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.

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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: http://bulletjournal.com/get-started/) 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.

 

 

Ignoring the decimal points (#mathismore than rules we just follow because we’re told so.)

followtherues

Okay, so right now, we are working on some skills with decimals.  The kids are inherently more comfortable with decimals than they are with fractions, so they are a little freer in their ability to consider how they might think outside of the ‘rules’.

Last week, we spent time building our knowledge of how to multiply decimals using Base 10 blocks.  Up to this point in their mathematical careers, their work with decimals has been focused on following the ‘rules’ they were taught in prior grades. Thing such as, multiply up, across, and then add, and then just put in the decimal points. “If there are two decimals in the question, then there are two decimals in the answer.”  They are a bit baffled when trying to explain why 3.0 x 0.4 is smaller than the first number, but they do know HOW to do the question — they just follow the rules.

Today, I posed this problem:

2.6 divided by 1.2

And I told them that, just for fun, I looked up what the textbook says to do. “To divide decimals, ignore the decimal points.”  I confess, I was, what? Just ignore the decimals? What does that mean? So, I posed it back to them. What does this mean? Why does this work?

Why is 2.6 divided by 1.2 equivalent to 26 divided by 12? What is really happening?

Eventually, we got around to having an interesting discussion about where the decimal is really ‘going’ when we do this. We aren’t really ‘ignoring the decimal’, what we’re doing is multiplying each portion of our statement by 10. We’re creating equivalent statements — we’re not really ignoring the decimals. We’re not really ‘magically making them disappear’.  We worked our way around to understanding that we’re multiplying both sides by 10 — creating equivalent statements.  The decimal doesn’t disappear – we are multiplying both sides by the same amounts, creating a proportional relationship.

The most important part of the discussion, though, was that the kids literally thought you could just ‘ignore’ decimal points — that you could just make them go away or ignore them. They had never considered that there was a mathematical reason behind what they were doing — they had just followed the rules.

This was and is one of the scariest aspects of being a math teacher — that students are willing to simply follow the rules without questioning WHY. And so I wonder, how do I build this culture in my classroom? How do we make our classrooms places where  1. teaching is more than conveying information and rules and 2. where students question those rules when they are presented? This is the challenge we face.

Embracing the struggle (aka #mathismore than a word problem)

I’ve been silent on the writing front recently. It’s been a struggle to think about formulating my thoughts, or to entertain the notion that what I might have to say is important enough to write about. January has been a struggle for me, in and out of the classroom.  But what it comes down to is this — I firmly believe in my approach to mathematics education, in my philosophy as an educator around assessment, around what learning is, around what my role as an educator is. In recent days, however, there have been a lot of little voices popping up though — maybe I’m wrong, maybe I’m doing it wrong, maybe I’m leading my colleagues astray, maybe I’m learning my students astray….so. many. maybes.

But what I have come to see is that, just like for my students, the struggle makes me stronger. It causes me to pause and reflect and to learn and ask questions and debate and discuss. Sometimes, that struggle means that I need to be willing to think differently, and sometimes that struggle means that I am affirmed in my thinking and given the courage to press on, despite opposition. I need to hold the line for my students.

We’ve been working a great deal on making our thinking visible. Recently, we’ve been working on combining our understanding of fractions, decimals, percentages and area. In order to do this, we’ve been working on Steve Wyborney’s Tiled Area Problems. (check out this awesome resource here: http://www.stevewyborney.com/?p=836 here.) Students are required to use their understanding of area, fractions, adding fractions, and proportionality and relationships in order to determine the area of the shape and then share their thinking. Students need to add fractions in a meaningful context, and then make conversions to percent and decimals to share their thinking in different ways.  img_0216

After we work on problems like this, I then post them on the wall outside my classroom. Every one’s work gets posted — not only the ‘best ones’.  I do this to celebrate and share everyone’s thinking, but also to show and share this type of activity with my colleagues. To me, this is working with fractions in a meaningful context. img_0219It’s not a word problem. It’s not a story. It’s not a task that says, “Add the fractions.” But it is a task where students designed their own problem, and then decided how they wanted to tackle it. They were involved from the very beginning — they did the math, they shared their thinking in whatever way made sense to them. Math is more than word problems in a textbook. It’s more than isolated practice of skills. It’s about making the best decision for the situation you’re in. It’s about embracing the struggle.