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.





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*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 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.





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*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 

Algebra 
  • 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

Algebra
  • 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

Algebra
  • 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

Algebra
  • Work with summations.
 
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