Classroom Top Four – #1 Course Evaluations

This is the first in a series of four posts that describes my favorite things that I’ve done in the classroom to improve my teaching. This post was cross-posted on my photo 180 blog

1. Course Evaluations

Have you ever tried to swim in a lap pool with your eyes closed? How long were you able to go without hitting a lane divider? I can get about 5 or 6 strokes in before I hit and need to correct my direction. I’ve done some triathlons, and one of the hardest parts of racing is swimming in a straight line. You can train all you’d like on lanes, looking down at a lane marker to go straight, but swimming in open water is a different challenge. The thing that worked best for me was to take some number of strokes, say 10, and then take a look to make sure you’re pointing in the right direction. As your muscles get more tired you tend to wander in different directions.

I’ve been asking my students for quarterly feedback for 4 or 5 years, and I’d put it in the top three changes that I’ve made that have most affected my teaching. I use the feedback to keep me honest. It’s hard to open up to anonymous feedback from teenagers, you think the worst is going to happen. But I’ve found that not only do they give marvelous feedback (“course” correction, do you see what I did there), but they tend to appreciate the addition of another data point that you give a damn, and that their input matters to you. There is so much good stuff that they have to say, and if you provide them time, space, and importantly, optional anonymity; they will hand you pure gold. It doesn’t have to be a long feedback form, my quarterly feedback form is only 6 questions:

Here are some quotes from this past feedback session, for some context, these are Juniors and Seniors in advanced math classes.

I love how in depth they think about how they learn best, and they definitely don’t all agree on their favorite methods. I love how they give me constructive feedback and compliments in the same response. I also deeply appreciate their pushback on thing that we need to work on as a class. And this isn’t some royal “we” going on here, they often see changes that they themselves can make to improve their learning (not that they always take themselves up on their own advice!)

An important part of this feedback cycle is to acknowledge their responses publicly. I like to try and get the gist of each question and write down my takeaways. I also think it’s useful to take a comment that I disagree with and explain my thinking. For example, there is a group of students who would rather I was more flexible with my reassessment policy. I explain to them that I wish I had a time turner because then I could provide each and every student as many opportunities as they needed to prove their knowledge on a topic.

I hope you can find a time to try something similar in your classes. It’s hard to not focus on a negative bit of feedback, but I’ve found that I’ve gotten ever so slightly better at seeing the big picture. You gotta bang into some lane dividers to keep your path.

Workflow: Processing -> Fiji -> 3D Print

This is an extension off a previous workflow series featuring Processing and Fiji





This 3D print has slices that represent a (linear) trip in the Julia Set from 0+0i to 0.32+0.64i. I wrote a Processing program to create the 400 Julia slices and then Fiji stitched the 2d images together into a 3D model. Here’s the interactive mandelbrot -> julia program that is featured in the last gif.  The printer is a resin based SLA model (prints from liquid goo!).

Play with Math

Learning ∩ Play ∩ Math ∩ Art

The connection between learning math and playing with math has been on my mind lately. Another connection that I’ve been thinking about is the interplay between math and art. This morning I decided to merge these two thoughts. I originally tweeted under the hashtag #playeveryday, but future challenges like these will be under the hashtag #playwithmath. Through exploring a graph in Desmos, can we both learn some math and make some interesting images?

Here is some great work by some tweeps:


Your Turn

What can you make? Can you learn some math by playing around?

Workflow: Desmos -> Selva3d -> 3D Printer (in 5 minutes!)

This is a continuation of the workflow series: part 1part 2, and part 3

This workflow was done entirely in the web browser and could be done on a chromebook. The only step that requires a more serious computer is sending the file to your 3D printer. Also, writing up this blog post took three times longer than the actual workflow!

Step 1: Desmos

Make a design in Desmos, turn off grid and axes and take a screenshot. Here’s the design that I worked with, and the screenshot that I used. I have the function listed twice so that the design was fully black, but I don’t think this would have mattered.
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Step 2: Selva3d

Go to Selva3d and register for an account. Upload your screenshot and it will extrude it for you. Download the 3D file.


Note, if you want slightly higher quality (for free), you can bring the image to inkscape, convert it to svg, and bring the svg file to tinkercad. This requires you to install software, but there are also online png -> svg converters. It’s one more step for a bit more quality.


Step 3: 3D Printer

Bring the file to your 3D printing software for final tweaks. After 1 hour of printing, here’s the result!


Sidenote, selva3d has some cool partners. Here you can very easily order a phone case with your design on it for $21.



Workflow: Desmos -> Inkscape -> Laser Cutter

This is a continuation of the workflow series: part 1 and part 2.


Have students create mathematical art in Desmos and bring that digital art into the physical world using a laser engraver. You can certainly bring any image from Desmos and engrave it into an object using a laser engraver, but this method allows you to cut out the actual line from the material so that all that is left is the shaded area.


Create a design in desmos that has lines which are crossing at points. This method will use the laser to cut on each side of a line, so that the material “inside” the line will remain.

Here’s an example: Wavy Sine Circles. Turn projector mode on.

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Take a screenshot of your image.


Open the image with Inkscape (free and open source vector image editor), and then select the image by clicking on it.


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Go to Path -> Trace Bitmap.

Select Edge Detection. This selection will make vector lines on each edge of our original image.

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Drag one of the images sideways so that you can delete the original image and just have the vector image left.

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Save the vector as a .svg file so that your laser cutter will be able to follow the lines directly (Vector vs Pixel images).

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This output looks a bit “lacy” for most materials that I’d use in the laser cutter. The lines are too narrow. No problem, do the same steps with the original desmos image, but this time zoom out before you take your screenshot. Desmos will always make it’s lines a certain width, so if you zoom out, the lines will be relatively thicker. Here’s the result after a quick zoom out.

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Laser Time

Bring your .svg file to the laser cutter that you have available (or to a vinyl sticker cutter). Here are some results (these are test cuts on manila folders):



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What do you think students could make with this workflow?



I’ve had some trouble with the output from inkscape giving me double lines, so that the laser is making two cuts really close to each other. Here’s a fix.

When you go to trace the bitmap, keep the selection on brightness cutoff.




Move the vector off of the original image and delete the original image (the one that is fuzzy).



Change the fill and stroke to the following: No fill, black stroke.


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Send to your laser cutter software. Much better!


If you had a friend taking this class next year, what advice would you give them?

I asked the above question, If you had a friend taking this class next year, what advice would you give them?, to the outgoing PreCalculus H group last year. I’m giving the incoming PreCalculus H their responses. Here’s a link to all the responses, and here are some of the absolute jewels. (No editing was done on this document, exactly how the students responded, mistakes and all.)

  • Use your classmates for help and work together
  • Good luck….even though homework’s aren’t graded make sure to actually DO them fully because if you don’t you’ll fall so behind….and retake as much as you can
  • To study A LOT and with a group of people so that if you have any questions you guys can talk about it together.
  • Also, you need to practice and study on your own if you do not understand something because it will catch up to you. You will not do well on the B quizzes if you don’t fully understand the material. If studying on your own doesn’t work, get extra help from a teacher.
  • There’s a lot of material you will cover in this course, and you want to be able to look back on it at the end of the quarter / year. This course will never be easy, but it is manageable. Reassess every opportunity you get, even if it’s just a 4.5. If your scared you’ll lower your grade if you get a 4.5, you probably don’t know the material as well as your grade shows that you do, and you’ll suffer at the end of the quarter because of it.
  • Take take homes seriously and do them with your friends, though never just copy what they have down. Though the skills on a take home are much more complex that what you need to do in class, they are skills you need to relatively comfortable with, and they actually do help a lot. If you don’t understand anything that you wrote down, stay after with Danderson and review the concepts. He can’t help you with the take home, but he can help you with the skills to do it (skills that you will need come midterm / final).
  • Don’t be afraid to rely on people. You aren’t going to be able to get through this class alone even if you are like a math whiz.


Workflow: Desmos -> Processing -> Fiji -> 3D Print

Note: this post is a more advanced version of the previous post and uses slightly different software. All this software is free and open-source.
In this post I’m going to outline how to use a workflow to go from a desmos sketch to a processing sketch to Fiji to a 3D file. The basic idea of the workflow is based on the idea of looking at a 3D object as a 2D image which is changed through time. Here’s a gif (source) that shows how we can view a cube (a 3D object) as a 2D slice.


So if we can create the 2D sketch, then we can create the 3D object with Fiji. Here’s a video that contains a walkthrough of all the steps involved in this process in more detail.

Desmos Sketch

I used Desmos as a tool for quickly prototyping the 2D sketch. The original idea was to make a wavy cup. So the cup would start with a circle base, and slowly change up the walls to have a wavy top. Here’s the desmos sketch that shows the cup being sliced from the bottom to the top:


(There is more detail on how this sketch was created in the video.)

Processing Sketch

Next, we can bring the sketch to Processing so that we can easily save a bunch of frames and have a lot of control over all the details. Here’s the live code (had to be modified for openprocessing because PShape isn’t supported in JS), and here’s the original code. Once again, much more detail is available in the video.


Next bring the 400 frames into Fiji. Fiji is often used for stitching 2D images from MRI machines into 3D objects.

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This is an optional step for shrinking the number of triangles used, and hence shrinking the model size.

3D Print

Shown with Makerbot, but every 3D printer has software that can do these steps.

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and two hours later:


More Wavy Cups

These both use a different period for the sine function, and the second adds in a sine inside the first sine based on the height to give it the “wiggle” back and forth.

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Spiral Ball

These were teased in a couple of tweets, the concept is the same as the wavy cup, but the size of each slice is controlled by a circle function. The desmos sketches are linked in the tweets.

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The black spiral ball uses consecutive fibonacci periods spinning in different directions, which is why it looks like a pine cone.

Fresnel Lens Sphere

Here’s the desmos sketch for this (admittedly weird model).

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Dragon Fractal

These were directly coded in processing, here’s the originating code to make the fractal.

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Let me know if you know of a different way of making this kind of 3d model! It works pretty well, but there are some rough edges with taking the 2D images to make the 3D model. Cheers!