Different kinds of Thinking: Ontario Math Achievement Chart

I’m evaluating some student work today and I’m struggling with the Achievement Chart for Mathematics (see page 28). In particular, this part of the Thinking category is bothering me:

An excerpt from the math achievement chart for Ontario

Take a look at the first point in “Use of planning skills”, called “understanding the problem”, which includes “formulating and interpreting the problem” as an example of that skill.

Now look at “Use of processing skills” point “carrying out a plan”, which includes “modelling” as an example of that skill.

Are these different? In my mind (up until now, at least), “formulating and interpreting the problem” has meant representing a situation mathematically so that we can apply our other math skills to solving it. Isn’t “modelling” in the context of “carrying out the plan” sort of the same thing? Representing components of the problem mathematically? Is the difference just when it happens (i.e. formulating/interpreting is initial planning, and modelling is during the act of solving)?

I’m not trying to be pedantic here; I’m having trouble distinguishing between the different components of Thinking when I’m trying to assess and evaluate my students’ work. I could use some external thinking on this issue (and math evaluation in general, I suppose).

Please comment; I’d love to talk to you if you have ideas about this stuff.

#CS teachers: I need help developing an inquiry task

I am trying to develop a task for my online ICS3C/3U Computer Programming/Science class, and I need the students to be deeply “engaged”.

I return again to my latest, favouritest definition for student engagement: “A student is engaged in learning when they feel a compulsion to pursue the learning apart from external motivations.” That is, not just for marks, and not just because Mr. G said they have to. I want them to want to learn.

So I want the students to pursue their own interests as much as possible. I believe giving students the freedom to choose will be a lot more likely to kindle that compulsion I want to see in them, and will result in deeper learning and, rather importantly, more fun for all of us.

My challenge is figuring out how to frame that task so that I give them

  • enough latitude to pursue whatever interests them (knowing that I can’t predict these things!)
  • enough guidance to help them formulate their plans (so they can learn from the experience)
  • enough restrictions to ensure that they are learning for this course (since I have to assess and evaluate them).

So I’m a little stuck. I’ve created an openly-editable Google Doc with their learning goals and my initial thoughts. I would be very, very pleased if you would take a look and add comments, ideas, criticisms… anything that can help me move forward with this. I’m trying to plan very carefully to minimize the chance that this wastes my students’ time and effort, and to minimize any unnecessary frustrations they experience.

Here’s the document; please share widely: Learning Computer Science Through Inquiry

Much thanks in advance, from me and from my students!

If-Then: are we thinking about it backwards?

I was working with a small group of big thinkers today. We were trying to formulate an “If-Then” statement: “If we [take some action] then [we will see this desirable result].”

Tarmo Poldmaa (@TarmoPoldmaa) was one of those thinkers. At one point, as we struggled with this [surprisingly difficult] task, he mused aloud, saying something like, “Are we looking at this backwards? Instead of saying, “If we take this action, we’ll see this result,” should we be saying, “We want this result; what actions must we take to get there?” (Note: I’m heavily paraphrasing to remove the context.)

At first I thought, No, that’s the same thing written the opposite way.

Then I realized it’s not the same.

I had just finished a conversation with another teacher about how we can’t change one facet of our practice and honestly expect a dramatic improvement in student engagement. There are too many separate problems which are contributing to the common symptom of disengagement; removing any one of those problems can’t be enough (although it can help).

So Tarmo’s thinking is really important here. Instead of fooling ourselves into believing that a single change can correct this issue, we should perhaps try to grasp the larger picture. We look instead to the end goal and determine what conditions need to be in place and what actions we need to take to achieve that goal.

To be clear: the If-Then structure is still valuable for a single action or group of similar actions, but it won’t be enough to describe solutions to a multi-faceted, complex problem. Unless your “If” has a lot of “ands”.

Why Aren’t Students Engaged?

An open math textbook with sheets of written notes on top.

Today I worked with all of the other e-Learning teachers in my board. We had some excellent discussions which tended to drift back to the idea of student engagement.

I don’t really like the term “engagement”; I don’t think it’s clear enough. People have very different understandings of the word in the context of student learning, and that can get in the way of real improvement.

For example, asking “what does student engagement look like?” yields responses like “they’re paying attention” or “they complete the assigned work” alongside “they are applying their learning to their own lives” and “they connect to outside sources for learning”.

Some of these are more accurate or useful than others, but they are all indications of student engagement, I guess.

But for me, student engagement is a question of student intent in their learning, which isn’t exactly an observable behaviour.

I like this definition: “A student is engaged in learning when they feel a compulsion to pursue the learning apart from external motivations.”

That last bit is key for me: why is the student learning? Because they need the marks? Because they fear consequences? Because they are enthralled by a question? Because they are curious?

So this leads me to the question: why are my students disengaged? The question holds as much value regarding e-Learning students as face-to-face students.

My colleagues and I identified approximately a bazillion potential causes of and contributing factors to disengagement. These are approaches or circumstances which prevent students from developing the compulsion to learn. Some of them we can work to improve; others we can’t.

But in the end I keep coming back to this idea: Good questions produce a compulsion to find answers, and the best questions are students’ own questions.

Look at me: I just spent an entire day grappling with questions alongside my peers, and here I am trying to make more sense of it at night. I have a dozen other tasks to complete before I can sleep, but I need to get this out right now so that I can pursue the answers. I am compelled.

So how do we foster a spirit of questioning and a culture of inquiry in the context of prescribed learning?

That’s my own challenge to work on tomorrow: how will I ensure that I’m giving students both the guidance and the latitude they need to pose great questions and then pursue the answers?

Accessing student interests in ICS3C/3U

I’m thinking about trying to tap into the students’ interests in my online computer science course. We’re at the point now where the [up-to-date] students have enough of the basics to be able to pursue topics of their choosing.

I was considering making a discussion forum/topic where they can share ideas they’re interested in, and then encouraging them to pursue that learning. I’m not sure how to word it, or manage it, or what exactly I hope to get from it, except that it’s important for students to take charge of their own learning. Maybe several students will have similar interests and be able to pursue them together.

Suggestions are welcome ;)

My students like the class blog (phew!)

I reminded my MDM4U students to visit our class blog today as I had posted a video this morning with solutions to some questions from class. I was pleased to hear that a few students have been heading there regularly. I played a bit of the video on the screen so they would understand what it was like, and then I asked for input about the sorts of resources they would like me to post.

First, most students want to have the videos. They’re nothing magical; they’re just me solving some problems on paper. A couple of students said they like the idea of being able to rewind me (since that’s hard to do live, in class).

Second, a few students also want a text format of some kind (a blog post, a PDF file, or something like that). I understand that; it’s faster if you just need a short bit of explanation. It takes a lot longer to type out a solution, but I could always write one and take a photo, I suppose.

The third response I took away was that posting videos like this is unusual in their experience, and they appreciate the extra effort on my part. Several students seemed to understand how much extra work goes into creating these supplementary resources, so that was nice.

At some point I’m hoping to post more student work for them to learn from instead of creating everything myself. I think they’ll learn a lot from doing it, and then a lot from each other. Videos would be awesome…

I care a lot about their success, and I’m hoping to give my students a better-than-average chance at being great at this stuff. A lot of the responsibility lies with them, but I’ll remove as many barriers to their learning as I can. I’m glad to hear that my work on the blog is being well-received.

Teachers DO care, Seth Godin

Seth Godin wrote a blog post yesterday which has been making the rounds in education circles. It’s called “Good at math”, and in it Seth makes a very good point in the first paragraph:

It’s tempting to fall into the trap of believing that being good at math is a genetic predisposition, as it lets us off the hook. The truth is, with few rare exceptions, all of us are capable of being good at math.

He’s right, without question. Most people are capable of becoming “good at math” the way most people are capable of becoming “good at reading” or almost anything else.

The problem with math education, he argues, is that there has been a focus on memorizing formulae in math classes instead of on the important stuff (I’m assuming he means modelling, problem solving, critical thinking, etc.). In particular, he says standardized tests are useless.

Before you continue, please understand that I agree with Seth. I agree that math education has been unbalanced in favour of rote learning and memorization, sometimes to the exclusion of problem solving and critical thinking, and is divorced from real-world, relevant application.

But one sentence in his post is really galling to me. One point he makes, almost casually, that has prompted this post.

He says,

It’s because you haven’t had a math teacher who cared enough to teach you math.

Again, I agree that the focus has been wrong, and that math education has been largely ineffective. But that’s not for lack of caring on the part of teachers.

Teachers are among the most caring people on the planet. We work like dogs, struggling beside our students, coaching, cajoling, encouraging, pleading. We can’t sleep for thinking about how to improve for the next day. Teachers agonize over their students misunderstandings, apathy, home lives, social problems, mental state, self-esteem, literacy, and all other aspects of their beings. We are charged with developing tomorrow’s citizens and we take it very seriously.

But it’s true that not all teachers are exemplary. Not every class is equally “engaging”. Teachers don’t all have the same arsenal of strategies at their disposal. We disagree about assessment, behaviour management, instructional paradigms, big ideas, learning goals,… pretty much everything, actually. There is little consensus among teachers.

Except for one thing: we care. We want the best for our students. We want to be better at our work.

So why has ineffective math education persisted for so long?

  • there is a cycle of successful math students becoming math teachers (this is true in every subject area, not just math)
  • it’s hard to employ new strategies
  • it’s hard to find mentors who have effective practices and can coach
  • there are approximately a bazillion other demands which take away from instructional practice, including other system- and school-level priorities

That’s a lot of reasons/excuses, but there are solutions.

Teachers must become aware that there are ways to improve their instructional practice. Some folks believe that the hard work is figuring out how to ensure compliance from students. “If only they would [insert good learning behaviour] then they would be successful” is a common statement. Developing this awareness requires exposure to better methods, whether through direct observation or other sharing.

Teachers need time and “permission” to try strategies which are new to them. Fitting everything into a math course is really challenging. Now ask a teacher to “take chances” with several lessons. It’s hard to convince people to try something new when you can’t promise it’ll work. In my experience, the best way to make the case is to remind them that what they’re doing now is only working for their strongest students anyway. That time spent “experimenting” can be gained back in the form of increased student understanding and skill.

School leaders need to encourage and support teacher learning in dramatic and meaningful ways. Make co-planning a priority. Visit classrooms. Learn about effective math instruction so that you can have rich conversations with your math teachers. Finding the time to develop as a school leader is possibly the most difficult task I’ve mentioned.

Seth’s blog post seems to have been aimed at the general public, reminding us that numeracy is as crucial as literacy, and that we have been damaged by narrow, ineffective teaching practices. That is a great message, and one that I think most people are hearing in his words. His claim that it’s because teachers don’t care is ridiculous.