Look for and make use of structure.
Supporting students in engaging in this mathematical practice is the major
objective of this week’s Friday Fave.
In
Match My Picture, we work first to create a need. When you’re writing equations for two
lines, it’s just fine to think about the slopes and y-intercepts
independently of each other.
When you’re writing equations for nine lines, it may start to get a bit
tedious. Plus, you may start to notice that these lines have something in
common. Not only that, but what is changing is changing in a predictable way.
It’s y=-2x plus something.
Hopeful that you’ve noticed the structure, we invite you to capture that
structure in symbols.
Finally, we show you how to capture it with our tools. We tell you about
lists. We invite you to use your newfound structure-capturing powers to match
more pictures, and to design your own structured decorative masterpiece.
Taking students from needing a structure, to representing it formally, to
using it creatively—this is what makes Match My Picture a Friday Fave.
While you’re thinking about matching, have a look at these activities too:
The Fave is writing these words at 9:29 on 9/29 (or the arguably palindromic
9:29 on 29/9 for those outside the US). That means it’s autumn where the
Fave resides, and in autumn one’s thoughts turn to lasers.
That’s right. Lasers.
Like how much easier it would be if you could just zap those fallen leaves
with lasers. Or how soon it will be winter and many people will have no choice
but to amuse themselves with indoor cats chasing laser pointers.
And what goes great with lasers (besides cats and fallen leaves)? Mirrors.
Which leads us to the Fundamental Theorem of Laser Pointers:
Lasers + Mirrors = Angle Play
That equation right there is the premise of this week’s Friday Fave:
Laser Challenge.
You set the angle of the laser and the mirrors, then click “Try
It!” and see the results of your work.
Negative angles, reflex angles, angles greater than 360°….try them all!
We offer several challenges of increasing complexity, and then we invite you
to design your own laser challenge. (You’ll need to show it’s
solvable before turning it over to your partner.)
So if your students are studying geometry, trigonometry, or physics, come have
them celebrate autumn—the season of lasers—with
Laser Challenge!
While our minds are on angles, here are a few more angle-based activities to
enjoy with a warm cup of cider (but sorry, no lasers).
I’m convinced the best choice we made when we started making digital
activities was assuming a human would provide some feedback on student
work. When technologists assume that
a computer will provide all the feedback, they constrain their
activities to the limitations of computers rather than their highest
aspirations for student learning.
Written responses, sketches, and opinions are either challenging or impossible
for computers to assess in 2017, which is why you don’t find them in a
lot of digital math activities.
Up until now, we have only offered automatic feedback on a very small handful
of item types – multiple choice, for instance – and then passed the remainder
of student work on to the teacher as a resource for class and individual conversations.
We’re certain that was the right place to start. Over the last several
months, however, we’ve asked ourselves what we can do to offer teachers
more automatic insight into student thinking without sacrificing what we love
about our activities.
So we now display one of these five icons on each screen:
Dash: It isn’t possible for students to do any work on this
screen. Save your time and attention for other screens.
Check: Everything on this screen is correct.
Cross: Something on this screen is incorrect.
Warning: Something on this screen isn’t merely incorrect but it
indicates the student may have misunderstood the question itself – intervene
ASAP.
Dot: This screen needs human interpretation.
A few notes about this process:
Lots of student work still falls into that final category. The most
interesting mathematical thinking is still very hard for a computer to
assess. All we can do is shrug and hand it off to much smarter humans.
We are very conservative in our application of correct checks and incorrect
crosses. For example, on our multiple choice + explanation items, we will
never display a check because we have no way of knowing if the explanation
is correct.
We had to write unique code to display these icons. That meant that across
our hundreds of activities and thousands of interactions between students
and math, a Desmos faculty member devised a unique definition of
“correctness.” Then she applied it, and two other faculty
members reviewed it, asking themselves, “Can we really be so certain
here? Is this diminishing student thinking at all?”
We can now offer teachers automatic feedback on all kinds of rich mathematical
experiences. For one example, we can ask students to
create a sinusoid for their partner. Any sinusoid:
The partner then graphs it algebraically.
And we’ll give the teacher automatic feedback, no matter what sinusoid the
students originally created!
We realize that a teacher’s time and attention are finite and precious.
We hope that the addition of these icons to our activities will help teachers
spend them wisely.
