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?

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?

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.

Desmos-Enhanced: The remodeled Pencil Sharpener Problem

Lately, I’ve found it tremendously enjoyable to revisit some of my favorite homemade problems and use Desmos to model them.

I decided to remodel the Pencil Sharpener Problem this time. If you’re not familiar, go check it out. Here’s how it goes.

Three boys are held after class for detention. I told they have to stay for a half-hour, but they can leave earlier if they can grind down 100 pencils by hand in less than a half-hour.

So, the three of them decide to take the me up on my offer and begin cranking the pencil sharpeners as fast as they can. Each of their top speeds is recorded on video. If we assume that they keep their top speed up the whole time and don’t slow down, then how long with their detention last?

This problem has created some fantastic student work. Enough so that it is almost tempting to force pencil and paper work.

However, I couldn’t resist the temptation to create a Desmos worksheet for it.
Besides, by now, Pencil Sharpener Problem is ready for an extension. How’s this:

It seems safe to assume that the boys will tire as they crank the pencil sharpeners over and over and over. How about we say that the each subsequent pencil takes 5% longer than the previous pencil. So, if first pencil took 60 seconds, the second one took 63 seconds, and the third one would take 66.15 seconds, and so on.

Would it still make sense for the boys to grind away pencils? Or should they just sit quietly for 30 minutes?

Direct Instruction vs. Inquiry: The What and the When

In my last post, I looked at the characteristics of high-quality classroom instruction and discussed why I felt like those were essential regardless of the model any given teacher used. There were some excellent comments left after I posted that, so I’d encourage you to go join the conversation.

What I didn’t discuss is the role of inquiry and the role of direct instruction. Each tool that gets wielded in a classroom is build to do a certain type of work. To maximize the effect, each tool must be used to do the job for which it was created. Direct instruction does one type of work. Inquiry does a different type of work. In order to highlight this difference, let’s consider a content standard.

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

CCSS 8.EE.C.7.A

Consider how we’d assess this standard. The students need to “give examples of”, which means they need to actively create something and explain why it’s the right kind of something. But, the explanation is predetermined. They can’t explain it anyway they want (according to the standard, at least). They need transform their example to match one of the stated forms.

So, the final assessment of that standard (if we choose to assess it to the letter, so to speak), would include three equations that the student created and then evaluated in a standardized way to support their claim that their equations had one solution, infinitely many solutions and no solutions respectively.

From my perspective, anytime the students are going to be expect to create something on the assessment, they will need some time to freely explore. You can’t assess a student on something they’ve not gotten the chance to practice. So, if you want them to create on the assessment, they need to practice creating. But we aren’t assessing their ability to create just ANYTHING. We want them to create strategically.

There’s also that standardized evaluation process they’ll use on the equations they’ve created. While there may be some value in allowing the students to explore a variety of different, homemade ways to tell what their equations are going to do, in the end, we are going to ask them all to do the same thing. They need to be taught this process.

Also, we need to make sure everyone is on the same page with the words “equation,” “solution”, and “variable.”

Hang on… I need a quote.

“[Highly-effective teachers] provided support by teaching new material in manageable amounts, modeling, guiding student practice, helping students when they made errors and providing sufficient practice and review.”

“Many of these teachers also when on to experimental hands-on activities, but they always did the experimental activities after, not before, the basic material was learned.”

– Barak Rosenshine

Based on his research, Rosenshine is saying that inquiry can work provided students possess the appropriate background knowledge.

He isn’t the only one to say stuff like this.

“[Content and creativity] drive each other. Students need a certain amount of content to be creative. Increased creativity drives deeper understanding of the content.

“Algorithms and problem-solving are related to one another. Algorithms are the product of successful problem solving and to be a successful problem solver one often must have knowledge of algorithms.”

– Dr. Jamin Carson

And also…

“Students need to be flexible problem solvers. We know that one thing that separates high-achieving students from low-achieving students in elementary school, is that the students who are successful can flexibly use numbers.”

– Dr. Jo Boaler

This idea can be found within a variety of researchers in high-quality math instruction. Students need to explore. They absolutely do. They need to freely explore and play with the math.

But in order for that to be effective as a learning tool, it really, really helps to have sufficient background knowledge. Be it the knowledge of algorithms helping to support and drive the problem-solving process, the math facts giving the elementary students flexibility, or in the case of our example 8th grade standard, a solid understanding of “variable”, “equation”, and “solution” to give the sufficient foundation on which to build their exploration.

So, for this standard, I would probably recommend a direct instruction introduction to the standard that ends with making sure that all students are clear on the three essential vocab words as well as the evaluation process.

Then, I’d move to an structured inquiry activity that led them through a chance to practice creating their own equations and evaluating them eventually leading them to make some generalizations about what equations look like when they have one solution, infinitely many solutions, or no solutions. I see the possibility for some small group discussions, reporting out… possibly a Google Sheet or some white boards and a gallery walk, etc.

And from my chair, this exercise through this standard demonstrates the bigger picture. It isn’t whether or not inquiry or direct instruction should be used in eighth grade.

It’s about what we are going to ask the students to do and which of those models supports the students best at which point during the instruction.

It’s not about which. It’s about what… and when.

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Quotes taken from:

Rosenshine, Barak (2012) “Principles of Instruction”, published in American Educator, Spring 2012 edition. Quote taken from Pg 12-19, 39. Quote taken from pg 12.

Carson, Jamin (2007). “A Problem With Problem Solving: Teaching Thinking Without Teaching Knowledge.” Published in The Mathematics Educator, Vol. 17, No. 2, Pg 7-14. Quote taken from pg 11.

StanfordSCOPE interview with Professor Jo Boaler. Quotes taken from times 2:40-3:20 in the video.

Bowling… and supporting struggling students…

We recently went bowling. It was my nephews’ birthdays. We brought the whole family. I have a three-year-old who doesn’t take well to being left out of the fun, and he shouldn’t have to. Bowling is for everyone.

But he’s little. Really little. A 6-pound ball is about 16% of his body weight. That would be like a 200-lb man throwing a 32-lb bowling ball. No one bowls like that.

So, we provided a support to make sure that he could meaningfully engage.

2015-02-15 12.34.15

It’s a ramp. He put the ball on the ramp. The ball rolls down the ramp and now he’s bowling.

My son just plain ol’ isn’t big or strong enough to roll the ball down the lane. His body doesn’t have the capacity to provide enough energy to the ball. That said, there are still things he can do. He can’t supply kinetic energy, but he can provide gravitational potential energy by lifting the ball up to the ramp. The modification is the ramp facilitating the conversion to kinetic energy that ends with the ball rolling toward the pins.

In this case, we haven’t removed the responsibility to pick the ball up, carry it toward the lane, and add energy to the ball. We’ve also not stolen from him the experience of seeing his ball head toward the pins, to compete with the other children, and to have fun.

All we’ve modified is the conversion to kinetic energy that he is unable to do. And if you think that he was feeling unfulfilled because we modified the activity, then think again. He was hopping around like he had springs in his shoes every time he knocked down some pins.

And the progression toward full participation was visible. My six-year-old daughter was bowling, too. No ramp. She would get a running start and heave the ball with two hands.

2015-02-15 12.26.46

Not exactly a textbook technique, but effective as long as we include a different modification: bumpers. (Full disclosure, the three-year-old had bumpers, too.) We were setting the standard that the the ramp isn’t HOW you bowl. It’s a modified way to bowl until you grow strong enough to no longer need the support.

I thought these things as we watched him bowl. And I considered the implications for our students who need their classroom experiences modified to be full participants. Like a student who is still developing fine motor skills needing someone to dictate his/her spoken words. Or a student who doesn’t sit still well being able to work standing up and helping that student develop the ability to produce quality work while standing.

Take the effort that the student is able to give and provide support in the areas that… well… they just aren’t there yet. The expectation is that the student will be given support in their areas of weakness at another time to facilitate their growth, but at the moment they are in your class, it is important for them to be able to participate as fully as possible with the classroom activities. There are learning outcomes that don’t depend on their weaknesses.

For example, there will be a time when that student works on developing the ability to work sitting down, but the teacher is currently leading an activity on the civil war, and “learning while sitting down” isn’t one of the stated learning goals for that activity.

Or consider a student who is in Algebra I, but has a weakness in basic computation. This is a fairly common weakness among our Algebra I students, wouldn’t you say?

If we could create a structure whereby those students are receiving the specific supports in the area of computation (perhaps a 15-min per day intervention right before school… or something), would it be possible to allow strategic use of a calculator to support students who are weak in computation. They’d still have to know what numbers need to be computed. They still need to be able to check their answers and apply them back to a situation, but the calculator acts as the bowling ramp. It supports their effort until they grow strong enough to heave the ball down the lane themselves.

I think it’s important that we consider ways to help our students find success in our classes despite their weakness. The alternative is watching those weaknesses become bigger and bigger barriers until our students are sitting in the back of the bowling alley watching everyone else have fun.