Gallons and Gallons of Pennies

Sometimes, “real-world” problems just go ahead and write themselves. And I say take advantage. Why be creative with the actual world can do the heavy lifting for you, right?

This floated across my Facebook feed. Pretty sure you’ll see where I’ve made some edits to the original texts.

Pennies Problem

Sequels could potentially include:

If Ortha wanted to exchange them for quarters, how many 5-gallon jugs would she need? You could do the same with nickels or dimes.

What would be the mass of each penny-filled jug?

What do you think? What other questions could come off this wonderful set up?

Exploring Reflections with Desmos Activity Builder

Geometric transformations take up a good chunk of the first quarter of Geometry (at least the way that I had it sequenced). The tricky part of teaching transformations in Geometry is the delicate balance between the non-algebraic techniques and understandings and their algebraic counterparts.

For example, consider the two following statements.

Two images are reflections if they are congruent, equidistant from a single line of reflection and oriented perpendicularly from said line of reflection.

The reflection of A(x, y) is A'(-x, y) if the line of reflection is the y-axis and A'(x, y’) if the line of reflection is the x-axis.

As a geometry teacher, am I to prefer one over the other? In my experience, they both present challenges.Students are often a little more enthusiastic about the first (which is why I start there), but can often be more precise with the second. The second requires the figure to have vertices with coordinates and the line of reflection be an axis. The first requires the figure be drawn (and fairly accurately, at that.)

So, in my attempt to learn how to use Desmos Activity Builder, I wanted to produce something useful. So, I made a Desmos Activity to bridge the transition from a visual, non-algebraic understanding of reflections to an algebraic one.

Full disclosure: This activity presupposes that students are familiar with reflections in a visual sense. It isn’t intended to be an introduction to reflections for students who are brand new to the topic.

Feedback, questions… all welcome.

The two key ingredients of real problem-solving

A quick word about dissent.

During a recent conversation with a teacher-friend I we stumbled into an area of conversation that allowed me to see dissent through the lens of leadership and problem-solving in a way that I hadn’t before.

Acceptance of dissent isn’t a new idea in leadership. Lots of writers talk about the need for leaders to appreciate it… here’s an example.

“Defining effective leadership as appreciating resistance is another on of those remarkable discoveries: dissent is seen as a potential source of new ideas and breakthroughs. The absence of conflict can be a sign of decay.”

– Michael Fullan (From Leading In A Culture of Change, 2001, pg 74.)

We were talking about problems that tend to have some pretty zealous advocates. For the sake of exploring a concrete situation, I’ll choose one for an example. How about student retention? This is a topic that can bring some energy out of some folks. It’s an important conversation, too. What happens when a student finishes a school year without meeting the minimum expectations to complete the grade/course they are in?

To push them forward would mean pushing the student forward into academic challenges that they likely aren’t prepared to tackle.

And making student repeat grades has just not been an effective solution according to ASCD, Education Week, John Hattie, etc…

So, when a district sits down to really solve this problem, they need to accept that they probably are going to need to choose a third option. Carelessly moving the student on is probably a poor choice. Making the student repeat the grade is also a poor choice.

The better option, the third choice, the one that will work better, is likely going to have to be crafted on site and with the resources available helping to guide the process.

This is where I began to see the need for two very distinct groups of people.

One group of people creates the boundaries… I’ll call them the idealists. These are the people who say, “We can’t retain them. We can’t. I don’t care what we do, but we aren’t retaining them. It doesn’t work.” Every issue has these people. Most of us can become these people when the issue at hand strikes us right. Luckily, they seem to be essential to the process. They also happen to be very frustrating to people who either disagree or just don’t see the issue as important.

The main issue with these folks is that zeal often doesn’t really solve problems. It creates boundaries for the solution, but (in the case of our example issue) simply eliminating retention doesn’t actually solve the problem of students falling behind. It just eliminates a series of potential solutions.

So, we need to bring in the dissenters… I think of them as the holders of the “yeah, buts…”

“We can’t hold them back.”

“Yeah, but they are still behind in their learning, so we can’t just move them on.”

Now… at this moment… as long as neither the idealist or the dissenter storms out of the room, the real problem-solving work can begin. The boundaries are set, the reality checker is in place and now the focus can turn to the ACTUAL problem In the case of promotion v. retention, it’s the fact that students are making it to the end of the school year not ready to move on.

And that takes some deliberate focus and patience. The zealous boundary-setters don’t want to hear about “yeah, buts…”. The dissenters tire quickly of the perceived inflexibility of the idealists. But I’m not sure real solutions to tricky, messy problems are more likely than when these folks can unify around a common goal.

American education (shoot, American culture as a whole) has a whole variety of problems that we are having trouble solving because the zealous idealists and the persistent dissenters have such a hard time embracing the valuable contribution that each other makes in the course of creating real solutions.

But real solutions… solutions that are effective and sustainable… probably require the active presence of both.