RAY: Everyone, almost everyone remembers the Pythagorean Theorem. A squared, plus B squared, equals C squared. And there are numbers like three, four and five; five, 12, 13 which satisfy that little equation.https://www.cartalk.com/puzzler/mathematic-mistake
Many hundreds of years ago a French mathematician by the name of Fermat said, this only works for squares. There is no A cubed plus B cubed, which equals C cubed. There is no A to the fourth plus B to the fourth that equals C to the fourth. Etc.
As luck would have it, a young mathematician issues a statement that he has three numbers which prove Fermat’s theorem is incorrect. He calls a press conference. Now, he doesn’t want to divulge everything right away. He wants to dramatize, build a little bit, does he not?
So he gives them all three numbers. But he doesn’t tell the power.
A equals 91.
B equals 56.
C equals 121.
So, it just so happens that at this little impromptu press conference, there are all these science reporters from all the po-dunky little newspapers that are around this town. And one of the guys, one of the reporters has his 10-year-old kid with him, because this happens to be a holiday. He’s off from school. And the kid very sheepishly stands up and raises his hand, and he said, I hate to disagree with you, sir, but you’re wrong. The question is, how did he know?
This one is short and sweet. And it totally depends on you. Share your nerdy math things with the students. Please. Students at any level should see how you enjoy the subject that you teach, and how you’re interested in math other than the math that you’re required by your school to teach. I can’t know what you are interested in, but I share a ton of math art with my students. Often it’s things that I’ve created, but that isn’t required. I share some of the best math art things that I find on twitter. A fair amount of my classes have a “oh and here’s something that my nerdy math twitter people were talking about…” moment.
Oh and if you have a twitter account and it’s shareable to your students (no inappropriate things for students), consider talking about it. I don’t have lot of current students who follow my twitter account, but I do have a fair amount of former students who follow me on twitter and while a hundred days go by without any contact, it’s an amazing feeling when they share something that they learned in college and thought about something that we did in class that was related.
Everyone is a fanatic about something. Share that with the students. Many of them are still growing into what they want to nerd out about and you can be part of that.
Ok, here we are after a full year between Classroom Top Four #2 and #3. Wow. Good news is that I have more refined opinions on this specific topic, partly because I changed schools and the friction of the change has made me more thoughtful about what I’d want in an ideal classroom.
This post is about the teacher use of technology in the classroom. Student use of tech is a different challenge.
My favorite setup:
- A projector. Large, clear, bright display on some flat surface on a wall. Large, clear, and bright is so important. Yet so hard to achieve in a school (seemingly). Oh and it needs a remote with a “freeze” button.
- An Apple TV connected directly to the digital projector. No network necessary (not using the streaming media features at all). HDMI only please, no VGA conversion. Sounds like a small nerdy request, but it’s important.
- An iPad with an Apple Pencil. This tablet easily mirrors to an Apple TV that in the same room (neither need internet access for this, technology magic makes it happen, but I’d want internet on the iPad).
- Notability app. Super powerful and easy to use software for making hand written notes, marking up images, and exporting pdfs to google drive.
- The full Apple setup totally isn’t required, but I think it’s the best current setup. I’ve been happy with an android tablet (that came with a real stylus), some random box to mirror it to the projector, and some app that did a similar task to notability. At each step there were some more hiccups, the mirroring connection was more buggy, the writing app was a bit more janky, and the export to google drive took 8 clicks instead of 3. I’m no expert though, there certainly may be improvements in all these areas.
- Writing with a real stylus on a tablet is super important. You need to be able to put your palm down on the tablet and have the tablet just read the stylus’s input. When it’s done right, it’s almost as natural as writing on a sheet of paper, but it has so many more benefits compared to paper. Also this is super important for:
- You need to have the students write on the tablet without thinking. It shouldn’t be a learning curve. You freeze the projector, give a student the tablet and stylus and say “solve this problem for me please” while everyone else in class is working on the same (or similar) problem.
- No Interactive White Board. Sorry Smartboard/Activboard etc. I don’t block half the room with my projected tablet. I can walk around the room and be present where classroom management requires me to be. I can give the tablet to a student so that they can (somewhat anonymously) share their work.
- Take pictures of student work and project them. Mark them up and discuss. No need for names, just “here’s some wonderful work that I captured from your class.” You know the kinds of kids that will be all “yo, that’s my work!” and the kids who’d rather be not called out. This gives them that option, and its soooo easy to do this in notability. Plus button, take picture, insert, crop and resize all takes < 20 seconds.
- Export your notes at the end of every class to a shared google drive folder and make sure the link to the google drive is in somewhere they can find it (best place for me? the about section of their google classroom). Done. Why not? Yes it doesn’t totally capture the class, but nothing ever does, and it’s a great solution for students who miss class, who’d rather not write and just pay attention, etc.
- Put Monument Valley on the tablet and give it to students who finish a test early and are just mindlessly scrolling instagram or snapchat.
- Use the Desmos app, or some other app and take screenshots of things that you’d like to mark up in the software.
What am I missing?
This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking.
Anna Blinstein and Michael Pershan asked us to respond to a prompt so we could have a set of posts for this Virtual Conference. The essay was “The Two Cultures of Mathematics,” by mathematician W. T. Gowers. Gowers breaks down pure mathematicians into essentially two categories based on their beliefs:
- The point of solving problems is to understand mathematics better.
- The point of understanding mathematics is to become better able to solve problems.
Caveat: I didn’t read the essay in much depth, and I’m sure that I’m purposely misinterpreting his conclusions. That’s ok with me, but don’t go rioting outside Gowers’ door because of this silly blog post.
I find it interesting for Gowers to define the culture of math and hence the culture of mathematicians with the *solving problems* lens. While it’s not an inclusive definition of what a mathematician, I’m ok with that, it’s a good place to start. But am I a mathematician by this definition? I don’t solve novel math problems, I’ve only ever solved problems that someone else has already solved (with exception of something personal, like how long can I make this $20 last at the horse track). Nor do I build theory or structure to help others solve novel problems. I teach teenagers how to use centuries and sometimes millenia old math to solve problems that have already been solved.
At best I’m a mathematician once removed, I teach those who might go on to solve novel problems on their own (although, as far as I know, none of the roughly thousand students that I’ve taught have done so). I don’t think that I’m one of the shoulders that these students are standing on if they become mathematicians, but I am lending a hand for them to boost up to a slightly higher position.
And I’m ok with this definition of a mathematician, it doesn’t bother me much one way or the other. I play with math. I make things with math. I teach math. But I’m certainly not solving new problems or creating structure to solve those problems.
Why am I ok with this? Because when I solve a problem that I’ve never solved before, it’s still new to me. It doesn’t take too much of the enjoyment away from me to know that millions of people have already solved this problem. Likewise, I don’t find it less enjoyable to hike a mountain that someone has already conquered: it’s still a new view for me to take in, a challenge for my body to take on, etc.
I also find a lot of value in solving a problem that someone else has already solved because then I can compare approaches to the solution. When I see a really cool math idea scroll across twitter, I often see if I can replicate that math with some sort of computer program. I’m not doing anything new to humanity, but it’s new to me, and that’s what’s important! Similarly, when we as math teachers can get kids genuinely interested in some variety of math (even if it’s not on the curriculum), we are making a difference. If it’s new to those kids, then that’s all that matters.
The #learntocodethroughmath project is something that I created sort of by accident. I see the audience for this project as someone who is somewhat comfortable with using Desmos to create math-artsy stuff with lists, functions, and parametric functions and is interested in learning how to code using Processing. So yea, pretty narrow. But hopefully fruitful and inspiring!
Here’s the original tweet that started things off:
Desmos to processing to paper in 4 pictures: https://t.co/b3tYN39SN8 https://t.co/gZuJEk5HMJ #learntocodethroughmath #plottertwitter #mathart
It's interesting making code for others to "learn" from. The length of the code is definitely longer to avoid shortcuts but keep clarity. pic.twitter.com/w3Kgv6B4kd
— Dan Anderson (@dandersod) March 9, 2018
Click on that tweet to see the three tweets in that thread. I’ll add more to that thread as time goes on.
Here is another set of tweets that ignited the project.
Desmos->Processing workflow in 3 pictures. pic.twitter.com/qwA3mtOwA7
— Dan Anderson (@dandersod) March 7, 2018
(To avoid twitter dying and killing my content, I’ll post the images from those tweets here too)
I’m baaaaaaack. My last post on this blog was six months ago, and that’s a shame. I’m shifting my “writing bucks” from the photo180 blog over to this blog. Some posts will feel a little photo180-y, but most will be more “fully featured”.
Removing friction from the learning cycle in a classroom is super important. Whiteboards, group work and overall positive classroom culture geared towards taking risks and helping each other out can go a long way. Dan Meyer recently blogged about the classroom move of rough-draft talk. Although it was a small change to my normal classroom routine, it paid off quite well. I asked the students to make a rough-draft of a trig graph that we were working on. This was still early days with trig graphing, but after a four minute rough draft session, the students were ready to talk about the graph. Every graph in the room had things to build off of, and every set of students had a productive conversation about what to draw. They were all coming from somewhere positive. This is a classroom routine that I think will stick with me.
I’m happy to let you all know that I’m a New York State finalist for the Presidential Award for Excellence in Math and Science Teaching (PAEMST). I worked on the application this past spring. I enjoyed the reflective nature of the application; it required a detailed description of a recorded lesson, and also a detailed explanation of my overall teaching philosophy. I’m not sure how many finalists have been selected in New York (probably 3 math and 3 science), of which they’ll give the award to at most two teachers from New York next year in Washington DC. Best of luck to all!
This is the second in a series of four posts that describes my favorite things that I’ve done in the classroom to improve my teaching. The first post on Course Evaluations can be found here.
2. Whiteboards and Furniture
On my classroom photo blog, photo180.recursiveprocess.com, I’ve written a LOT about my class’s whiteboard use. In this post I’ll share and expand on what changes have happened in the classroom because of the shifts towards whiteboarding and better furniture.
There were two kernels to the start of the use of whiteboards in my classroom: Frank Noschese’s The $2 Interactive Whiteboard and Alex Overwijk’s Vertical Non-Permanent Writing Surfaces and Visible Random Groupings. (VNPS and VRG).
— Dan Anderson (@dandersod) November 2, 2012
I encouraged our high school to buy enough whiteboards for every classroom in our high school.
Peter Liljedahl has done research on the use of VNPS, and some of it can be found here. Here are some things that I’ve noticed after my class has changed over to working on problems on whiteboards (standing or seated).
- the students are quicker to start a task (supports research)
- they are more willing to take risks (supports research)
- they work longer on a task (supports research)
- they communicate more with partners (supports research)
- students who struggle on a topic are actively supported by their group members with little prompting from teacher
- students who normally fly through a problem (often making mistakes along the way) are more careful and make less mistakes
- related, students who have a strong knowledge base get even stronger, because of the required communication between peers, they are explaining their thinking and trying to formalize their thinking through making generalizations and conclusions
- if they are vertical whiteboards, then I can more quickly address conceptual errors by seeing an error from across the room
- groups self check their work against other groups and naturally compare their work to the neighboring groups
- through more intentional gallery walks (though this is tough with larger classes) and sharing by taking pictures and putting them up on the projector, every student can see the numerous different ways that are available to solve a problem, and can make some judgments about which they prefer
I’ve really enjoyed using VNPS to both challenge students with stronger knowledge and support those with less understanding. Confirming some of the research, I’ve noticed a massive positive shift in my classroom when I’ve changed to this kind of work in class. It’s been important for properly challenging students who have less understanding on a topic because they are more willing and able to jump into a problem that they would have struggled with by themselves. They have a built-in support system of the teacher, but more importantly they also have a support system with their partners who might have more understanding in a topic. Their partners can explain things in new and different ways compared to a teacher because they themselves have just learned the topic, as opposed to relying on the teacher who learned the topic before they were born!
The other important part of Liljedahl’s research is the use of randomized grouping. Because the groups change based on a transparent and random nature every two weeks, every student gets the opportunity to work with just about everyone else in their class. Students who have more understanding need to effectively communicate their thinking to a peers with slightly less understanding. As anyone who has taught someone else knows, you often don’t truly understand a topic until you can explain it to someone else.
This is a subtle thing, but after being in a room with tables and chairs that move around easily (and can be separated), I think I would have trouble going back to a room of these:
It’s been very freeing for the encouragement of group work to be able to quickly go to “battle stations with whiteboards” (although this picture doesn’t show whiteboards, it’s the only pic I could find)
If they’re looking at the projector screen (or often if they’re doing problems on the seated whiteboards mostly by themselves) then we’re in this setup:
I’ve tried groups of 2 and 4, but have settled in on groups of 3 after their written feedback asking for their preferences.
You probably have much more control over the first of these two changes, but both of them have been very nice shifts to my classroom. For those who have played with either of these two classroom ideas, what have you found to be helpful? What are the next things that I should try with regards to whiteboards or furniture?
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.
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!).