# 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.