Good Article: Neyland's Playing Outside

Generally speaking my best teaching has not resulted from a kind of effortful deliberateness. It has come when I have allowed modes of attentiveness and intuitiveness to come together in a kind of play that goes in its own direction; a direction that, when viewed in retrospect, turned out to be propitious. This is when I was the most effective. This is also when I was the least focussed on predetermined pedagogical paths and on self-conscious carefulness. Good jazz players and good teachers make mistakes, and have bad days, of course. This does not alter my firm belief that we are at our best when we are improvisers, not when we are corporate linear planners.

Jim Neyland wrote this toward the end of his delightful article Playing outside: An introduction to the jazz metaphor in mathematics education.

To me, good teaching is like conducting an orchestra of learning. The teacher and students are responding to one another in such an organic way that it's hard to imagine how the teacher makes her decisions.

If you think math is the one subject that just can't work this way, perhaps Neyland's opening will change your mind a bit:

The trouble with mathematics is that it looks a little more logical and consistent than it is. Mathematics has a universally recognised exactitude. It also has an inexactitude that tends to remain concealed. Mathematics contains what we might call ‘illogical truths’; that is, truths whose apprehension requires, not logical reasoning, but insight and imagination. These ‘illogical truths’ are experienced as surprises; as malformations. ... The person who studies mathematics is more than a logician. The study of mathematics has an inescapable element of unpredictability. And, mathematics endlessly escapes being captured within the logical frames of any mathematical structures.

[Thanks to Malke Rosenfeld of Math in Your Feet for pointing me to this inspiring article.]

New Toys & Games: Perplexus & Trango

When I go to Michigan to visit my family and friends there, I always stop by Mackinaw Kites and Toys in Grand Haven. They have the best game selection I've ever seen, and always have lots of interesting games and puzzles out, with salespeople ready and eager to demonstrate.

This time I got another set of Katamino for our friends up north (whose daughter loved it), a 3D puzzle for the boy I tutor (it reminds me of Rubik's cubes, which he loves), a magnetic monkey jumping toy which my son is enjoying, and Perplexus, which my son and I are both puzzling over. Perplexus is a clear plastic ball with an amazing 3D pathway inside that you try to keep the ball traveling along on. Because it involved eye-hand coordination, my son is better at it than I am. He likes that.

My brother got my son a game called Trango. My son hasn't played it yet, but I played it with our friends up north, and enjoyed it. Each turn you play either 2 or 3 pieces, according to the roll of the die, by connecting them to the other pieces already played. You get points every time you make one of the 4 shapes shown on the box (triangle, hexagon, chevron, and spade). You can add one piece to a patch already on the board and get points for the new shape you made. Playing this really got us looking for shapes.

Happy New Year

My celebration of the new year has often involved writing a list of the things I'm grateful for. (I've slacked off a bit in recent years, partly due to single parenting, partly due to the fact that I shifted from paper journals to writing on the computer.) Here are some of my gratefuls for this past year. I'm grateful..

1. ... to Jeremy Stuart and Roy Robles for the lovely video they made of my math salon.
2. ... to everyone who commented here, and to the 39,350 readers of this blog.
3. ... to everyone who submitted a chapter for the book Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet, and to Maria Droujkova and others who have helped with the book.
4. ... to everyone who attends my math salons.
5. ... to my morning class in the fall, for their determination and good spirit.
6. ... to my college district (and union!) for the chance to have a year sabbatical - it was marvelous!
7. ... to James Tanton for his book, Math Without Words.
8. ... to paperbackswap.com and betterworldbooks.com, for making my book-buying habit more sustainable.
9. ... to Julie Brennan, for the wealth of sharing that happens on Living Math Forum, the community she created, and to Maria Droujkova for the same at Natural Math.
10.  ... to Chris Evans for reading and commenting on ever chapter in the book.
11. ... to Linda Palter for drawing marvelous pictures for the book.
12. ... to DawnMarie, who watches my son when I go off to do my math projects.
13. ... for the internet.

I'll leave it there for now. My personal gratefuls list often went over a hundred items. I'll come back later and add some links. (I'm on vacation at my parents' house - no wireless and a very uncomfortable keyboard have made blogging difficult. I'll be home on January 5.)

Math Mama's Advice: Tips for Helping Older College Students

Received in email a few months ago:

Dear Miss Sue,

I discovered your Math Mama blog about a month ago, and have been having a most enjoyable time going through the archives.  I am a math tutor at a small Los Angeles community college (love it!) and am always looking for new insights and tips in conveying algebra 1 and 2 concepts.  I have already got some good ideas from your site - do you know of any others? 

The students I have who need the most help are older ladies who are back in college/first time who have had unpleasant math experiences in their youth. I'm a bit of a nurturer/hand-holder, so I like to break things down as simply as possible. 

Thanks so much!

Paula

[I asked her permission to reply here.]

Dear Paula,

I had quite a few older women in my 10am class this semester, and I had them in mind as I thought about your question.

I think it's important to address their fears directly. I recommend Overcoming Math Anxiety, by Sheila Tobias, and Mind Over Math, by Stanley Kogelman and Joseph Warren. (I buy used copies online for $3 or $4, and sell them to students. I used to lend them out, but I lose about 10 books a year that way, so I figured selling them was more realistic.) I also recommend the audio track I created. It's a guided meditation, and I recommend they listen to it every night for a few weeks.

I think helping them lead from their strengths might be more important, though. I try to help each class become a community. Some groups take off with it, and others don't. The older students know what they want, and are ready to go with it. This particular class has become an amazing community. They come in over an hour early (we are SO lucky the classroom is empty before their class!) and study together. They have another student lead them, and even though I like getting questions in class, they feel freer to ask questions in their group. They don't accept not getting it, and will work together until they do get it.

If you tutor one-on-one, you could still help this dynamic along by introducing these students to each other. Have you heard that 'the one doing the work is the one doing the learning'? That would mean that you learn more from tutoring than they do - unless you can get them helping each other.

I asked my students what other advice I might offer you, and they said that working together was key. They talked about keeping each other going when it got tough.

Perhaps if you recommend some of your favorite online resources for them to check out, they'll discover things that excite them. Many of my students really liked watching math videos. Check out mathtv.com, khanacademy.org, and (my favorite) James Tanton's videos.

Let me know anytime your students are particularly stuck, and perhaps I (or the folks who read my blog) can help. Thanks for writing.

Warmly,
Sue

Anyone want to offer other advice to a tutor of older students?

My Math Alphabet: F is for Factoring

My Math Alphabet: F is for Factoring 



Factoring Numbers
Factoring numbers is usually pretty straightforward.  If we're trying to find the factors of a number n, we check each prime up to the square root of n*. There are easy ways to check whether 2, 3, and 5 are factors, and about 73% of randomly picked numbers will have at least one of these factors. (I just now figured that out. Challenge #1: How?)  But once we've pulled out those factors, or purposely chosen a number that doesn't have any of those factors, it gets just a bit harder; with the help of a calculator, we just check whether n/p is a whole number. Here's a cute bit of math trivia: The first three numbers I'd check with a calculator - 7, 11, and 13 - multiply to make 1001, so for example, 137,137 factors to 7 ^. 11 ^. 13 ^. 137. (137 does not have 2, 3, 5, 7, 11, or 13 as factors, so I know it's prime.)

Although this search is relatively simple, it depends on knowing the list of primes to check (2, 3, 5, 7, 11, 13, 17, 19, ...), and that gets lots harder to produce for really big numbers. I remember reading the Scientific American article in 1978 about the mathematics behind a new form of cryptography, that involved multiplying two very big prime numbers to get a  number that would be too big to factor (in a reasonable amount of time). This scheme, now dubbed RSA after the 3 mathematicians who worked it out, is part of what helps keep online information private. Of course, what qualifies as a big enough number changes over time as computers become more powerful. Here's a list of the first ten thousand primes.

Factoring numbers is good practice for improving your number sense, and mathematicians sometimes find that a meditative exercise. If you have a calculator handy, it's pretty quick to check out the factoring of the number in this xkcd comic (the square root of 1453 is 38.118..., so you only have to check the primes up to 37). That's challenge #2.

Some mathematicians are good at arithmetic and would check out possible factors in their heads. I'd rather not myself, but G.H. Hardy and Ramanujan, perhaps like most number theorists, would have liked that sort of thing. A much-quoted story has Hardy saying his taxi  number, 1729, was dull, and Ramanujan responding that it was a very interesting number because it's the smallest number expressible as the sum of two cubes in two different ways. Hmm, I just factored 1729 (challenge #3), and I think it's interesting in another way - its prime factors are in an arithmetic sequence.



Factoring Polynomials
We spend a lot of our time in beginning algebra courses on factoring polynomials, and some teachers question whether that's a good topic to spend the time on. One reason for factoring polynomials is to find solutions to equations. But computers and calculators can answer those questions so easily, perhaps this topic become has become archaic. (And besides, most polynomials can't be factored. I remember a lovely post about that, and cannot find it. Can someone point me to it?) Here's a cool picture by Dan Christensen, of all the complex roots of polynomials with integers coefficients of degree 5 or less. (More here.)

In the comments to dy/dan's post Grocery Shrink Ray, one person wrote:   (comment #13)

Is teaching factoring the best use of time in the classroom? Is it the best topic we can teach students, at this point in their mathematical learning? WCYDWT can’t answer those questions, but it can raise them and get us thinking. In the case of factoring, I think these questions are fair ones. It’s true that I’m asking some pretty leading questions here; as you might be able to guess, my answer would probably be “no”. In my view, factoring is not terribly critical in real life, and this happens to be correlated to the fact that it’s hard to come up with WCYDWT problems (or real-world problems) to motivate learning factoring.

I replied:

Factoring may not have real-life applications, but it leads into cool deeper math topics. I see it as very important in algebra.

 So, what are those cool deeper math topics? 

One is Pythagorean triples, which I've posted about before. But here's a mathematical use for factoring I'd never seen before: George Sicherman wanted to find a way to create a pair of dice that would have the same sums as regular dice, with the same probabilities (7 comes up most often, for instance).  He used polynomials (and their factors) to find his unusual dice. (Found at Plus Magazine.)

What are some of the other cool mathematical uses of factoring?

_____
*If f is a factor of n, then f ^. g = n. One of these two numbers must be less than the square root of n, and the other more, unless n = f ^. f.

Math Salon Today

If anyone reading this blog lives near me (Richmond, California) and wants to join us, please do. We're making snowflakes, nameflakes, and stars, and thinking about symmetry.

You can call me at 510 (Do spambots care about phone numbers?) 236 (I hope not, but if they do, they'll miss this, right?) 8044, for directions.

Math Game: Risk Your Beginning Algebra Skills

I had a blast on Thursday playing the game show host for our game of ...  Risk Your Beginning Algebra Skills¹. I showed my students the fabulous prizes they could win (Blink, pentominoes, or 3 burnable cds), and used my announcer voice instead of my Math Mama voice.

I'm giving an early final on Tuesday so they can have two chances (the official final exam period is the next week). We had to start reviewing this week, which is usually sort of boring for me. This totally changed it for me, though I can't guarantee it was a better review for them. I do think the high energy keeps people going.

Most of the problems featured in the game came from their practice finals, and they were supposed to do all of the problems on their practice final before coming to class on Thursday. But of course most of them hadn't. We have a high school housed at our college called Middle College High School, and I have a number of high school students in two of my three classes. They have a class called Early College Seminar, where they get help with their homework. So they had all done their practice finals. And the winners of the game in the first two classes were all MCHS students.

I didn't want to just have one of the students who always does well be the winner, so I got prizes for two winners. One for the high score, and one for the 'most improved'. For that, I gave each person a 'multiplier' to multiply their final score by. I got the multiplier by taking 100 divided by their grade so far (a number under 100). The best students multiplied by 1. People who haven't done their retakes had multipliers anywhere from 1.4 to 8.

In the fist class, they both chose Blink. In the second, they both wanted pentominoes. In the third class, a boyfriend and girlfriend both won, so they got Blink and pentominoes.

Here's the gamesheet²:

You can risk anywhere from 0 to 100 points on the first question. Add the points to your total if you're right and subtract them if you're wrong. Now you can risk anything up to your new total on the next question. Before solving each problem I reminded people to make sure they had risked their points. I think the winners were the big betters, rather than the best math students. If you double your score on every right answer, 8 right gets you to almost 20,000.

I'll try this at the beginning of next semester for reviewing some of the stuff I think they should already know.




_____
¹I blogged a few weeks ago about how this game was used for a lecture on pi.
²The .doc file is here: https://drive.google.com/file/d/0B4Lou9CsLnQxcFhVNHJuV3dSVVE/view

Gift Ideas

My local friends get strawberry jam from Swanton Berry Farms, that tastes like fresh strawberries. And I usually make a pot of holiday mushrooms (recipe below). But those aren't math gifts...

I like getting used books for people I know will enjoy them, and I buy them through Better World Books whenever I can. (They get library rejects back into circulation, and contribute to literacy causes.)

Books
The Cat In Numberland (4 to adult, review)
You Can Count on Monsters (4 to adult, review)
The Man Who Counted (4 to adult)
Quack and Count (1 to 5 years old)
Anno's Counting House (4 to 7 years old?)
How Hungry Are You? (4 to 10, reviews of last 3)


Games
Blink
Set
Katamino (review)
Blokus


Puzzles
Tangrams
Chocolate Fix
Rush Hour

Snap Circuits

Keva or Kapla blocks (expensive)

These are only my most favorite math things...


Holiday Mushrooms
most of a bottle of Burgundy wine
4 or 5 pounds of mushrooms
3 sticks of butter
3-5 bullion cubes or tablespoons of 'Better Than Bullion'

Wash the mushrooms. Only cut stems off if they're really hard. Put in a big pot with the rest. Cook for 8 hours. Your house will smell good, and they taste fabulous! (Don't forget the toothpicks.)

More from James Tanton, on Twitter

Yeah, I have a Twitter account, but I seldom check it out. I don't see the point, usually. But I just discovered that James Tanton (jamestanton) is tweeting interesting problems. Hmm, can I get those delivered to my Google Reader, or something?

Republic of Math blogged about James' question regarding n plus square root of n: Can it ever round to a perfect square? (It's answered at the blog post, so you might want to play with it before clicking through.)

When I went to Twitter this morning, James' latest question was:

60houses in a row. 3 roof colors cycle in 3. 4 door colors cycle in 4. 5trim colors cycle in 5. Every house unique roof,door,trim color set?

He had to really squeeze to get that one in the 140 characters, didn't he? 

Thanks for feeding us some good math, James!

My Math Alphabet: E is for Eigenvectors and Eigenvalues

E is for Eigenvectors and Eigenvalues

This post is about fear.


Part 1. Fear.
Those words, 'eigenvector' and 'eigenvalue', sounded scary to me for years. I expected the concepts they reference to be hard. And so they were hard. These words / concepts come up in a course called linear algebra. The rest of the course was easy for me, but I struggled with these 'eigen' ideas. So when I was preparing to teach linear algebra for the first time, I got nervous again. It took a while to embrace the idea. Now that stuff doesn't seem so hard. But I remember that the word threw me off, and I know to take it easy when I teach it.

Square roots throw off my algebra students, partly because there's a weird new symbol involved. And partly because the concept that goes with it doesn't mean much to them. (I should have started more slowly than I did this semester, with finding sides of squares that have various areas. I'm kind of zipping through roots and rational expressions, because I wanted to focus more on our last book chapter, using the quadratic formula and graphing parabolas.) They know the √81 is 9, but if they're trying to simplify √162, they'll correctly write √81*2, and then proceed to change that √81 into √9, and then into a 3. It's all just moves in a game that they don't quite understand. And the game is scary.

I remember being very uncomfortable with Greek letters in my first calculus course. There were way too many of them. How was I supposed to memorize twenty-something new symbols?! So I'd read things by saying squiggle every time I saw a Greek letter. That doesn't work if there's more than one of them in the statement you're trying to read. I still slow way down when I try to read things with unfamiliar Greek letters. (Alpha, beta, gamma, delta, and theta are fine. Phi and mu are ok. I'm too lazy to go find the command to write them properly.)


If we always start with interesting problems, instead of with definitions and notations, will we intrigue people enough that fear will never come up? Or is grading going to always introduce the fear factor? What's a good problem for getting students thinking about square roots? Has fear ever come between you and your interest in math?


Part 2. Eigenstuff.
If you don't even want to think about eigenvectors and eigenvalues, stop here. But I'm going to attempt to write about them in a way that makes them feel less scary.

Eigen means something like 'innate' or 'its own' in German. That doesn't sound so bad. They're about not changing in a certain way, so what's a name I could make up that emphasizes the concept of staying the same? (I want to make up my own name for these as a way to make them my own. I think that will help me like them better.) What about home-vectors and home-values, as in 'staying at home'? It's similar to homo- which means same (homogenous, homophone, and homomorphism) and to homeo- which means similar (homeopathy, homeostasis, homeomorphism*), and that's good. It feels homey to me, and that's good too.

Linear algebra deals with vectors (think arrows) and operations on them. For the vector, let's imagine (1,2,3) in R³, our usual 3-dimensional space. (1,2,3) points 1 unit east, 2 units north, and 3 units up. An important concept in linear algebra is linear transformations, which take a vector or group of vectors, and stretch, rotate,  or reflect them.

Start with a particular linear transformation A, which is represented by a matrix. The transformation happens by multiplying the  matrix A times the vector (say x), getting Ax. Given A, are there vectors that don't change direction, but just stretch or shrink? If there are, those are called eigenvectors (home-vectors), and each one has an eigenvalue (home-value) associated with it, that describes how its length changed. The eigenvalues/home-values usually get labeled with Greek letters, lambda most commonly. I'll use s for stretch (or scalar, if you like). The definition gets us Ax=sx, which says that when the transformation A works on the eigenvector/home-vector x, you get x multiplied by a number (that's what scalar means), so that it just stretches.

Wikipedia has more, of course. (I liked seeing the applications, though most of it was beyond me. And I liked this: They pointed out that if the dimension is 5 or more, there is no method for finding exact values, and round-off error can make numerical methods problematic "because the roots of a polynomial are an extremely sensitive function of the coefficients".)

What's a good problem for getting students thinking about these ideas, before we ever say the E-word?


_____
*Homomorphisms and homeomorphisms are two different things, both mathematical. See wikipedia.