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

y=3x+5

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.

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).

linearSystem.png

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.