Trying to find math inside everything else

Making It Stick

After going to Anna Vance’s session on Make It Stick, I implemented some of the ideas she presented and thought I found reasonable success with them. However, as I hadn’t read the book, it was a little half-hearted and could be improved. When I was looking through my library’s education e-book collection and saw it there (amidst a sea of worthless looking books, save Other People’s Children, which I also checked out) I decided to pick it up.

A few things stuck out to me, some of which I tweeted, but the one that I keep thinking about is the Leitner System, which they describe thusly:

This struck me for a few reasons. First, I love the idea that the “flashcards” don’t have to be what we typically think of as flash cards, but rather representations of anything we need to practice. Second, it’s a system that is learner-led, so if I can get my young mathematicians onto the system, they can run it themselves. (And extend it to other parts of their lives.)

So my thought became thus: how can I weave this system into my classroom? Here’s my thoughts. I’d love some feedback.

  1. Create a system of boxes (folders? tabs?) – I’m envisioning four in a set – for each student.
  2. At the end of each lesson, have the class write on (an) index card(s) something from that lesson that they think they should know. (This practice of summarizing their learning is also mentioned in Make It Stick.) It could be a knowledge fact (the definition of a polygon), a skill (solving a linear equation), or something broader (what are some ways systems of linear inequalities are applied?). If it is a skill or broad question, it should not have a specific example. (So they shouldn’t have a card that has them solving 3x + 2 = 8 every time they see it.) Then put those cards in box 1.
  3. Their standing HW is to practice whatever is in Box 1 every day. If it says something like “Solve an equation,” they need to generate their own equation, then solve it. (Generation is also mentioned by Make It Stick as a way to increase stickiness.) When they get it right, move it down a box. When box 2 is full, practice those the next session, and so on.
  4. On Fridays, give some time in class for students to practice, especially their box 2 or 3, if they didn’t have the time to do that at home. Then give the usual quiz.
  5. After taking a quiz, they should then reflect on what they did and didn’t know, and if there is something they didn’t know that isn’t on one of their cards, make a card for it right then and put it in box 1.
  6. To qualify for a quiz retake, all the topics for a quiz need to be on cards in Box 3 or 4. Otherwise, they need to study more before they can retake. (This would mostly be an honor system, as nothing stops them from just putting the cards in there.)

Does that sound feasible? What needs improvement?

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(Doesn’t that title sound like it should be a YA fantasy book?)

I shared this Desmos Activity Builder I made on Twitter, but never wrote a post about it. Oops! Let’s do that now.

First, here’s the activity.

This activity is basically adapted from Sam’s Blermions post that I had helped him think about but he did all the hard work of creating questions and sequencing them. I’ve used the blermion lesson in the past and it went well, but that last part lacked the punch I was hoping for, having to work with analog points and compiling them together myself. But then I thought – wait, Desmos AB does overlays that will do this beautifully, if I can just figure out how to get the Computation Layer to do what I want. So I set to it.

The pacing helped pause things to conjecture about what chormagons are – I stopped them at slide 5 so they can make those conjectures. I then advanced the pacing to 6 for the “surprise,” and 7-8 to make further conjectures. Look at some of theirs below.

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Even showing them everyone’s, I tended to get conjectures about the shape their points make – a trapezoid, a pentagon, a heptagon, etc. (One person seemed to guess the truth, but that was uncommon.)

Then I showed them the overlay.

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I saw at least one jaw literally drop, which totally made my day. We talked about why it might be a circle, and what that means for these shapes, then I introduce the proper terminology “cyclic quadrilaterals.” (I taught this lesson as an interstitial between my Quads unit and my Circles unit.)

Speaking of proper terminology, you may notice I called them Chormagons here, as opposed to Blermions. That’s important – Blermions is very google-able, it leads right to Sam’s post! But Chormagons led them nowhere. (And they were so mad about that!)

Now Chormagons will lead right here. So this is an important tip if you use this DAB: make sure you change the name of the shape!

Grading Talking Points

Two main things I wound up talking about at MfA Summer Think were talking in math class and grades. One thing we talked about in regards to grades is that students (and parents) often flip out when introduced to a new grading system that is different from what they are used to, even if by the end of the semester they come around and say that they are glad it was done that way.

I thought, then, instead of just springing my grading/SBG system on them, that we could reflect on what grading systems really mean and what they should do first, to prime the transition. So I created a grading Talking Points (with help from my Twitter mentions for some statements).

Free-writes

At the MfA Summer Think, I went to a Teacher’s Poetry Circle. It was pretty great. Below are what I wrote during the two free-write times, slightly edited/punched up.

Why do I teach?

To learn about math
To learn about people
To learn about cultures
To learn about relationships

To rebel in small ways
To rebel in larger ways
To comprehend a system that was not designed with our best interests at heart

To spread joy
To share knowledge
To forge connections and broaden horizons
To create experiences that linger in hearts and minds

To help others reach their true potential
To help myself reach it, too
To help us all figure out how this world works
To help us all figure out where to go next

Hard

This is hard
but there are harder things.
Changing the world
Dismantling structures that oppress
that
is hard.

But maybe that’s what this is,
just at a smaller scale?
Maybe “hard” is just a matter of scale.
Can we scale up what we do?

Maybe it is impossible –
the square-cube law restricts us all
and our attempts to scale up
collapse
under their own weight.

Sometimes a law must be broken
To do what is right.
Why not this one?
Why not push ourselves to the edge
of the possible?
Will we fail?
Will we fall?

I allow myself to fall
because only by falling can you see
the true heights and depths of where you were and where you can go

From the air, you can see everything.

My school has been trying to better create conditions for productive struggle in our classes, because a lot of students have taken a very receiving stance. So early in our Area and Volume unit, I decided to use this task from Illustrative Mathematics.

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The task is a 7th grade task, and so involved nothing new for my high school geometry students – just area and perimeter/circumference. But the task has a lot of parts, not all of which are obvious from looking at it. So I gave them task, and then I was “less helpful.” In fact, I barely spoke during the lesson, only quietly clarifying things, but reflecting their proximity questions back towards themselves and their other group members.

Almost every group that attempted the task solved the problem on their own. (I followed up with an extension where they designed their own stained class on the coordinate plane and found the price using the same pricing, for those who finished quickly.) I had a group of three girls who don’t usually feel very confident in my class feel like rock stars after figuring the whole thing out themselves.

A few days ago, I saw this tweet:

I thought it really applied here. While the content was still related to what we were learning in high school geometry, the opportunity to solve a complex task with little scaffolding was really helped by using a task from an earlier grade. I recommend it.

I had a pretty good lesson recently that I wanted to share. It was at the end of my quadrilaterals unit, and so we were working on coordinate proofs. I love coordinate proofs because you can get so much information from just a pair of coordinates, which lends itself to lots of different ways of solving the same problem. Add to that how many different ways there are to prove something is a square, and we have the start of something good.

I gave the students the above sheet, starting off with some noticing/wondering about the graphed figure. Then I assigned each table a different method to prove that the quadrilateral is a square. Each group was off to their whiteboards to get started.

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It was really great to see each group discussing the problem so intently, and it reminded me how easy it is to facilitate discussion when up at the vertical whiteboards. Afterwards, the students went around in a gallery walk to compare their proofs to the other methods. They analyzed how they were similar, how they were different, and thought about which method they might prefer in the future. (Some comments included things like preferring method 2 because it only involved slopes, even though it involves more lines.)

The whole lesson went so smoothly and had tons of intra- and inter-group discussion. Need to use the structure again.

Whole Class Test

I teach an SAT Math Prep this year, which has been an interesting challenge. We basically started off with lessons on all the different content in the exam, then had a long section on tactics (which can be framed as test-taking tactics but I noticed are often just tactics for solving problems in general, which was nice). But we reached the end of those, and the (in-school) SAT is a month away. The obvious thing to do is to just keep doing practice exams, but that can get a bit boring, for both me and the students. Plus, the class that meets Tues/Thurs hasn’t had very many graded assessments this marking period, so I needed to give them something.

I had decided that grading them on correctness in a practice SAT is not appropriate. I had told them this before, and they knew their grades on their assignments were more for things like how they applied the tactic we were learning. But last class they walked in and I gave them a Part 3 exam (the non-calculator part) and told them it would be graded – but there would be a plot twist. For right now, just take it individually, except this half of the room should start from the back and go forward. Oh, and you get 5 fewer minutes than normal.

While they were working, I went around on my whiteboards and put up the numbers 1 through 20 well spread out, and an ABCD for 1-15. (I wish I had taken pictures!) This started to get them suspicious. When time was up, I told them my grading scheme: it was out of 5 pts, and they lost a point for every question they got wrong. So if you got 15 right, that’s a 0. But! They had the remaining 20 minutes of class to work together and figure out what the right answers should be. And if anyone got less than 15, the whole class lost a point – forcing them all to work together. (With limits, of course – they won’t be penalized for that kid who went to the bathroom for 15 minutes during this, for example.)

A suggestion I made to them was to go around and make votes for their answer for each question. A clear consensus might mean that that is the right answer. However! Don’t be afraid to put your answer down even if everyone else’s is different. I’ve seen questions where only one person got it right. I told them they need to convince each other of what the right answer is.

Let me tell you, I heard so many great conversations as they and I went around the room. Because it’s the SAT, no one gets them all right, so everyone is being pushed to make a convincing argument that their answer is right. Students who weren’t sure got explanations from others. It was delightful!

About halfway, I noticed a clear consensus for about 15 of the 20 questions, but the middle 5 were really quite split. So I lead the class in sharing out their reasoning for some of those questions – never saying what the right answer was, but again letting them convince each other.

It was a nice collaborative effort – I highly recommend it.