It’s the same reason I think he’s an awesome person. He is invested in getting people to see the joy that is inherent in doing mathematical work, and the creativity associated with it. To the point that years ago, when he happened to be in NYC, he came and talked to my Algebra II classes about aperiodic tilings (and brought a ton of these tiles for my kids to work with).He coauthored/illustrated a fabulous adult coloring book, Patterns of the Universe, which inspired one of my students last year. (And led me and a colleague down a crazy fun rabbit hole this summer.)
And more recently, this year, when I started curating a mathart show at my school, he immediately agreed to be a featured artist and sent not only artwork, but some extra tiles that I requested just because I knew our lower school kids would enjoy playing with them. (A walkthrough video of the gallery is here.) Here are some of the tiles he sent… Yup… fractal tiles!
One of the pieces he sent for the show were these curvahedra. I had fun making and playing with them (and the sheets that the pieces popped out of).
So feel free to check out his kickstarter if you are interested…
As an added bonus, I emailed Edmund gads of questions about his work, his process, and his curvahedra. And below is our Q&A!
I hope they feel curious, but relaxed, looking to see what can be discovered and enjoying the pleasure of discovery. Many people report that it can also be quite a mindful experience, which I had not expected.
There are a whole collection of different balls and eggs that are the most obvious things to make. Donuts are the next option. I have made some exotic pieces which branch and rejoin. I discovered when making pieces for the kickstarter that you can make some nice cones. In theory though there are few limitations, anything that can be made as a mesh can be made from Curvahedra. As meshes make many of the creatures and objects in CGI the limits are hard to find!
I liked mathematics initially because I could do it, but that never gave passion. What fired my love of the subject was the way you could create little worlds and then explore them. Once the rules were fixed you had constrained the behaviour, yet that did not mean that you knew what could happen. I often feel that the most valuable thing I learnt from my PhD was flexibility of thinking. In mathematics you have to be flexible to think about what you can control and how that works with other rules. In this vein Curvahedra is a system that can do many things. The fun is in discovering the most beautiful and making mathematical rules to help find that.
I wanted to make 3d objects on a machine that cuts essentially in 2d, a laser cutter. I knew that paper could bend, so that was my goto material. I tried a lot of options, not always successfully. Then with the right connector and the right idea suddenly I had Curvahedra. It was a revelation when I made the first ball.
It has really helped my understanding of space and surfaces. In particular getting an idea of the curvature of a surface, and when that is rigid or not. That understanding helped when I was studying 2D crystals with a Physicist. I have also used Curvahedra to investigate minimal surfaces, these are surfaces that minimise strain and cannot be improved by local change. This is exactly what the paper seeks out, you impose some geometry and the paper pieces find the most efficient form.
They litter my house, many hang from string in my windows. Luckily my wife likes them too, she has been really supportive helping put together the kickstarter and will be playing a key role in running it. I could not have done it without her.
The notions of curvature and minimal surfaces that I discussed above, as well as other concepts from multivariate calculus are really shown off well by the system. In fact many of these advanced topics can be easily introduced to elementary students with Curvahedra. There is also some excellent topology to discover. On the other hand that mathematics is maybe a little too advanced for a general audience. There is a lot of geometry to be done. The regular polyhedra turn up naturally and general ideas about space will come from playing with the system.
I would hope they would both use it in the same way. As a low direction activity exploring, where people initially make a basic ball and then are just left to it. This is how I usually run sessions with the system. As people discover things that can be done it is natural to try to find out if you can discover everything that could be done. That gives natural mathematics questions. I always feel that when people have a question and mathematics is the answer they are far more motivated to look at the mathematics.
I love all sorts of visual mathematics and geometry more generally. The discovery of nonEuclidean geometry was such a revelation and is incredibly beautiful to me. It is hard to list favourite ideas though, as it is all so cool. As an undergrad I swore off differential equations, for example. I did not look at any mathematics involving them for years. Then I finally started to see how they linked to geometry, made art with them and did research and now I love the ideas especially in Differential Geometry. The best answer is often that my favourite maths is what I have just been studying! For mathematicians, I have been fortunate enough to spend time with John Conway and both he and his work are phenomenal. The humour and insight in Archimedes’ writing is also a great favourite. For geometric insight Alicia BooleStott is incredible, she was able to work out all sorts of properties of four dimensional figures that I could not approach without a computer. Similarly Felix Klein’s visual insight is amazing. Finally you cannot be a geometer, interested in the more elementary side of the subject, without loving the work of Donald Coxeter.
Max Bill is my favourite artist of any sort and I also love the work of Kenneth Martin. They both worked in concrete art which attempts to create form out of nothing, rather than refining essentials from something real as in abstraction. That is a notion that works very well with mathematics. Moving in the direction of mathematics, George Hart has many inspiring pieces, his use of symmetry and shape to create intricate structures definitely played a role in the development of Curvahedra. Finally I am very lucky to be able to work regularly with Chaim GoodmanStrauss who brought me to Arkansas. His work which he describes as the illustration of mathematics was probably the single biggest inspiration for me. For a time I felt like all my best ideas were stolen from him.
I want more people to get their hands on Curvahedra. Every place I have taken it is seems to inspire and delight and I wanted to spread that further. Sadly I doubt that Math Art will make anyone rich, but I do hope it can get into more hands and inspire more love and enjoyment of the subject.
All sorts of geometry and math toys and games, and time to explore them. Things waiting to be explored by the students on their own terms rather than to satisfy a rigid curriculum. Though the problem with ideal worlds is that they also have ideal students! This might not be practical in all sorts of ways.

[1] To be very, very clear: I asked Edmund if I could promote his kickstarter. He did not approach me. And I am also getting nothing from him for posting this. In fact, I’m about to pledge money to his kickstarter!
I’m guessing if you’re reading this blog, you’ve had that happen to you.
For me, this internal joy is everything. Why I love math, below all the flowery adjectives, is because of the emotional impact it has on me when things click.
It’s an internal thing. A private thing. But what if it didn’t have to be. [1]
I had this idea last year, and I’m terrified and thrilled about introducing it on Tuesday. This idea is designed to specifically make this joy visible and public.
Here it is…
A bell. I’m going to ask kids to ring the bell on their group tables when they have some sort of insight or discovery or revelation that simply causes them joy. I want their internal joy to become external. I want my kids to recognize that this joy is something they should be conscious of and recognize it when it happens. I want a class culture where moments of joy are acknowledged and celebrated.
When a bell rings, what happens? Nothing. Kids continue working. Near the end of class, I will ask kids “anyone who rang a bell, can you share a bit about that moment and insight that you had? and can you describe that feeling with one adjective?” Or maybe I’ll call on someone who rang a bell? I do know that I don’t want the bell to interrupt the flow of class and thinking. Will I have the bells out every day for that one class? I will probably have the bells out a lot after I first introduce them, to start building the classroom culture around them, but then I can see me probably make more conscious choices about when the bells will be out based on what we are doing in class.
Will this work? I don’t know. I’m going to roll it out in only one of my classes. Why? Today I had a killer class. The kids were persevering and having so many gaspable moments. It was ridiculous. I went into the math office after just to tell the other teacher of the class to get psyched for his next class (he hadn’t taught that lesson yet). But because this happened, it is a perfect time to have a conversation with my kids about the joy of mathematics. (I’ve had the bells since school started… I just hadn’t found the right time to introduce them.)
I can see a number of things happening.
What do I suspect? Truth be told, I’m super excited about this, but I think #3 is the most likely outcome. It’s hard to be consistent with something that doesn’t get off to a solid start, because then keeping it up even though it isn’t working well feels fruitless, and finding ways to fix things and change course is way tough.
However I will say that I started using, with this class, the red/yellow/green solocup strategy for groups to selfassess where they are in terms of their own progress, and it’s been amazing. So that gives me hope for this idea working with these kids. Wish me luck!
[1] Okay, sometimes it isn’t private. I love when a group high fives when they figure something out. That happens when it is something hardearned. Something they worked for. I also remember years ago a kid getting so worked up about understanding the sum of angles formula for sine that he literally fell in the floor. So sometimes the joy is visible. But I suspect that a lot of the joy that kids feel (when given the right kind of tasks, which put them at a place where they can have those hardearned moments) often has a momentary and fleeting nature. I hope a ding! can give voice to those fleeting moments.
The poster for the show. The name, by the way, came from a #MTBoS tweep!
In this post, I wanted to share with you the gallery virtually. (In a future post, whenever I get a chance to breathe, I’d love to talk about the programming we made around the show and possibly a bit about how the show actually came into being.)
Here’s a walkthrough video of the gallery:
Here’s our vision statement for the show:
The website for the show is here. The five artists featured in the show are: George Hart, Edmund Harriss, Veronika Irvine, the Oakes Twins, and Paul Salomon.
And lastly, here are photos of many of the pieces:
I’m sure that someone has come up with this visualization before. It feels so obvious to me now. That that didn’t make me any less excited about coming up with it! I immediately showed it to two other teachers because I was so enthralled by it. #GEEKOUT
I am thinking how powerful a gif this would be. Start out with 1. Have two arrows emanate from that 1 (one arrow saying times x and one arrow saying times y) and then it generates the next row: 1x 1y. And again, two arrows emanate out of both 1x and the 1y (arrows saying times x and times y). And generating 1x^2 1xy 1xy 1y^2. Then then a “bloop” noise as the like terms combine so we see 1x^2 2xy 1y^2.
And this continues for 5 or so rows, as this sinks in.
Then at the very end, some light wind chime twinkling music comes up and all the variables disappear (while the coefficients stay the same).
Of course good color choices have to be made.
Who’s up for the challenge?
Okay, I’m guessing something similar to this already exists. So feel free to just pass that along to me. Now feel free to go back to your regularly scheduled program.
Of course I was hitting a wall with my own work, so I started thinking about this. While watching Jeremy Jones’s video, I started thinking of what was happening graphically/visually with standard deviation.And I had an insight I never really had before.
So I made an applet to show others this insight! I link to the applet below, but first, the idea…
Let’s say we had the numbers 6, 7, 7, 7, 11. What is the standard deviation?
First I calculate the mean and plot/graph all five numbers. Then I create “squares” from the numbers to the mean:
The area of those squares is a visual representation of how far each point is from the mean.[1] So the total areas of all those five rainbow squares is a measure of how far the entire data set is from the mean.
Let’s add the area of all those squares together to create a massive square.
As I said, this total area is a measure of how far the entire data set is from the mean. How spread out the data is from the mean.
Now we are going to equalize this. We’re going to create five equal smaller squares which have an area that matches the big square.
We’re, in essence, “equalizing” the five rainbow colored squares so they are all equal. The side length of one of these small, blue, equal squares is the standard deviation of the data set. So instead of having five small rainbow colored squares with different measures from the mean, the five equal blue squares are like the average square distance from the mean. Instead of having five different numbers to represent how spread out the data is from the mean, this equalizing process lets us have a single average number. That’s the standard deviation.
I’m not totally clear on everything, but this visualization and typing this out has really help me grok standard deviation better than I had before.
I created a geogebra applet. You can either drag the red points up and down (for the five points in the data set), or manually enter the five numbers.
https://www.geogebra.org/m/EatncEg2
My recommendation is something like this:
Okay, I literally just whipped the applet up in 35 minutes, and only spent the last 15 minutes coming up with these scaffolded questions. I’m sure it could be better. But I enjoyed thinking through this! It has helped me get a geometric/visual sense of standard deviation.
Now time to eat dinner!!!
Update: a few people have pointed out that the n in the denominator of the standard deviation formula should be n1. However that would be for the standard deviation formula if you’re taking a sample of a population. This post is if you have an entire population and you’re figuring out the standard deviation for it.
[1] One might ask why square the distance to the mean, instead of taking the straight up distance to the mean (so the absolute value of each number minus the mean). The answer gets a bit involved I think, but the short answer to my understanding is: the square function is “nice” and easy to work with, while an absolute value function is “not nice” because of the cusp.
In Advanced Precalculus, I start the year with kids working on a packet with a bunch of combinatorics/counting problems. There is no teaching. The kids discuss. You can hear me asking why a lot. Kids have procedures down, and they have intuition, but they can’t explain why they’re doing what they’re doing. For example, in the following questions…
…students pretty quickly write (4)(3)=12 and (4)(3)(5)=60 for the answers. But they just sort of know to multiply. And great conversations, and multiple visual representations pop up, when kids are asked “why multiply? why not add? why not do something else? convince me multiplication works.”
Now, similar to my standard Precalculus class (blogged in Nominations, Part I, inspired by Kathryn Belmonte), I had my kids critique each others’s writings. And I collected a writeup they did and gave them feedback.
But what I want to share today is a different way to use the “Nomination” structure. Last night I had kids work on the following question:
Today I had kids in a group exchange their notebooks clockwise. They read someone else’s explanations. They didn’t return the notebooks. Instead, I threw this slide up:
I was nervous. Would anyone want to give a shoutout to someone else’s work? Was this going to be a failed experiment? Instead, it was awesome. About a third of the class’s hands went in the air. These people wanted to share someone else’s work they found commendable. And so I threw four different writeups under the document projector, and had the nominator explain what they appreciated about the writeup. As we were talking through the problem, we saw similarities and differences in the solutions. And there were aha moments! I thought it was pretty awesome.
(Thought: I need to get candy for the classroom, and give some to the nominator and nominee!)
The best part — something Kathryn Belmonte noted when presenting this idea to math teachers — is that kids now see what makes a good writeup, and what their colleagues are doing. Their colleagues are setting the bar.
***
I also wanted to quickly share one of my favorite combinatorics problems, because of all the great discussion it promotes. Especially with someone I did this year. This is a problem kids get before learning about combinations and permutations.
Working in groups, almost all finish part (a). The different approaches kids take, and different ways they represent/codify/record information in part (a), is great fodder for discussion. Almost inevitably, kids work on part (b). They think they get the right answer. And then I shoot them down and have them continue to think.
This year was no different.
But I did do something slightly different this year, after each group attempted part (b). I gave them three wrong solutions to part (b).
The three wrong approaches were:
And it was awesome. Kids weren’t allowed to say “you’re wrong, let me show you know to do it.” The whole goal was to really take the different wrong approaches on their own terms. And though many students immediately saw the error in part (a), many struggled to find the errors in (b) and (c) and I loved watching them grapple and come through victorious.
And with that, time to zzz.
In all of my precalculus classes (I teach two standard sections and one advanced section), my kids are being asked to do tons of writing. A few who have had me before in geometry are used to this, but most are not. And honestly: getting down what mathematical writing is, and how to express ideas clearly, is hard.
So what do I do? I throw them into the deep end.
On day two of class, I ask them to write an answer to a problem for a seventh grader to understand. On the third day of class, they come in, and are given the name of the student who comes after them alphabetically (and the last person is given the name of the first person alphabetically). Then they read these instructions:
Everyone moves to the desk of the name they were given. Then I project on the board:
And I give students to read through a different student’s solution. They have to make sense of it — pretending to be new to the problem. And then they critique it. Eventually, probably after 35 minutes, I left them return to their seats. They read over the comments. I talk about why the feedback is important. And how specific feedback is useful (so “good explanation” is less useful than “your explanation of how the groups were made was easy for me to follow”). And then we continue on with class.
Here are examples of some postits (front of a few, then back of a few):
To follow up: that night for nightly work, I gave students a writing problem — a simple probability problem. My hope was that this would help them pay attention to their explanations. I collected the problem and read through the writeups.
They weren’t so hot. Most of them didn’t talk about why and some didn’t have any diagrams or visuals to show what was happening with the problem. So I marked them up with my comments. (They got full credit for doing it.) The next day I handed them back and shared my thoughts. I also shared a copy of a solid writeup — one that I had created — along with four or five different possible visuals they could have used. (I realized –after talking with Mattie Baker about this — that I couldn’t really get my kids from point A to point B unless they saw what point B looked like, and what my expectations were.)
At this point, I wanted to figure out if they were taking anything away from all of this. So I created a page with three questions. A formative assessment for me to see what my kids understand and what they don’t about the content. But I also asked them to take all the feedback they’ve gotten about writing and explanations, and explain the heck out of these problems. Here’s an example of one of the problems (one I’m particularly proud of):
I collected them today. I haven’t looked through them carefully yet, but from a cursory glance, I saw some thoughtful and extensive writeups. And even from this cursory glance, I can see that these two activities — plus all the conversations we’re having about explaining our thinking in class — have already made an impact.
Yes, they’ve gotten some ideas of what a good writeup looks like. They know diagrams can be helpful. They know words to explain diagrams are important. They know the answer to why is what I’m constantly looking for when reading the explanations.
But more important to me is the implicit message I’m trying to send about my values in the classroom. I think a lot about implicit messaging to communicate my values, especially at the start of the year. And I am confident my kids know with certainty that I value all of us articulating our thinking as best as we can, both when speaking but also when doing written explanations.
However I had a really good day today. I saw my advisory and two of my four classes. I even went to some of the varsity volleyball game after school!
This post isn’t about my kids or my classes. It’s going to be about some things I’ve done at the start of this year.
(1) Inclusivity. I read a book about trans teens this summer. We had a lot of conversations about pronouns last year. We as a school have taken gendered pronouns out of our mission statement. Last year I included this in my course expectations:
But this year, in my gettoknowyou google form that I give to kids, I asked for their pronouns.
Chances are, I probably am not going to get any different answers that what I expect this year. But I’m not including this question for the majority of kids. I want to be ready when that first kid gives me pronouns that differ from what I may expect. I want that kid to know they can find comfort (not just safety) in our room. And I want all kids to know things that I value. And I think this question sends that message — no matter who the kid is.
That’s the idea behind it. Who knows if my intention is how the kids will understand/interpret it?
(2) Mattie Baker and I were working at a coffeeshop before the year started, and he showed me his class webpage, which had this video (which I’d seen before) on it front and center:
I loved how *real* this video felt to me. Not like something education schmaltzy which makes me want to roll my eyes. I then went searching for a twin video that explicitly talks about the growth mindset. I had a dickens of a time finding one that I felt would be good for students to watch, but didn’t seem… well… lame. I found one:
So as part of the first set of nightly work, I’m having kids watch these videos and write a comment on them in google classroom. (So others can read their comments.) As of writing this, one class has already had two kids post their comments (even though I don’t see them until next week). I read them and my heart started singing with happiness. I have to share them:
Two videos aren’t a cureall. But having kids realize how important having a positive cando attitude, and how important it is to look at math as a skill to be developed (rather than something you’re innately born with and is fixed) is so important to me. I have to remember to be cognizant about how important this stuff is, and how important it is to reinforce daily.
(3) In both of my classes today, one student said something akin to “I first thought this, but then I talked with Stu (or listened to the whole class discussion) and I changed my mind.” I stopped both classes and made a big deal about how important that was for me. And how those types of statements make my heart sing. And why they make my heart sing. So they should say those sort of things aloud a lot. Okay, so I said it once in each class. How can I remember to say it a lot more? In any case, it was a teacher move I was proud of.
Oh oh another teacher move… I saw when one student was sharing their thinking with the class, but not everyone was facing the student. And I remembered Mattie Baker and Chris Luzniak’s training from this summer (on dialogue in the classroom). I told everyone they had to face the person that was speaking. I need to remember next week to make this more explicit — and talk about (or have kids articulate) what they should look like when actively listening to someone. And why it’s important to give this respect to someone. They are sharing their thinking — which is a piece of them — with us. They took a risk. We need to celebrate that. And try to learn from their thoughts. Doing anything else would be a disservice to them and to our class. (Okay, clearly you can tell I’m thinking through this in realtime right now by typing.)
(4) Robert Kaplinsky has created a movement around opening classrooms up. I personally hate being observed. Before someone comes, I freak out. Of course as soon as I start teaching, I absolutely forget that they are watching. Totally don’t even recognize them as an entity. In fact, I think I often teach better, probably because I’m subconsciously aware I’m being watched so I’m hyperaware of everything I’m doing. But leading up to it is horrible. And I also hate the idea of “surprise visits” because… well, who likes them?
That being said, I know that getting feedback is important, and I know that in my ideal school, classrooms wouldn’t be silos. So I joined in. Not for all my classes… I need to dip my toe in gently. But I posted this next to my classroom door:
Next week or the week after, I’ll probably put this up as a “do now” and ask kids “what do they notice/wonder?” about it. Then I’ll tie it into a conversation about growth mindset and the videos they watched.
(5) For the past two years, I’ve been teaching only advanced courses. (In fact, because of that, I asked to teach a standard course… I have taught many, and it was weird to not have that on my plate for two years.) And I heard from someone that a few kids were nervous about having Mr. Shah because “he teaches the really hard courses? will he be able to teach us?”
I know that my first few classes with these kids need to show them that I am different than they expected. I also was proud of this paragraph I put on the first page of their course expectations…
(6) I met my advisory for the first time. Seniors. The thing is: we’re ramping up our advisory program to be more meaningful. Advisors are going to be with their advisees for four years. We are going to be the initial point of contact for many things. And we want to be there to support and celebrate our advisees in a way that we haven’t been able to in our previous set up.
But for all this to happen, I need to form relationships with my advisees. Relationships that go beyond pleasantries. In our training for our new advisories (amazing training… I think I should write a post to archive that thinking before I forget it… done by the Stanley King Institute) we did an exercise. We found someone we didn’t know (I found a new teacher at our school). We had to think about something meaningful to us, and something real (not something like our favorite sports teams… sorry sports fans)… and then talk about it with our partner for EIGHT MINUTES. Anyone who has been a teacher knows that speaking about anything for eight minutes straight is tough. It feels like eternity. While that person is speaking, the other person is actively listening. They can say a few words here and there, like “oh yeah…” or “totally,” but it wasn’t about having a conversation.
Normally I’d roll my eyes at something like this. But at the end, I felt like I got to know this new person at our school pretty well… actually, considering we only had eight minutes, amazingly well… and we bypassed all the initial superficial stuffs. That stuff, like movies and books and stuff, we’ll get to later. Yes, it was awkward. But yes, it worked.
So here’s how I adopted it for my advisory. I met with them today to do a bunch of logistics, and then I took them to a different room. I had cookies, goldfish, crackers, and a cold drink for them. And I explained this exercise. And I said: “I want to do this with you. I want to get to know you.” And so I took out the notecards I prepared, and I shared stuff about my life with them. And they were rapt. I told them about stuff going on in my family that was exciting and stuff going on that was tough, I told them “things I wish my students knew” (this is such a great way to flip “things I wish my teacher knew”). I told them my total anxiety for the start of the school year and why I had it, and I told them my total excitement for the start of the school year and why I had it. I even said: “I never feel like I’m a good enough teacher.” When I was saying that, I wondered how many kids think “I never feel like I’m good enough.”
A photo of my index cards are here… but I only used them as launching points. I didn’t want to be rehearsed.
I’m a guarded person, and I made sure never to cross the line between personal and professional, but when I finished, I sensed some (all?) of them were processing that a teacher opened up to them in this way. A few thanked me for sharing with them.
I wanted to set up an initial connection, and send the message I want you to know that I’m not an advisor in name only… I’m opening up to you because I want you to believe that when you’re ready, you can open up to me. They’re seniors. They have a lot figured out. But I hope they know I’m here for the stuff they don’t have figured out.
In the next week and a half, I have 10 minute meetings with all of my advisees individually. I told my kids they are going to talk to me about what’s meaningful to them for 8 minutes. I acknowledged it would feel awkward. I told them they didn’t need to open up in any way that made them feel uncomfortable. But I wanted them to speak about whatever is meaningful to them. We’ll do favorite books later. Now I’ll get to know them on a more personal level. [1]
(6) As you might have noticed from #4 above, I’m trying to be better about formative assessments. I want to make sure I know what kids are thinking, and where they are at, and use that to refine or alter future classes. I haven’t tried this out yet (today was just our first day!!! I only saw two classes!!!), but I made a google form for exit tickets.
This is a #MTBoS sample version, so feel free to click on it, and fill out fake feedback to get an idea of the form.
Pretty awesome idea, right? I didn’t want to have a bunch of preprinted slips (something I knew I wouldn’t actually do).
(7) I took a page from Sara Van DerWarf’s playbook. I didn’t do this on the back of name tents, but I have a separate sheet that they’re filling out. For my two classes today, I asked them to share something about themselves that would help me get to know them as something other than a kid in our math class. Some kids gave a lot, some a little, but I learned something about each one of my kids. As I’ve mentioned, I’m terrible with names. But what’s nice is on this sheet I created, I put photos (they’re in a school database for us to use) and knowing something about them is helping me remember their names. It’s odd and unexpected and lovely. Kids interested in arts/photography/social justice/sports/debatingarguing/nature/etc. I liked writing that little note back to my kids. I don’t know what question I’ll ask next. I may ask them “Math is like…” (like James did). For the penultimate one, I should definitely take a cue from Sara and ask them to ask ME a question.
I have a few more ideas for posts percolating. I hope that I get the time and motivation to write them. But it’s nice to be back!!! SCHOOL IS IN SESSION!
***
[1] This is what I included in my email to my advisees:
I know it may feel awkward, but when you meet with me during this meeting, you’re going to speak for 8 minutes about things that are meaningful to you. So something more than a listing of your favorite books/movies. If you need help thinking about this: what makes you tick? what makes you gasp? what are your thoughts about senior year and the future? what could you not imagine doing? what are you feeling? what keeps you up at night? These are all questions that might help you find things that are truly meaningful to you. I found it really helpful to have an index card of things when I was talking with you, because I was nervous. I suggest doing that!
I’ve been struggling with coming up with a precalculus unit on polynomials that makes some sort of coherent sense. You see, what’s fascinating about precalculus polynomials is that to get at the fundamental theorem of blahblahblah (every nth degree polynomial has n roots, as long as you count nonreal roots as well as double/triple/etc. roots), one needs to start allowing inputs to be nonreal numbers. To me, this means that we can always break up a polynomial into n factors — even if some of those factors are nonreal. This took up many hours, and hopefully I’ll post about some of how I’m getting at this idea in an organic way… If I can figure that way out…
However more recently in my play, I had a nice realization.
In precalculus, I want students to realize that all quadratics are factorable — as long as you are allowed to factor them over complex numbers instead of integers. (What this means is that is allowed, is allowed, but so are and and . (And for reasons students will discover, things like won’t work — at least not for our definition of polynomials which has real coefficients.)
So here’s the realization… As I started playing with this, I realized that if a student has any parabola written in vertex form, they can simply use a sum or difference of squares to put it in factored form in one step. I know this isn’t deep. Algebraically it’s trivial. But it’s something I never really recognized until I allowed myself to play.
I mean, it’s possibly (probable, even) that when I taught Algebra II ages ago, I saw this. But I definitely forgot this, because I got such a wonderful a ha moment when I saw this!
And seeing this, since students know that all quadratics can be written in vertex form, they can see how they can quickly go from vertex form to factored form.
***
Another observation I had… assuming student will have previously figured out why nonreal roots to quadratics must come in pairs (if p+qi is a root, so is pqi): We can use the box/area method to find the factoring for any notnice quadratic.
And we can see at the bottom that regardless of which value of b you choose, you get the same factoring.
I wasn’t sure if this would also work if the roots of the quadratic were real… I suspected it would because I didn’t violate any laws of math when I did the work above. But I had to see it for myself:
As soon as I started doing the math, I saw what beautiful thing was going to happen. Our value for b was going to be imaginary! Which made a+bi a real value. So lovely. So so so lovely.
***
Finally, I wanted to see what the connection between the algebraic work when completing the square and the visual work with the area model. It turns out to be quite nice. The “square” part turns out to be associated with the real part of the roots, and the remaining part is the square associated with the imaginary part of the roots.
***
Will any of this make it’s way into my unit on polynomials? I have no idea. I’m doubtful much of it will. But it still surprises me how I can be amused by something I think I understand well.
Very rarely, I get asked how I come up with ideas for my worksheets. It’s a tough thing to answer — a process I should probably pay attention to. But one thing I know is part of my process for some of them: just playing around. Even with objects that are the most familiar to you. I love asking myself questions. For example, today I wondered if there was a way to factor any quadratic without using completing the square explicitly or the quadratic formula. That came in the middle of me trying to figure out how I can get students who have an understanding of quadratics from Algebra II to get a deeper understanding of quadratics in Precalculus. Which meant I was thinking a lot about imaginary numbers.
That’s what got me playing today.
I’ve been having so much fun with it.
Here’s the problem. Look at the spiral below…
We see that 1 is located at (0,0).
We see that 2 is located at (0,1).
We see that 8 is located at (1,0).
If we continue this spiral in this manner, can you come up with a formula for the coordinates of the kth number?
So what I want to know is if we consider the number 2016, can we come up with a way to precisely define where it is? What about 820526487?
One easy way around this is to write a computer program that just brute forces our way through it. So here’s the constraint: I want a closed formula for the xcoordinate and ycoordinate. That means no recursion! No if/then statements! Just an equation that relies on k only.
You know one of the most frustrating things? Going down a path and feeling good about it, even though it is pretty complicated. And then having a new insight on how to attack the problem (which *just* happened to me now as I typed up the problem and look at the image I created for this post) [1]. And realizing that approach might yield it’s secrets so much easier!
In any case, I thought I’d share the problem because it’s given me so much enjoyment thus far. If you do get obsessed and solve it, please feel free to put your answer in the comments. I have a feeling there are a variety of valid solutions which look very different but yield the same answer.
[1] What this reminds me is how slight changes in representations can lead to new insights! Before I was using this image that Megan sent me:
If you want to see that I did solve it, check out this Geogebra sheet. It won’t give away how I solved it (unless you download it, look at how I defined each cell, and then reverse engineered it).
https://www.geogebra.org/m/cXhwYh3P
So yeah… 2016 is at (22,13), and 820526487 is at (14322, 4784).
I am so proud of myself! I came up with a closed form solution!!!
I am going to put a “jump” below here, and then show what my solution is, and write a little about it. So only read below the jump (meaning: after this) if you want some spoilers.
NOTE: There is a small error in the solution below… I defined M=floor(sqrt(k)0.001). It turns out that almost always works until you hit huge k values near perfect squares (I know, I know). However the real solution has M=floor(sqrt(k1)). That fixes all the problems, I think. You can read below about the solution, and then how I caught my error, and how I fixed it.
Okay, so this was an insane problem. To make the formula for the coordinates where any k value exists simpler, I defined a new variable M [for Megan, who nerdsniped me with this problem!] which was equal to: M=floor(sqrt(k)0.001).
I know, I know, it’s so ugly! And what I think is so insane about this is that now I see so much simplicity and structure in the equations. It doesn’t look like gobblygood to me. But looking at this as an outsider who maybe didn’t attack the problem, I wouldn’t know how in the world someone would come up with this. It feels impossible!!!
I am going to try to give a little sense to the structure of this solution, without going into all the ugly details about how it came about. (I would actually love to show the process, but it would take a mininovella to trace my thinking. And I have a terrible memory. Though when I ran out of paper at the coffeeshop I was working at, I did start working on a googledoc, so some of my thinking is archived.)
Let’s look at the pieces of the solution structurally:
There are three parts to the solution: the underlined bit, the swirl bit, and the zigzag bit.
Let’s look at the underlined bit. What this part of the solution does is find us a coordinate near the right coordinate. To see this concretely:
For any k value, the underlined part will yield one of the yellow squares in that diagonal.
So for numbers like k=51, 55, 62 (the grey numbers), the underline bits put us at the coordinates for 57.
And for numbers like k=101, 114, 120 (the purple numbers), the underline bit puts us at 111.
Let’s say k=106. If we only had the underlined part of the solution, that would yield the coordinates (5,5), which is where 111 lives. If we say k=113, and we only had the underlined part of the solution, that would also yield the coordinates (5,5), which is where 111.
So the underlined part of the solution gets us close to the answer, but not quite fully there.
Once we’re at a yellow square, to get to the real coordinates, we need to move up/down or left/right from that yellow square. So for example, if I wanted 106, that would be me moving right 5 units from the 111 square. Or if I wanted 113, that would be me moving up 2 units from the 111 square. Or if I wanted 61, that would be me moving down 4 units from the 57 square. Or if I wanted 56, that would be me moving left 1 unit from the 57 square.
So all we need to do is figure out a way to move us up/down/left/right appropriately.
What the swirl bit of the equation does is precisely that. It calculates how far up/down/left/right I need to move (where up and right are positive values and down and left are negative values…).
However there’s a problem! How do you alter the equation so you are only moving up and not also right? You don’t want to move both directions from the yellow square — only one of them! (So, look at the 106 or 113… we only moved in one direction from 111, not both.) If we only had the underlined part and the swirl part of the equation (but not the zigzag part of the equation), we would always start at a yellow square near our true answer, but then we would move both horizontally and vertically. That’s a problem. To get to any of our values, we only need to move horizontally or vertically.
Thus, the stopgap for that is in the zigzag part of the equation. That part of the equation looks super complicated. But in actually it’s not. Let’s look at a similar equation (which is where I came up with it):
That’s basically what the zigzag piece means in the xcoordinate equation. It’s an on/off switch. It yields a value of 1 when we want to move in the xdirection, and it yields a value of 0 when we don’t. Because the swirl and zigzag piece are multiplied together, we have two options…
If the xzigzag piece is “on”, we are saying: start at the yellow square and move horizontally the number of units that the swirl piece of the equation says to move!
If the xzigzag piece is “off”, we are saying: start at the yellow square and do not move horizontally.
Similarly for the ycoordinate equation, we have the exact opposite zigzag piece…
What I mean by opposite is that if the xzigzag piece yields 1, the yzigzag piece yields 0 (and if the xzigzag piece yields 0, the yzigzag piece yields 1).
So similarly:
If the yzigzag piece is “on”, we are saying: start at the yellow square and move vertically the number of units that the swirl piece of the equation says to move!
If the yzigzag piece is “off”, we are saying: start at the yellow square and do not move vertically.
And since the two zigzag pieces are “opposite,” what we only ever move horizontally or vertically, but never both.
An example.
Let’s look at a specific example. Let’s consider k=113.
Evaluating k=113, we get the following for the x and ycoordinates:
x=(5)+(2)(0)
y=(5)+(2)(1)
Where the red number is the value of the underlined part, the green value is the value of the is swirl part, the blue value is value of the zigzag part.
Notice the underlined part put us at the coordinates (5,5) (the yellow square with 111). That fixed us near the true value of the coordinate. Then the swirl part said we need to move either up/down/left/right two units. And the blue part says: actually, we only want to move vertically, so ignore the horizontal motion.
Let’s do it one more time with a different k value. Let k=56, we get the following for the x and ycoordinates:
x=(4)+(1)(1)
y=(4)+(1)(0)
Where the red number is the value of the underlined part, the green value is the value of the is swirl part, the blue value is value of the zigzag part.
Notice the underlined part put us at the coordinates (4,4) (the yellow square with 57). That fixed us near the true value of the coordinate. Then the swirl part said we need to move either up/down/left/right one unit. And the blue part says: actually, we only want to move horizontally, so ignore the vertical motion.
Although the original equation looks super complicated, it may seem less complicated now! There’s some thinking behind the structure!
This being said, I am absolutely sure there are myriad equations which will work, and mine is probably one of the uglier ones. I started down a path that got a bit circuitous, and I have a feeling that if I were to try this again in an attempt to come up with a simpler answer, I would be able to do so!
So when verifying my work on geogebra, I was hesitant about the fact that in order for me to make things work out when writing my solution, I had to shift things around a bit. So I had that weird thing called M which was floor(sqrt(k)0.01). I had to subtract that 0.01 because I was having issues with perfect squares, but nothing else.
Turns out that was a bad idea… I started thinking about if that could mess things up for high values of k which happen to be near perfect squares (which, visually, means at the corners where you switch from going one direction to another as you walk along the spiral). And this is what I found… it does mess things up. Check it out…
These two red dots represent 20166 and 20738. You can see they are just a bit away from the location they should actually be at!
So I changed that 0.01 in my equation for M, so M=floor(sqrt(k)0.000001), and everything turned out well.
It’s a little annoying, because I wanted a perfect closed for solution.
Wait. WAIT. I have a fix I think… I am going to define M=floor(sqrt(k1)). The whole idea of having M=floor(sqrt(k)0.001) is that for values of k which were perfect squares, I wanted M to be one less than the number being squared… so I subtracted out a little piece to make the floor function round it down. But then problems occurred.
Instead of arbitrarily subtracting 0.001 (which causes trouble), I can get the same shift by simply taking the square root of the previous integer (so instead of the square root of k, I take the square root of k1). This does the same thing… I think…
Let me check… AND… IT WORKS! And it looks slightly more elegant!
The fix works perfectly (I hope). Here it is in final form (I hope):
Our presentation is here:
The crux of the presentation is that we (not just Tina and me, but many in the #MTBoS) have done a lot to make the #MTBoS community more welcoming and accessible to newcomers (the ExploreMTBoS initiative and mentoring program, the mathtwitterblogosphere website).There are conferences (#TMC) and tweetups (all over the place). There is a #MTBoS booth that travels to various (often NCTM) conferences and is manned by #MTBoS participants, to spread the word!
Other #MTBoS created things are available that are useful for teachers who don’t participate in the #MTBoS. There are books that have been written by #MTBoSers (e.g. Nix The Tricks, The Classroom Chef). There are website that are created by #MTBoSers and used by teachers everywhere (e.g. Visual Patterns, Which One Doesn’t Belong, Fraction Talks, Estimation 180, Would You Rather, Open Middle). There are podcasts (e.g. Tales from the Chalkline, Infinite Tangents). There are webinars and the Global Math Department Newsletter which rounds up and distills stuff from the community.
There are a number of smaller #MTBoS intiative that have happened pretty organically: A Day in the Life initiative, and the Letters to a First Year Teacher initiative, and Virtual Conferences.
And there were fun community building things, like Harlem Shake (Tweep Version) and Twittereen (and the now defunct, for those who remember, “Favorite Tweets”).
All of this is to say: for those who are interested, there are many ways to help the community. You just have to find something you love about the #MTBoS, and then come up with a way to create/share/expand it with others. (That often involves breaking the idea into smaller chunks, getting other people on board to help, and actually holding each other accountable.)
The #MTBoS doesn’t have a set of leaders. It only works because of the members. You don’t need to ask for permission. You don’t need to have been tweeting or blogging for months/years. You don’t need a “huge” project. You simply need to decide you want to do something, and do it.
That is what our session was about. We shared some ideas that we had for places the community could grow, and ways people could actually do it, and then had people share their own thoughts and ideas.
Personally, the projects I’d love to see someone take on:
(1) Department presentations: I’m all about “packaging” something to make it easier for others to use. So I’d love for a group of people to create 34 “Introduction to the #MTBoS” presentations/workshops that math teachers can give to their departments. They can be different styles/lengths, and can have different activities involved. (For example, I made my whole department sign up for the GMD newsletter. At another presentation, I made a #MTBoS scavenger hunt, where different finds/activities were worth different points.) Then, anyone who wants can choose one and adapt it to make it work best when they want to evangelize the #MTBoS to their inreallife colleagues! [Note: A number of #MTBoS presentations have been archived in the comments here.]
(2) A #MTBoS video: I saw PCMI (a math teacher conference I’ve been to) created a video to “sell” the program. I would love it if there were a #MTBoS video which captured the essence of what the community is. Maybe 3060 seconds. Something professional that evokes feelings and excitement, the emotional essence of #MTBoS, rather than outlining what it all has to offer… Capturing lighting in a bottle, that is what I suppose I’m asking for. But if this can be done well, well… I think it could serve our community well.
(3) So you want to have a tweet up…: A number of people have held tweetups by now. I think it would be good if there could be “instructions on how to organize a tweet up” — from how to find people and contact them about attending to how to find a space to hold it to what to do at a tweet up. Again, perhaps two or three different “packages” for what tweetups could look like! This might make it easier for someone who might want to organize their own tweet up!
(4) NCTM article: I’d love for someone to write an article about the #MTBoS community for Mathematics Teacher (or another NCTM journal) – to share what the community is about, how it has affected someone’s teaching practice, and to show ways for others who might be curious how to get involved. There is also a call for articles for the 2018 Focus Issue which is on Tool Kits for Early Career Teachers which I think a really wonderful article about #MTBoS could be beneficial.
I wonder if two newbie #MTBoSers and two experienced #MTBoSers could collaborate on writing it! I am personally interested in having this happen because I think it is a way to spread the word through more traditional channels, and might just pique the interest of a lot of teachers!
(5) Getting Goofy: In addition to things to expand the reach of the #MTBoS, I think there is room for so much more goofy things that can happen (today I saw a tweet that said #keepmtbosweird, copyright @rdkpickle). I don’t know what this might be, but some sort of goofy community building event like twittereen or the great hedgehog sweater run or needaredstamp. A massive picturebased scavenger hunt? A virtual trivia night? A stupid funny poster contest?
(6) Appending #MTBoS to Existing Conferences: A number of people who are going to conferences (e.g. CMC south, Asilomar, NCTM) are planning 2hour meetups with #MTBoSers. I think it could even be #MTBoSers arrive a day early or stay a day late and have a miniget together (or even a super mini conference in the hotel!). I’d love a “package” that outlines how to organize one of these meet ups.
(7) Get more contributors to the One Good Thing blog: I love the One Good Thing blog. I would love for there to be more regular and semiregular contributors. The more voices we have when talking about the joys of teaching math, the better. It has helped me out so much during my saddest and most down days, when I open the blog and see old things I’ve written. And I love reading the joyous elations that other teachers have.
I had one more idea that I have decided I am going to take on… For those who remember them… I am going to bring back Virtual Conferences. I loved the idea of them, and the person who hosted them is no longer doing them… so I’m going to bring them back from the dead!
The ideas above are things I’ve been mulling over. The ideas that came up in our meeting, or on twitter afterwards (using hashtag #ExpandMTBoS) are below (in the pictures or in the storify):
These ideas include involving Reddit, making a landing page website/app, creating a MTBoS logo, having teachers tell more of their stories, etc.
Choose something small, like presenting the community to your department or manning the #MTBoS booth at NCTM. Choose something huge, like creating your own conference, or website on (topic x), or writing a book. Or choose anything in between. But if you have the time and inclination, think of a way you can help #ExpandMTBoS!
If you have an idea of something you want to do, tweet it out with the #ExpandMTBoS hashtag. Get people to help you! And make your idea a reality!
Sara VanDerWerf (her blog) gave a keynote that was reminiscent of a keynote last year. She said “What are you an evangelist for?” (For her, one of those things is Desmos, because of the equity and access it allows her kids.) Once you know that thing — the thing you are willing to go to bat for, the thing you want to spread — you should think consciously about how to best evangelize it. That might include having an elevator speech ready for you to give, and being conscious of the different audiences you may be talking with about it (students? parents? teachers? admin?). Being an evangelist isn’t just being passionate… it includes enacting that passion by finding ways to share “the best… with others who can benefit.”
Sara’s fabulous calculator museum (mausoleum!)… all your calculators are dead… all hail Desmos!
I know I am an evangelist for the #MTBoS. However in terms of math content or math teaching, I don’t quite know what I’m an evangelist for… yet. All this reminds me the end of this blogpost I wrote last year after TMC, where I was trying to figure out what my “brand” was (and came up emptyhanded). But I have faith that with enough time, I’ll figure it out.
Speaking of evangilism… Jonathan Claydon (his blog) shared a “my favorite” about Varsity Math, a community he’s built up at his school. I’ve had a teacher crush on this guy for years. There’s something about his energy and style and humor, and the fact that he is good at something I am not (yet) good at (being a “relational” teacher)… he’s a must follow. In any case, Jonathan is an evangelist for changing the way kids look at math at his school. Although ostensibly his goal is to increase the numbers of kids taking AP math classes and increase the AP scores of these students, he’s doing it by building a supportive math community — one that feels like a club. He is doing this by creating “shared experiences.” He knows he has succeeded if he can get kids to say “I love (varsity math). (Varsity math) feels like family. You couldn’t understand because you’re not in (varsity math).” The only way the last statement could make sense is if an entire culture is built around (varsity math). Of course what goes in the parentheses is open. Read about his project here. See a photo of @rawrdimus here:
This “my favorite” spoke to me. I’ve been consciously working at my school about raising the math department. Not in terms of teaching and learning (I don’t have much say in that), but in terms of getting kids engaging with math outside of the math classroom. I brought the New York Math League contest to school, I’ve worked (with another teacher) to concertedly increase the number of students taking the American Math Competition each year (from around a dozen to seventy+). I found a nonstressful virtual math team competition that students can compete in so that they can fit it in their busy schedules. I have coadvised math club for years. I started Intersections with a science teacher, a mathscience journal for students to submit their works to (it’s now four years old!). Lots of things… I want spaces and times for students to engage in math outside of the classroom. But with all of this, I don’t see a culture of kids who geek out about math. There isn’t a community or culture around doing mathematics at my school. And Jonathan’s talk helped me realize that I have to think intentionally about building a community. It is more than “if you create it, they will come.” It isn’t the event or space that I design, but the “shared experience.” What does this mean? What does this look like? I don’t know yet. But perhaps having a studentcreated chant before each virtual math team competition, bonding field trips (math movies? museum of math? math scavenger hunt?), swag as proud identifiers, a wall of fame…
[Update: I was having trouble figuring out what precisely I want to accomplish in my school. And today is day 3 of a crazy math frenzy day where I’m having fun exploring and writing lesson plans and playing around and coding and getting stuck and getting unstuck and having frustration and elation — so much elation. And then I read this post by Annie Perkins, which talks about a sortofcrisis I’m having (posts here and here). And in my current haze, I see the glimpses of what I want to achieve. Why do I want kids to engage with math outside of the classroom? Because it’s beautiful and fun to play with and just play mindblowing cool. But they don’t get that in the classroom — at least not regularly enough. Jonathan created a community of kids who were vested in AP math. I think I want to figure out how to create a community of kids that love to (a) be exposed to interesting/strange things about math, and (b) play with math and explore it. Less “math team tricks” and “competition problem solving strategies” and more pure unadulterated fun. Things like this fold and cut problem that I did in geometry. Or generating and analyzing their own fractals. Taxicab geometry. And I think lecture might be okay for some of this — a lecture on infinity or Godel’s incompleteness theorem. Or following some internet instructions on how to build a planimeter out of a sodacan to calculate the area of a blob just by tracing around it. Or going as a group to a math lecture at the Museum of Math. Or learning about higher dimensions. Whatever! I want to get kids to geek out about how cool and fun math can be. I want a math is cool community, where there is a culture of nerdsniping and geeking out and regular mindblowingness. The truth is I probably don’t have time this year to come up with a plan to execute this to make it happen this year. I also think that the lack of free time that kids have in their schedules might make any plan of mine totally impossible. But I think it’s worth brainstorming… maybe not for this year… maybe for next year.]
At the Desmos preconference, I learned about three things
(1) “Listening to graphs.” This feature was included for vision impaired students, but I think many of us teachers started dreaming up other uses for it. To get a sense of it, check out this piece (done by Rachel Kernodle and James Cleveland) playing “Mary Had A Little Lamb” (click on image):
To play (at least on a mac), press COMMAND F5 (which enables voiceover), go to the fifth line and press OPTION T (to tell the computer to “read” the graph with sound), and then press H (to play the graph). When it’s done, you can turn off voiceover by pressing COMMAND F5 again.
Some thoughts… Have the audio for some periodic and non periodic functions, and have kids do an audio function sort? Play audio of graphs (without telling kids that) and have kids do a notice/wonder (before sharing what they are listening to). Have kids identify if a graph has a horizontal asymptote for end behavior from an audio file? Have kids identify which graphs might have a vertical asymptote from an audiofile? Play sine and cosine (or secant and cosecant) and have kids not be able to tell which is which (because they are just horizontal shifts of each other). Have kids devise their own piecewise functions and play them, while other kids have to graph them. Create a piecewise function and have a student who enjoys singing to sing it? I am not convinced that anything I’ve thought up could help a deeper understanding of any topic, but I also don’t think it could hurt. Some kids might really get into it and enjoy playing with math…
(2) Card Sort: You can create card sorts in desmos now! Check a bunch of them out (that were created at the Desmos preconference)! Or if you just want to go to one of them, click on the image of Mattie Baker’s card sort on visual sequences:
To gain this functionality on your desmos account, go to teacher.desmos.com and click on your name in the upper right hand corner, click on LABS, and then turn on Card Sort.
(3) Marbleslides: You can create your own marbleslides in desmos also! Turn it on in labs (see above). Then you have the capability of building your own! If you don’t know about marbleslides, check out this marbleslides activity made by the desmos folk on periodics. At least to me, the use of marbleslides is to help students understand function transformations… so I can see it useful for helping kids gain fluency in transformations. (Anyone see another use for marbleslides, that I’m missing?)
Hedge talked about how she uses SnagIt to display student work. She takes a photo of student work on her phone, and using an app called FUSE, transfers it to SnagIt (on the laptop) — as long as both are on the same wifi network. Here’s her blogpost showing it in action! It costs money ($29.95) but I trust Hedge!
I attended PCMI years ago, and I recall Bowen and Darryl using this technique (kids working on problems, taking pictures of different approaches) to facilitate discussion to bring different ideas together. Nearing the end of a session, they would project pictures of student work, people would explain their thinking. Bowen and Darryl would sequence the pictures in a thoughtful way. They wouldn’t focus on those who “got the answer” but on various approaches (visual/algebraic) — whether they worked to get the answer or not. I liked that so much, and I suspect SnagIt could allow that to work for me in that way.
Fouss gave a wonderful hour long session on making trigonometry handson for students. Instead of telling us what she did, we got to do some of the activities, and that was powerful. There were activities I’ve read about that I thought “eh, okay, but it would be more efficient to do X, Y, and Z” and then I did them and I saw how the act of doing them could be helpful. Here are three that we got to do: understanding radians with smarties, creating a unit circle with patty paper, and creating a trig wheel to help kids practice converting between radians and degrees and visualize what the size of the angles look like.
All her materials are linked to from her presentation, and are easily found on this folder on her google drive. I have to scour them to find my favorites. I did love the radian activity. If you make the radius of the unit circle 7 smarties long, then you can have a good discussion on whether 3 radians is 180 degrees or not… (21 smarties won’t quite make it to 180 degrees… but 22 smarties will fit snugly… nicely giving the 22/7 approximation for . Nice!)
Some of the ideas linked to from her presentations that I want to steal:
(a) Trig Stations
(b) Two Truths and a Lie (useful for more than just trig!)
(c) #TrigIs (useful for more than just trig!)
(d) If I choose to do ferris wheel problems, this ferris wheel comparison [but modified to be more challenging]
(e) Desmos’s Polygraph for Sinusoids and Marbleslides for Periodic Functions
(f) If I teach trig identities, use this matching game (and have kids check their answers once they are done by graphing on desmos!)
(g) Headbandz, trig edition! (for graphing trig functions)
Joel Bezaire presented a great game that can be used in warmups to help students see relationships and patterns. His video on it is here, showing the game and how it is played:
Kathryn Belmont (@iisanumber) gave a great way to have kids really put forth effort on openended assignments without using grades as a stick. She will ask kids to do this assignments, and then put their work on their desks. Each student gets posts its, and as they wander around the room, they put postits on the works they see… They write two accolades for good things, and two ways to push back or improve the assignment. The way I envision this in my classroom, not everyone will see everyone else’s work, but everyone will see 56 other students’s work. After the walk about, the teacher says: “Do you have any nominations”? Jake might reply “I would like to nominate Kiara.” If Kiara feels okay about being nominated and “accepts the nomination,” the teacher takes Kiara’s work and puts it under the document camera. Then Jake might say, “Kiara did … and what I thought was so awesome about it was …”
(Her slides for her minitalk are here. A video of her talk is here.)
The teacher is no longer the sole audience member for the work, and kids are defining what good work looks like. In Kathryn’s classroom, she saw a huge increase in kids putting in effort in these openended assignments. (I can see this being useful in my own class, especially when I do my explore math miniexplorations.)
I went to a session by Jessica Breur (@BreurBreur) which was fantastic. Although it was only one hour, I wish it were a morning session. She wants to have teachers establish a culture where students:
She highly recommended Cohen’s Designing Groupwork (a book which I have but haven’t read).
To start, over the first week or two, students will be doing lots of groupwork activities. And at the end of them, they will (in their smaller groups) focus on what the group “looks like” “sounds like” and “feels like.” They don’t necessarily need to focus on all three at once — students could focus on “sounds like” during one activity and “feels like” on another. After the week is done, the class comes up with a set of norms in these three categories — where they talk about what successful/good/fun groups look/sound/feel like.
We did a lot of handson work trying out some of these groupwork activities — and she has included all of those activities in her slides. Here is one of my favorites:
This is the red solo cup challenge. A group of 3 or 4 is given 6 red solo cups, stacked inside each other, placed face up on the table (so like a regular drinking up face up). The students are given a rubber band with four strings tied to it (even if 3 students are doing this, keep the four strings). Student must put the solo cups in a pyramid formation. If they finish that, there are other configurations that Jessica includes in her presentations (or students can design their own challenge for others!). Afterwards, the group reflects.
Similar tasks can be done, like 100 Numbers, Saving Sam, Four 4s [but making an emphasis that we want as many ways to generate the numbers 120, not just one for each], Master Designer, or Draw My Picture.
For more “mathy” things, you can do a Chalk Talk/Graffiti Board– where students answer questions before a unit to activate some old ideas. For example, “What do you know about the number zero?” [In fact, any sort of talking point/debatey statement can be used here.] Kids write anything and everything they know on a poster in their group of four. Then hand the posters up and students walk around and read other students’ responses (if time, writing their own comments down). Finally, for closure, you can ask students aloud or using exit slips “What are two things you didn’t think about that you saw on the graffiti boards?” Another more mathy thing is a donut percent task. An example is here but I’m confident it could be modified for trigonometry (values of trig functions, identities, etc.) or rational functions (equations and graphs) or any number of things! The idea behinds this is that each person in the group is given four slips of paper, and as a group, four complete donuts have to be created.
Sounds simple? But here’s the rub… group members must follow the rules below to each get their own donut completed.
You should keep a poster of the 8 Standards of Mathematical Practice, and every so often during activities or groupwork, ask students which ones they are using.
Once norms are established at the start of the year, you consciously need to be doing activities that practice the norms. Be intentional about it. (If you find that kids aren’t listening to each other, find an activity that promotes listening.)
I loved this session. However what I need now are a set of activity structures that I can fit actual mathematical work into. So things which develop understanding, or practice solving something, etc. And it would be nice not only to have the activity structures, but the activities themselves all in one place (so, for example, activities for Precalculus!).
My morning session was called “Talk Less, Smile More” and was led by Mattie Baker and Chris Luzniak. In the session, they provided various structures to promote math talk in the classroom. I am going to outline some of the ideas that I can see myself using in my classroom.
DEFENSE MECHANISMS & CLASSROOM CULTURE: Most importantly, to get talk in the math classroom involves getting over student defense mechanisms. Students fear being seen as stupid, and they fear being wrong. In order to do this, you have to lower the stakes so kids can temporarily bracket their defense mechanisms to create emotional safety. These could be by doing things like chalk talks (silently writing responses to questions, and responding to other student responses) or doing notice/wonder activities where all responses are honored. Many of the ideas that Chris and Mattie shared in the session do this, by providing a structure for talking, and a bit of a safety net (often where no response is right, or students are required to give a particular answer and justify it).
When implementing it, you have to be consistent and do these structures fairly often. Start simple, and then get more complicated with the statements/questions. Give a lot of energy and excitement — especially if a student gives a wrong answer or a right answer (“Oh wow, what an interesting thought… let’s explore that…”). If students turn to the teacher and say “Mr. Shah, what about…” sit down and redirect it to the class. (Remember the teacher is not the center… this is about getting kids to be the center!) As teachers, we have to watch our own facial expressions (a.k.a. don’t make a face when you hear a totally wrong answer). You can avoid this (if it’s a problem for you) by looking down at a clipboard when someone is responding.
At the end of a class or a portion of a class with a lot of mathematical talk, do “shout outs” (shout out something they learned, or something someone else said that helped them). And ask kids (to fill out on a card) what they took away from class today (and what questions they still might have). Or “I used to think ____, but now I think _____.”
To give students some crutches when talking, have posters with these simple statement starters to help them (on all four walls):
TALKING POINTS: In this session I first got to experience Talking Points. I’ve read about them on Elizabeth Statmore’s blog (see links on the right… a bunch of talking points are hosted in one of her google drive folders). But the truth is: I wasn’t sure how much I could get out of them. Now that I’ve participated in one, I feel differently. This is how they work:
(1) students in a group of 4 get n statements. The first round involves one person reading the first statement, and then say “agree/disagree/unsure” and then explain why they chose that response. They must give the reason. The next person does the same, then the next, then the last. The important part about this is that no one can comment on another person’s reasons. They can just state their own reasons. They can match someone else’s reasons, but they have to be stated as their own.
(2) The second round involves the first person saying “agree/disagree/unsure” (after hearing everyone else’s thoughts) and then they can give reasons involving other people’s thoughts. Others do the same.
(3) The third round is quick and short. Each person says “agree/disagree/unsure” and gives no reasons. Then someone records the tally of the responses.
Here’s an example of what talking points can look like (when they aren’t about math content):
Talking points can also be math content related. Instead of “agree/disagree/unsure,” you can use “always/sometimes/never” or some variation that works for your questions. In our miniprecalculus group, we brainstormed some talking points around trigonometry:
After participating in talking points, we as a group came to the following realizations:
To debrief:
CLAIM AND WARRANT DEBATE: In a math class, you want students to justify themselves. To build that justification as central to the class, you can introduce the notions of an argument which is essentially a statement (a claim) made with sound reasoning (a warrant). (This language comes out of the speech and debate world.)
When responding to a question, a student must stand up (even the teacher should sit down) and say “My claim is _______, and my warrant is ________.” If the student messes up, that’s okay, just have them do it again. You have to build this structure as essential to answering questions. (To reduce the fear, you can give students some think time to write something down, or talk in a pair, before doing the claim/warrant step.) When doing this, I am not going to have kids volunteer… I am going to cold call using the Popsicle Sticks of Destiny (names of kids on popsicle sticks… I draw one randomly…).
When introducing claim/warrant, make sure you not only teach the structure, but also have kids who aren’t speaking face the speaker and put their eyes on them. Be explicit about the expectation. You can also have kids summarize another student’s point to make sure they’re paying attention. (If you catch a kid not following the audience instructions, you can walk over near them… if not, you can tap them on the shoulder… or kindly talk with them after class about how “it’s really polite to…”)
To build this up and create this as a routine and class structure, you should do claim and warrant debates every day or every other day at the start of the school year. Use the language “claim” and “warrant” on assessments too!
Types of questions you can ask to get kids started with this:
The best movie is ______.
The most important math topic is ______.
________ is the best method for solving the system y=2x and y=x+1.
[show a Which One Does Belong and say] ______ doesn’t belong.
Notice that each of these don’t have a “right” answer. It lowers the barrier of entry for kids.
One powerful type of question one can create are “mistake” questions. For example:
To extend claim/warrant, you can also create “circle debates” which truly forces listening. One person states a claim/warrant, and then another person summarizes that claim/warrant and then makes their own claim/warrant. This continues. It will sound like: “What I heard is that this statement is sometimes true because …. My claim is ____ and my warrant is ____.” I think only very open ended questions would be good for this structure.
Another powerful way to extend claim/warrant is to engage is “pointcounterpoint.” Let’s say the statement is: “Would you rather have crayons for teeth or spaghetti for hair?” The first person makes a claim/warrant, and the second person (no matter their true feelings) must disagree and make the opposing claim and give a warrant. Then the third person opposes the second person. Etc. It forces students to think of other points of view. In a question like “_____ is the best way to solve this system of equations” it forces students who might only approach a system in one way to consider other methods and justifications for those other methods.
CREATING DEBATEY QUESTIONS/STATEMENTS: Use the following words:
In the session, we took all types of questions (e.g. Graph ) and came up with debatey questions based on it (in this example, we said “what number would you change to change the graph the most?” or “what’s the best way to graph a sine function?”). I’m not yet good at this, but I found that even with a little practice and people to bounce ideas around, I’m getting better. We had fun in my group trying to come up with debatey questions based on this random “do now” that Chris and Mattie found online:
I thought it would be impossible, but the group came up with tons of different ways to convert this to a debatey statement: (a) without solving, which is easiest to solve? (b) which would you give to your worst enemy? (c) which are similar? (d) rank from easiest to hardest? (e) a 5th problem that would fit this set of equations would be ____ (f) a 5th problem that would not fit this set of equations would be ______ (g) which one doesn’t belong? (h) give 4(x+3)=6 and ask what the most efficient way tot solve it? and then follow up with “how could you change the problem so that method is not the most efficient?”
After a month or two, the use of claim/warrant may die down. If kids get the idea and are justifying their statements, that’s okay! It’s not about the structure as much as the idea behind the structure!
QUICKWRITE: I love this idea because I make writing integral to my classroom. You give kids a prompt and you tell kids to write nonstop for 2 minutes without editing. They have to continually write. Examples:
It can help with vocabulary, but most importantly, I see this as a way to get kids to stop overthinking and looking for “the right” answer, and just write down anything and everything without selfediting of their thinking. It’s like a condensed noticing/wondering done individually. I can be used before a debate — to give kids time to think. Or perhaps depending on the question, kids can “shout out” one part of their quickwrite? But doing it at the start of the year — to help kids get comfortable writing in math class in an nonthreatening, nonevaluative manner — is such a great idea!
RUMORS: This idea was stolen from Rona Bondi at alled.org. On a notecard/paper, everyone write a response to a question or a couple questions (the one we used is “what is our idea setup of our classroom?” but I think it could be used at the end of class with questions like: “One thing I find easy to understand in this unit is… One question I still have about this stuff we’ve been working on…?” or “The most important mathematical idea from today is …?” or “The best way to approach graphing trig functions is…”).
After everyone is done writing, everyone finds a parter and reads their card, the other person reads their card, and then they discuss. There is a time limit (maybe 60 seconds). Then they swap papers. Everyone finds a second partner, and they read the card in their hand to the other person, and they discuss what is written on those cards (not their own cards) and then swap. This goes on three or four times. This forces listening, it allows ideas to slowly spread, and the papers can be kept anonymous.
ONE INTENTIONAL MISTAKE: [update: a la Kelly O’Shea] Each group of students gets a giant whiteboard and a problem (it could be the same problem as other groups or a different problem). They are asked to solve the problem making one “good” mistake (so nothing like spelling names wrong, transposing a number, or labeling the axes wrong). They then present their solution to another group — playing dumb about their mistake. The other group should ask good questions to help students get at the error. Questions like “don’t you need to add 3 to both sides” is too direct… You need to ask questions which lead the group to see and understand the mistake. So perhaps “what is the mathematical step you used to get from line 2 to line 3, and why is it justified?” might be better.
So without further ado, this is my letter from this year.
***
May 27, 2016 – June 2, 2016
Dear STU,
It’s Friday evening, 9:53pm, and I’m at home listening to Kurt Cobain and his guitar. I know what you’re thinking, and sorry, nope: no showtunes today. Nearby is the book I just put down. It takes place in the nineties, the U.S. is entering a protogrunge phase, and Nirvana is a recurring theme. The nineties is also when I was in high school and so every so often — usually when one chapter ends and I take a mental pause to regroup — I’ll get flashes of forgotten high school memories. You see, I have a terrible memory. It’s almost comical how much I don’t retain. Almost. So those moments where some feelingrich memory is drudged up — the heartpounding anticipation of a wildlyliked senior picking up friendless newtotown sophomoreme in his car to go to a mock trial practice, or the awe of being perched on the roof of a house with a friend where every word carried into the night sky crackled with deeper meaning — I let them wash over me. Recalling them with any vividness get rarer and rarer as the years pass. (That’s something no one tells you about growing up. Your experience of the world dulls — from vibrant neons to faded pastel watercolor. Your memories become mottled with gaps, like a desiccated leaf chewed up by hungry pests.)
Why am I telling you this? As I now reminisce about me in the nineties, I know you are reminiscing about your lives too. Packer will become a temporary line on your resume, and then — soon into your working lives — not even that. (No one includes high school on their CVs.) You’re moving on, growing up, and you’re losing something and gaining something. You are adults and you are not adults. You are who you are and you are not (yet) who you are.
As you know, in physics there is a wave function. It’s a probability function describing all the possible states of some system. For example, is a particle here or there or waaaay over there? And — here’s the kicker — that wave function is the best that we can do to describe things. The system isn’t knowable in any better way. The function within it has all these possibilities, some more probable than others but still, oh so many possibilities. “How many?” I imagine asking you one day in S202, and in unison I hear you all replying “Infinite!” And left alone, the infinite possibilities undulate in time, directed by Schrodinger’s equation. Until one instant it isn’t. It collapses. All possibilities reduce to one actuality. Why? How? The why is easy: someone tries to find out more about the system… a measurement is taken. (A box is opened to peek at the cat.) And in that measurement the wave — and all the possibilities — is destroyed. (The cat is either alive or dead.) The how is harder: how does a collection of probabilistic states turn into a single state? That it happens is known, when it happens is known, but how it happens is unknown.
You — right now — are infinite possibilities spread out before you. Right now, you can’t even know what they all are, but they exist. The way you move through the world, the choices you make, the person you strive to be, those all shape the landscape of those possibilities over time.
Like you perhaps, I had grand designs when graduating high school. There were so many things I wanted to accomplish, so many things I wanted to learn. But one thing I did know — the thing that had the largest chance of becoming true — was that I wanted to become a high school math teacher. I truly never know if that nugget surprises students when I share it with them. I always think it does, because in my time at Packer, I’ve only had one student tell me they wanted to be a teacher (and now they are!). But here’s the thing: even then, I knew I loved math. Not in a small way, but in a way where I could work on problems for weeks and be in pure bliss. In a way that when I figured something out, I would force my poor mother listen to me outline how I cracked the mathematical nut — even though she had no idea what my excited explanations were all about. I wanted desperately to share with the world that feeling, of the frustrating and seemingly intractable journey ending in deep insight and a joyous satisfaction. I couldn’t not share that love with others! I wanted others to have that joyous satisfaction too.
I told my teachers this. And one — the one who looms larger and larger as I get further and further away from high school — got this about me. It was Mr. Parent, my junior and senior year English teacher. He occupies a special place in my limited memory because he was the first person I met who truly and fully embodied the life of the mind. The engine that drove this man was intellectual curiosity, and to bear witness to that sort of person – and his unbridled passion – had a lasting impact on me. At the end of my senior year I bought him a book and wrote him a letter explaining how much he meant to me. In that letter, I offered up a quotation by Richard Feynman, physicist and boyhood hero:
I was born not knowing and have had only a little time to change that here and there.
If someone asked me what I wish for my students, I would answer with a pat: “to be good, and to be happy.” I can’t speak to being good part. That’s for you to figure out. But I suspect for you seven, because in you I see parts of me, one path to lasting happiness is to continuously follow your intellectual curiosity. That is our common bond, and one that I have been grateful to have had the opportunity to bear witness to from the first day of class until the very last day. Because we share that, I hope that you remember in the most bleak of days: there is something magical about the world around you. Keep an eye out for the magic. It appears as questions… and there are so many questions! How can we – billions of years later – know about the earliest moments of the universe? Where does matter come from? How can the world be probabilistic (quantum) in nature when everything feels so causal? How do we know about the smallest worlds we cannot even see? Why are there rainbows on the surface of an oil spill? How do rubber bands work – how do they come back to their original shape? How can we – on this planet – know how far things are, and that there are other galaxies out there? How is it that the natural world somehow can be encoded through simple and elegant mathematical formulas? Does that imply that math is somehow encoded in the universe, and it is being discovered rather than invented? Does the fact that we keep on digging in mathematics and are still drawing connections among disparate subfields imply that there is some grand unifying structure undergirding everything mathematical and physical?
Mr. Parent walked up to me on my graduation day and handed me a letter in return — a letter I treasure to this day, keeping it ensconced between the pages of my yearbook. In response to Feynman, he returned one of his own devising: “Stephen Hawking speaks of the thermodynamic, psychological, and cosmological arrows of time that define existence as entropic movement from past to future in an expanding universe. And that seems to define the hero’s journey: the personally expanding possibilities revealed in a courageous life bounded by and aware of entropic time.” I personally read this as an intellectual quest: you – dear students – are in a world that is growing in knowledge and is constantly reshaping itself around you. And you – dear students – have only a lifetime to enjoy it. And I mean “only a lifetime” because the world is vast and time runs short.
As you quest, don’t be afraid of failure. Let failure be a marker of pride, because you tried. You know me, I don’t know much about sportsing, but I do know that you miss 100% of the shots you don’t take. Set the bar slightly higher than you think you are capable of achieving and work extraordinarily hard. Harder than everyone else around you.
Wave functions collapse. But the possibilities of our lives only collapse when we are no more. You are an infinity of possibilities, remember that. You have so much time, and so little time. Make it meaningful.
Always my best, with sincerest best wishes,
Sameer Shah
Another teacher and I started this journal four years ago (this is the fourth issue). And each year we took less and less of a role, as we trained leaders, taught them to organize themselves, and got them to look for new members. Next year, though, we’ll have to be more hands on because we have only a few people who were on the staff for one year only. The new leaders have some great ideas for next year!
With that, I’m out. Hey, I did say it was a super short blogpost.
Earlier in the year, I showed my student a video of an augmented reality sandbox that I stumbled across online. She showed interested in making it. It takes in a mapping of a surface (in this case sand in a sandbox) and projects onto the surface colors representing the height of the sand over time (so red is “high” and blue is “low”). The cool part about this is that the projection changes live — so if you change the sand height, the projection updates with new colors. Level curves are also “drawn” on the sand.
Here are some videos of it in action (apologies for the music… I had to put music on it so the conversations happening during the playing with the sand were drowned out):
The student was going to design lesson plans around this to highlight concepts in multivariable calculus (directional derivative, gradients, gradient field, reading contour maps) but ran out of time. However upon my suggestion, during her presentation, she did give students contour maps of surfaces, turned off the projector, had students try to form the sand so it matched the contour map, and then turned the projector on to have students see if they were right or not.
During the presentation, one student who I taught last year (but not this year) said: “This is the coolest thing I’ve seen all year!” and then when playing with the sand: “I AM A GOD!” Entrancing!
In my first year of teaching this course, a student was entranced by lissajous curves when we encountered them. These are simple parametric equations which create beautiful graphs. I then suggested for his final project that he create a harmonograph, which he did. Seven years later, I had another student see the original video of my student’s harmonograph, and he wanted to build his own! But he wanted his to have a rotary component, in addition to two pendulums which swung laterally. So he found instructions online and built it!
Here are some of the images it produced:
And here is a video of the harmonograph in motion:
(You can watch another video here.)
During the presentation, the student talked about the damping effect, how the pendulum amplitudes and periods had an effect on the outcome, and how lissajous curves were simply shadows of lissajous knots that exist in 3space. Because of the presentation, I had some insights into these curves that I hadn’t had before! (I still don’t know how mathematically to account for how the rotary pendulum in the student’s harmonograph affects the equations… I do know that it has the harmonograph — in essence — graph the lissajous curves on a somewhat rotating sphere (instead of a flat plane). And that’s interesting!
A student was interested in creating tools for teachers to illustrate “big” multivariable calculus ideas… Contour lines, directional derivatives, double integrals, etc. So she made a set of five of super awesome teaching manipulatives. Here are three of them.
The first is a strange shaped cutout of postercardboardish material, with four animals hanging from it. Then there is string connected to a magnet on top, and another magnet on the bottom. If you hold up the string and you aren’t at the center of mass, the mobile won’t balance. But if you move the magnet around (and the student used felt around the magnet so it moves seamlessly!), you can change the position of the string, until it balances. This is a manipulative to talk about center of mass/torque.
Another is a set of figures that form “level curves.” At first I was skeptical. The student said the manipulative elow was to help students understand countour plots. I wanted to know how… Then the moment of genius…
You can change the height of the level curves to make the “hill” steeper and steeper, and then look straight down at the manipulative. If you have a shallow “hill,” you have contour lines which will look far apart. If you have a tall “hill,” you have contour lines which look close together.
Finally, a third manipulative showcases the tangent plane (and it can move around the surface because of magnets also). I can see this also being useful for normal vectors and even surface integrals!
Two students decided to work together on a project dealing with cartographic mapping. They were intrigued by the idea that the surface of the earth can’t perfectly be represented on a flat plane. (They had to learn about why — a theorem by Euler in 1777.) They chose two projections: the Gall Peters projection and the Stereographic projection.
They did a fantastic job of showing and explaining the equations for these projections — and in their paper, they went into even more depth (talking about the Jacobian!). It was marvelous. But they had two more surprises. They used the 3D printer (something I know nothing about, but I told them that they might want to consider using to to create a model to illustrate their projections to their audience) and in two different live demos, showed how these projections work. I didn’t get good pictures, but I did take a video after the fact showing the stereographic projection in action. Notice at the end, all the squares have equal area, but the quadrilaterals on the surface most definitely do not have equal area.
An added bonus, which actually turned out to be a huge part of their project, was writing an extensive paper on the history of cartography, and a critical analysis of the uses of cartography. They concluded by stating:
We have attempted, in this paper, to provide our readers with a brief historical overview of cartography and its biases. This paper is also an attempt to impress upon the reader the subjective nature of a deeply mathematical endeavor. While most maps are based around mathematical projections, this does not exclude them from carrying biases. In fact, we believe there is no separation between mathematical applications and subjectivity; one cannot divorce math from perspective nor maps from their biases. We believe it is important to incorporate reflections such as this one into any mathematical study. It is dangerous to believe in the objectivity of scientific and numerical thought and in the separation between the user and her objective tools, because it vests us, mathematicians and scientists, with arbitrary power to claim Truth where there is only perspective.
Beautiful. And wellevidenced.
One student was interested in fluid dynamics. So I introduced him to the Navier Stokes equations, and set him loose. This turned out to be a challenging project for the student because most of the texts out there require a high level of understanding. Even when I looked at my fluid dynamics book from college when I was giving it to him as a reference, I realized following most of it would be almost impossible. As he worked through the terms and equations, he found a perfect entree. He learned about an equation that predicts the change in pressure from one end of a tube of small radius to another (if the fluid flow in the tube is laminar). And so using all he had learned in his investigation of the field, he could actually understand and explain algebraically and conceptually how the derivation worked. Some of his slides…
It was beautiful because he got to learn about partial differential equations, and ton of ideas in fluid dynamics (viscosity, pressure, rotational velocity, sheer, laminar flow, turbulence, etc.), but even needed to calculate a double integral in cylindrical coordinates in his derivation!
This student works in a lab for his science research class — and the lab does something with lasers and quantum tunneling. But the student didn’t know the math behind quantum mechanics. So he spent a lot of time working to understand the wave equation, and then some time trying to understand the parts of Schrodinger’s equation.
In his paper, he derived the wave equation. And then he applied his understanding of the wave equation to a particular problem:
He then tackled Schrodinger’s Wave Equation and saw how energy is quantized! Most importantly, how the math suggests that! I remember wondering how in the world we could ever go from continuousness to discreteness, and this was the type of problem where I was like “WHOA!” I’m glad he could see that too! Part of this derivation is below.
Overall, I was blown away by the creativity and deep thinking that went into these final projects. Most significantly, I need to emphasize that I can’t take credit for them. I was incredibly hands off. My standard practice involves: having students submit three ideas, I sit down with students and help them — with my understanding of their topics and what’s doable versus not doable — narrow it down to a single topic. Students submit a prospectus and timeline. Then I let them go running. I don’t even do regular formal checkins (there are too many of them for me to do that). So I have them see me if they need help, are stuck, need guidance or motivation, whatever. I met with most of them once or twice, but that’s about it. This is all them. I wish I could claim credit, but I can’t. I just got out of their way and let them figure things out.