Sloppy notation doesn’t seem to be reducing understanding of solving linear systems

A couple of weeks ago I wondered here:

Is sloppy notation for solving linear systems reducing understanding?

The TL;DR is “no, not really”. There are other problems besides notation.

Using subscripts to denote a specific point isn’t something Grade 10 students seem super-familiar with, in spite of their supposed experience with the slope formula:

m = \frac{y_2 - y_1}{x_2 - x_1}

More than three quarters of my students simply neglected to use those subscripts when solving systems. They wrote solutions without following the model I presented to them in class.

The ones who did use the notation had a stronger understanding of the concepts/strategy on average. I don’t believe the use of good notation was the cause of that improved understanding; rather, students who understood the concepts were more likely to use the [more complex] notation I presented.

There were two main barriers to understanding in this unit.

First, students do not connect the graphical and algebraic representations of lines. If presented with an equation like


most students can tell me the slope and the y-intercept. But until I ask for those parameters, or until they actually graph the line, they typically don’t visualize that line at all – it’s just a bunch of symbols.

This lack of crossover between representations means that students are not making sense of their own work and judging the reasonableness of their solutions.

Second, students are neither skilled nor fluent with solving linear equations. They do not always remember the inverse operations, and they rely on phrases and tricks to complete these processes. They have trouble because phrases like “move it to the other side and make it negative” doesn’t work well for multiplication and division, and they forget to apply an operation to each term in an equation.

It’s kind of the same problem as the first. There is a feeling of flailing about in the classroom, of trying to apply poorly understood or misunderstood rules to a fairly complex process without even being able to confidently test whether the result is correct.

So notation isn’t the issue. If you have kids in grades 8 or 9, make sure they can solve equations quickly and accurately, including those with fractions. If you have kids in grade 9 make sure they practice graphing lines and determining equations based on graphs. They’ll be in much better shape when learning the more complex techniques in Grade 10.

Teaching cell phone photography

I’ll be working tomorrow with a group of Grade 8 students. We’re going to talk about how to take good photos with cell phone cameras. I’ll be giving them a handout to try to help them understand the ideas, and we’ll practise too.

Here’s the document if you want it: cell-phone-photography

Student move: a fauxkay

fauxkay (“foe-KAY”)


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:



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






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.”

She looked steadily at me. “I didn’t know the answers.”

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
  • 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.