Tutoring News

My weekly tutoring gig with "Artemis" is going well. When we started, he didn't know much algebra, didn't really understand distributing. He's learned distributing mostly through experimenting, and has learned an amazing amount of algebra through playing with his TI-84. (I wrote about the first time we played with the calculator here.)

He loves coming in with graphs on his TI and getting me to guess the function. Here's the one he started with last week. (I got it. Can you?) I showed him some similar graphs to see how well he could do. He got them all. (I told his mom I figured we were into what would normally be the third year of algebra, after about two months of untutoring.)

I never have a lesson prepared for him, and we often go into college algebra sorts of topics (it's called math analysis in high school), riffing off those graphs. Some weeks I worry that we won't have a path in to good topics just by our inspirations. But it always seems to work out that we do something solid.

I know all kids wouldn't move this fast, but to me this demonstrates the power of play. That's all we're doing, playing with ideas...


My Other Student
R comes for tutoring about once a month. (I think it's a special treat for him.) He and I have been playing with Kenken and logic puzzles. He decided last week that he wanted to make his own Kenken puzzle. Yeay! That sounded like something Maria Droujkova would have suggested, but that I don't think of so readily myself. So we did it. We started with one box, and figured out what had to be true. Our puzzle isn't standard because we didn't give a total for many of the boxes. If you think about what has to be, you'll figure them out.

Here's the puzzle R and I made...

[I used Excel to make it pretty like this, and used print to save as a pdf. I learned that my blog doesn't want pdf's - it wants jpegs. So I saved the pdf as a jpeg. I don't see a jpeg option in Excel, so that might be the best way to do it...]

Anyone else want to try their hand at making their own kenkens?

Math Stories

I posted, back in June, about my favorite math books. They're ordered from youngest to oldest readers, and the first one is Quack and Count, by Keith Baker:

This is a board book, so it's good for the youngest child who will sit and listen to a story. But it stays good because it's so luscious. Great illustrations, fun rhythm and rhyme, cute story, and good mathematics. 7 ducklings are enjoying themselves in every combination. “Slipping, sliding, having fun, 7 ducklings, 6 plus 1.” (And then 5 plus 2, etc.) It would be great to have a book like this for all the number pairs that make 8, and one for 9, etc.

I teach the 'big kids' (8 to 14, most 10 and under) at Wildcat, the 'freeschool' my son attends, and another parent teaches the 'little kids' (5 to 7). She has to move from one house to another this week, and I just agreed to sub for her. I have enough trouble getting myself to move down from my college professor level to teach the 'big kids' well. I had a moment's panic when my son said, "Do easy things, Mom." Then I remembered Quack and Count.

I have 7 rubber ducks I borrowed from my son, so we can act out the story with the ducks. And then... Hmm... We have lots of active boys, and maybe a number story like this about cars would appeal to them. Maybe if I wrote it, they could illustrate it. I know they aren't big on extended writing, but maybe they'd get into thinking up other number stories.

Here's my first draft:

Crash and Count!

5 little race cars in a row
Count those race cars as they go

Racing fast and having fun
5 little race cars, 4 plus 1

5 little race cars, 3 plus 2
Looked like a crash, but then some flew!

Now they need to miss the trees
5 little race cars, 2 plus 3

5 little race cars, 1 plus 4
Most of them have crushed their doors

Turn off the engine, climb out fast
5 little race cars stop at last


[Edited to add: I just realized that this story verges on plagiarism. I copied the pattern given by Quack and Count as closely as I could. That seems totally cool for creating math lessons, but maybe not so cool for a story I'm posting online. So consider this a take-off on Baker's work.]

Number Logic

Jonathan, over at JD2718, has posted a logic puzzle he created, using lies and truths about numbers. I enjoyed it. But then our discussion turned to trying to create them.

I worked for a long time, starting from the answer I wanted and a bunch of things that were true and false about it. I couldn't get it narrowed down enough using just 5 clues. So I tried another number, and another. I think I've finally got a puzzle that works, but it doesn't feel as elegant as Jonathan's. Tell me if you get one possible answer. (But don't post the answer, please.)

And I'd love to know if anyone else has any success making these up.

Who am I?

There are four true and four false statements about the secret number. Each pair of statements contains one true and one false statement. Find the trues, find the falses, and find the number.

1a. I am the sum of two squares.
1b. I do not have any repeating digits.

2a. I'm even.
2b. I have exactly two digits.

3a. I'm prime.
3b. I am one less than a triangular number.

4a. I have just one digit.
4b. I am the product of consecutive primes.

Richmond Math Salon

Saturday, December 12
2 to 5pm

• All ages welcome. (Fun for kids 3 to 90.)

• Explore math in a fun, safe environment, where no one will judge you.

• A family math event: You and your kids can explore math in a way that works for each of you.

This month we’ll be making snowflakes, talking about symmetry, and playing with logic puzzles.

This monthly event is currently held in a small home (in Richmond, CA), so please RSVP if you plan to come. The Richmond Math Salon is hosted by Sue VanHattum, a math professor at Contra Costa College.

Interested? Email me at suevanhattum at hotmail, or call me at 510-236-8044 (before 8pm).

Schedule of salons for 2010: Jan 23, Feb 20, Mar 20, Apr 17, May 15, Jun 12, Sep 18, Oct 16, Nov 13, Dec 11.

Why I haven't been posting...

I use an iBook G4 laptop with DSL connection at home. A week and a half ago, I finally said yes to one of those pesky 'please let us update your software' messages I get regularly from Apple. My computer got trashed.

I will probably buy a new computer, but haven't done so yet. I think I'll be buying a desktop (mac, still), and getting my laptop cleaned up, fixed up, etc. Meanwhile I'm limping along, logging in wherever I can get wireless connection, since my ethernet connection doesn't work.

Anyone who wants to discuss this with me is welcome to phone me at my cell phone: 510 367 8 zero eight (one more than four). More gory details below. I've been spending more time reading actual books, hanging out in my yard, and cleaning house since the mini-disaster. It's all good, but I need to get this settled...

Today I'm seeing tons of good math posts, and am wishing I could join in more fully. But there's a 2 hour limit on my connection here, so I'm trying to just catch up on my blog reading. When I do come back, I'll have a few posts I've been working on offline. ;^)



=====
Gory details:
Apple techies graciously spent hours with me, even though I have no tech support contract, after I almost started crying about it being Apple's fault my computer was trashed. They and I agree that software cannot destroy hardware. But they think my hardware is at fault. (My ethernet connection broke back on April 14, and I bought a usb ethernet adapter and got my taxes out on time. So I use a usb port for my ethernet connection. One techie talked about the problem with the ethernet connection 'migrating' to the usb ports. Huh?!) My hard drive is almost full, which may have contributed to the problem. Besides no capacity for etherternet connection, I can't print. Etc

When Too Much Becomes Too Little

Joseph Ganem wrote a great article for the APS (American Physical Society) News, titled "A Math Paradox: The Widening Gap Between High School and College Math". Ganem is a physics prof at Loyola University Maryland, and works with incoming students at orientation. He's also the father of 3 children (aged 14, 17, and 20), and helps with their math homework.

I want to share 3 quotes from his article, and then encourage you to follow the link to read the whole thing.

He satrts here:

We are in the midst of paradox in math education. As more states strive to improve math curricula and raise standardized test scores, more students show up to college unprepared for college-level math.

After some examples from his kids' homework, he has these questions:

So if eighth graders are taught math at the level of a college sophomore why are graduating seniors struggling? How can students who have studied college level math for years need remedial math when they finally arrive at college? From my knowledge of both curricula I see three problems.

I'll let him tell you the 3 problems, so I don't steal all his thunder. His conclusion seems just right to me: (President Obama, Secretary of Education Duncan, Are you listening?)

All three of these problems are the result of the adult obsession with testing and the need to show year-to-year improvement in test scores. Age-appropriate development and understanding of mathematical concepts does not advance at a rate fast enough to please test-obsessed lawmakers. But adults using test scores to reward or punish other adults are doing a disservice to the children they claim to be helping.

It does not matter the exact age that you learned to walk. What matters is that you learned to walk at a developmentally appropriate time. To do my job as a physicist I need to know matrix inversion. It didn’t hurt my career that I learned that technique in college rather than in eighth grade. What mattered was that I understood enough about math when I got to college that I could take calculus. Memorizing a long list of advanced techniques to appease test scorers does not constitute an understanding.

Please read his article in all its glory, over at APS.

Help! Questions about You Tube...

Back in May, my algebra class humored me with a discussion about what made our class work as well as it did. One student did a video of the discussion. She uploaded that onto YouTube and then deleted it from her camera. YouTube rejected it because it was too long. Is there any way I can recover that? I had assumed not, but I see one scene from it on the list of videos, along with the notice that it was rejected.

Also, can I moderate or prevent comments, or delete them? My students were brave enough to get videod while explaining math problems, and now someone has made a nasty comment.

Math Teachers at Play #19

Are you wondering where MTAP #18 went? Here's the story (contest-winning entry from Lisa Downing), and we're sticking to it!

The Odds were at odds with the Evens. It never seemed fair to them that two Odds made an Even but two Evens didn't make an Odd. Fifteen fired the first salvo by stepping into the order twice. Sixteen managed to jump in, but then Eighteen disappeared. Seventeen and Nineteen were prime suspects. The Numerologist stepped in and told everyone to get back in the right order or ELSE. Unfortunately Eighteen was still missing. The authorities launched an investigation but there were so many factors involved that they never could get to the root of the problem.

Riddle: What do 10011, 23, and 13 have in common?

Gorgeous! The photo above, of an anglerfish ovary magnified 4 times, was taken by James E. Hayden, and won 4th place in the Nikon Small World photomicrography competition. You can see lots more winning photos here. More about Hayden at the end.


Getting down to business, allow me to welcome you to a math buffet. I think there will be something here to tickle everyone's palate. (Photo by skenmy.)

• Maria Andersen is at it again! She's redesigned her math for elementary teachers course, and shows us some of the results in Transforming Math for Elementary Ed. This post is full of links to work her students did, some of it exciting stuff.
• John Golden offers us a game called Area Block modeled on Blokus (one of my current favorites to play with kids). I haven't had a chance to play it yet, but I am definitely looking forward to it. (Here's a bit of math teachers at play trivia. John and Maria work within 30 miles of each other in my home state. What state is it?)
• Denise walks us through an algebra word problem, translating from English to "mathish". Nice! And Jason wants our help with teaching students how to read a few different types of problems, in "When vocabulary isn't the issue" and "A reading experiment". The puzzle given in that second post looks fun!
• Pat Ballew gets us thinking about geometry with his "Notes on Cyclic Quadrilaterals".
• Do we discover math or invent it? How we answer that question affects how we talk about the math of the ancients. Dan MacKinnon reviews a book and discusses this intriguing issue.
  
• A short review of The Little Book of Mathematical Principles is at We Overstep.
• Pumpkin Patch offers us another game, Sum Math.
• The descriptions don't always look accurate to me, but you may find some goodies among the "100 Incredible Open Lectures for Math Geeks". Some are audio only, some are videos.
  

If you haven't stuffed yourself yet, here are a few more tidbits I noticed over the past two weeks.

• Two mysteries solved, one a murder, and one the opening chord of Hard Day's Night.
• Sometimes there's a hard way to solve a problem, and an easy way...
• Does a bullet that's fired from a gun land at the same time it would if it were dropped? Built on Facts gives us some equations and Myths Busters weighs in on YouTube.
• On thinking before you compute.
• Like comics? Indexed on confusion as a function of information, and a graphic novel about Bertrand Russell.

_____________

Here's the interview Nikon did with James E. Hayden. I wrote and asked him how he uses math in his work. He said:

As for the math - it is, of course a large part of doing any kind of research. In a lot of my regular work, we work with analysis programs that quantify different aspects of the images we capture. Everything from simple counts (how many cells in the field of view?) to time lapse analysis - how fast are the cells moving? In what direction? At what angle to the original movement? Does the rate of movement change over time? Do we need to quantify the interaction of the cells in some way? Is there a change in the fluorescence intensity values? And other interesting things like that. My assistant, Fred Keeney (who won an Image of Distinction in this year's competition) has been taking computer programing classes to help us automate these kind of analyses.

Submit your blog article to the next edition of math teachers at play using our carnival submission form. Past posts and future hosts can be found on our blog carnival index page. (Our schedule is changing to once a month. Denise at Let's Play Math! will host the next carnival on November 20.)

Contest: Write a number story by Wednesday

The Math Teachers at Play blog carnival came out twice as #15. Since then we've had #16 and #17. We'd like to iron out the numbering, and so the upcoming issue will be #19.

I am personally sponsoring a contest for the best little (ie, very short) story written about how the numbers got mixed up this way. The winner gets their story included in the next MTaP, which comes out here on Friday, and gets a $10 gift certificate at Better World Books. I get to be the judge. :^) It could be funny, mysterious, intriguing, whatever will be memorable.

Deadline: Midnight (PST) on Wednesday

Why? Partly to have fun with our glitch, partly to 'make math our own' as Maria Droujkova likes to say. And partly because I came to math on the internet through 'living math' - Julie Brennan's (trademarked) term for math through stories, and for the list she runs for mostly homeschoolers and the site she maintains with gazobs of ideas for how to engage kids in mathematical thinking through math stories and finding math in any story.

The Internet is a big treasure hunt!

I'm laughing at myself right now. I wonder if I can make this funny for you. Before I explain I should tell you ... my memory is so bad... (I once got a card that had an old woman saying that on the front. Open it and see, "How bad is it?" in a bubble. She replies... How bad is what?) And maybe I should claim that my brain takes a bit to get into gear in the mornings?

Last Thursday I posted a bunch of links, and included:

I can't remember now why I did this, but I'm still intrigued... I went to Wolfram Alpha and typed: factor 1782^12+1841^12. It's just a bunch of big numbers. Why do I like it?

On Saturday night Joshua Zucker replied:

why 1782^12+1841^12? I don't know why to factor it, but of course it equals 1922^12 (just try it on your calculator, not at Walpha of course!)

Well, I wasn't thinking of the consequences, and I believed him. I was impressed that he could use his calculator to factor the huge number you'd get from 1782¹²+1841¹². I didn't reach for my calculator, being comfortably ensconced in my recliner. No, I reached for Google, and I googled Josh, because I was curious about what math he might lead me to.

I found a comment he made on Cut the Knot many years ago (in 2000). So I started exploring Cut the Knot, and thoroughly enjoyed some discussions about how math uses words differently from their common usage.

Eventually I remembered my original quest and googled "1782^12+1841^12". The very first thing I got was:

Fermat's last theorem. Statement that there are no natural numbers x, y, and z such that x^n + y^n = z^n, in which n is a natural number greater than 2. ...

[Fermat's last theorem was proved in 1995 by Andrew Wiles. There are lots of whole number solutions to x² + y² = z², like 3² + 4² = 5². But the theorem says there are no solutions to x³ + y³ = z³, nor to equations like that with higher powers.]

Huh? But Josh said 1782¹²+1841¹²=1922¹²?? Next entry I clicked on was about a Simpson's episode:

In the 1995 Halloween episode of the award-winning animated sitcom The Simpsons, two-dimensional Homer Simpson accidentally jumps into the third dimension. During his journey in this strange world, geometric solids and mathematical formulas float through the air, including an innocent-looking equation: 1782¹² + 1841¹² = 1922¹². Most viewers surely ignored this bit of mathematical gobbledygook.

On the fan discussion site alt.tv.simpsons, however, the equation caused a bit of a stir. “What’s going on, he seems to have disproved Fermat’s last theorem!” one fan marveled, referring to the famous claim by Pierre de Fermat—proved just months earlier—that for any exponent n bigger than 2, there are no nonzero whole numbers a, b, and c for which a^n + b^n = c^n. The Simpsons equation, if correct, would be a counterexample to the theorem, meaning that the proof had been wrong.

Ahh, now I get it! And I finally had a vague memory of reading a post somewhere about how 1782¹² + 1841¹² and 1922¹² look exactly the same if you evaluate them on a calculator. (Try it!) That must be why I had originally gone to Wolfram Alpha with this.

Meanwhile, here's the other treasures I found:

• A silly math discussion on nrich
• On simpsonsmath.com, Lisa Simpson inspires Girls Just Want to Have Sums
• I learned how to use html to do the powers right (angle brackets around sup and /sup, before and after)
  

Is anyone else giggling, or is the humor lost in translation?