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.


The Beauty of Geometry

Every now and again, I take the opportunity to simply opine on the beauty of geometry. Math gets a bad rap because of it reputation of being cold, lifeless, functional and academic. (Some folks aren’t helping this by proclaiming that the arts are what we do to enjoy school and math is what we study to get paid later on in life.)

Don’t get me wrong, there certainly are academic ways of discussing art, music, iconography, fashion, design. There’s technique and vocab to all of these areas. Students of these disciplines are still students who must study, but their exploration isn’t saddled with the atmosphere that math is. In math, it is often believed, the box is set; the boundaries drawn. The math frontier is closed. There is no need for exploration when there is nothing to explore.

I’ve always felt like geometry has the capacity to challenge those notions. Kolams, quilts and origami help students understand the aesthetic value of straight lines, precise measurements, perfect circles and right angles. Sometimes, you have to build them to complete your understanding of them. That process alone can bring with it its own supply of feedback. When you are trying to create something visually appealing, often times, the eyeball becomes the expert in the room, not the teacher. Attention-to-detail and technique become valuable without encouragement.

At a recent professional learning opportunity, I was given some time to play with KEVA planks. So I did. The planks are all congruent rectangular prisms. So I placed one on the table. Then I placed a second on with a slight rotation (the diagonal intersection points were designed to sit right on top of each other, but the counter clockwise rotation was determined by the next block being placed so that the vertices were placed on the preceding blocks’ short-segment midpoints. It ended up being about 10 degrees.)

That all sounds pretty mathy (and probably somewhat unclear since I’ve never had to verbalize the process before). But the resulting tower is pretty cool-lookin’ (at least I think.) I simply love when objects with straight lines and right angles are arranged to look like curves. This can happen in algebra as well. As a teacher, of course I don’t know if my students will share my fascination, but fascination isn’t the goal. It’s tough to measure and, besides that, it’s fickle.

I’d encourage you to look for opportunities to change the cold, academic atmosphere surrounding the math. How can we warm this wonderful subject up? We used to take advantage of those tricky days right before a long break and do art projects. Thanksgiving Origami, or build a Christmas (or holiday… or wintertime) scene using nothing but triangles.


What ideas do you have?

Maker Geometry – What can blocks do for you?

So, today I got to play with blocks called Keva planks. A set of Keva planks are nothing more than a whole bunch of congruent wooden rectangular prisms. You can build towers, mazes, bridges, and… of course… geometric shapes.

So, a task came to my mind. Let’s suppose we were going to take the planks and make the smallest possible cube.

  1. How many planks would we need for the combined widths to match the length of a single plank?


The answer’s five, by the way. So, each face is 5 planks wide (which is a single plank long)

Looks kinda like this…


So, naturally… now we can calculate stuff. Like surface area and volume. But to do that, we’ll need some numbers.


So, in my haste,  I clearly did a lousy job of lining up the blocks on the measuring tape. Sorry about that. Not exactly a deal breaker, but annoying.

But since maker materials are becoming more common, I figure you might prefer your students pull their own measurements.

There are a couple of ways that I could see variations of these: different shapes (different kinds of prisms, for example). I could also consider challenges like, given x-number of blocks, who can build a figure with the largest volume or a flat shape with the largest area?

Keva blocks seem like a low-risk, high reward manipulative simply because the start-up would be so quick. What could you do in your class with a set?

Latest Updates to My Geometry Course

My geometry course… funny. It isn’t really my geometry course anymore. Little by little it’s becoming more of a curated set of links that make me wish I still had a classroom. That’s what community will do for you and the innovative math community more than takes care of people. I’ve noticed some excellent learning experiences that my geometry course has been missing. So, I’m adding them. All three are Desmos activities.

Parallelograms in the Coordinate Plane by Jared – This one asks some very interesting questions and has a nice format in which to do it. Due to the algebraic nature of proving this parallel, parallelograms are a nice target for a Desmos activity.

Transformation Polygraph by Julia Finneyfrock (@jfinneyfrock) – Seriously. Check this activity out and then daydream about all the outstanding vocabulary the students would be forced to use while competing with each other.

Working with Dilations by Mr. Rothe – This one is particularly handy because my unit on dilations and scale factor wasn’t my favorite. This is a very robust activity (36 slides), so time might become an issue, but depending on how you run it, I feel like there’s a ton of value embedded in the different learning activities.

If you give any of these a try, please let me know how they go.


4 Desmos Activities I’m adding to My Geometry Course

It’s been about 14 months since I last taught Geometry, but that hasn’t kept me from keeping a keen eye out for high-quality activities to add to My Geometry Course.

Here are the goods that I’ve found looking through the searchable offerings on Desmos.

Ryan Brown offers a nice introduction to polygons using a do-it-yourself Polygraph activity. This strikes me as a nice kick-off to an early unit in Geometry. Serves as a good vocab review. For high schoolers, it would likely serve as a nice low-entry point to Polygons and wandering around listening would give a nice window into what your students could possibly missing as you are about to embark on a new unit.

Kate Nowak offers a nice algebraic introduction at circles in her Activity-Builder offering.  I like this because the common core circles standards tend to be a little more algebraic than some of the other units, so we put it toward the end of the year as we are ramping up the students for their next challenge (which is often Algebra II). Using specific vocabulary like “proportional” and having the student drag points around the graph sets a good tone for what students can expect in a second semester circles unit.

Pizza Delivery by Scott Miller is a very interesting take on reflections. I am actually bummed that I’m not teaching Geometry any more because I’d like to see what this one looks like. You’re going to want to look at this one.

Mathy Cathy offers a nice introduction to reflections as well. I really like the questions she chooses and the tasks seem approachable as reflections are explored for the first time.

Don’t forget Geometry when teaching Algebra

Right now, Michigan educators are trying to sort out the implications of the state switching to endorsing (and paying to provide) the SAT to all high school juniors statewide instead of the ACT as it had previously done. The SAT relies very heavily on measuring algebra and data analysis. This leaves plane and 3D geometry, Trig and transformational geometry under the label “additional topics in mathematics.” The new SAT includes 6 questions from this category. Compared to closed to 10 times that from more algebraic categories.

This comes up in every SAT Info session I lead. Should we just stop teaching geometry? All Algebra? Could geometry become a senior-level elective? If it’s not going to be on the SAT, then what?

These questions reflect a variety of misconceptions about the role of testing in curriculum decisions. In fact, these are the same misconceptions that are driving an awful lot of the decisions that are being made. And in this case, I think there’s more at risk than simply over-testing our students.

It would be a real shame to see Geometry become seen as an unnecessary math class. Because it’s not.

And to illustrate that, I’m going to tell you a story about a conversation I had with my daughter. She’s 6.

She wanted to try multiplication. She’d heard the older kids at school talking about it. So, I taught her about it and let her try some simple problems to see if she understood. And she did, for the most part.

So, I started to not only make the numbers bigger, but also reverse the numbers for some problems that she’d already written down. She had already computed 2 x 3, and 4 x 2, and 5 x 3, but what about 3 x 2, and 2 x 4, and 3 x 5. Were those going to get the same answers as their reversed counterparts?

She predicted no. So, I told her to figure them out to show me if her prediction was right or wrong. To her surprise, she found that switching the factors doesn’t change the product.

“Daddy… why does that happen? It should change, shouldn’t it?”

Translation: Daddy, how do you prove the Commutative Property of Multiplication?

How would you prove it?

The geometry teacher in me thinks about area when I see two numbers multiplied. We can model (and often do) multiplication as an array. It’s just a rectangle, right? A rectangle with an area that is calculated by it’s length and width being multiplied.

What happens if we rotate the rectangle 90 degrees about it’s center point? Now it’s length and width are switched, but it’s area isn’t. Because rotations are rigid motions. The preimage and image are congruent. Congruent figures have the same area.

If l x w = A, then w x l = A. 

Geometry helps prove this. Geometry also helps support a variety of other algebraic ideas like transformations of functions within the different function families. Connections between slope and parallel and perpendicular lines. There’s also the outstanding applications of algebraic concepts that geometric situations can provide. Right triangle trig, for example, is often a wonderful review of writing and solving three variable formulas involving division and multiplication. (A consistent sticking point for lots of math learners.)

As a teacher who spent years watching how Geometry presents such an environment for real, effective and powerful mathematical growth, eliminating it will leave a lot of holes that math departments are used to geometry content filling.

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?