# Improving the evaluation of learning in a project-based class

I’ve been struggling for a few years with providing rich, authentic tasks for my computer science students and then having to evaluate their work.

My students learn a lot of skills quickly when solving problems they’re interested in solving. That’s wonderful.

I can’t conceive of a problem they will all be interested in solving. That’s frustrating.

In the past, I have assigned a specific task to my entire CS class. I tried to design a problem that I felt would be compelling, and that my students would readily engage with and overcome. The point has always been to develop broadly-applicable skills, good code hygiene, and deep conceptual understanding of software design. The point is not to write the next great 2D platformer nor the most complete scatterplot-generating utility.

Unfortunately, I could never quite get it right. It’s not because my tasks were inherently weak; rather it’s that my students were inherently different from one another. They don’t all like the same things.

I believe that students sometimes need to do things that are good for them but that they don’t like to do. They sometimes need the Brussels sprouts of learning until they acquire the taste for it. But if they can get the same value from the kohlrabi of learning and enjoy it, why wouldn’t we allow for that instead?

So I’ve tried giving a pretty broad guideline and asking students to decide what they want to write. They choose and they complete a lot of great learning along the way. Their code for some methods is surprisingly intricate, which is wonderful to see. They encounter problems while pursuing a goal that captures them, and they overcome those problems by learning.

Sounds good, eh?

Of course, they don’t perform independently: they learn from each other, from experts on the Internet, and from me. They get all kinds of help to accomplish their goals, as you would expect of anyone learning a new skill. And then I evaluate their learning on a 101-point scale based on a product that is an amalgam of resources, support, and learning.

Seems a bit unfair and inaccurate.

I asked for suggestions from some other teachers about how to make this work better:

• ask students to help design the evaluation protocols
• use learning goals and success criteria to develop levels instead of percentage grades
• determine the goals for the task and then have students explain how they have demonstrated each expectation
• determine the goals for the task and then have students design the task based on the expectations
• find out each student’s personal goals for learning and then determine the criteria for the task individually based on each student’s goals

I’m not sure what to do moving forward, and I’d like some more feedback from the community.

Thanks, everyone!

# Some advice I give my students

I teach high school. I say these things to almost every class.

“Don’t trick yourself into thinking you understand something you don’t.”

“Write it down.”

“Be gentle with each other.”

“You won’t look back in ten years and wish you had been meaner in high school. No matter how nice you think you are now, when you’re older you’ll see it differently. So be kinder than you think you should be now.”

“Let people like what they like. If they’re not hurting anyone it’s fine. I don’t need you to like the music I listen to, but I do need you to let me like it.”

“Everything is fascinating if you’re curious.”

# 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.

# 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

# 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.”