Wednesday, April 13, 2016

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.

Monday, March 7, 2016

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


Problem Solving


 
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


Science

Statistics


Estimation & Elementary


Math for Parents

Social Justice



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.

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!)







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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!)

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 NaturalMath.com.)




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.





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*This link goes to bookfinder.com, 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).

 

 
Math Blog Directory