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.





_________
*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).

 

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)

and

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!).

 

Arithmetic

 

Geometry

  • 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.



Announcements

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.)  

Sunday, August 23, 2015

The algebra needed to read about climate change...

This article (at occupy.com), on a lawsuit from a group of young people demanding that we do what it takes to recover from climate change, looks very interesting. One line seemed either wrong or surprising to me, though.

We must immediately commence carbon emissions reductions of 6% each year until the end of the century. Timing is crucial. If we wait until 2020 to begin emissions reductions the annual requirement is 15% per year.
Starting only 5 years earlier, they are saying that we can do 2/5ths as much reducing each year, for 85 years instead of 80, and get the same result. It seems too dramatic. I want to think about how to analyze it. I don't yet know what assumptions I can make.

  • Should I compare total emissions from now until 2100? (I think so.)
  • Should I assume emissions are growing exponentially from now until 2020 in the 2nd scenario? (I think so.)
  • What else would I need to know? (Are there other factors that make this more complicated?)
This seems like a perfect question for pre-calculus. Too bad I'm not teaching it this semester.

I think I got it. I think this assumes that we are currently increasing our carbon emissions at a rate of about 20% a year.  We are not. It's more like a tenth of that - about 2.5%. (Government source here.)

If you want to do some real math, think about what you would do before continuing.

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I figured it like this. I count this year's carbon emissions as 1. If we decrease 6% a year, that means we have 94% of the previous year's emissions. So the total emissions from now until 2100 is
S=1+.94+.94^2+...+.94^84. This simplifies to S = (1-.94^85)/(1-.94). Note that the .94^85 is so close to 0 that we can ignore it. We Get S=1/.06 =  16.666. So the article is saying that for the next 85 years, we can emit 16 times this year's emissions.

If we increase until 2020, we would start with higher emissions, H. 15% decrease per year leaves 85% of the previous year's emissions. Our sum would be
S=H+.85H+.85^2H+...+.85^79H = H(1-.85^79)/(1-.85) = (almost) 1/.15 = 6.666.

16.666 - 6.666 = 10. So somehow we get 10 times this year's emissions within the next 5 years. If our emissions are currently increasing so that our emissions next year is r, then
S = 1 + r + r^2 + ... +r^5 = (1-r^6) / (1-r) = 10. I asked wolframalpha.com to solve  this and got r = 1.2, for a 20% increase per year.

I asked John Golden to check my work. He used a continuous increase model and got close to 9%, mush lower. But still not low enough to match what's happening.

So it seems that either the article has a typo, or my mathematical model is not including everything it should. Humanity seems to be at a tipping point. Can we change our ways of making decisions, from capitalism to something else, in time to save ourselves from our foolishness? I would like everyone to be able to do this sort of math.

Saturday, August 22, 2015

Linear Algebra Question

On Thursday we arrived at Theorem 1 in David Lay's Linear Algebra and Its Applications:
"Uniqueness of the Reduced Echelon Form
Each matrix is row equivalent to one and only one reduced echelon matrix."

The proof is in an appendix, which is a bummer, because this class feels like it could build from first principles nicely up to all its glory. The proof involves material from chapter 4, and I have to fight my way through it. Isn't he worried about being circular?

I was thinking out loud in class. I said (more or less):
If the system is consistent, it has a particular solution set. You can read the solution off from the reduced echelon form, so it can only give you one answer. [In class I wasn't thinking about free variables, and whether those could be different somehow. I was just thinking about problems with one unique solution.] We know it gives the right answer because
we've already shown that elementary row operations create row equivalent matrices, which have the same solution set.

What about an inconsistent system? I'm not sure about that. If you can break his theorem, I'll give you extra credit. 

Well, I just broke his theorem, I think. (I hope none of my students are reading my blog yet.) Given the system

Have I broken his theorem? Should he have said this instead?
"Each matrix representing a consistent system of equations is row equivalent to one and only one reduced echelon matrix."

Friday, August 21, 2015

Random Grouping Cards and Slips

I have just finished my first week of class.

I have finally used Myra Snell's Random Grouping Cards, to put students in groups. I've been wanting to do this for the past year, and finally got over my inertia problem. Research shows that putting students in visibly random groups gets them participating more. (Visibly means they don't wonder if the teacher made it non-random.)

Myra's cards work for a class of 32 students or (a bit) fewer. If you class is bigger or much smaller, you'll need something different. I couldn't figure out an easy way to get mine onto her format. So mine are Random Grouping Slips. I have sets for 16, 23, 32, and 48 students. You cut off the first column, and then slice apart the rows.

I was intrigued that I could not (easily) get 24 student slips. The last one would have put two people together in the last group who had been together before. The way I set it up was based on 16. There was no simple way to make it smaller.

I ended up with classes with 20, 40, and 28 students, so I've made those too now. They're organized a bit differently. I don't like the time it takes to cut them on the paper cutter. Hmm...

Some of the students complain, but I think I am already seeing more of a community forming among the whole class. I'll be watching for ways in which this changes classroom dynamics.

I have also finally begun to implement the Gallery Walk I learned about at the CAP (California Acceleration Project) conference from Myra. I hope to write about that soon.

All three of my classes seem to be going well.
 
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