Math Performances

 I'm teaching a summer algebra course for incoming students. It's our intermediate algebra course stretched to two semesters, my first time teaching it. Part of it uses or parallels Stanford's How to Learn Math course. We work on number flexibility, patterns (a lot of visual patterns), graphing, proportional thinking, and linear functions and systems. It's part of our Oliver Wilson Scholars program, which works on community building, transition to college, and college success strategies, plus a start on math and english coursework. Each course has an assigned tutor who attends class and works with the learners directly for tutoring sessions. Lyndsey is our tutor and they are doing a smashing job.

In general, I love the graphing stories/lessons (web archive) from Dan Meyer. Of late, I've been doing them via Adam Poetzel's Desmos activity, but did find a current site with all the videos. For this class, I wanted something more experiential. Back in the days of TI calculators we had the motion detectors and I loved those walk the graph lessons. (How can we not have a motion detector app for smart phones?) Lyndsey thought of having groups perform for each other. I came up with four scenarios for them to do, and one that I could demonstrate. Distance of the head above the ground, distance from the edge of the whiteboard, distance between two people, and distance between a person and a ball.

Warm up was to try to guess the joke from this comic. They hadn't heard any of the expressions, but were able to use the 2nd and third graphs to guess at them.

I kept the 15 second time frame. I wanted movement that would be traceable at a classroom scale. I asked them to make it repeatable, and had them perform it twice. Once so people could see what was happening, and once for data. Instead of a hard 15 seconds, we had someone in the course count out loud. The most complicated performance was a minute with slow counting, but it didn't seem to bother anyone.

Here's the handout, with a couple questions revised after doing it. The screen for demonstration was pretty viewable. I wrote a script that I followed.

3 seconds pull down slow

Let go 2 seconds

Pull down fast 1 sec

Stay 3 sec

Put up half way 1 sec

Stay 1 sec

Slow up 4 sec

I got a little messed up with the counting, instead of 1 to 15! And there are some surprises trying to do anything in real life, so the practice was different from the data round.

We compared data, graphed the result, and compared the graph to what happened. One notice was that the graph was kind of the opposite of the motion, it went up when the screen came down. This is definitely a challenge with qualitative graphs, wanting the graph to somehow be a picture of what's happening. If you have a chance to have learners graph what happens to height above the ground when climbing and coming down a slide, you'll see what I mean.

Though I was nervous, new lesson, new course, it worked better than I could have imagined or had any right to have it go. They came up with creative movements, and it seemed to really help with an intuitive meaning to the graph because of the connection to the kinesthetic.

Here's the folder with all the videos. 

The sequencing worked out pretty well. I think the head above the ground was the easiest to understand, and the graph parallels the motion in a natural way. Distance from the edge of the whiteboard was easy to understand. Distance between people was fascinating, and I'd love to do that in a college algebra or calculus class to graph both positions and then the difference. The basketball was really surprising! I expected a much slower and varied trasfer, but the rapid passing gave them a lot to notice and made data collection a challenge.

They selected one of their graphs to turn in, answering an additional question about how does the graph show the action. Maybe I need to come up with a better way to ask that, but there were some good responses. A lot of understanding about data to graph, and some good understanding of the graph properties. Some confusion about minimum and maximum, which makes sense to me since there were two sets of data, really, the x and the y. 

The follow up the next day was Four Corners, a game for practicing coordinate graphing. Not a hit, but good practice. A couple pairs got really into it.

I'll definitely try this activity again, so if you have ideas to improve it or just to try... hit me!

World Tessellation Day '26

 Some of my favorite tessellations from this year!

To me the 2 greatest current tilers I know are Ghee Beom Kim and Miki Imura, who both share a lot on Facebook.

Kim: often works with simple shapes, great fractal connections.

Imura:

Although Regolo is still killing it.

Most of Acid Lich's work is more abstract, but this was fun with the Einstein tile.

Hana Murray may be closing out her daily pattern block work, but she has made hundreds!

I've enjoyed a lot of David Houlton's work this year (in the FB Tiling and Tessellation Group). So clean!

We'll close out this section with the ultimate in tiling personal protection from Dr. T:

Not the same as these geniuses, but a few of my favorites from what I've made this year. All of these are interactive - links go to GeoGebra.

Two rhombus tilings.

Based on a tiling seen in Renesse, France.

From a two tile tessellation.

From last night!

Looking for more? Check out what I share on Bluesky or last year's post. Or make your own: Steve Mayne made an amazing tiling app this year.

We'll close with this sweet tessellation from one of Xavier's learners. Not only is the art amazing, but look at the great rotation action!

AL, Comical

 

AL, Logical, available on Kickstarter until March 5th!

Perhaps the question that have come up the most in discussing AL, Logical with people:

Why a comic book?

Time passes between panels, and it's especially good when we are seeing problem-solving. Something happens. She's THINKING. 

The reader's perspective can be quite different than in prose. Xavier (coauthor, pencils, inks, colors and letters!) is especially strong on showing action instead of telling about it. When the house is moving, we don't see the house changing, we see the elder god seeing the house changing (we see it in its eye). It's humanizing.  Comic books help you to take the perspective of the people in the story. We see AL the way the mathematician sees her, and vice versa. 

And then there are all the joyful little visual tricks he does, like "hey there" backwards in Chapter 4. Fun, but also serves the story. It also offers the reader a chance to notice. Also in Chapter 4 there are a lot of clocks. What's happening with them? 

A big part of our impetus for writing the book is that I see math as being all about problem solving, and he sees art as being all about problem solving. And he does a lot of problem solving throughout the book, which gives readers two ways into AL's problem solving experience.

In a sense, the comic book format literally allows the reader to see math how we see it. 

This might be just because we are a comic book loving family (well, 3/4 of us and one patient parent), but comics are a story form that invites rereading. Even a 70ish page comic like this one is about like three issues of a comic book. Readers have taken from a half an hour to an hour to read it, almost inviting you to pick it back up, look at specific pages, see if the beginning makes sense with the end... go where your attention takes you. Partly because of the story, and mostly because of Xavier's art, the comic really offers a lot of chances to make connections as a reader, and to notice AL making connections in her thinking and experiences.

There are some other math comic books (see the list at the end of the AL, Logical page on the blog), but this one is pretty distinct from those. As fond as I am of them! In our comic, we really worked to make it about the narrative, not a math lesson in disguise. While still having fun, real mathematics.

Please take a look, and help us spread the word.

PS> My previous blog post about the book goes into some of the relevant frameworks behind the math content and the math processes we get to see in the story.

AL, Logical

 The mathy graphic novel I wrote with my high school art teacher son Xavier is up on Kickstarter!

The Kickstarter has a lot of information about the story, so I'll geek out here a bit more.

We started with the protagonist: a middle school student who knew she wasn't a math person. The haunted house idea came along pretty early. And a pretty cosmic haunted house with some Cthulu near relatives peeking in.

One of the frameworks we built the story around are the Van Hiele levels of geometric reasoning. This let AL develop and tackle more complex problems as the story progresses.

Though AL does more than geometry as she's tackling problems from all different kinds of math.

A lot of math comics are directly trying to teach some specific content. This comic is about trying to show the experience of doing math. The framework that helped the most for this is from Tracy Johnston Zager's Becoming the Math Teacher You Wish You'd Had. These chapters...

3: Mathematicians Take Risks

4: Mathematicians Make Mistakes

5: Mathematicians are Precise

6: Mathematicians Rise To A Challenge

7: Mathematicians Ask Questions

8: Mathematicians Connect Ideas

9: Mathematicians Use Intuition

10: Mathematicians Reason

11: Mathematicians Prove

12: Mathematicians Work Together and Alone

... really give you a great feel for what doing math is.

All this might make it sound dry, but the focus was on the story. Very inspired by stories like The Phantom Tollbooth, and our love for comic books, we were glad to see that the beta readers found it fun.

More about the story and other mathy comics in the AL, Logical tab, and of course at the Kickstarter!

Who WIns?

 I'm teaching our Statistics and Probability for K-8 Teachers for the first time. Had excellent support and suggestions from Jenna Laib, colleagues Jon Hasenbank, David Coffey and Hope Gerson, and from Stephanie Casey and ESTEEM folks.

Working out bit by bit what we're going to do. Luckily I get to teach it again next year...

We just finished our first week, and I loved how the Day 2 lesson worked out, so I wanted to share it and think about it a bit. I miss the reflection of blog writing about my teaching!

On Day 1 we had started exploring measures of typical. We got out the unifix cubes, and did a bit on how could we make the distribution more fair. It took several rounds of give aways, but we got there. Some tubs of 252 and some with 251. Then each person built a stack as long as their first name. We organized from shortest to longest and thought about how to answer "How long is a 323 student's first name?" We had initial estimates, then talk went to median and mean. We retooled and did full legal names. Much bigger range, but a surprisingly dominant mode. So there was another consideration. Our emphasis was not the number you said to answer the question, but why you would say so.

Each day one of the teachers leads a Slow Reveal Graph, and Day 2 Tessa started us out on Disney Princess Baby Names, so energy was pretty high. If they had to name kids with Disney princess names they'd go with Belle and Aurora. Yvette's great question was what can you say about this without calculating?

Good discussion. I added in that since all of them had 7 elements, we could just look for a total instead of dividing. Some good call backs to Day 1's discussion of mean and balancing or distributing.

Then the main activity. They used NRICH's great millisecond timer tool and each collected 5 points of data for trying to hit 10 seconds exactly. They each found their median and mean, and thought about what would make a better measurement of who was best at estimating 10 seconds. Uniformly, each table decided on the mean. I raised up the idea of how some sporting events use your best score. They decided that they wanted to include outliers, that consistency matters.

The NRICH page that got me started on this idea had a set of data for discussion, which they credited to the great Don Steward, from his awesome Median blog.

Anna	Ben	Charlie
31	36	37
26	19	32
32	39	24
27	36	32
29	20	30

Typically great prompt from Don. They all found Ben had a mean of 30, though they all thought he was the worst of the three. People were divided over whether Anna or Charlie should win. People liked Anna's accuracy, but others were compelled by Charlie's spot on 30 seconds. They had time to propose a new summary statistic to answer the question of who is best.

One table proposed |mean-30|+|median-30|+range, low score wins. Another table proposed just the best score, tie broken by 2nd best time. Voting was mostly in favor of the more complicated statistic. But definitely the Charlie fans wanted best time. 

And then the breakthrough moment! Someone said 'What if we just totaled up how far each guess was from the target?' Absolute deviation! I tried to play calm and cool, and gave them time to think about it.

Revoted, and people went 22/24 for the absolute deviation. My inner teacher was screaming.

They reevaluated their own times using this metric, and decided on a table representative for the class champion. Then, a class championship, three rounds. Very exciting, started close, and then someone ran away with it.

In their summary, lots of great discussion about variation, mean, median and their limitations.