What do you choose to learn about when you’re not at school?

I had an interesting talk with the student today about a variety of topics related to schooling and education. I asked her one question which has been staying with me throughout the evening so far. 

“What do you learn about when you’re not at school? What do you learn about because you want to, not because you have to? What are you curious about?”

I think each person’s answers can give some insight into what their passions are. Curiosity is an incredibly valuable commodity, and nurturing it is some of the most important work we do. Let’s help foster the inquiring mindset while being careful not to steal the passion by imposing our structures. 

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Student research I want to read

My Grade 12 Data Management students complete a research project as part of the course. They create a questionnaire to help answer a question they’re interested in, or to look for relationships that bear further study.

While working with one student today to help develop that question, we talked about how she started to take guitar lessons but didn’t stick with it. She said that she regretted stopping the lessons, but that a part-time job and her own laziness got in the way. She expected that she would have been quite skilled by now had she put in the time over the past year.

I asked if she thought that experience was common, and how the music school offering the lessons might have made it easier for her to stay with it. She figured that a lot of people start lessons but don’t continue, and she had some suggestions for improvement that seemed reasonable as well.

Wouldn’t it be nice to know what the most common reasons are for quitting music lessons (and for sticking with lessons), and if there is a correlation between some other variable and perseverance? For example, do students without jobs stick with music lessons longer? What other factors play into persistence?

She might pursue this area of research for her project. I hope she does. I want to know the conclusions she draws from it, because it might have an impact on the way I teach math and computer science.

Learn-practise-perform cycle limits learning in CS

Like many courses, the beginning of my current computer science e-Learning class looked like this:

  • Teach small skill
  • Teach small skill
  • Give feedback on practice work
  • Teach small skill
  • Teach small skill
  • Give feedback on practice work
  • Evaluate performance task

This separation of learning from graded performance is intended to give students time to practise before we assign a numerical grade. This sounds like a good move on the surface. It’s certainly well-intentioned.

But this process is broken. It limits learning significantly.

If the performance task is complex enough to be meaningful, it requires a synthesis of skills and understandings that the students haven’t had time to practise. In this case I’m evaluating each student’s ability to accomplish something truly useful when they’ve only had the opportunity to practise small skills.

If instead the performance task has many small components which aren’t interdependent, students never develop the deeper understanding or the relationships between concepts. In this case I’m evaluating each student’s small skills without evaluating their ability to accomplish something truly useful, which isn’t acceptable either.

And there isn’t time to do both. I can’t offer them the time to complete a large, meaningful practise task and then evaluate another large, meaningful performance task.

The barrier here is the evaluation of performance. It requires a high level of independence on the part of the student so that I can accurately assign a numerical grade.

So I’m trying something different.

Instead of these tiny, “real-world” examples (that I make up) to develop tiny, discrete skills, I started teaching through large, student-driven projects. I got rid of the little lessons building up to the performance task, and I stopped worrying about whether they had practised everything in advance.

The process looks more like this:

  • Develop project ideas with students and provide focus
  • Support students as they design
  • Provide feedback through periodic check-ins
  • Teach mini-lessons as needed for incidental learning (design, skills, etc.)
  • Summarize learning with students to consolidate

I couldn’t design a sequence of learning tasks that would be as effective as my students’ current projects are. They’re working hard to accomplish goals they chose, and they’re solving hundreds of small and large problems along the way.

They couldn’t appreciate the small, discrete lessons I was teaching with the small, artificial stories. They didn’t have the context to fit the ideas into. It was only when the project was large and meaningful that my students truly began to grasp the big concepts which the small skills support.

And now I don’t have a practise/perform cycle. It’s all practice, and it’s all performance. It’s more like real life, less like school, and it’s dramatically more effective. It’s much richer, much faster learning than the old “complete activity 2.4” approach.

Evaluation is very difficult, though.

My students told me what’s going on in my class

I talked to my data management kids today about the not–so-great class we had yesterday. We pushed all the desks aside and put our chairs into a (sort of) circle for this conversation. I explained how frustrated I was with the lack of feedback I was getting during class, and that I was concerned that my goals did not align with their goals for the course.

I asked them why they were taking the course, and what they were hoping to get out of it. My speculation last night was partly on target: their primary goals are to get a high school diploma, with a good mark in this course, so that they could get into “the next thing” (university programs for most of them). Some mentioned that they thought statistics would be helpful for their planned program. Overwhelmingly the course is seen as a means to an end. It’s not 110+ hours of learning; it’s more like a long tunnel they must pass through to get on with life.

This is what I was afraid of, and yet sitting there with my students I can’t blame them. Our school system (through post-secondary as well) trains them to focus on achievement, which is measured by task completion and marks. Our system doesn’t (can’t?) train them to value learning over these other goals, because the system itself doesn’t value learning over task completion and marks.

We had an honest conversation about what really matters in a math class. We talked about how they all learn exactly the same things even though they don’t all have exactly the same plans for the future. We talked about how we have a “just-in-case” curriculum: you must learn these skills just in case you need them someday.

And the most frustrating part for me was that they all know that a lot of what we do in class doesn’t really matter in the sense that it doesn’t really change them. They haven’t been improved by learning how to use the hypergeometric probability distribution. They will forget it when the exam is over because it doesn’t matter much to them. It’s not something that they’ll use, likely. And if they need it, it’ll be because they’re steeped in all the math that goes along with it.

But not everything we do is like that in my class. Some things do matter. And I’m feeling a bit guilty tonight because I think I should have focused the course a bit differently, spending more time on the parts that will change my students. We’re only a few weeks from the end of the course and we don’t have the luxury of a slow, thoughtful pace that the remaining topics deserve. I can’t fix that now, but I can work on it for next year.

I grabbed the Chromebook cart and sent my kids to a Google Form with three paragraph-response prompts:

  • Start
  • Stop
  • Continue

They each wrote anonymously about what they think we should start to do in our class (perhaps an approach they like from another class), stop doing (approaches I’m taking that aren’t working for them), and continue doing (class components they don’t want to lose if I change things). Their responses were fascinating, and I’m going to read them over a few more times to make sure I get it all. It was pretty clear they don’t want any more audio clips, though :)
Our conversation also revealed that I misinterpreted their silence as a lack of interest or understanding. What I learned from them today was that there were portions of yesterday’s class that they did enjoy, but I couldn’t see it. They didn’t provide feedback I was expecting and I didn’t adjust my teaching to suit their needs. It was a difficult conversation for me (and probably them), and it took some time, but it was worth it. I understand my students better now, and I think I can be a better teacher.

It’s not all fixed, but I don’t feel quite like I did yesterday. I’m going to go to class tomorrow with a plan to improve my teaching and their learning at the same time.

I wouldn’t be disappointed if I weren’t working so hard

Facepalm

I taught a class today that didn’t go well. Actually, it went pretty badly.

I tried to engage in a discussion with my students that required critical thinking about statistics in the media, and they mostly didn’t engage with me. It took me a long time to plan out the lesson, carefully choose my resources, and prepare everything to guide them to a deeper understanding an appreciation.

And it mostly crashed and burned. You can read the play-by-play on the class blog, if you want (link).

Each day I write a post explaining what happened during class for anyone who missed it, and for the reference of those who were there. Today I shared my frustration with their stance in the room. From that post:

You’re not here to “do school”. You’re here to develop skills and learn to think critically. Calculating medians is not a way to develop your brain. Completing tasks is not the point.

I need you to be able to analyze, interpret, draw conclusions, and make decisions based on data. Any spreadsheet can calculate medians, but Excel can’t tell you whether three minutes of exercise is enough each week or whether e-cigarettes are a good thing.

I’m fully aware that our school system tends to prioritize finishing activities over real learning. Math can be particularly vicious because of the number of discrete, technical skills required to even begin to “see the big picture” of how everything relates and works together.

But I’m trying hard to break outside of that mode. Really hard. I’m trying to make real learning the priority. And I’m not above admitting that I made a mistake here. This lesson wasn’t designed well, or I didn’t prepare my students well for this approach today, or maybe both. But here’s what I need next:

I want what’s best for each of you, and that means actual learning, not just task completion. If there’s something you need in this class to make that happen and I’m not providing it, I need you to tell me. Today didn’t work, and I don’t want a repeat performance tomorrow. None of us does, I hope. Help me out.

I really mean that. I was so much more disappointed today because of how much planning and time went into this failure. And worse: I don’t know what I’ve learned from the experience. I’m now counting on my students to tell me what they really need to meet the goals I’ve set out for them.

Off Topic On Purpose

This morning Gregory Taylor (@mathtans) tweeted to me:

We back-and-forthed once, which I hope displays properly here:

And he was right. So here I am, even though I’m tired, because I like to write. It might even relax me a little.

In a rut

I notice that my last four posts have all been about work. That can be all right, I suppose, but I try to keep this blog more varied than that. Unfortunately, when I decided to write a post tonight only work-related ideas popped into my head. Tragic. I’m living and breathing this job, and it’s consuming my entire brain.

I don’t want to be in this brainrut all the time; I don’t think it’s good for me, my brain, or my family.

Rut-jumping

My plan is to intentionally write some posts in the next while that are not related to my job in any way that I can perceive.

Therefore I’m signing off entirely for the night, even though I’m sure there are emails waiting for responses, because I’ve already worked 60 hours this week and it’s only Thursday. I’m going to escape into something I enjoy and turn off my teachermind until tomorrow morning.

Good night.

Which kind of calculator promotes good algebraic thinking?

I teach high school math. Students bring scientific calculators to class, or they sometimes have to borrow one from me. I have two types available: immediate execution calculators and formula calculators. I’ve been wondering lately whether one type of calculator is better for learning algebra than the other.

Here’s how they work (see Wikipedia for a longer explanation: https://en.wikipedia.org/wiki/Calculator_input_methods).

Immediate Execution

The TI-36X Solar 2004 version immediate execution calculator.

These calculators work by performing calculations along the way as you type in values and operations. For example, you can evaluate the expression

\sin(3 \times 45)

by typing 3, multiply, 45, then the sine key. As you press operations and operands the calculator will evaluate what it can according to the rules of order of operations, or BEDMAS. For binary operands (those taking two values to produce a result, like multiplication), you put the values in order. For unary operations (those taking just one value, like squaring or taking a sine), the value must be present on the calculator screen when you press the operator key. These calculators usually have a bracketing feature to allow the user to work through complex expressions without using memory storage.

Formula

The Sharp EL-510R formula calculator.

These calculators work by waiting until the user has typed in a complete expression to evaluate, then evaluating the entire expression. The order of button-pushing is pretty much as the symbols are written in the expression, making them easier to use for a lot of folks. Once a value is calculated, it’s stored in an “answer” variable in case it’s needed for the next evaluation.

Algebraic Expressions and BEDMAS

When we write out algebraic expressions, we have a number of conventions to follow. The most important convention is order of operations, which people usually learn to remember with the mnemonic BEDMAS or PEDMAS:

  • Brackets (Parentheses)
  • Exponents
  • Division and Multiplication
  • Addition and Subtraction

When evaluating (simplifying) an expression, you first simplify the smaller expressions inside brackets. Then you evaluate exponents, then division and multiplication in the order they appear, and finally addition and subtraction in the order they appear. It’s useful to think of brackets as isolating sub-expressions, which then follow the same rules. It’s also useful to think of this order as the “strength” of the operation: multiplication is a stronger operation than addition, so it holds its operands more tightly together, and it gets evaluated first.

When a student is learning order of operations, it often feels like a set of arcane rules. There is no reason, from the student perspective, that it has to be this way. In fact, it didn’t really need to be this way, but the convention was established and now it’s important to abide by it (if you want to be understood, that is).

How a calculator helps (and hinders) learning arithmetic

People often lament that today’s youth can’t perform basic arithmetic in their head. It’s unfortunately true; I often see students reach for their calculator to evaluate 35 \div 5 or even 4 \times 6 . These are facts which prior generations had drilled relentlessly and now have available as “instant” knowledge. Younger people typically haven’t spent enough time practising these computations to develop facility with them. This is partly because the calculator is so readily available.

(Aside for parents: If you have kids, please do make them practise their age-appropriate facts. It’ll help them in the same way practising reading makes things easier)

This will draw a lot of heat, I’m sure, but I think calculators do have a strong place in even K-6 learning. They let students explore quickly without the burden of computation getting in the way of non-computational learning. It’s the same effect that web-based, dynamic geometry software can have on learning relationships between figures, lines, etc. (if you’re looking for awesome dynamic geometry software, try GeoGebra – free and wonderful).

But calculators are a hindrance when students are learning to compute fluently. They allow a student to bypass some of the thinking part of the exercise. Don’t let students (or your kids) use a calculator when they don’t have to. Only use them when students need the speed for the task they’re completing.

How a calculator helps (and hinders) learning algebra (?)

Here’s the part I don’t know about, but I’m speculating about.

I think immediate execution calculators require students to understand the algebraic expressions we write, where formula calculators bypass the thinking part of evaluating expressions.

As with arithmetic, if practising evaluating expressions is not part of the learning, and might be getting in the way of the goals for learning, then either type of calculator is fine.

But as students are developing their understanding of algebra and the order of operations, the immediate execution calculator displays the results of operations as they are evaluated, while the formula calculator obscures the evaluations in favour of a single result.

When a student types 5, add, 6, square, equals into an immediate execution calculator, they see the value 36 as soon as they press the square button. There is a reminder that the square operation is immediate. Similarly when a student wants to evaluate \sin(30+45) they must type 30, add, 45, equals*, then sine, emphasizing that the bracketed portion has to be evaluated first (i.e. before the sine function is applied).

*A student can use brackets, which is equivalent to pressing equals before sine. Also, I hope anyone using the sine function knows that 30+45 is 75 and doesn’t need a calculator’s help for the addition.

Is there research?

I perused the InterTubes to find research into this question, but either it’s not out there or I’m not skilled enough to find it.

I want to know whether one calculator is better than the other for a student who is learning to evaluate expressions.

Has no one looked into this? Help?