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.)  

Sunday, August 23, 2015

The algebra needed to read about climate change...

This article (at, 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.






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 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.

Saturday, August 8, 2015

Links on Saturday (lots for First Day)

First Day 

 First Week

Other Good Stuff

Sunday, July 12, 2015

Playing with Math: Can you write a review?

Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is on Amazon now! But we don't yet have any reviews. If you've gotten a copy of the book, can you write a review on Amazon? We would be so grateful.


Friday, July 10, 2015

Links on Friday

I'll be leading a Math Jam for eight days just before Fall semester starts, helping students prepare to succeed in Beginning Algebra. My eight topics:
  1. Number Sense
  2. Fractions
  3. Negatives
  4. Algebra
  5. Percents
  6. Graphing 
  7. Slopes
  8. Problem-Solving  

For fractions, I plan to do a bit with Egyptian Fractions. Here's a site that looks good for that. I looked at the Beast Academy site to see if they had anything good. I found 5 things I liked: one game and two puzzles using the area meaning of multiplication, one puzzle on ordering of decimals,  and one game like Taboo for communicating about shapes.

Thursday, July 2, 2015

Playing with Math: Inspiring Online Conversations

First sighting of a comment on a mathematical blog post that was inspired by seeing the content in my book...

Jonathan Halabi writes jd2718. His post, Puzzle: Who am I?, became one of the puzzles in Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers.

Today Lara H replied to his post:
I came across this puzzle in the book “Playing with Math.” I found a different solution based on a wrong assumption I made at the beginning of solving the puzzle. I was thinking that a number with 3 digits also has 2 digits so I made both of those statements true and came up with 4097, which works for all the other conditions.

I responded with:
I’d say ‘different interpretation’ instead of ‘wrong assumption’. I wonder how many solutions the puzzle has using your interpretation. (Pretty exciting to see my book has inspired new discussion on Jonathan’s blog post!)

We are hoping that the book will inspire online conversations. This is the first drop of what we hope will eventually become a deluge.  

Saturday, June 20, 2015

Book Review: The Archimedes Codex

I bought this book because I wanted to understand more about Archimedes' role in the ancient development of calculus ideas. When I got it, I was worried it would be another book I wouldn't want to wade through. I was so wrong!

The Archimedes Codex, by Reviel Netz and William Noel, is fascinating. Like much good science writing these days, The Archimedes Codex reads like a detective story. It is gripping! Netz writes chapters about Archimedes, his math, and translation issues. Noel writes chapters about the travels of the manuscript, and the attempts to use modern technology to get better images of Archimedes' writing.

In 1998 Christie's auctioned off this battered medieval manuscript which on its face was a prayer book, but also contained traces underneath of Archimedes work, which had been scraped off. It sold for two million dollars to an anonymous bidder. William Noel, of the Walters Art Museum in Boston, followed the story and emailed the agent of the buyer. The buyer agreed to work with the museum to attempt restoration of the manuscript. Most experts expected little from the work, since the manuscript was in such bad condition. But the project, which took years, brought to light previously unknown work by Archimedes.

Archimedes had explored the idea of infinity more carefully than had ever been realized. He also did work in combinatorics, which no one had even suspected. The math is pretty easy to follow, and it's amazing. I've dogeared about a dozen pages, so I can read passages to my calculus students.

This is perfect summer reading. Enjoy!

Monday, June 1, 2015

Imbalance Abundance Puzzles (We're in the New York Times!!)

Paul Salomon posted some delightful puzzles a few years back, I got in touch with him about including them in the book, and now his puzzles are featured in the New york Times' Numberplay column!

I met Gary Antonick (who writes Numberplay) in person a month or two ago at a lovely meeting of math popularizers. We were both excited to meet each other*, and he asked if he could share some of the book's material in his column. Of course I said yes.

I knew the column was coming today, but forgot to look until I saw Mike South's Facebook post mentioning it. Mike writes great math explanations on Living Math Forum, but doesn't blog. I wanted to include something of his in the book, but didn't manage it. (Here's Mike on thinking about zero.)

Gary included a great photo that goes so well with the puzzles, I want to make up a new puzzle to go with it. Hmm.

If you don't already have your own copy of Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, you can buy one here.

*I finally got to meet the fabulous Fawn Nguyen in person, too! What an exciting day that was!

Wednesday, May 27, 2015

Preparing for the Fall Semester: How to Get Students to Participate More

Last summer, at a conference for the California Acceleration Project, Myra Snell used a cool way to set up random groups of participants/students. I wanted to use it in my classes, but just didn't get around to it. It seems, after almost 30 years of teaching, that it has become hard to change the way I run my classroom.

But I did change one thing this past semester. I noticed, while sitting in on a colleague's Calc III class, that I really appreciated the notices he wrote on the board at the beginning of each class. So I began to do it too. Maybe I could implement a few more good habits by watching other teachers during the second and third weeks of class.

Coming back to those random groups... I recently read research that found two effective strategies for getting students to participate more. One is visibly random groups. 'Visibly' means that they can't suspect the teacher of manipulating the group memberships. Myra's method is clearly random, looks easy to implement, and allows for up to four different groupings per class day. You have a slip for each student, with a number, a letter, an animal, and a food on it (for example). Those slips are set up so that no one is with any of the same other people more than once. I've asked Myra for her slips, but last night I was eager to think about it, and created my own. I don't know if this is the best way to do it, but I think it will work. Myra's slips had the 4 terms in a square and mine will be all in a row. I don't think that's a problem.

The second strategy which made a difference in student participation was student use of vertical whiteboards. The researcher(s?) compared paper and whiteboard, used vertically and horizontally. [Unfortunately, I can't find the research I originally read, which mentioned both the visibly random groups and the vertical whiteboards.] I'd like to try this out with the class I'll be teaching for the first time this fall, a compressed version of beginning algebra (first half of the semester) and intermediate algebra (second half of the semester). It's officially the same courses we've always taught, but I get to use a different curriculum, and will be using something project-based. I'm excited about implementing this.

Tuesday, May 26, 2015

Machinery, Lines and Circles

On Facebook, someone posted an animation of how a sewing machine works. It wasn't enough to help me understand how the top thread manages to get around the bobbin mechanism. I searched on youtube, and nothing helped. This article on math and the sewing machine made me think for a moment that I was getting it, but I still am not. How is that bobbin mechanism held in place in a way that allows the thread to get around it? (Do you see how the top thread moves past the whole back of the bobbin? How is that possible?) They say that the bobbin is held snugly inside its case, but how is the out part attached?

I think I need a transparent sewing machine, so I can really see how this is working.

On thing leads to another (especially online!), and I ended up at this site from a museum for mathematics, called The Garden of Archimedes, in Florence, Italy, where I encountered this very simple statement about the difference between constructing a circle and a line - something I had never thought about before.

The simplest curves are doubtless the line and the circle. To draw circles, one uses a compass. It's sufficient to keep a constant distance between the tracing point and the centre, and one obtains a near-perfect circle, even with a primitive compass. At first sight, one would think that tracing a segment is also a very simple operation: you just need to use a ruler or pull a string taut. In fact, things don't work exactly like that. In order to draw a good straight line with a ruler, one needs the ruler itself to have a "straight" side, but the value of a ruled line depends on the ruler that was used to make it. So, who made the first ruler? To apply the same method to the circle would mean, for example, to take a coin and trace its edge - the circular profile would be "intrinsic" to the instrument itself.

It would be better to apply to the straight line the principle used to draw the circle, rather than vice versa.
Inatead of using a ruler or straightedge, can't you use the "pull a string taut" method, with something a bit less flexible than string? Maybe something that freezes into position? Hmm... Apparently that's not the avenue that was followed. You can find out the fascinating history of the solutions people found for this problem by going to the Garden of Archimedes site.

Tuesday, May 19, 2015

Teaching My Son (Post One of Many?)

I started out really believing in unschooling. (Advice to self: Beliefs are dangerous.) My son has attended a free school, where he didn't have to attend classes (K-2), and then was homeschooled at the homes of friends (I'm a single parent), in groups of 2 to 8, with very little required of him. He has learned a lot over the years, but not in the conventional ways. If you're an advocate of unschooling, that may not sound like a problem at all. But for him it was. He thought he was 'behind' in reading, and felt bad about that. He totally avoided math because of how far behind he thought he was. He thinks he's dumb because he hasn't done the conventional academics.

Now he wants to go to a 'normal' school. So I signed him up for 8th grade at a charter school his friend goes to. (I've heard great things about it, and it is supposed to be project-based.*) Part of going to a regular school means catching up on all the 'regular subjects.' So I've begun requiring him to do 'academics' daily. (He asks if he has to. I say yes. He then shows subtle signs of relief. He really wants me to make him do this. This blows my mind.)

About a month ago, we started with 15 minutes of reading and 15 minutes of handwriting practice each day. I don't care about his handwriting. He does. He is so embarrassed about it that he resisted signing in for his trampoline class. A few weeks ago, I added spelling (his desire), geography (identifying the states), and math. This week we're adding science and an essay on the history of bikes. My opinion is that the only things he really needs to catch up on are math and writing (essays, stories, ...). It helps that we're doing this, because he also needs to become more aware of conventions - how to write dates, what schoolwork looks like.

For math, we're using Beast Academy. We started with book 3A. Yes, the 3 means third grade. We don't mention it, but he knows this is "supposed to be" for younger kids. Beast Academy has challenging work, though, and if we make it though all eight of the levels (3A-D and 4A-D), I think he'll be pretty well-prepared to join a class of 8th graders. I will look over the 'standards' for 5th to 7th grade later this summer, and see what might be missing from what we're doing. I have made a math plan for the next 14 weeks, leaving out some of the topics in the Beast Academy books (perfect squares, variables, counting, logic, probability). I'm sure they are excellent, but my goal was to find a way to pare it down, so he gets as much as possible of the foundational skills he'll need, in the short time we have before he starts 8th grade.

The first day that we did math, he was sitting next to me, saying his answers, waiting for me to confirm before he'd write them down. I did. (What he needs, as he takes on this huge emotional challenge, is support. Once he feels more secure, I'll be able to say things like "How can you decide whether that answer is right or not?")

On the second and third days, I noticed that his wrong answers were usually one off. To me, that meant he wasn't noticing things I notice about even and odd numbers. I printed out something from the nrich site that looked good. We haven't tried it yet.

It's very fun for me to be planning out his math curriculum. But this is very stressful for him, so our work time can be full of conflict. Once he buckles down and gets started, I get to quietly support him. Mostly I just confirm his answers. He is already seeing progress, and feeling good about it. I am trying to use Denise's technique of buddy math, offering to do every other problem myself, and then talking my way through it. He seems to prefer doing the problems himself most of the time, but let me do one problem last night.

The lessons he's working on are about finding perimeters. It has been a great way for him to work on adding numbers, with something extra thrown in. Most of the shapes have more than six sides. While we were working last night, I told him I noticed that he picked numbers that add to ten, which is a good strategy. I said that some people call those ten-bonds. He said he didn't know them all. I asked him for numbers that add to ten, and he got a bunch. The ones he hadn't mentioned, I asked him about: "Eight and ...?" He was surprised that there were only 5 pairs, I think.  (At least two different stories in Playing with Math address this issue - Prison Math Circle and The Math Haters Come Around.) When he was unsure of 5 plus 8, I told him that I sometimes forget that one myself, and one way to figure it out is to move 2 from the 5 to the 8, so you get 3 and 10.

I am exploring the balance between telling (ten bonds) and helping him to discover (hopefully we'll do that with odds and evens). I am so happy to be doing this, and marveling at how hard it was for me to see that he actually wanted me to make him do it.

*Yes, I agree that charter schools are being used to mess up the regular public schools. Difficult situation all around.

Thursday, May 14, 2015

Moebius Noodles is Delightful

Moebius Noodles is headed into its second printing soon. For the past few days I've been reading it over carefully to offer suggested edits. What a delightful task I gave myself! It has been so fun to remind myself of all the activities for young children Maria Droujkova and Yelena McManaman have put together.

Their suggestion for creating an iconic times table got me dreaming. How can I get my son (who "hates" math, unfortunately) inspired to take photos for a times table collection? I was dreaming of a website that would show the whole table on one page, with each photo pretty small. And when you hover over a photo, that one would show up big. I don't know how to do that, though...

Here's a photo (from that feels like it belongs in the Grid section of Moebius Noodles, except that there's no pattern to the pieces. Well, the rows and columns are a bit wonky too. Hmm...

Kate Nowak posted this on Facebook. The question that came to her mind (among other less mathy questions) was ... How do you count these?

[Edited to add: In the comments, Joshua described a very cool pattern he saw, and suggested that it's like 9 plus 4 is 13, which looks like my diagram below.]

Moebius Noodles has four sections: Symmetry, Number, Function, and Grid.

The mirror book introduced in the symmetry section is so simple, and so cool to play with. Just get two small rectangular mirrors (at a dollar store), tape them together along one side, and use with photos or drawings, to see lots of symmetrical designs.

My favorite game in the function section is Silly Robot. The grownup plays the robot, and follows orders exactly (while always trying to find a way to mess up the intention of the orders).

If you know anyone with a child from one to eight who'd like to find ways to play around with mathematical ideas, Moebius Noodles is a great resource.

And my book, Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, is delighting readers across the U.S. (and hopefully around the world).  Here are a few photos of happy readers. Send me a photo of you with the book, and I'll add it to my collection (especially if you live far from me!).

This is shaping up to be a very fun summer...

Tuesday, May 5, 2015

Aunty Math

Welcome to the world of
Aunty Math
Math Challenges for K-5 Learners

One of the reasons I put together a book was my fear that good online writing often just disappears. One of the sites I had really liked - and thought of including somehow in the book - was a site with stories from Aunty Math (Aunt Mathilda). It disappeared before I could contact the author. And for years, I thought it was just plain gone.

This evening I searched for Aunty Math, and found that someone had managed to get to this site through the Wayback Machine. It is now available as an archive. Check out all eleven past challenges. I think you'll enjoy them.

I would love to be in touch with the author, Angela G. Andrews. I googled her, but I don't see an email address. I'll just thank her here for her lovely stories. Thanks, Angela!

My book, Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, won't disappear. If you want a copy to appear in your mailbox, order one now.

Saturday, April 18, 2015

The Book is Beginning to Arrive!!

Dylan Kane (@math8_teacher) just posted this photo on twitter a few hours ago. It's the first sighting of Playing with Math in a crowdfunder's hand!

We got a message from one more crowdfunder a few minutes later that her copy had arrived. The books are coming!

My living room has stacks of books along one wall, sent to me by the publisher. I signed and repacked about twenty of them this morning, to send out to our $100 and over contributors.

It is so exciting to know the book is finally in people's hands, after 6 1/2 years of work.

Want to have the book in your hands? Order a copy now.

Tuesday, March 31, 2015

A New Site for Critical Thinking (

Which One Doesn't Belong? Many of us have played with puzzles like that since we were very young. Most of those puzzles had one right answer. Christopher Danielson has been championing versions of this where every item could be the right answer. He's created a 16-page shapes book for young children, built on this principle. And he recently took it out to classrooms around Minneapolis, learning much about kids' understandings of shape.

Christopher's enthusiasm has engendered enthusiasm across the MTBOS (math twitter blog o sphere), and tonight I was able to attend a Big Marker online event discussing a new website dedicated to these puzzles:

What fun!

And so one more nifty tool is added to our techno toolbox for math class. (I have been loving for a few years now, and use and whenever I get a chance.)

Saturday, March 21, 2015

Algebra Skills Needed for Calculus

Sam Shah posted his list here. I loved his list, but wanted to rewrite it a bit for myself. (Also, Sam finds it more effective to review the algebra ahead of time, while I think it's more effective to review once we see the need in our exploration of calculus.) I am posting this now, so it's available as an answer to this question on math educators stack exchange.

I teach my calculus course in an order that I think will help students learn. I have four units:
  • Unit 1 includes history, graphing functions, slopes of tangent lines by approximation, algebraically finding the derivative using the limit (which we do not carefully define yet), seeing the similarities between velocity, rate of change, and slope, average versus instantaneous velocity, derivative from a graph, (estimated) derivative from a table of values.
  • Unit 2 includes derivative properties needed for polynomials, graphing, limits and continuity, trig derivatives, and optimization.
  • Unit 3 includes chain rule, derivatives of exponential functions, implicit differentiation, derivatives of inverse functions (ln x, tan-1x), and related rates.
  • Unit 4 includes integration (finding area under the curve), anti-derivatives, fundamental theorem of calculus, and substitution method. If there is time we include volumes of rotation (which I think is a perfect ending for the course).

Algebra Skills needed for Unit 1 

  • Determine the equation of a line given two points, or a point and a slope, or a graph of a line, 
  • Find the average rate of change over an interval given a function or its graph, 
  • Clearly express what is happening to an object given a position versus time graph, 
  • Evaluate f(x+h) for any given function f(x), 
  • Rationalize the numerator (to find the derivative of the square root function) , 
  • Simplify complex fractions (to find the derivative of the 1/x function). 

Algebra with Calculus Concepts 
  • Approximate, using two points close to each other, the instantaneous rate of change at a point, given a function or its graph, 
  • Explain clearly why the procedure you used gives an approximation of the true instantaneous rate of change, 
  • Sketch a velocity versus time graph given a position versus time graph, 
  • Construct the formal definition of the derivative by modifying the definition of slope, 
  • Apply the formal definition of the derivative to simple polynomials and to simple square root functions.

Algebra Skills needed for Unit 2

  • Multiply out the expression (x+h)n (necessary to understand the proof for the derivative of y=xn),
  • Identify the holes, vertical asymptotes, x- and y-intercepts, horizontal or slant asymptote, and domain of any rational function,
  • Sketch the basic shape of a rational function,
  • Identify an equation for a rational function given a sketch of the function,
  • Explain clearly what a hole and an asymptote are,
  • Construct the equation of a piecewise function given its graph,
  • Sketch the graph of a piecewise function given its equation,
  • Work with inequalities,
  • Give both triangle and circle definitions of sin x, cos x, and tan x, and explain how they’re related,
  • Evaluate sin x, cos x, and tan x at all multiples of  π/6 and  π/4, without a calculator,
  • Understand trigonometry identities, including and sin(x+h)=sin x cos h + sin h cos x,
  • Accurately graph y = sin x and y = cos x.

Algebra with Calculus Concepts
  • Graph a polynomial or rational function, showing its maximums, minimums, and inflection points,
  • Follow complicated logic (in the definition of limit).

Algebra Skills needed for Unit 3

  • Understand composition of functions,
  • Use logarithm properties to “break apart” a single logarithmic expression into simple logarithms,
  • Understand properties of exponents,
  • Be able to graph exponential and logarithmic functions.

Algebra with Calculus Concepts
  • Think in terms of composition of functions to determine outer and inner functions, in order to use the chain rule.

Algebra Skills needed for Unit 4

  • Work with summations.

Friday, March 13, 2015

Copy Number One of Playing with Math

At 3:30 this afternoon, UPS knocked on the door and delivered copy number one of Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers!

It is beautiful!

Now we put in the full order. Books coming soon...

If you haven't ordered your copy yet, you still can.

Friday, February 6, 2015

Linkfest for Friday, February 6

Before I share all the delicious goodies I've stumbled on, news of the book is in order:

Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is just about done with page layout - and it's looking so beautiful! I am sending in the last proofreading corrections today, and will do the last fixes to page number mentions as soon as I've seen the final copy. Then it's off to the printers, then all the copies get shipped to the publisher, and finally get sent to the hundreds of people who ordered copies during the crowd-funding last summer. If you're eager for your own copy and weren't around for the crowd-funding, you can order now. (You know I'd be tickled if we sell out our first printing quickly!)

The Links
  • Two Truths and a Lie: Get calculus students to make up stories from their lives, using the idea of rate of change, and matching given graphs. Brilliant, Shireen!
  • I like this for a first day activity! (I just figured out how to link to this on my google calendar to remember to look at it in August!) Getting the students involved in discussing what education should be, and what productive failure might look like.
  • Explained Visually has animated graphics for trig functions, exponential growth, statistical processes, and more. Fun.
  • Beautiful teacher story. “You just listened, so then I could figure it out.” 
  • This post asks: Is there room for math that isn't hard? The post and comments are both interesting reading, and I'd enjoy seeing more comments. The blog is called Math Exchanges, and their more recent post, Over or Under, is great too.
  • About half a year ago, I joined in the crowd-funding for the math game Prime Climb. It arrived in early December (or was it in Novemeber?) and we played it at my holiday party. People definitely enjoyed it. Now I've heard about another game being crowd-funded. Three Sticks is a geometric game, developed in India. It looks fun. For a $35 contribution, you get the full set (and escape the very high shipping charges).
  • The math in the solutions may be too hard to follow, but this problem is charmingly simple: Your hallway is one meter wide, and turns a corner. What is the greatest base area of an object that can be carried flat through the corner?
  • I'm not so good at making things (origami, etc), but these pretty mathematical sculptures do look fun.
  • Every textbook I've seen that includes conic sections shows the conic, and then shows another definition, and never connects the two. This blog post makes some of the necessary connections. (Anything on Dandelin's spheres catches my eye.)
  • Tricky puzzle. (Do you like that sort of thing?) The 7 at the bottom is NOT a typo.
  • I'm always happy to hear about new math circles. Here's one in Santa Cruz, in the news.
  • Estimation questions are a great way to build number sense. And Andrew Stadel has a twitter feed just for that. This week included a few questions about these Lego Lions: How many legos? How long to build? How many legos tall?

A Question
I'm teaching Linear Algebra, and I find it a bit odd that linear transformations by definition don't include lines like y = mx+b (with b not 0). A student asked the significance of the word linear (she thought it was a silly question, and I assured her it definitely was not silly), so I started searching online. I noticed this site, which defines a linear transformation for statistics - differently from the linear algebra definition. It looks like the two definitions contradict one another. Any ideas about how standard this statistics definition is, or pointers to discussions of this difference in definition?

[Oops! I lost a few weeks on the #YourEduStory challenge. Maybe I can get back to it. My pre-calc class is going better than usual. My calculus students loved having all those handouts in a coursepack. And I love thinking about all the connections in linear algebra. This week's topic: Define "learning" in 100 words or less.]

Sunday, January 18, 2015

My Favorite Teachers and Me

The #YourEduStory blogging challenge question of the week:
How are you, or is your approach, different than your favorite teacher?

I don't have just one favorite teacher. I have lots. Long, long ago, before I started teaching, I made a list of my favorite teachers:
Mr. West, high school biology, and then anatomy and physiology
Ms. Purvins, high school Shakespeare teacher
Mr. A, high school poetry teacher
Mr. X, UM philosophy prof
Ms. Y, UM history of feminism prof
Gisela Ahlbrandt, EMU math prof
 There were probably more on the list at the time. These are the ones I still remember. (And I'm losing the names. Yikes!) When I made the list, I noticed something interesting. There were about equal numbers of men and women on the list, but they were very different sorts of teachers. The men were good performers, and the women were good facilitators. A few did both well (the poetry guy and Gisela). I wanted to do both well. I thought about taking some drama courses to improve my performance skills. I did that while teaching in Muskegon, and realized I needed a different sort of course. Performing in a play is a lot different than performing as a teacher. Improv might be good for me. Hmm... I also learned a lot about facilitation over the years.

I know now that the best performers make students happy to come to class, but that's not enough. We need to get students actively engaging with the material for them to learn much. (Mr. West did that in lab, even though I remember his great lectures.) If you don't know the research done by Eric Mazur on this, check it out.  (This video might include the best parts of the hour-long video I watched a few years ago.)

How is my approach different than theirs? I think it's only in the combination that I'm different. I try to pull in all my students (like my Shakespeare and history of feminism profs did). I ask them multiple times each class to show me with thumbs up, down, or sideways how well they understand what I've just explained. I call on students randomly. (Because teachers tend to call on male students more.) I come in as excited as my bouncy philosophy prof. I suggest my students try strange experiments, like my poetry prof did (he had us write at a cemetery and a mall). I try to be as accepting and as challenging as my best teachers were.

Math Circles at Nueva School

Nueva School, in Hillsborough, south of San Francisco, puts on a math night three times a year, with multiple math circles, along with a puzzle and game room. Nancy Blachman invited me to lead two math circles last night, one for 2nd and 3rd graders and another for 4th and 5th graders.

2nd and 3rd grade Circle
This circle met for just 30 minutes. I know that the Collatz conjecture is dependably fun for kids this age, so that was our main activity. I asked the kids what they thought mathematicians do, and got a reasonable answer, but saw that there wouldn't be time for useful discussion. So I said a bit about math being like a game for mathematicians, and how fun it was to come up with a new puzzle.

In 1937 (I just said it was about a hundred years ago), Lothar Collatz came up with this puzzle/game:
  • Pick a number.
  • If it's even, cut it in half. Write your new number.
  • If it's odd, triple it and add one. Write your new number.
  • (We drew an arrow from each number to the next.)
  • Repeat until you get back to a number you've already written.

Collatz conjectured (guessed) that the sequence would end up at 1, no mater what number you started with, but he couldn't prove his conjecture. Mathematicians have tried to  prove this for over 75 years, and it is still an open question. (It is very likely to be true. Using computers, people have tested every number up to and past 5 quintillion.)

As I expected, the kids loved it. At the end, I showed them a "mind reading" trick.
  • Pick a number from 1 to 31. Don't say it, just keep it in your brain.
  • (I pretend I'm sucking their thoughts over to my own head.)
  • Now show me which of these five cards it's on.
  • (I barely glance at the cards.)
  • Your number is ___.
After we did it a few times, I had the parents cover their ears and told the kids how it worked. I had  the five cards on the board, and half-size index cards for them to make their own cards. They loved it.

4th and 5th grade Circle
This circle met for an hour and a half. My plan was to analyze Spot It with them. (I've written at least 4 posts on using Spot It for math circles. Search on Spot It to find them.) We started out playing the game for about 15 minutes, which they all enjoyed.

The problem was, half of them had done this last year in their math class at Nueva! Luckily, one girl had come early and I had shown her the number trick. I asked her if she wanted to teach it to the others. She did.

I split the group in two, and she showed her group the number trick, while my group started thinking about the game. I had one boy who answered every question very quickly, and asking him to slow down didn't help. So, after we had figured out that there would be 57 different pictures, I got out the half-size index cards and suggested they make their own decks, with 4 pictures per card. Or, if they weren't into drawing pictures, 4 numbers per card. They worked hard at trying to make a deck where each card matched every other card on exactly one picture.  Towards the end, they wanted to play with the number trick too.

About halfway through the girl who led the other group came over and said, "The number trick is done." So I joined their group for a bit, and asked, "Why does it work?" A few parents were there, thinking about it with their kids. I should have asked them to work with all the kids (about 6 of them), but didn't think to say it. A few kids wandered away, to the puzzle room, no doubt.

The kids who stayed worked hard on the problems and had fun. I had a great time.
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