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.
We’re excited to release our latest activity into the world:
Transformation Golf.
Transformation Golf is the result of a year’s worth of a) interviews
with teachers and mathematicians, b) research into existing transformation
work, c) ongoing collaboration between Desmos’s teaching, product, and
engineering teams, d) classroom demos with students.
It’s pretty simple. There is a purple golf ball (a/k/a the pre-image)
and the gray golf hole (a/k/a the image). Use transformations to get the golf
ball in the hole. Avoid the obstacles.
Here’s why we’re excited to offer it to you and your students.
Teachers told us they need it. We interviewed a group of eighth grade
teachers last year about their biggest challenges with their curriculum. Every
single teacher mentioned independently the difficulty of teaching
transformations – what they are, how some of them are equivalent, how they
relate to congruency. Lots of digital transformation tools exist. None of them
quite worked for this group.
It builds from informal language to formal transformation notation. As
often as we ask students to define translation vectors and lines of
reflection, we ask them just to describe those transformations using informal,
personal language. For example, before we ask students to complete this
challenge using our transformation tools,
we ask them to describe
how they’d complete the challenge using words and sketches.
The entire plane moves. When students reach high school, they learn
that transformations don’t just act on a single object in the plane,
they act on the entire plane. We set students up for later success by
demonstrating, for example, that a translation vector can be anywhere in a
plane and it transforms the entire plane.
Students receive delayed feedback on their transformations. Lots
of applets exist that allow students to see immediately the effect of a
transformation as they modify it. But that kind of immediate feedback often
overwhelms a student and inhibits her ability to create a mental concept of
the transformation. Here students create a transformation, conjecture about
its effect, and then press a button to verify those conjectures.
Elsewhere in the activity we remove the play button entirely so students are
only able to verify their conjectures through argument and consensus.
Students manipulate the transformations directly. Even in some very
strong transformation applets, we noticed that students had to program their
transformations using notation that wasn’t particularly intuitive or
transparent. In this activity, students directly manipulate the
transformation, setting translation vectors, reflection lines, and rotation
angles using intuitive control points.
It’s an incredibly effective conversation starter. We have used
this activity internally with a bunch of very experienced university math
graduates as well as externally with a bunch of very inexperienced eighth
grade math students. In both groups, we observed an unusual amount of
conversation and participation. On every screen, we could point to our
dashboard and ask questions like, “Do you think this is possible in
fewer transformations? With just rotations? If not, why not?”
Those questions and conversations fell naturally out of the activity for us.
Now we’re excited to offer the same opportunity to you and your
students.
Try it out!
