds of years by skilled mathematicians — why would we just drop our students in there as a starting place? It means we do homework tasks that involve critical thinking and reflection and write tests and other tasks that are authentic and relevant to what we have learned and how we best learn it.

With any change, there is always a dip before there is a rise. Do I think our approach to mathematics instruction needs to change? No, not necessarily. Not if we are striving to teach in a way that is relevant and meaningful and engaging for our students. Not if we are encouraging our students to be active participants in their learning. Do we need to change our curriculum? Maybe. As others have said, the front part of the document contains a wealth of mathematical skills that we all use every day. The rest, however, reads like a checklist of things we have to check off, in that order. I don’t think teaching with those checklists in mind is a helpful practice. Rather, considering what the big concepts are, and then helping our students develop the skills and understanding to go deeper in those concepts is how our curriculum needs to shift.

I do a lot of professional reading — and many books have influenced my thinking and understanding: Jo Boaler’s Mathematical Minders; Hattie et al’s Visible Learning for Mathematics, just to name the two most prominent. And of course, I’d be lost in my journey without the amazing people I work with, both ‘in real life’ and on Twitter.

At the end of the day, it’s still a journey. We are all still learning and growing. And this means that we are all changing. The way I will teach mathematics this year will be different than the way I taught mathematics last year. That doesn’t mean that I think that how I taught last year was wrong; it just means that I’m still learning, that I’m striving to do things better. At the end of the day, it’s not about ME. It’s about those 25 (give or take a few) precious young people who will come into our classroom on Tuesday morning that I have the amazing honour to work with each day for the next ten months. It’s my responsibility to do what’s best for them. And I can’t wait.


Accountable Assessment


So, I have long been frustrated by the assessment practices in my classroom (read: all over the world). If I had my way, I’d eliminate grades all together in every level of education. True, I didn’t always feel this way. But as I have progressed in my understanding as an educator, I have slowly come to see that the grades, the numbers, the labels have a drastic impact on the learning of my students, and not a good one. I teach Grade 8 and love it. I love everything about Intermediate students — from their emerging personalities and sense of humour, to their day to day changing emotions. No two days are ever the same in this profession.

For the past few years, I have half-heartedly tried to be a ‘grades-free’ classroom. This mostly consisted of me recording marks in my mark book, and only giving feedback on work that I handed back to students. This was a baby step, and now, I’m ready to take more of a plunge into being truly grades free.  However, at the end of the day, I do have to put marks on report cards, and so, guided by the ideas that I have embraced from Pernille Ripp’s Passionate Learners (follow her at @pernilleripp), I am going to try something brave. I am going to ask my students to be equally, if not more, accountable for determining their assessments in my classes. Students will, ultimately, be responsible for choosing the grade that goes on their report card. Now, this sounds absolutely ridiculous, even to me. How on earth can they do that? How does that prepare them for high school? For real life? But, let’s face it, students do actually determine their own grades already by their attitudes, their efforts, the learning they demonstrate each day, and their conversations. However, usually I’m the person that is responsible for putting the number on the learning, not them. This year, I’m turning the system on it’s head.

This will require work. This will require learning and conversations, and debating and discussion and working collaboratively with my students to help them make effective decisions about showing and sharing their mastery of their learning. It will require open communication with parents and administration. But I believe that it will be worth it. Think of the learning that will happen!

What I’m struggling with right now is how to organize and document this assessment. Obviously, I can’t just use a spreadsheet or grid to just record marks. I need documentation. So, I’m working on a way to record this information. Every students’ mastery will look different and may come at a different time. This will require flexibility and creativity. I welcome any and all ideas, comments and thoughts on this.

Diving in: Math Blogs

img_0296I have always wanted to have my students write math blogs, but have never really been sure where to begin or what they should look like or what on earth we’d even write on them, and so I have always held back. But today, I ‘ripped off the proverbial band-aid’ and, without really knowing what on earth it was going to look like or even really how to set them up, we launched our Term 2 math blogs using Google Sites.

My vision for these sites is for my students to keep a record of a few things: 1. Their own reflections on their learning and who they are as mathematicians and 2. To create a portfolio of work that demonstrates their growth and understanding and mathematical journey. I have visions of photos posted with their thoughts and explanations on what they have learned about a concept and what they have learned about themselves. I don’t really know how it will unfold, but I do want them to write regularly and to use it as a tool for their own self-assessment.

Practically, I know that this will require time and energy from me to guide their writing and provide feedback. But I think this investment is worth it — it’s giving my kids a voice in their learning and is, in my opinion, a skill itself that is becoming somewhat of a lost art. The lost art of writing on a regular basis for reflection, and growth and learning, which is what I try to do here with my own humble little blog. Maybe no one else will read it. That’s okay. The practice of writing it helps me be a better learner, a better educator. I hope the practice of writing will help my students become more confident in their work as mathematicians.  Time will tell.

Why I stopped writing lesson plans

img_1409This sounds like  some sort of “confessions of an experienced teacher” that I’m hoping my principal doesn’t read, based on the title. But, actually, I would hope that my administration would support this idea. I know what you’re thinking….how on earth would it be okay with your principal that you don’t write lesson plans?

So, I am a 100% type A personality — I love organized binders with pages in sheet protectors that are all typed out. I love Sudoku because it’s satisfying to fill in boxes with numbers. I write in my daybook with coloured pens and take notes by hand using coloured markers, and, despite the appearance of my house, I love things to all fit together the way they should. Up until the beginning of this year, I always followed my high school teacher training, writing up unit plans, and then writing out detailed lesson plans with objectives and curriculum expectations and assessments and questions and….so much detail. I did this so I had a record of what we were going to do, and then I shared them with colleagues, and never worried about being away. It was satisfying. It was like laminating stuff–It made me happy to see all those pages in sheet protectors in my binder. 🙂

But here’s the thing — I would do all that work, and then I’d look at it before I’d teach, and then, instead, I’d just go with the flow of my classroom, talking and discussing and learning together organically. Don’t get me wrong, we were still covering my learning objectives and working on the task or idea that I had written down, but it didn’t always go with the flow that I’d carved in stone in my lesson plans. And then I’d have to go and adjust them and re-write the next ones and print them out again and so on…. Even though I was planning all that with my students in mind, it still felt really…restricting.

So, I made a change. Instead of writing out lesson plans, I started creating slide decks that were my daily plans — math warm up activities, problems, records of student thinking, collaborative slides written by my students, and we go from there. The kids all have access to the slides, so they can go back and refer to something, or look at a problem again, or see each other’s work. If I don’t like the order of something, or I find another activity that I think will work better, I just slot it in. I shift things around. It’s organic. It’s always changing, its active. It’s real life in my classroom. And then I share the slides with my grade partners, whom I continue to mentor, and they work from them too. So now my planning involves creating a unit scope and sequence — what concepts do we want to cover? What big tasks do I want them to do, what problems do I think will work here? What assessment pieces am I looking for?  And we go from there. It changes all the time based on my students and what they need.  I keep notes of what worked and what didn’t work in my Bullet Journal (If you haven’t used a Bullet Journal, and you’re a notebook person, check them out here: and take pictures on my iPad of student work that I then reflect on. (And the rubber duck keeps me company…. 🙂

I don’t have a binder full of page protectors this year. I don’t have neat organized lesson plans. I have a messy, always changing, authentic record of our learning together. It’s unsettling sometimes, but it’s also exciting. It feels like what learning is supposed to be.


I am not a conveyor of knowledge.

I love my job. I am proud to tell people that I’m a teacher, and that I’m a math and science teacher. I love to talk mathematics, I love to talk science, and above all, I love to talk about my classroom. In recent days, though, I’ve really begun to shift the way I think about what it is that I do each day. At one point in my career, I would have been nervous to think that I didn’t have all the answers for the questions that were going to come my way. Now, I embrace the questions. I have some answers, sometimes, and sometimes I don’t. And that’s okay — see, those moments when I’m posed with a question that I haven’t anticipated, or that I don’t know the answer to, is whaen my learning happens. And it’s when my students’ learning happens too.

Recently, I told my students that I didn’t think that, as their math teacher, or as their science teacher, that it wasn’t actually my job to teach them math or science, but rather to provide the conditions in our classroom to help them be the best learners they could be, within the context of the math and science curriculum. The reality of it all is, if my students have a question about something, they aren’t going to think, ‘Oh well. I have a question. I guess I’ll have to wait to talk to my teacher. I don’t know what to do!”.  No, instead, they are going to ask Google, or Siri, or..gasp…wikipedia. They are going to look up an answer, and go with it. The challenge is, they need the skills to know whether or not the answer that the Internets have provided is an accurate one. It has been quite astounding to me the number of times I have had kids submit work or overhear them talking and realize that they’ve googled an answer and actually have no idea what it means.

So, I’m moving away from being “the conveyor of knowledge” in my classroom, to being a “co-learner”.  This means that my time is spent participating in the building of knowledge with my students. I work each day to provide the conditions for learning — it’s my goal to teach my students how to be good learners, within the context of the curriculum areas that I am responsible for. It means that sometimes, as part of the lesson, I tell kids to google something or to ask Siri, and then we write down what those sources have to say, and analyze it. We ask questions about it. Most of the time we say, “What does that actually mean!?”. And then the kids realize they are creating and sharing the knowledge building. I am not simply transmitting ideas to them — it’s creative, it’s messy, it’s shared. Sometimes we argue. Sometimes we don’t arrive an “answer”, or “solution”, but we always learn something, about the content, and about ourselves. It’s exhilarating!

“I need help! Someone come!”

On Friday in my math class, we worked on creating sonobe units to create sonobe cubes that we are using to explore ratios, geometry and proportionality. Earlier in the week, we did some work around this topic with some boxes that I had already created, and, after that, some of the students expressed interest in creating their own. So, Friday found us creating origami boxes. (As a side note, as I taught my class, I simultaneously taught a colleagues class down the hall via Google Hangout which, I must confess, was pretty cool.) I have tasked my students with creating two boxes of differing sizes so that they can continue to explore the relationship between volume, surface area, similarity and congruence as well as the ever important and always present math process expectations.

Having the students make their own units and then boxes was a really valuable experience. Some students found that it came easy to them; while others found it more challenging. Watching the natural collaboration occur in my room was something that I hadn’t anticipated. Of their own accord, students who felt confident got up and moved around the room helping their peers to create their units or create their boxes. Some students who worked with me last year and who had experience with the boxes, became co-teachers with me, and then moved back and forth between classrooms, assisting in my colleagues class as I continued to teach both classes via the video link.

At one point, a student called out, “I need help! Somebody come!”. And someone DID come. And it wasn’t me. It was a peer; a friend. They automatically became teachers. THIS is my goal as an educator: that my students take ownership over their own learning, and then become part of the cycle of learning for others. It was a beauty to behold and made this math teacher proud.


Exploring Proportionality – Day 1 (aka why we shouldn’t teach math in isolated units and strands).

This week, we began exploring proportionality as a our focus concept for the next several weeks. Much like most of what we have talked about this year, my students seem to have a procedural understanding of ratios — as in, they can define the word ratio and solve problems in the section of the textbook called, “Ratios.” However in the task that we looked at in class, I didn’t use the word ‘ratio’ or ‘proportionality’ once. All I said was the following:

  1. I made a sonobe cube out of 6 x 6 origami paper.
  2. Then, I made a sonobe cube out of a larger size paper.

Your task: Can you determine the dimensions of the paper I used to build the second cube using only what you know about the original size paper and whatever other information you’d like to gather about the two boxes.

I was surprised by the variety of different strategies that came up as they worked through this problem. As a whole, they approached the problem by gathering a variety of different dimensions from each of the boxes — the side length, the volume of each box, the size of the triangles, the area of the different faces of the two boxes.  They then worked to try to related the dimensions to each other to try to figure out what how the two boxes were related to each other.

Even though almost every single student can tell me what a ratio, and almost every student can complete isolated ratio questions, they didn’t make the connection between their prior work with the procedures related to ratios. One of the things I wonder the most about is whether or not they would have immediately tried to use a ratio if I had given the problem immediately following a lesson ratios.

That’s one of the things I that have noticed about student learning. Students demonstrate understanding of a topic during a unit of study — the teacher teaches a lesson within that unit, students answers questions and do tasks related to the topic, because they know that’s the topic that they are learning about — the might be applying the concept to a ‘new’ question, but they know what topic they are supposed to be applying because they know they are in the ‘unit’ of ratios, or adding fractions, or algebra and so they draw on that knowledge. And then we move onto the next unit. And when we do that, it’s like all the things they have learned get filed away in the back of their minds, as they expect that since they are done with the fractions unit, they don’t need fractions any more until next year.  This is the biggest argument I have come across for considering teaching math in concepts and not linear, isolated units. Students need to spiral through and around math concepts over and over in contexts that are meaningful and thought-provoking. Students need to explore the connections between ideas and concepts in ways that they are able to draw upon their prior foundational knowledge.

Of course, assessment of this type of learning is a disaster at times — At the end of the day, I’m told I have report on my student learning in  isolated strands, and it’s hard sometimes to separate out all of this rich learning into separate strands. Until we change our reporting practices, I will untangle the web of ideas and learning that comes from this approach to mathematics. It’s something I’m willing to struggle through, though, for the sake of my student’s learning and understanding.