Saturday, January 16, 2016

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!)

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

Saturday, January 2, 2016

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!

Sunday, December 27, 2015

Does your kid hate math? Try a new angle.

Long before I became a parent, in my teaching (of community college students), a number of them told me how bad they were at math even though their mom or dad taught it. I figured the parents pushed too much or something. (Blame the parents much, do we?) I ‘knew’ I wouldn’t do that.

Well, I don’t think I pushed. But my son hates math, and is consequently way behind his peers. (He unschooled for years and there was no ‘behind’. But he chose to go to a regular middle school this year, where the other kids have mostly had the standard schooling.) So when two people I respect got into a meaty conversation about this, my antennae popped up. They’ve allowed me to share this conversation, which occurred in a closed group on Facebook called 1001 Math Circles. (Ask to join if you’d like - group description: A place to share and discuss your #mathcircles plus learn more about the Natural Math principles! Run by Shelley Nash and Maria Droujkova of

Lhianna: Hi. I'm a homeschool mom of daughters 7 and 13. I absolutely love math and creative problem-solving and my oldest daughter hates it. My failure to transfer my love of math to her drove me to find better ways of teaching and sharing the beauty and excitement that I see. I found out about Math Circles and have done the summer training camp with Bob and Ellen Kaplan for several years now. I run Math Circles around Philadelphia as time and opportunity allow. I love getting inspired by all the great ideas of a wonderful math community like this one. Thanks for letting me join!

Maria: Lhianna, welcome! The Kaplans’ community is wonderful. Maybe we can have a live chat sometime about your circles? When someone hates math, there is usually what I call a grief story. Even with homeschooling, our children can get enough grief "second-hand" from us, or from the society... When I ask people who hate math what happened to them, they usually do know, and tell their stories. Do you know what happened to your 13-year-old? And what does your 7 year-old like to do? It's such interesting age for girls!

Lhianna: My 7 year-old loves logic problems. (The island of knights and knaves kind. I have a special fondness for all of Raymond Smullyan's books!) She likes unit origami (especially the sonobe units). And she seems fascinated by anything to do with parity. Also building with geometric shapes of all kinds.

I think my 13 year-old has a deep fear of getting things wrong in any subject and in general in life. In other subjects she finds ways around it. But it is especially devastating for mathematical exploration. You really have to try many different avenues and be able to look at your failures and analyze them to arrive at a solution in math. Math is about exploring what is unknown to you and she can't stand that. She prefers the familiar.

It has been an interesting journey for me. I started thinking how lucky she is to get an exploratory background in math. I then realized my own shortcomings that, while I loved to explore math, I hadn't been able to communicate that idea to my child. Which led me on a wonderful journey of discovering Math Circles and many more amazing people and sources full of creative ideas about learning math.

But as my daughter continued to hate it (and trying to do math with other people too, not just me), I also learned that math is not for everyone like I originally thought. It's ok now that she doesn't like math! That is a homeschooling journey to learn and accept this. (When she does do some math she is perfectly able to learn and understand the concepts. She just has zero interest and will not voluntarily spend any time on math study).

I am currently dragging her through "The Art of Problem Solving" book series so she can have enough math to go on to higher education. (And it's a pretty decent series for a textbook!) I am very much an amateur. I am constantly learning and open to new ideas. Any suggestions would be greatly helpful.

Maria: Lhianna, thank you for sharing. Yes, I am with you - love of math for its own sake isn't for everyone (just like any other area); but I do feel that everyone can feel good doing some math-rich activities in their own ways. I see a pattern in your interaction with math and with your 13 year-old. Do most of your math activities center on problem-solving?

In contrast, have you ever tried math activities that don't involve problems, solutions, answers, or unknowns? There are activities where you: (1) only work with what you know, and (2) don't seek any answers or solutions. When I say that now, can you picture 4-5 examples of activities that I am talking about?

Lhianna: Not off the top of my head. What kinds of activities are you thinking about?

Maria: Logic is so lovely! Smullyan's books made a difference for many people. Camp Logic, which we published this year, is one of our most popular books, too. I just sent three big boxes of it to groups. Next year, "Bright, Brave, Open Minds" will be out, by Julia Brodsky - there are very lovely logic activities in there, too. Here are a few things to try from that book:

Lhianna: I see my 13 year-old use math in other activities (she really likes to cook and make up her own recipes which involves experimentation and therefore doubling and tripling many measurements as well as analyzing the ratios of one ingredient to another). Is this what you are talking about? Or math games? She likes to play SET.

Maria: Lhianna, so the goal is to find math-rich activities that: (1) are not problem-solving, and (2) center on what you already know, and yet (3) are open and can be made uniquely yours. Let’s see if we can find a fresh angle on what your daughter can try…
  • Storytelling. You tell what you know; you make the story interesting, fun, pretty, and may invent details, but you know your story (and math therein). Vi Hart videos are like that. Or storybooks like The Cat in Numberland.
  • Illustrations. Take something you know. Illustrate it with a picture, comic, video, toys, interpretive dance smile emoticon Basically, represent it by some medium you like. A lot of math comics are illustrations of math jokes, for example.
  • Programming. Take a formula or pattern you know and use, and make your computer (spreadsheet, solver, etc.) do it for you.
  • Scavenger hunt. Find some math idea you know (e.g. ratio) in what you like (e.g. Star Wars, your favorite park, or your room). Or find a lot of math ideas in one book, movie, room... Make a curated collection. There are a lot of those online. Have you tried that sort of approach? How did it go?
SET is a very good game too. To use this as an example of doing what you like and know... We do this activity where we make our own set of SET cards from scratch, using our own shapes and themes. On the one hand, it's something you know. On the other, the amount of delicious a-ha moments you have along the way is just incredible!

Lhianna: Great idea! Thanks. And thanks for the advice. I will start looking for activities and examples that follow along the lines of familiar but open. I appreciate the new perspective.

Maria: I would love to hear what else you find, because you have such a thoughtful approach to the whole thing! Moving the focus to, "Love SET, like Vi Hart videos, like Tangram puzzles..." (from, "hate math").

Do you have a kid who hates math? Do any of these ideas sound like something you might want to try out with them?

Thursday, December 24, 2015

Question for my Readers

Lately, when I'm trying to write a post, I often get shifted over to some sort of ad. Does that happen to any of you reading my posts? If it does, I may move my blog over to Wordpress.

Wednesday, December 23, 2015

The Roots of Calculus - Archimedes

Archimedes did a lot that nowadays looks like calculus...

He determined the value of pi very precisely, by starting with a hexagon inscribed in a circle, then a 12-sided polygon, then he kept doubling the number of sides until he got to a 96-gon. A procedure like this is called the 'method of exhaustion', and it looks a lot like what we do nowadays with limits.

I am embarrassed to admit that I couldn't figure out how he did it. (I think I was focusing on area, and that might be harder.) I just found a great video by David Chandler (whose youtube channel is Math Without Borders).

Here's a summary:
Start with a hexagon inscribed in a circle of radius 1 (giving diameter of 2). The perimeter of the hexagon will be 6. This gives a lower bound on pi, which is the ratio of circumference to  diameter. We know the circumferences is bigger than this perimeter of 6, so pi is bigger than 6/2 = 3.

If you cut one of the triangles that made the hexagon into two, you get a radius that crosses a side of the hexagon at right angles. You can use the Pythagorean Theorem (twice) to find the new side length. Repeat 3 times and you're at the 96-gon. Archimedes had none of our technology, and little or none of our algebraic symbolism, so the calculations were much harder for him. We can do all this on a spreadsheet, and up comes pi (if you have a column for the perimeter over the diameter). So satisfying!

If this doesn't make sense, watch this lovely video. Thank you, David!

Archimedes did a lot more than find a value for pi! What's your favorite bit of calculus that started out with Archimedes?

Monday, December 21, 2015

Fun Mathy Books

Is it too late to suggest good holiday gifts? Here are some books I think you might like.

This is Not a Math Book, by Anna Weltman

Patterns of the Universe: A Coloring Adventure in Math and Beauty, by Alex Bellos

Mathematical Mindsets, by Jo Boaler

Intentional Talk: How to Structure and Lead Productive Mathematical Discussions, by Elham Kazemi

Dan MacKinnon wrote a lovely review of a book I hadn't heard of before, at his blog, Math Recreation. Here's the beginning of it...
In The Puzzle Universe: A History of Mathematics* in 315 Puzzles (TPU), Ivan Moscovich stretches the concept of puzzles to encompass almost anything that combines curiosity and playfulness (playthinks is his preferred term for this more general category of puzzling items). No surprise - these playful curiosities are inherently mathematical. In an informal and accessible way, Moscovich details the development of these puzzles, revealing their surprising family resemblances and the deep mathematics behind their playful exterior. [read the rest at Dan's blog...]
And of course, there's my book, Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, along with all the other cool books at Natural Math.

*This link goes to, which will point to other sites. It's the best way I know of to find the least expensive copy available. (My other links point to the sites that were cheapest at bookfinder on the day I wrote this.)

Lots of Links

For months I've been saving cool things in tabs in my browser. I think I was up to over 80 tabs when I started cleaning up yesterday. Here are the goodies...

Games, Puzzles, & Problems

I have to admit that I skip the Intermediate Value Theorem when I teach Calc I (please tell me if you think I'm short-changing my students), but here are two great posts about it. If you ran a race at an average pace of 3:07 per kilometer, did you run any single kilometer in exactly 3:07? (from Scientific American) and an activity using Desmos (from Christopher Danielson).


Sunday, October 25, 2015

Math Teachers at Play, #91

Number 91 feels like we're closing in on 100. The last time I hosted MT@P, we were at #71 and I managed to include 71 posts. I wasn't quite that ambitious this time. (Old math posts don't go stale. You might enjoy browsing through a bunch of the old Math Teachers at Play blog carnivals. And don't forget our partner carnival: the Carnival of Mathematics.)

If there are 14 people in a group, and each shakes hands with each other, there will be 91 handshakes. (Can you see why?)

91 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13
(which makes it triangular)


91 = 7 * 13
(the middle and last numbers in the sum above)

Will this always happen for triangular numbers?

Games & Puzzles

  • Shannon Duncan, a 6th grade math & science teacher, shares 4 Reasons to Promote Math Success through Games at the MIND Research Institute blog, illustrating her ideas with some of the games she has her students playing. I especially like the first point - making a mind-body connection.
  • John Golden (@mathhombre) shares Angle of Coincidence at his blog, Math Hombre, about an angle identification game he's developing. Ask your students to playtest it and give him feedback! John also wrote about the start of the semester, and included a game called In or Out?  that looks fun.
  • Jeff Trevaskis shares a Multiplication Tic-Tac-Toe Game at his blog, webmath. 
  • Carole Fullerton shares Number Tile Puzzles at  her blog, Mathematical Thinking. 
  • Gray Antonick interviewed Paul Salomon in the New York Times Numberplay column, about his Imbalance Puzzles, one of many puzzles and games featured in Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers (my book, published in April!).





  • Stephen Cavadino (@srcav) shares Parallelograms at his blog, cavmaths, on a student's creative way to find the area of a parallelogram.
  • Ioana I Pantiru (@LThMathematics) shares Playing with Paper Folding at her blog, Life Through a Mathematician's Eyes, showing the steps of an origami construction. In her post, Maths Class Everywhere, she asks readers to take her survey of math classes around the world. 
  • Curmudgeon shares Circles on a Lattice, at their blog, Math Arguments 180. I wonder if this would make a good problem for a math circle...  
  • Greg Blonder, a professor of manufacturing and product design, shares Trisecting the Angle With a Straightedge, at Plus Maths.
  • There have been lots of posts in the past few months about classifications of pentagons (here's one), because a new (15th) type of pentagon that will tile the plane was recently found. Here's a good background post, from before the discovery, from the Mathematical Tourist.


It's All Connected


Ideas for Learning ...

  • Kate Snow (@katesmathhelp) shares How to Teach Your Kids to Read Math  at her blog, Kate's Homeschool Math Help. I'm still trying to teach my college students how to read math, with some of the same tips.
  • Manan (@shalock) shares Becoming Mathematically Fluent at his blog, Math Misery.
  • Shecky (@sheckyr) shares True Deep Beauty ... at his blog, Math-Frolic, about the how our understanding of math deepens.
  • Chris Rime is making monthly math calendars (Algebra I, II, and Geometry), available as doc or pdf at his blog, Partially Derivative.

... And Teaching

  • Tom Bennison (@DrBennison) shares How to enjoy your NQT Year at his blog, Mathematics and Coding. [I had to look up NQT. It means newly qualified teacher, and in England and Wales, you are "inducted" in your NQT year, (generally) your first year of paid teaching.] I like his suggestion to make time for doing some math(s) yourself.


I'm going to the Joint Mathematics Meetings in January in Seattle. I'd love to connect with other bloggers who are going. There's a math poetry reading plus art exhibit on Thursday evening at 5:30. You can get all the details from JoAnne Growney's Intersections blog.

Friday, September 18, 2015

Joint Mathematics Meetings - Seattle in January

I think I'd like to present. I've never done that at the JMM. I'd like your help. Here's (my second draft of) what I've written for my proposed abstract: 
Have you seen your students disengage from your calculus class in the first week as they struggle with the technical topic of limits? They don’t see the point, get mired in the algebra and can become alienated. I will share why I save limits for later and start out with an exciting and historical approach using slope and velocity.

But perhaps your textbook, like mine, follows a traditional approach? I will also share how I used parts of two Open Education Resources (OER) by Matt Boelkins and Dale Hoffman, along with a few pages I created, to make a coursepack for my first unit. [Link to modifiable materials provided at talk, or by email.] Their materials gave my students the support they needed in our excursions off the traditional textbook’s beaten path.

I’ll help you see why there’s a better order to the topics. (It’s not just the limits.) And I’ll show you one way to make Calculus fun for yourself and your students.

You can use the experiences I share in my talk as inspiration to help you get started remixing OER to develop your own approach and materials. Using these materials in a coursepack alongside the required text may also be a way to show your reluctant department that they don’t need the $200-plus conventional textbooks.

  •  Have I said enough to make it clear what I have to offer?
  • What more should I say?
  • What should I change?
  • Would you come to my talk?

(My deadline is in 4 days.)  
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