A couple new ways to visualize the math

What could we do if we gave the students more control over how they presented their learning to us?

A simple Google search for some random geometry topic… let’s choose angle pairs… reveals a whole collection of visual images meant to serve as posters, visuals, flash cards, etc.

And most of them are really, really boring. Our students could do better than that. But they might need a tool to do it. Here’s two.

Canva

What I love about these two tool is that they are really, really easy to use. Free to get started (and, quite frankly, perfectly satisfactory without leaving the free version) and easy to share.

Now, you might be asking, “why would we want our students to spend time making this stuff?” Fair question.

Remember, to make something helpful to others, they need to learn it themselves. And for some students, being able to make something awesome-looking can help to add some motivational value to some bits of content that are difficult to jazz up. (Angle pairs, for example.)

Thinking of something like this…

And, of course, it won’t work for all students. So, you can keep the quiz handy for the students who would prefer to show you what they’ve learned that way.

Building (math and science) Knowledge

There are different types of knowledge. To know how a square is defined is different than knowing how to identify one. These are different that knowing how to draw one with ruler and paper. Or construct one with compass and straight edge.

A square is a pretty simple example, but when it comes to ramps, bridge, exploring mass and rotation, there’s being able to answer questions about it, and then there being able to build object to create examples, non-examples, and solve challenges. And it can help to have the right tools.

My office recently got a set of Keva Planks. Now, I’ve gotten to see these blocks in action a number of times and they are pretty cool. First, there is almost no learning curve. The blocks are all exactly the same. There’s no connecting pieces or fasteners or adhesive. There’s nothing to them. But with them you can build ramps and towers, polygons and prisms. They set up easily, the clean up easily.

Full-disclosure: They aren’t free. (Cheapest place I’ve found them is Amazon… 400 blocks for \$90). Ordinarily, I don’t make a point to advocate for expensive tools, but in this case I think it could be money well spent based on the needs of the location. Also, they are durable and shareable. One department or grade level team could probably make good use with one to share.

Anyway, it’s worth looking in to as they allow us to explore types of knowledge that we otherwise might not be very well equipped to explore. (Particularly with those Next Gen engineering practices that are starting to become a reality in many states.)

For another look, here’s my latest podcast: Instructional Tech in Under 3 Minutes #5 – Keva Planks.

A long-awaited solution (featuring Geogebra)

So, a while back, I posted “Circles from Cedar Street” with an intriguing (at least to me) picture designed to kick off the conversation about rotations.

Then, I left it alone. Like, literally. I didn’t solve it. I should have solved it. I let you down.

I especially let guys like Dan down, who also found the problem intriguing. Sorry about that.

But I realized today that it’s never too late to make it right. So, using Geogebra, here’s is one possible solution to the Circles from Cedar Street problem from back in August of 2013.

Why I love this picture #2

A while back, I discussed why I loved this photo. It’s possible that I have a series starting here (sort of like I have with “Real or Fake”. Anyway…)

Today’s photo is a beautiful example of tech integration because the tech is actually integrated. Integrated with what? Well, in this case, with manipulatives.

See? The student has the problem presented online and will record her answer online, but there clearly is no expectation that the work will be fully digital. Which is a good thing because it can be difficult to ignore how effective manipulatives can be in helping student model and visualize mathematical topics (in this case, 3-D images. She’s building a rectangular prism given the front, stop, and side view.)

That’s why I love this picture.

Perplexity and how it appears…

Here’s a video (by Derek Alexander Muller) I think you should watch.

The critique of #flipclass aside, I’m intrigued by the way the narrator describes the value of “bringing up the misconception”. It’s almost like a thorn that creates some discomfort that only learning will relieve. This gets close to Dan Meyer’s use of the word “perplexity”.

From Dr. Meyer: “Perplexity comes along once in a while. What is it? It’s when a kid doesn’t know something, wants to know that thing, and believes that knowing that thing is within her power. That right there is some of the most powerful learning moments I’ve ever seen – so powerful that it’s really hard for me as a teacher to mess those up.”

There’s power in perplexity. I’ve seen this in my classroom on multiple occasions. It’s important to remember that there’s three distinct parts to creating what Dr. Meyer is describing. First, there needs to be something worth knowing. Second, you have to create the want. And finally, we need to empower the students so they feel enabled to know that thing. What Dr. Alexander suggests is that becoming aware of your misconception seems unsettling (leading to claims that videos were confusing), but also leads to more learning. The discomfort fed a drive to resolve the discomfort.

The tricky thing is that misconceptions are a tool you can use when they are available. Science provides a particularly fertile ground for misconceptions because so much of it is drawn from experiences many of us have regularly. Alexander uses the model of a ball flying through the air. This video uses the phases of the moon and the seasons.

The potential for misconceptions is necessarily lightened when there’s no misconceptions, so the quest for perplexity in math needs to take on a different look, proper planning and timing, and different strategies for when perplexity isn’t an available option. (Preconceived notions are just as good at times. After all, we ALL think we know something about squares!)

It’s like Dr. Meyer says, those wonderful perplexing moments only come along once in a while. We foster those moments when we have them, try to create as many as we can and we do our best every other time.