# fauxkay (“foe-KAY”)

## Noun

The utterance of “OK” by a student to whom you have just explained a concept but who does not actually understand what you said. It is sometimes said very slowly so as to express their uncertainty. When the uncertainty is not clear, an unwary teacher may be fooled into believing the student has grasped the concept.

Example: Jordan wouldn’t tell Mrs. Jones directly that he didn’t get it, but his fauxkay at her explanation was clear enough.

# Is sloppy notation for solving linear systems reducing understanding?

Whenever I prepare to teach a concept in my class I typically flip through my previous years’ stuff (notes, handouts, blog posts) to see how I approached things before. I also look in the textbook, especially to see the notation that is used.

I’m about to start teaching my MPM2D class how to solve linear systems. I’m going to start with the “Substitution” method, so I’ve been browsing the archives.

Solving linear systems is challenging, in my experience. Students tend to lack a good understanding of why we can substitute expressions for variables, and then they assume it’s simply arcane magic they need to memorize (albeit temporarily).

Today it occurred to me that our notation might be causing some of the problems, and improving clarity might improve understanding.

## The Graphical Interpretation

When we are solving a linear system of two equations, we are looking for the point(s) of intersection between the lines represented by the equations. Graphically, we want to find the point at which the lines cross each other. Students generally are fine with this idea. They get the picture (haha).

The trouble starts with algebra (doesn’t it always?).

## The Algebraic Interpretation

We have two linear equations, and we are looking for the point(s) which satisfy both equation. That is, we want to find an $x$-value for which both lines have the same $y$-value.

The equations are statements about how the $x$– and $y$-coordinates of points on the lines are related. For example, here are two lines:

$y=2x+5$

$y=\frac{1}{2}x-7$

These lines have a single intersection point, at $(-8,-11)$. So -8 is the $x$-value for which both lines have the same $y$-value, -11.

## The Substitution Method

To find this, my textbook, my past self, and my colleagues would all employ the Substitution Method, which says that if you can isolate a variable in one equation, you can substitute the corresponding expression in for that variable in the other equation to find the intersection point. This works because we are looking for the point for which both equations are true simultaneously.

In the example above, the first equation tells us that $y=2x+5$. If we apply that restriction to the second equation, by replacing $y$ with $2x+5$, we get

$2x+5 = \frac{1}{2}x-7$

This is the substitution for which the method is named. We now have an equation with one unknown, $x$. Solving this equation tells us that at the intersection point we have $x=-8$. We can now substitute $x=-8$ back into either original equation and solve to get $y=-11$.

## My Concern

Maybe it doesn’t really matter, but I’m concerned with how we often write out the algebra without explanatory words and with imprecise notation. I don’t mean during a lesson; in that first, iconic example in class we write out words like crazy, trying to make all the magic connections apparent. I mean during later solutions – practice exercises, one-on-one help, and student work.

I think we should be saying this instead:

Suppose there is a point of intersection $(x_1,y_1)$. Then

$y_1=2x_1+5$ and $y_1=\frac{1}{2}x_1-7$

$2x_1+5 =\frac{1}{2}x_1-7$

$\frac{3}{2}x_1=-12$

$x_1=-8$

$y_1=2x_1+5$

$y_1=2(-8)+5$

$y_1=-11$

So the point of intersection is $(-8,-11)$.

See the tiny, subtle difference? We’re picking a specific point and calling it $(x_1,y_1)$.

The reason I’m wondering if this might be better is that students often ask me why they can substitute $2x+5$ in for $y$ in the other equation. And their concern is reasonable, because $2x+5$ isn’t always equal to $\frac{1}{2}x-7$; that’s only true at the intersection point.

We write out the sentence “When the lines intersection we have $2x+5 = \frac{1}{2}x-7$” and go from there during our lesson or on our notes. That’s what my textbook does. But we rarely require this “extra” writing, and instead let a page of symbols replace clear communication and thorough thinking.

## Should I try it?

Do you think I should use $(x_1,y_1)$ and see what happens? Does it even matter? I’m not just being pedantic*; I’m hoping that better communication will lead to better understanding.

Let me know if you’ve done this before, if you think I should try it, or if I’m out to lunch on this one.

*I’m also being pedantic, I’m sure.

# Too honest for EQAO

I administered the Grade 9 EQAO Assessment of Mathematics this semester. It’s a provincial, standardized test that students write for two hours across two days, an hour per day. Part of the test is multiple choice, and part is open response (longer, written solutions).

In the weeks before the test I practised with my kids, gave advice, and tried to make them comfortable while encouraging them to do their best. I told them to try every question, saying things like “You can’t get marks for work you don’t show!”, “You never know what you might get marks for!”, and “If you don’t know a multiple choice answer you should guess.”

One of my students left three multiple choice questions blank.

The EQAO Administration Guide expressly forbids drawing a student’s attention to an unanswered question. So I collected her work.

Afterward I asked her about it. “Why didn’t you answer those questions? You could have guessed; you might have gotten some right.”

I felt (and feel) terrible about it.

Not that I didn’t prepare her well for the assessment. I feel terrible because I realized that I asked my students to lie

I asked them to guess “if necessary”, to hide their lack of knowledge, to pretend that they knew things they did not. Because I want them to get good marks, and I want our school to do well.

That is a terrible thing to ask, and for a meaningless reason.

My student didn’t just guess. She didn’t play this ridiculous game. She showed integrity.

And I’m really proud of her for that.

# 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

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.

# Stop being selfish: adult brains need helmets too

I spent part of Sunday afternoon riding bikes with my kids on the Hub Trail in Sault Ste. Marie. It was a good time, and we all enjoyed ourselves. Here we are about halfway through our 5K trip, taking a break on a bench:

We passed a few other cyclists along the way. The Hub Trail parallels Queen Street, a major road, for a portion of our route. Queen Street now has a bicycle lane, for which I’m sure many cycling enthusiasts are thankful. Unfortunately, my non-scientific survey of those pedaling people indicates they either (a) vastly overestimate their skills, or (b) vastly underestimate their mortality.

Hardly anyone was wearing a helmet.

Queen Street has a posted speed limit of 50 km/h, which means most cars are hovering around the sixty-click mark. At that pace a cyclist in the bicycle lane can’t do a whole lot to prevent injury if there is an accident, whether it is caused by a motorist or the cyclist.

Unless, of course, that prevention was enabled prior to the journey. Like, you know, putting a helmet on.

We know helmets work. We even have a law that says kids have to wear them. Why don’t adults have to wear them too? What is it about adult brains that is less valuable than kid brains? As far as I understand it, adult skulls aren’t significantly more durable when negatively accelerating due to asphalt.

Some people claim some sort of obscure “right” to not wear a helmet, in the same way that it’s an obvious right to ride a motorcycle sans cranial protection at 70 mph on an interstate. “It’s my life” and all that nonsensical garbage. Your decision to not wear a helmet is outrageously selfish. When you make a tiny mistake, or when a driver makes a tiny mistake, your decision to not wear a helmet might amplify that error from minor to catastrophic.

“I grew up riding my bike without a helmet, and I turned out fine! We weren’t so reckless because we knew we didn’t have helmets on!”

The fact the you’re alive and uninjured says nothing about all of the people who have been injured or killed in these sorts of accidents. Your random survival is a single data point. I’m happy you’re alive, but I’m so disappointed that you’re being so careless with your life and the future of that anonymous driver you might share an accident with.

While I’m talking about it, start wearing a helmet when you’re skating with your kids. I don’t care if you played hockey. Accidentally hitting your head on the ice is stupid.

(If you do a quick Google search about the efficacy of bicycle helmets to prevent injury you’ll find a lot of ideologically-driven pages saying that they don’t really work. Read a lot further than that if you want the truth. Helmets reduce injury in the event of a collision. The larger, more complex question is whether legislation is effective in preventing overall injuries.)

# ﻿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.