I’ve been thinking a lot about how we look at academic standards (what we call curriculum expectations here in Ontario) lately. Each person who reads a standard seems to read it through their own lens. That is, as we read a standard, we attach what we believe is important to that standard based upon our prior experiences. With this in mind, it might be worth looking at a few important parts of what makes up a standard (expectation) in Ontario. Each of our standards have some/all of the following pieces:

**Content**students should be learning**Verbs**clearly indicating the actions our students should be doing to learn the content and demonstrate understanding of the content- A list of
**tools and/or strategies**students should be using

Each of these three pieces help us know both what constitutes understanding, and potentially, how we can get there. However, while our standards here in Ontario have been written to help us understand these pieces, many of our students experience them in a very disconnected way. For example, if we see each expectation as an isolated task to accomplish, our students come to see mathematics as a never-ending list of skills to master, not as a rich set of connections and relationships. There are so many standards to “cover” that what ends up being missed for many students is the development of each standard. The focus of teaching mathematics ends up as the teaching of the standards instead of experiencing mathematics. We give away the ending of the story before our students even know the characters or the plot. We share the punchline without ever setting up the joke. We measure our students’ outcomes without considering the reasoning they walk away with… while many students might be able to demonstrate a skill after some practice, it’s quite possible they don’t know how it’s helpful, how it relates to other pieces of math, and because of this, many forget everything by the next time they use the concepts the next year.

To help us think deeper about what it means to experience mathematics from the students’ point of view, Dan Meyer has been discussing building the “intellectual need” with his whole “If Math Is The Aspirin, Then How Do You Create The Headache?”. Basically, the idea here is that before we teach the content that might be in our standards, we need to consider WHY that content is important and how we can help our students construct a need for the skill. He has helped us think about how our students could experience the long-cut before our students ever experience the short-cut.

Let’s take a specific example of a specific standard:

– **construct perpendicular bisectors**, **using a variety of tools (e.g.,**

** Mira, dynamic geometry software, compass) and strategies (e.g., paper folding)**…

If constructing perpendicular bisectors is the Aspirin, then how can we create the headache? How can we create a situation where our students need to do lots of perpendicular bisectors? Well, I wonder if creating voronoi could be a possible headache. Take a look:

A Voronoi diagram is a partitioned plane where the area within each section includes all of the possible points closest to the original “seed” (the point within each section). So, how might students create these? If they already knew how to create perpendicular bisectors, they could simply start by placing seeds anywhere on their page, then create perpendicular bisectors between each set of points to find each partition. However, Dan Meyer points out how important it really is to spend the time to really develop the skill starting from where our students currently are:

**“In order for the CONSTRUCTION of the perpendicular bisector to feel like aspirin, I’d want students to feel the pain that comes from using intuition alone to construct the voronoi regions. This idea ties in other talks I’ve given about developing the question and creating full stack lessons. I’d want students to estimate the regions first.**

**Here is a dream I had awhile ago that I haven’t been able to build anywhere yet. Excited to maybe make it at Desmos some day.” **

If you can, I’d recommend you take a look at Dan’s dream. It really illustrates the idea of building his “full stack” lesson. If we think back to the original standard again,

– **constructperpendicular bisectors**, **using a variety of tools (e.g.,**

** Mira, dynamic geometry software, compass) and strategies (e.g., paper folding)**…

it might be worth noting the specific pieces in orange. I wonder, given a lesson like this, how much time would be spent allowing students opportunities to consider strategies that would make sense? Or, how likely it would be that our students would be told which strategies/tools to use?

Below you can see the before and after images from a student’s work as they attempt to find perpendicular bisectors for each set of points.

Tasks like this do something else as well, they raise the level of cognitive demand. Take a look at Stein et., al’s Mathematical Task Analysis Guide below:

While most students might experience a concept like this in a “procedures without connections” manner, allowing students to figure out how to create voronoi brings about the need to accurately find perpendicular bisectors, and consider how long each line would be in relation to all of the other perpendicular bisectors. This is what Stein calls “Doing Mathematics”! And hopefully, the students in our mathematics classes are actually “Doing Mathematics” regularly.

**As always, I want to leave you with a few reflective questions:**

- How often are your students engaged in “Doing Mathematics” tasks? Is this a focus for you and your students?
- If you were to ask your students to create voronoi, how much scaffolding would you offer? If we provide too much scaffolding, would this task no longer be considered a “doing mathematics” task? How would you introduce a task like this?
- Creating a perpendicular bisector is often seen as a quick simple skill that doesn’t connect much with other standards. However, the task shared here asks students to make connections. Can you think of standards like this one that might not connect to other standards nicely? How can you build a need (or create the headache as Dan says) for that skill?
- Are you and your students “covering” standards, or are you constructing learning together? What’s the difference here?

I’d love to continue the conversation. Write a response, or send me a message on Twitter ( @markchubb3 ).

Thanks for sharing this Mark. I’ve been exploring Voronoi diagrams our unit on geometric constructions. Also connecting this to the incentre and circumcentre of triangles. If you create a Voronoi from three seed points, perpendicular bisectors meet at the circumcentre of the triangle created by the three points. If you cut out the triangle and pour salt on it, the ridges created by the salt trace the angle bisectors and meet at the triangle’s incentre.

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I did not know this. I’ll definitely try this out!!! Thanks for sharing.

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Thanks for the ping, Mark. In order for the CONSTRUCTION of the perpendicular bisector to feel like aspirin, I’d want students to feel the pain that comes from using intuition alone to construct the voronoi regions. This idea ties in other talks I’ve given about developing the question and creating full stack lessons. I’d want students to estimate the regions first.

Here is a dream I had awhile ago that I haven’t been able to build anywhere yet. Excited to maybe make it at Desmos some day. Thanks again.

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Thank you for your comments and the link. A few thoughts:

1. I didn’t share at all HOW we could develop the question in detail, and the link you’ve shared explains nicely how we can use our intuitions before we jump to any mathematical strategies.

2. If you don’t mind, I’d like to add your piece into the body of this to help us consider the HOW.

3. While I did discuss the notion of scaffolding and the inverse relationship between scaffolding and “doing mathematics”, I might go back to the pieces in our standard that might be helpful [using a variety of tools (e.g., Mira, dynamic geometry software, compass) and strategies (e.g., paper folding)] along with the pieces that aren’t there (intuition).

Thanks again for taking the time to read through this. Much appreciated!

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Why did you pick onPerpenicular Bisector. Is it because it is a problem. Yesit is problem.The kids just dont get it. Interesting. This always baffles me. I may introduce the concept at the first of the year and see how many can retain it later in the year. It does not happen. What can i say. I love spending much time working in Geometry.

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It would be interesting to have the students determine what types of 2D shapes are created, how many of each and whether or not the same numbers/proportions are found on every Voronoi.

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I realize this post was quite a while ago, but I just came across it. Back in summer 2010 we had a group of about 30 urban high school students (grades 9-11) on our college campus half-days for 5 weeks of science and math experiences. Voronoi diagrams were the theme for the first week of math which was titled, “How are the spot patterns in a giraffe’s fur, the cracks in drying mud, and deciding where to locate the next Pizza Hut all related?” We started with considering behavior of territorial birds (connecting to the previous science week) and boundaries for assigning students to schools. We used the ideas of centers of interest (schools, nests, dens), boundaries, regions,and nearest neighbors. After experimenting just a bit with an applet and trying to make some sketches on paper the students were sent outdoors in groups – either to the parking lot with string & sidewalk chalk or among the trees with rope & garden staples – to act it out and find the boundaries with given centers of interest (traffic cones or trees). It was amazing to observe how the perpendicular bisector construction arose organically out of the bodily actions necessary to map out the problem solution! – not that the students knew beforehand or even recognized it if they’d had prior geometry class. And this was just the first day. Other days included a trip to the zoo to take photos of giraffes, voronoi games & Scott Snibbe’s interactive art, and applications where Google maps and the zoo photos were imported into Geometer’s Sketchpad. Students examined the math, formalizing some of the geometry, learned about the history and applications to various fields or science, and did extensions to Delaunay triangulation. I’d love to have the context to be able to implement this work again.

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