The Cat in Numberland is Back in Print

One of my favorite mathy kids' books is back in print. In The Cat in Numberland we visit Hotel Infinity, with its infinite rooms, all full and yet they always seem to be able to make room for new guests.

Good for ages 5 to adult. (It's $19.95 plus over $9 shipping. I think they should charge less for this slim volume, but this book is so wonderful, it's worth it.)

Fun Question from Michelle at Prairie Creek Community School

"Recently in math, we were working on the Deka Tree, a tree that has 10 trunks with 10 branches with 10 twigs with 10 leaves.  One trunk, one branch, one twig, and one leaf is cut off...how many leaves are left?"

I found this question in Michelle's post about doing Forest School. And she talked more about it in this post.


I have over fifty tabs open with interesting goodies. I hope to find time soon to sort them out and share...

Calculus: From Secant Lines to the Tangent

Our semester began on Monday. I'm teaching Calculus I (as always, because it's my favorite class), Statistics, and Algebra for Statistics. All three classes were a joy to teach this week, even though I was a bit underprepared because of the chaos in my personal life.

On Thursday I was working on wrapping up the exercise from Active Calculus that the students had been working on since Tuesday. I showed the velocity curve we'd been exploring on Desmos, and limited the domain to the appropriate times, 0 to 3 seconds (which I learned how to do with the face project I described in my previous blog post). I had a little trouble remembering how to make a secant line attached to one stable point and one moving point, but I got it. (And helped the students get it. This took some hard thinking for many of them.)

Then I had a wonderful surprise. When I pulled the moving point over the stable point, the line disappeared and Desmos said "x= undefined or undefined" (not sure where their stutter came from...). I gasped. I hadn't expected that, and it was a perfect way to start talking about this problem calculus has of needing two points to figure slope, but needing to use just one point from the function in order to have a tangent. I got to talk about Newton and Bishop Berkeley and fluxions and infinitely close. It was great fun for me. On Monday I'll find out how much the students got out of it.

Calculus: Reviewing Functions with Desmos

I like to dive into the calculus ideas in my calc I course, so I do not start with review. (I use a just-in-time approach, reviewing what we need when we need it.)

But I know that the students' understanding of functions is weak and needs to be brought to mind. So I was excited about having them outline their own faces in Desmos as a homework assignment, which I learned about at Twitter Math Camp from Deb Boden (@debboden).





Here are my instructions:

Desmos Graph of Yourself

1. Set up an account on desmos.com. (It’s free.)
2. Upload a selfie into desmos. (Click the + in the upper left corner of the desmos calculator screen to add your image. Photos with you facing front are easiest to use.)
3. Use various functions to outline features of your face. (At least: lines, arcs of circles and ellipses, parabolas, and trig functions. Try including exponential and log functions, hyperbolas, and cubics.)
4. When you’re done, you can hide your photo to display the icon you’ve created. You can also hide the axes by clicking on the wrench in the upper right corner.
5. Add a link to your completed desmos work on our class google doc: [Link removed, for student privacy. I didn't need google, actually. They could have submitted directly to Canvas. But we may need the google doc for our viewable collection.] (My icon is linked there. You can check it out to figure out how to do this.)
6. We’ll share these in class and see how many classmates everyone can recognize.

Every time I got stuck, I googled my question. For example, "Desmos function restrictions" helped me make short pieces of the curves I used. If you are still stuck, start a discussion item here.


Rubric
50% Required function types  (lines, arcs of circles and ellipses, parabolas, and trig functions) 10% each (Extras can bring this score up to 60%)
15%   Good Match with Photo
15% Visually Engaging
20% for using Desmos and Canvas (our "learning management system")

(I didn't post the rubric until after they turned in their work, but I will next time.)

31 out of 44 students turned it in. I am loving Canvas, which our college just started using. (We used d2l before and I hated it. Yes, I have strong feelings about things.) I took hours grading this, but once I get good at it, I think I could do this in about an hour. Canvas made it easy. And now I know that a third of my class is having trouble. So I know I need to intervene somehow. Good information to have.

Here are some of my favorites...

My Favorites: Becoming Invisible & Math Relax

On the last day of Twitter Math Camp 2016, I got to do a ten-minute presentation about two of my favorite teaching ideas. This is a quick way for me to share the links.

Becoming Invisible  is a great collection of things you can say when you're trying to hand the floor over to the students.

Math Relax is my audio track to help students get over the anxiety some of them feel during math tests. After my talk, people mentioned some lovely anxiety-busters, including giving the student a nice stone to play with. I might just make a basket of magic calming stones...

Kahoot

I heard about Kahoot from my colleague, who heard about it from his wife who teaches third grade. It's a game site with lots of content already available.  I looked up logarithms yesterday, found a kahoot* I liked, and played it with my pre-calculus class.

[To find a kahoot you like, choose Public Kahoots in the black bar at the top, search on a term like logarithms, click on"Only show Kahoots made by teachers?", and search the list. I've been looking for the ones with high counts on the favourites list, but I might find better criteria later. Once you find one you like, favorite it right away. There doesn't seem to be an easy mechanism to get back to it later.]

We are about 2/3 rds of the way through the semester. The energy is a bit low about now. This game livened things up and kept us focused on mathematical ideas. The students loved it.

This evening, I made a pretty simple kahoot to go along with my murder mystery, which we're starting in precalc right now. I'll use this kahoot next week, when we're farther along in the murder mystery.



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*A kahoot is a gamified quiz. Each question is set up with multiple answers. Students use a pin shown on the screen to sign in using their phones. They get points for right answers based on how quickly they answer.

A Huge Bunch of Lovely Links

I have so many tabs with cool math posts, I don't know if I can possibly get them all into this collection. (I never seem to have enough time to finish, and then more goodies accumulate.)


Math & Teaching Ideas I might use

• I want to show this to my pre-calc class as part of their intro to trig: sailboat leans to get under bridge
• Wild Maths has a lovely collection of questions with photos: Move it to prove it
• Snug angles, from Sam Shah. He's doing it with a geometry class. I'm wondering if it might be good at the beginning of the trig unit.
• As I prepared for Math Jam, our 3-day pre-semester math boost, I found lots of cool ideas here: Math Assessment Project (assessment doesn't sound promising, but these activities have lots of open-ended questions)
• I just learned that if you start with the harmonic series, which diverges, and take out all the terms with 9s in the denominator, you'll get a converging series. Too weird. I don't understand it yet, but I sure want to. (for Calc II)
• Geometric construction on sciencevsmagic.net. (I knew about this site, but reading this post made me decide to use it in Math Jam to get them playing around.)
• Mobius Hearts. Too fun not to do. (I didn't use it, though. Too overwhelmed this past month to do anything new...)
• Sam Shah made this fabulous website, Explore Math, that pulls together gobs of cool math resources from the web. He has his students pick one (or was it a few?) to play around with and report on. I believe I'm going to do this in pre-calc.
• What is proof? Here's a good conversation about proving the Pythagorean Theorem with visuals. Includes the best video I've ever seen, on my favorite proof.
• Trig Fairy Tales (having students write them) 
• Infinity is so weird! (infinite ping pong balls in, infinite ping pong balls out, how many left in?)
• Infinite sums and China's demographics
• Algebra Aerobics Stick Figure in Geogebra

Problem Solving

• Finding out how far you can drop an egg without breaking it
• Systems of equations, using a problem with no solution
• On problem solving, with videos. I might give the absolute value problem in precalc, as a challenge.
• Doubling surface area, a good question 
• What's the longest time you've ever spent solving a problem?
• Flipping pancakes 

 
Using Desmos

• An introduction to desmos
• Linearization in Calculus, an amazingly detailed lesson using desmos, with commentary about how students did with it
• I do a unit in trig called Days Of Our Lives, using minutes of daylight on each day of the year as data, and getting students to construct an equation for it. This Moon Illumination project someone made on desmos using the activity builder looks like something I could imitate. (Where did they get their data? Who made this?)
• Desmos art project

On Teaching

• Michael Pershan, on writing about teaching
• How to help people remember what they learn (using retrieval practice)
• How do you respond to wrong answers? This post helps me think about that.
• A good summary of Dweck's Mindset research  
• Ben Blum-Smith on the strategies used at PCMI. "when students are talking to the room it is always students that Bowen and Darryl have preselected to present a specific idea they have already thought about. They never ask for hands, and they never cold-call. This means they already know more or less what the students are going to say." And then Elizabeth responded. I loved her katamari.
  
• Using sentence starters for math conversations with 4th and 5th grade students 
• Fraction talks 
• Getting students talking to each other
• Getting students not to fear confusion
• Physical activity during lessons improves learning (research with elementary students, but I imagine it would help my college students too. Yikes! I don't like this perspective: "the researchers found no differences on reading scores. They think activity works better for subjects with a lot of memorization and repetition." Math should not have lots of memorization!)
• More movement and math...
• If I were a high school teacher, I'd seriously consider this. Metacognition and homework
• On Metacognition (download pdf, interesting part for me is sections 3 and 4) 

Science

• Simulation, mathematically modelling how chemistry and growth work together

Statistics

• The missing 11th of the month
• Linear Regression and Movies
• Why no one is average

Estimation & Elementary

• How many blocks will equal an apple? (3-act lessons, with video) 
• Number Talks
• Pre-algebra: Working with signed numbers

Math for Parents

• Parents’ Math Anxiety Can Undermine Children’s Math Achievement
• Fractions may be elementary (previous topic), but the idea of fractions is also the first math concept that messes a lot of people up. Here's James Tanton's new collection on fractions. 
• When did you stop playing around with mathy ideas?
•  A video about what kids learn in the early grades about addition and subtraction (Please let me know what you think!)
• Finding the Beauty in Math

Social Justice

• On responding to people's surprise that I'm a math teacher
• How affirmative action makes for a better physics education
• "Why Black kids don't like math..." 
• The master's tools... (Dr. Danny Martin's talk at NCTM conference)

Playing with Math

• As usual,  this game (called this game is about squares) is more about logic than about math. What I'm finding interesting is how impossible it seems, and then when I (and others) go away and come back, it can suddenly seem so easy.
•  Tracy Zager wrote a great post on evaluating math fact apps. Lots of good ones are mentioned in the comments. [My comment: I would really love to be able to find this app online so I can recommend it. I have this game on my phone. It seems to be called 1 Whole. There are rectangular shapes that fill with liquid. You push one toward another and they go together if the sum is less than or equal to one. You watch the liquid rise. If it’s 1, it goes away and you get points. You keep going until the screen is full of things that won’t combine (sum > 1). There is no time pressure, the conceptual basis seems strong to me, and mistakes aren’t allowed. No penalties, no bad sounds, it just won’t work. I think it’s pretty good. I wish I could find it online. Cna anyone help me?]
• Kids like doing the simple math involved in thinking about the Collatz Conjecture. [Start with any number (whole,  >1). If odd, triple it and add 1. If even, cut in half. Repeat. Does this always end up at 1? Conjecture is 'yes'.] Mathematicians don't know the answer, but they like to explore the question in sophisticated ways. Here's a post on what sorts of functions come close to modeling the number of steps it takes to get to 1 from each number.
• This game would have made it into my book, I think. Cross Over looks like it has enough strategy to entertain us jaded adults, and it's for addition and subtraction practice. Coolo.
• Not math. Go. Learning to play go. 
• New game for iphone (really, it's logic not math), Ringiana 
• I love surreal numbers. I need to come back and read this more carefully when I have more time to play with it. 
• A silly little game. Totally violates Tracy's criteria (nothing timed). But mathy folk may like it. How many primes can you identify in a minute (with no mistakes)?  (Use y and n for y and no.)

Books

• Here's a great list of fun math books, compiled with a 14-year-old in mind, but almost all good for adult mathophiles too. I think this list came from the same question and has a different set of books.
• My publisher is having a sale. All 5 books published by Natural Math for $50 total. What a great way to expand your playful math collection.

My Favorite Course (to teach): Calculus

Why is calculus my favorite? Let me count the ways ...

1. It tells a story.
2. It has cool historical connections,
3. ... and great connections to science.
4. It's a good time to help students start to see what proof means.
5. I keep learning more.

Calculus Tells a Story...
...if we let it. And the conventional textbooks don't. So I used two different creative commons texts (Boelkins and Hoffman), some of my own materials, and a few things from some of my favorite bloggers, and I made a coursepack to use for the first three weeks. I gave a talk about it at the Joint Mathematics Meeting a week ago. As part of my preparation for that, I made a new blog page. Click 'calculus' above, and you'll see all of my materials, including the slides from my talk, links to the creative commons texts I used, and lots more.

What stories does calculus tell? It takes one of the central concepts from algebra, that of slope, and twists it so it will work for curves. To do that, we need to consider two points that are "infinitely close together," whatever that means. So we have to delve into the weirdness of "infinitely close." Once we get good at all that, we can find out where things reach their maximum and minimum values, and use that to graph all sorts of curves. We also use that to optimize, to get the most volume with the least surface area (when building boxes), for instance. And then we play with finding areas of strange shapes, and how that's connected to slopes.



Calculus has cool historical connections, and great connections to science.
Archimedes figured out all sorts of things that are really a part of calculus (call it proto-calculus), and used the 'method of exhaustion' which is a foundation for what we now do with limits. Newton and Leibniz are credited with inventing calculus, even though lots of what we do in Calculus I had already been figured out. The main thing they discovered was what we call the Fundamental Theorem of Calculus, which says that areas and rates of change are inverse functions. It makes sense that two different people invented calculus because it was needed at the time for the science questions that were being considered: lenses and light, paths of planets, gravity, angle to shoot a cannon, volume of the Earth. And then it took 150 years to get that limit thing just right, and another 150 years (in 1960 Abraham Robinson invented non-standard analysis) to prove that Newton's original conception (of fluxions) wasn't so far off.



It's a good time to help students start to see what proof means.
Did you realize that the two 'formulas' we all know for circles are very different sorts of creatures?  The first, C=2*pi*r, is really just a restatement of a definition. pi is defined to be C(ircumference) over D(iameter), so it takes 2 or 3 algebraic steps to get to C=2*pi*r. But the other, A = pi*r², should be proved. The simplest almost-proof comes from cutting the circle up and rearranging it.



I keep learning more.
I learned two cool things while preparing for that talk: Newton had a clearer conception of limits than we usually think,  and Archimedes' calculation of an approximation for pi was easier to follow than I would have imagined, and really simple and beautiful (in our modern notation).

And to make this post a fun one for all you MTBOS folks, here's the worksheet I designed to share with my calculus class (.doc and .pdf), leading them through Archimedes' first few steps as he worked toward the 96-gon to approximate pi. Go ahead, try it and put your answer for the 96-gon in the comments. (I couldn't find it anywhere else online!)

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*(There's a better way to show word docs, right? Someone tell me. I should know that after all these years of blogging!)

Newton and the Notion of Limit (he knew more than I thought he did)

Preparing to give a math talk has been very educational for me. I posted about ten days ago about finally figuring out how Archimedes calculated pi with his 96-gon.

Now I just found out that Newton wrote more about limits than we're usually led to believe. In 1687, Newton wrote:

"Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...."

This quote comes from Bruce Porciau's paper, Newton and the Notion of Limit, in Historia Mathematica. He gives much more evidence that Newton understood the limit concept pretty well.

I guess I can still say that it took the best minds in all the world 150 years to come up with a precise definition of limit. But Bishop Berkeley's complaint ...

"And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?"

... now seems to me more the product of a small mind and less the careful quest for precision of a mathematician. Now I lean more toward thinking Newton (and Leibniz?) got it, but it took 150 years for a mathematician to create a precise definition that would convince all the other mathematicians.

Joint Mathematics Meetings in Seattle this coming week

I leave on Wednesday for the Joint Mathematics Meetings in Seattle. I'm giving a talk there on using creative commons textbooks in calculus. Friday, 1:20pm, room 620. I'd like to meet online friends there!