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 of 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 lernertandsander.com/cubes) 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?

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 (wodb.ca)

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: wodb.ca

What fun!

And so one more nifty tool is added to our techno toolbox for math class. (I have been loving desmos.com for a few years now, and use visualpatterns.org and estimation180.com 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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