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.

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Using video to capture quadratic motion

In my MBF3C class today we observed quadratic motion and modelled it with http://www.desmos.com, and online graphing calculator. I’ve recreated the steps here (with some fake data).

First, I went to Desmos and projected a blank Cartesian coordinate system onto the screen at the front of the classroom.

Desmos_blank
Then, I found a black rubber stopper (I teach in a science classroom) a little more than an inch across.

I asked for two volunteers who would be willing to throw things at each other. That was easy.

They practiced lobbing the stopper back and forth to each other in front of the screen, trying to get the black rubber to crest visibly near the top of the grid. Eventually they were confident they could do it.
I held my iPhone in landscape orientation and recorded a slow-motion video. After a few attempts I felt we had a successful toss, and the students returned to their seats without injury.

We scrubbed through the video slowly and recorded the x– and y-coordinates for each major tick of the x-axis.

Then we plotted the points in Desmos (using the Table feature):

Desmos_table

We then graphed a generic quadratic using the vertex form and its parameters (y=a(x-h)^2+k). Desmos provided the sliders for each parameter:

Desmos_basic

As a group the students decided to make a negative and small, to flatten the curve, then they adjusted h and k to fit:

Desmos_transformed
Some interesting stuff to note about the process:

  • Even using 120fps there were places near the edges of the curve for which it was hard to see the coordinates (blurring and gaps).
  • The vertex wasn’t on the y-axis, which was surprising to the students.
  • The glare of the projector made the grid a little hard to see.
  • We had to have the lights out to make the grid visible at all on camera, and the dim lighting made the video a bit grainy.
  • The parabola we fit to the data worked really, really well.

What’s next

I’m going to perform some more motion tasks with them to get more quadratic data, and we’re going to do some curve-fitting to model and predict things (for example, how far can you throw a ball off a 10th-story roof?).

I’d like to have a large, physical grid on the wall or something so that I can have the lights on when we record video.

I want students to record video and analyse it. Lots of them have iPhones, and I bet some of the other phones can take good, crisp video. If not, there’s some learning there too (about interpolation if nothing else).

I’ll try some other phenomena also.

A video you can use

Here’s another video we took, if you want to use it: