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.

 

 

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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.

More than Algorithms

math-algorithms

If you had asked me, when I was a math student, what math was about, I would have told you it was about memorizing different procedures and formula and then figuring out how to use them in different problems. I survived math class by memorizing, or trying to, and it was terribly difficult and distressing. I was good at memorizing the steps, but couldn’t figure out how to use the steps if the question I was asked to do wasn’t exactly the same as the problem I’d been shown initially. I had no concept of why we were doing what we were doing…I just tried to stay afloat and keep up with my peers. As a result, math class was challenging, and I began to tell people, “It’s okay. I’m not a math person. I’m a science person.” And people would be like, “Well, how does that even work? How can you be a science person and not a math person?”. The truth of the matter is that, I did pretty well as a math student, in the end, with my collection of Bs and As up through my Grade 13 year, and I passed my first year university math class, even though most of the time I felt like I really had no idea what was happening.

When I became a teacher, I had no intention to teach math. I was going to be a high school science teacher and that was that. However, as the course of my career would have it, I found myself with a Grade 8 Homeroom position, and that required me to teach math. So I set about teaching math the only way I knew how– how I was taught math. And one day, as I stood in front of my students and looked at them, I knew that most of them were feeling what I had felt as a math student — the struggle to memorize procedures was real. And it had to change.

I’ve come a long way since that fateful moment those 10 years ago, and now I find myself in the surprising position of being a mathematics educator-leader. I am passionate about mathematics education, and assessment and work each day to share that passion with my students and my colleagues. I am a respected member of the leadership team and am called upon regularly to provide leadership, coaching and mentoring for teachers in my building and elsewhere. How did this happen? What changed?

What I came to realize is that mathematics is more than a collection of algorithms to memorize and use; it’s more than formulae and word problems and trains leaving stations and the cost of 60 watermelons. It’s more than strands and units and tests and quizzes. It’s something more more than all of those things. It’s beauty and art and magic and patterns and conjectures and reasoning and justifying and collaboration and communication and ideas and concepts that weave together in surprising and wonderful ways. As I realized this, I began to change the way I viewed teaching math, the way I talked about mathematics to my students, their parents, and my colleagues. My goal now each day is to engage my students in the wonder and beauty of mathematics through debate, discussion, the building of fluency and flexiblity, and tackling problems head on, even if we don’t have any idea how to begin. We argue, we debate and we question. It’s glorious. It’s exhilarating. It’s exhausting. It’s the best part of my day.

It is my intention to use this blog to document this messy, exhausting, exhilarating journey in the hopes that someone else might embark on the journey.

Meet Jerry

win_20170102_14_49_57_proHere I am. I have been a blogger in days and years past, and it always seems to start off gangbusters, and then fade away. However, I have been convicted recently to start up my writing again, even if just for my own personal record of my learning. I teach mathematics and love every minute of it. I also teach science, which I also love, but my road to being a mathematics educator and leader has been one of personal and professional self-discovery, and so my role as a maths leader is what I really love to do each day.

Pictured above is Jerry. Jerry belongs to a student in my class who has had a profound impact on me as an educator in general, and as a mathematics educator specifically. This is the first year in my role as math leader that I have really begun to dive into the ‘meat and potatoes’ of authentic mathematics education. I spent the last several years reading and learning and sharing, and this is the first year I have had the opportunity to put my learning and passion into practice in a classroom. I reorganized my long range plans by concept, and have built my program about depth of understanding of mathematics, rather then procedural memorization of mathematics. We spend a great deal of time talking about math, and sometimes fighting about math. We tackle big problems for several days at a time. Our goal is always to get to the WHY of the mathematics.

It’s been a challenging shift for most of my students. They are in a place where they are accustomed to linear math units, focusing on procedures and steps and memorization. They are used to traditional assessment and homework. I started off by throwing those things out the window. The kids had a lot of questions; and so did their parents. And it was a battle. It still is some days. There are days where I wonder if what I’m doing is the right thing, whether or not I’m doing my students a disservice in someway. I wonder if I am leading my colleagues in the wrong direction.

So, back to Jerry. As I mentioned above, Jerry belongs to one of my students. The reason why Jerry is my starting point is because of what Jerry represents to me. His owner is a student that reminds me each day that I’m on the right track. He sincerely wants to know and understand the WHY of what we do each day, he pushes me to think about mathematics and my work as a math teacher in new ways. When I wonder if “I’m getting it right”, he reminds me that the hard work is worth it. He reminds me that we are all learners together. I am exceptionally grateful for this each day. And so I could think of no better place to start than with giving credit where credit is due.

This is why I do what I do.