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.

My Math Alphabet: D is for Dance

D is for Dance

People say music and math are closely related, but deep down, I don't really get it. I'm not so good with music. Let's say I'm a slow learner. I can sing pretty well, if it's a song I've sung lots and lots of times. I can play the penny whistle well enough to get compliments now and again. (Penny whistle is probably the easiest instrument there is.) I tried to learn guitar for years, and was always mediocre (at best). I was terrible at the timing.

Then there's dance. I love both music and dance, but I'm slow at both. My favorite sort of dance is contra dance. If you go to a contra dance, you're welcomed in, and taught how to do it. You don't have to bring a partner, and no one scowls when you mess up (well, almost no one). Good thing folks there are patient, because ... I'm a very slow learner.

So I never would have thought, on my own, about looking at the connections between math and music or dance. But I'd seen a number of questions about this on the Living Math (Yahoo group) and Natural Math (Google group) lists, so when Malke Rosenfeld emailed me about her Math In Your Feet program, I paid attention.

I sent a message to both those lists, with a link to her blog, and figured she'd help a few people with their questions.

But when I read her blog, I got excited myself. It turns out she works in public schools, and is trying to help the schools see how important movement is for kids' brain function. Yes! (My son, who never has to sit at a desk, would wither if he were stuck sitting in a classroom for hours on end.) She also addresses some cool mathematical questions in her work with kids. Things like: How many times do I have to repeat this 'jump and turn' to get back to where I started?

I think this post is my favorite for getting a sense of what she does. If you're like me, this blog will stretch your notion of how to approach math. You might also like this collection of math and dance links. (It's a wiki, please add to it if you're moved to.)

One more story about me and dance... At contra dances, when the band (there's always a live band) takes a break, waltz music goes on the sound system, and it's time for waltzing, which I. Could. Not. Do. I'd try to count - 1 23, 1 23, 1 23 - but it never helped. Part of my problem is that I like to take B.I.G steps when I try on my own to dance to that music. But there's another problem. The proper way to waltz involves 3 counts forward and 3 counts back. Two years ago, at the Queer Contra Dance Camp, a marvelous dancer showed me a different way to count, 1 23, 4 56. Somehow, that made all the difference. Now I can waltz - more or less.

My Math Alphabet: C is for Calculus

C is for Calculus, Which Gets Me Jobs

Calculus helped me get both of my full-time community college positions, and it was calculus that got me back to teaching in the first place. I love calculus.


The Slope of a Curvy Line
Back in the early nineties I was doing user support for a progressive internet provider, known as PeaceNet, EcoNet, Institute for Global Communications, and a few other xNets. We worked out of a small office in San Francisco, and I did phone support all day, helping people all over the country get online, while also trying to write manuals and learn more in between calls.

In January of 1995 I flew to Seattle to visit some friends, and on the way back home I sat next to a man who was a Native American AIDS activist. We talked about many things, and at one point I explained to him what calculus is:

You know how in algebra, you graph lines, and find the slope? The slope tells the rate of change, which is important in lots of real-life applications, but most of those don't make straight lines. Finding the slope for a curvy line is what calculus does.


If we draw a tangent line to a curve, we can define the slope of the curve to be the same as the slope of that line. The problem is that a tangent line to a curve only touches it in one point, and you need two points to find the slope. So we cheat. We use the point of tangency, and then for a second point we look at secants  (lines that touch the curve in two places) through that point, with the second point sliding closer and closer to that first one, so that the secant line is twisting closer and closer to the tangent line.

I drew lots of diagrams on our napkins, something like this (though of course I couldn't animate mine). And he told me I should be teaching. He suggested I get a job at an Indian college. I missed teaching, so I looked up the Indian colleges, and thought about it.  I was too tired of moving to new places to go for it, but that got me started on looking into community colleges, and that summer I applied at a number of colleges back in Michigan, where my family is.


What's It Good For?
One of my interviews was at Muskegon Community College, only an hour away from my family in Grand Rapids. During my teaching presentation, JB, who was on the committee interviewing me (they were all pretending to be my students) asked, "So what's this good for? Anything?" I asked what he was interested in.

JB: "Rocks." (He's a geology teacher.)

SV: "Hmm, well, would it be possible to know the shape of a layer of rock underground, and want to know its volume?"

JB: "Yeah!"

SV: "And would it be likely for the shape to have a circular shape, so it would be the same in any direction from a central point?"

JB: "Oh yeah, that's common."

SV: "OK. What if it were shaped like this..." And I drew a hypothetical rock layer formed by two parabolas crossing. We imagined it spinning around the y-axis. I then explained how to use something the textbooks call the 'shell method' and I call the 'tin can method', to figure out the volume.They were impressed, and I got the job.


Related Rates
I worked there for six years, and was happy with my work. But I wanted to adopt a child, and it became clear I wasn't going to be able to adopt in Muskegon. (As a single pagan lesbian, I just wasn't the sort of family the social workers were looking for there.) I decided to try to move back to Ann Arbor, or back to the Bay Area. During my interview at Contra Costa College, I had to do an unplanned lesson. I got to choose between linear algebra and calculus, and then they would give me a topic. I considered doing the more advanced linear algebra topic, in order to impress them, but decided it might backfire if I couldn't explain it well enough. I stuck with calculus, and they asked me to explain related rates.

That's one of my favorite topics (I know, lots of pseudo-context in those problems, but I think they're neat!), and I enjoyed getting to play with it. I had thought my planned lesson went well, but later found out they weren't particularly impressed with that, and it was my impromptu talk on related rates that got me the job. They must have seen my eyes light up as I started to explain how one rate of change was related to another, and how we could solve our problem using those relationships.

I adopted my son a year and a half after I moved back out here; I'm now the happy mama of an 8-year-old. I've been at CCC for nine years now, and I love the diversity of my college. Thank you, calculus!

Deriving the Quadratic Formula: James Tanton's Twist

I've always enjoyed showing students how to derive the quadratic formula. I don't test them on it, so the stress level is lower. And it's late in the term, so they appreciate a break from the pressure, and most really do try to get it. I get a few making those appreciative sounds that happen when the lightbulb goes on, and that makes it especially fun.

But it's hard slogging through some of the weird steps. Here's the standard derivation, if you haven't done it in a while. Check it out, and imagine trying to explain it to people who are pretty fragile around math.

So the math education gods were smiling on me last week, and the day before I brought this topic to my students, I interviewed James Tanton, who (out of the blue) showed us his twist on this. (Thanks, James, for helping me with my lesson plans!)

If you just can't find the time to watch the video, it goes something like this:

ax²+bx+c=0 

We want a perfect square in the first term, 

so we multiply both sides by a:                  a²x²+abx+ac=0

We want the second term to have a factor of 2, and to keep the first term a perfect square, 

so we multiply both sides by 4:            4a²x²+4abx+4ac=0

We almost have what we see in the box above, but we want b² and not 4ac, 

so we do a little adding and subtracting:   4a²x²+4abx+b² = b²-4ac

Now factor the left side:

(2ax+b)² = b²-4ac

Taking the square root of both sides steals away a solution, so we include a plus or minus:

2ax+b = ±√ b2-4ac  

Subtract b from both sides and divide both sides by 2a, and you've got it. Much prettier than the standard derivation, I think.

I used this in class last Thursday, and I think it went much more smoothly than the standard version. I always do it twice, so I did it again on Monday. Students said they got it, and some liked it. I haven't spent enough time on completing the square (our days together are numbered, at this point in the term), so I don't expect their understanding goes very deep, but it's a start.

I'll do this again next semester in my intermediate algebra course, with a better grounding in completing the square. I look forward to it. Maybe all our conics problems will be easier, with James Tanton's brilliant help.

Number Tricks Show the Power of Algebra: Subtract by Adding

I gave a mastery test last Tuesday, and my first class had worked hard to get ready for it. I wanted to show them something fun the next day, instead of diving into our next unit: roots, completing the square, and the quadratic formula. I really enjoy the buildup of all that, but it takes some serious work on their part. I wanted to start with something more .. light-hearted.

So I showed them a number trick I learned recently. (Was it at AMATYC? Was it online? Or was it somewhere else? I haven't a clue.)


The Trick

If you don't want to do a subtraction, say one with lots of borrowing, you can add instead.

Let's try an example:
Lots of borrowing needed here, and two borrows in a row can get confusing. So instead we'll do this...

1. Find the "9's complement"
2. Add
3. Subtract one from the highest-value digit (leftmost position) of the result, and add that one in the ones place.

To find the nines complement, subtract each digit from 9, and write the result in place of the digit.

Now we add, and do a little fussing...




Why it Works
The real fun, with Helen on the plane and with my students, was showing why it works. (One student wanted to know why they hadn't been taught this before. I said I thought it was too much of a trick, and you might forget just how to do it, and math shouldn't be that way. But maybe it would improve some people's subtraction enough to be worth it for them.)

Let  x-y represent the subtraction problem. Let's start with supposing y is 3 digits (or less), and see if we can find a way to generalize, which is harder to write down. Find y complement, which I'll call y_c, by finding 999-y. If y_c=999-y, then y=999-y_c. And

x-y 

= x- (999-y_c) 

= x-999+y_c 

= x+y_c-999 

= x+y_c- (1000-1) 

= x+y_c-1000+1. 

Done.

I like that! And, more importantly, so did my students. I used to think number tricks were silly, but the ones I did at the start of this semester, along with this one, have convinced me to change my mind about equations versus expressions.

I used to feel like equation solving was the heart of algebra, but the algebraic explanations of number tricks start with an expression representing the numbers, and then involve simplifying until something comes out which really does explain why the trick works. Simplifying expressions suddenly seems much cooler than it used to. (I used to also think you couldn't check your answer when you simplified an expression, but all you need to do is put in a random number to the original and your answer, and make sure they match. Try 2 or 10 for an easy number, since 0 and 1 aren't good choices for catching mistakes.)

Math 2.0 Webinar Series: James Tanton

This evening at 9:30 eastern time, 6:30 California time, I'll be interviewing James Tanton as part of the ongoing Math 2.0 webinar series created by Maria Droujkova. I've reviewed a few of James' lovely math books here, and have used his videos in a few of my posts. He also sends out a monthly puzzler to people on his email list, runs math institutes at St. Mark's School where he teaches, and more.

Here's what James says about himself on his 'about' page:

James Tanton (PhD. Mathematics, Princeton University, 1994) is a research mathematician deeply interested in bridging the gap between the mathematics experienced by school students and the creative mathematics practiced and explored by mathematicians. He is now a full-time high school teacher and does all that he can to bring joy into mathematics learning and teaching.
James writes math books. He gives math talks and conducts math workshops. He teaches students and he teaches teachers. He publishes articles and papers, creating and doing new math. And he shares the mathematical experience with students of all ages, helping them publish research papers too!

We'll have fun exploring math with James. Click here to join us. Come early if it's your first webinar, so you can get situated.

AMATYC Conference in Boston

Well, there wasn't enough math to suit me. But a few of the sessions were great. I especially liked two of the Saturday workshops.

One was on the history of pi, and the presenter, Janet Teeguarden, set it up as a game of Risk (her own version).  We got a sheet with 25 multiple choice and short answer questions, and put our answers down. As she presented, she'd stop at each question and ask us to put down how many points we wanted to risk. We started with 100 points, and each question allowed us to increase that if we were right. One person was sure of his answers on ten questions and didn't try any others, so he got 102,400 points. I was only sure on a few answers, but tried a bunch. I got up over 10,000 points. The game kept me listening much more intently to a presentation that I might have mildly enjoyed otherwise. (And I'm not good at listening to presentations that don't have audience participation, so I might have drifted away, even though it was interesting.)

Although this presenter used Powerpoint for her presentation and the game (it made a cool swooshing noise as each answer was shown), I think I could do something similar with just a worksheet and the board. I think I'll try this when we're reviewing for the final exam. I could give them a practice final to do as homework, and then play Risk with them as I go over it.

[Edited to add...]  Here are the first few questions of the 'Risk Your Knowledge of Pi' game:

	Risk	Points
		100
1. What is the formal definition of pi?
2. In what book will you find the following? “Also he made a molten sea ten cubits from brim to brim, round in compass, … and a line of thirty cubits did compass it round about.”
3. What ancient Greek mathematician determined that 3 10/71 < pi < 3 1/7?
4. He used circles with inscribed and circumscribed polygons to make this estimate. How many sides did his final polygons have?

I risked 100 on question 1, got it right, and entered a total of 200.  I risked all 200 on question 2, got it right and had 400. I think I risked all 400 on question 3 and got it right, for a total of 800. I risked 50 on question 4, and got it wrong, for a total of 750.


The next session was something about improving antiquated word problems. The presenter (a textbook author) talked about how silly some problems are, and how he tweaks them to make them more relevant. He said: "Functions are the heart of intermediate algebra." I've been looking for a way to tie all the topics together, and thought that might work for me. I'm excited to look at the text again, and see whether a function focus would tie most of it together.

The problem we worked longest on started out something like:
The 10,000 seat stadium will sell out for the rock concert. The better seats are $65 and the cheaper seats are $45. How many of each ticket type must be sold to bring in revenues of $500,000?

Silly question. Why do we want that revenue in particular? If we change it to a function question, it becomes more interesting: Create a function that takes number of $65 $45 seats as input, and produces revenue as output. There are all sorts of things that can be done with this:

• Graph this function.
• What does the y-intercept represent?
• What does the slope represent? (Hey, the slope is negative. More tickets means less revenue? How's that possible?)
• What x and y values make sense?

On Thursday (backing up), I went to a fun workshop on using the abacus. Now I'd like to spend some time trying it out with kids.


I also visited a math circle on Sunday morning and had a great talk with one of the presenters afterward, just before I caught my plane home. The woman I sat next to on the plane liked to talk, and that made the time go faster for the first half. After 3 hours, she'd run out of stories and neither of us was looking forward to the 3 hours left to go. I offered to show her some math, and she said sure. I showed her binary numbers by starting with a magic trick. (Hey, I think I have to revise my post on binary. I left that out!) You normally start with 5 cards, each showing some of the numbers from 1 to 31. I just wrote them on my paper. She got into it, and I showed her a way to subtract by adding (decimal first, then binary). Our flight wasn't over yet, and she said she liked algebra, so I showed her some of the Pythagorean triple patterns. We were landing as we finished that up. Thank you, Helen, for being an eager student - you made my flight so much more fun!

It's good to be back home, and I'm pumped up for the last few weeks of the semester.

Blogger Ethics: Book Reviews

Some ethical considerations for my blog are obvious to me:

• I'm always honest, and do my best to be fair. 
• I choose not to take any ads, because that's what I prefer in my internet experience. 
• I don't even read most of the email solicitations I get, from people requesting to do guest posts, or asking me to link to their '10 Best X' sites. I have asked a colleague to do a guest post because I loved what she wrote on a small population email list, and I considered letting another blogger do a guest post here because I liked his blog (we just never worked out the details). To me a guest post must be from someone whose work I know and respect.
• If I'm reviewing anything I got for free, I mention that fact.

But I've had a question in my mind for a while about book reviews, and haven't answered it yet.

Once in a while, when I see an interesting new math title, I request a review copy from the publisher. If it's good, I read it and review it, mentioning in the review that I got the book for free. (There's one book at least that I liked but haven't yet reviewed. I want to find time to look it over again before I review it.)

But if I don't like the book, my feeling is that the best thing to do is just not write a review. I think other people might like the books that I don't like; our reactions to books can be as varied as our reactions to any art.  If I say that I don't like a book, that requires evidence to back up my displeasure. But I don't want to bother with all that if I'm not fond of the book. I think I'd like to pass those books on to another blogger. Perhaps they'll find a reader who likes them better.

Is this the right thing to do, or is there some reason I ought to review the books I'm not fond of?

AMATYC Conference in Boston, Next Week

I'm headed to Boston on Wednesday for the annual conference of the American mathematical Association of Two Year Colleges. Let me know if you'll be there too.

My Math Alphabet: B is for Binary

On, On, Off, Off, On = 16+8+1=25

B is for Binary

I loved teaching about binary numbers when I used to teach basic computer courses. What's going on inside that box? Lots of little electronic switches are going on and off, and that's enough to represent any number (and anything else you put in a translation table). The switches are not the mechanical sort you see to the right; they're microscopic.

On=1, Off=0. Start on the right with the ones position (2⁰), double each time for the twos (2¹) place, fours (2²), eights (2³), sixteens (2⁴), and further. One byte in a computer is 8 bits (bit is short for binary digit), so the highest-valued binary digit in a byte is worth 2⁷ or 128, and a byte can take on any value from 0 to 255. For instance, I'm 54 - that's 32+16+4+2, or 00110110.

Rick Regan really digs binary, and his blog, Exploring Binary, says it all. Much of it is too technical for me, but I've enjoyed a number of his articles. How I Taught My Mother Binary Numbers looks like a good starting point, along with One Hundred Cheerios in Binary. His post on the look and sound of binary numbers is delightful, as is The Binary Marble Adding Machine.

Here's one more fun site for binary numbers: Computer Science Unplugged.

Outside of math, I think we too often think in binary terms - good or bad, left or right, boy or girl, etc. I like it when I can break out of those boxes and find a third path. But in math, binary is fun.

One-Way Functions: What Are Some Simple Examples?

Tanya Khovanova gives a great simple example of a one-way function, which is becoming dated. She's looking for other examples. Can you help her out?

A one-way function can be used to encode a message. To decode the message, we'd want to go backwards, using the inverse of the function. It's very hard to compute the inverse of a good one-way function. Anything involving financial transactions online should use a secure encoding.

Alphabet Books, and A is for Abacus

I used to like making alphabet books for my son. He and I made a book together we called the Cool Car Alphabet Book. We mostly used Wikipedia for photos, and managed to find a type of car for every letter. I use a program called ClickBook to get the pages to come out right. (Pages 1 and 2 go with 19 and 20, and then 3 and 4 go with 17 and 18, etc. Staple and fold, and you've got a book!)

Today I got the urge to do a math alphabet. It's been done before -  G is for Googol: A Math Alphabet Book looks pretty fun. But I thought I'd have fun doing it, and maybe I'll come up with something a bit different. I think I might write a short blog post for each letter. (Sue, do you know how to write short blog posts?) Some letters look more fun than others. I hope you'll enjoy coming with me on my journey from

A is for Abacus and
B is for Binary, through
Z is for Zero.

A is for Abacus

Many years ago my mom got me an abacus for Christmas, just like the one you see here. It had a little booklet with it called Bead Arithmetic, with lessons on all the basic operations. I learned to add and subtract, and began learning to multiply. I had fun with it, but didn't go very far. (I never learned to divide, and have no idea how one would find a square root on it.) I still have it, though I haven't played around on it for years.

This is a Chinese abacus (suanpan). The five bottom beads are ones (times a power of ten) and the top beads are fives. To start, you clear it by pushing all bottom beads down and all top beads up. To represent 57, the second rod from the right would have one top bead down for the 50, and the last rod would have one top bead down and two bottom beads up, for the 7.

If I remember correctly, the procedures for adding and subtracting sometimes used all of the beads on a rod, although the final form of a number never uses all five bottom beads or both top beads. (I just now learned from Wikipedia that this bead configuration can actually be used for hexadecimal numbers (base 16), since you could represent any number up to 15 on each bar.)

Expert abacus users can perform arithmetic on an abacus faster than most of us can do it on a calculator. Here's a sweet video of some kids in Japan taking classes to learn how to do mental math while visualizing the abacus (the Japanese abacus is called a soroban).

It might be fun for kids in this country to learn to use an abacus when they're working on addition and subtraction problems. Feeling the beads while thinking about the numbers would be  grounding. Elementary teachers might enjoy learning to use the abacus along with their students, approaching arithmetic from a new angle. I think the best thing parents can do to help their kids with math is to learn some math themselves. If you're up for it, get yourself an abacus. Here's an online abacus to play with for now.

My son goes to a mini-school with just 5 kids in it, currently held in our friend Felicia's home. After I started writing this, I asked her if she'd like to teach some math on the abacus and she said sure. Yesterday I went to Oakland's Chinatown and bought 6 of them, for $6.50 each. (Available online here.) I'm hoping she gets the kids intrigued enough that they'll try to teach their parents!

Scientific Notation: Big Numbers and Small Sizes

Next year I'll do scientific notation a few days earlier, right after we start working on exponents. This semester I followed the book's strange order just so that there'd be more time between mastery tests - students needed a break.

On Wednesday, I introduced scientific notation. We practiced converting between standard notation (writing numbers the usual way) and scientific notation. Most textbooks I've seen give rules that involve the words left and right. Being a bit dyslexic myself, I don't find those very helpful. I have my students tell me: big numbers have ten to a ... positive power, and small numbers have ten to a ... negative power. They were relieved to have a slightly easier topic, and enjoyed our introduction.

On Thursday, I wanted to work with them on multiplication and division, and on recognizing what to do in story problems. The size of the numbers makes using common sense hard, so I emphasize making up their own parallel problem, with the same structure but easier numbers. (Thanks, George Polya, for all your good ideas.) How many of this tiny thing in this biggish space? Let's think about how many 2 inch things in an 8 inch thing - oh yeah, divide biggish space by tiny size to get how many. (We got lots of practice on unit conversions in these problems.)

As I prepared for that class, I lamented my lack of internet in the classroom. I wanted to show students a number of sites. To help them understand big numbers, I started with the National Debt, which is currently around $13 trillion. I don't know about you, but I think my brain loses it somewhere between a million and a billion. ($1 million = a house in the hills, $1 billion = 1000 of those houses?) I can sort of see what a million dollars is, but a billion is just huge, and so is a trillion. So how do we get a feel for the difference between one huge number and another? This site shows hundred dollar bills.  A million dollars fits in a briefcase, a billion takes ten warehouse pallets, and a trillion ... the picture reminds me of the photo of the Better World Books warehouse.

(Terrible resolution. It looks better when I see it in my email. I got this copy from Google images. Did I mess it up somehow?)

I showed them a million plastic cups by getting it on my screen before class and then walking around the class, showing them my laptop screen. With proper internet capabilities, I also would have shown my class the Powers of Ten film (9 minutes long) and the Universcale.

Since I can't easily have the internet tell stories for me, I mostly had to do it myself.  I just read The Ghost Map, by Steven Johnson, and folks in the math department were dressing up as detectives for Halloween. So I told the story from this book of the detective work John Snow (a medical doctor and researcher) did in 1854, as he gathered evidence for his theory that cholera was spread through drinking water. Most scientists and doctors at the time thought cholera was spread through bad air (miasma), but cholera causes severe diarrhea, so Snow suspected drinking water. A severe outbreak of cholera hit the Broad Street neighborhood of London in late August, 1854, and over the course of just a few days hundreds of people died. Snow figured out that the Broad Street pump (no drinking water in the homes) was the cause of the contagion. I told this story in class, and we looked up the size of the cholera bacteria (a student's cell phone got us that much internet at least): 1.5 microns, which is 1.5x10^-6 meters.

I asked how many it would take to make a line of them across the room. We figured that out, and then found how long a line a trillion of them would make. We also figured out how long a line the burgers sold at McDonald's would make (over 100 billion sold).

Two out of three classes really got into it. (The afternoon class is a tough sell.) I had a great time with a topic that's usually been much less fun.

[SBG] More, Shorter Tests; Less Textbook

More, Shorter Tests
Before the recent spate of blog posts on SBG, I had already switched partway to something similar. I gave mastery tests on teh most important concepts, along with my regular tests (two chapters at a time). This semester I decided to switch over to mastery tests almost completely (plus a final exam).

This past summer I looked over the official syllabus for our beginning algebra course (at a community college), and decided what I thought was most important.  (The official syllabus seems to just follow the chapters of the texts we use, instead of laying out what's really important.) Then I thought about how it fit together. I think the long lists of standards some algebra classes have to cover are a problem. I wanted something shorter; I wanted to be able to easily describe what we do in the course. I decided that, like a play, the course has two main acts, along with prologue, intermission, and epilogue:

• Prologue. Pre-Algebra Toughies. Fractions, Integers, Distributing, Order of Operations. (Chapter 1 in most texts.)
• Act I. Linear. Solving Equations, Graphing, Systems of Equations. (Chapters 2 to 4 in many texts.)
• Intermission. Exponents and Scientific Notation.
• Act II. Quadratic. Multiplying and Factoring Polynomials. Solving (Quadratic) Equations. Graphing Parabolas. (With a side trip to Roots. Chapters 5, 6, 8 and 9 in our text.)
• Epilogue. Everything else there's time for. Inequalities, Rational Expressions (chapter 7), Proportions. (I think I can have more fun with these when they're frills at the end.)

Then I decided on the mastery tests:

1. Multiplication Facts
2. Pre-Algebra Toughies
3. Solving Equations
4. Graphing Basics
5. Graphing Applications
6. Systems of Equations
7. Scientific Notation
8. Factoring Quadratics (and solving)
9. Solving and Graphing Quadratics

It looked good on paper, but what I found out after I started was that I did want to break it down more. Now I'm thinking of most of the mastery tests as collections of subtests. Students can retake any subtest, and I'm keeping scores for each of those in my gradebook (an Excel spreadsheet). For example, the graphing test has 3 parts: Equations to Graphs, Points to Equations, and Visual (estimate the slope of a line without identifying points). The first two parts each have two problems with two or three parts. This is the longest test I've given so far. The only tests that aren't broken into subtests are Multiplication Facts and Graphing Applications (identify rate of change and y-intercept with units, and explain their meaning in a sentence).

Many of my students say they aren't able to come to my office hours, so I'm ending class twenty minutes early each Thursday to make time for retests. I make a new version of each test each week. I think next year I'll limit retests to two or three days a week so there's less time between the first person seeing a test and the last person taking it (my attempt to limit the cheating). If you'd like to see my tests, let me know. If it's one or two people I can email you. If it's lots, I can post them.

On the graphing test, I made up problems for one person who had gotten everything but y-intercept questions right. I gave her an equation in slope-intercept form, an equation in standard form, and a problem with two points. In all 3 she just had to tell me the y-intercept. Otherwise, people just take the standard retest.

I used to spend a lot of time figuring out the partial credit. Now I don't give much partial credit. Small mistakes lose some points. Bigger mistakes just make the problem wrong. The time I've gained in grading I spend making new versions of the tests. I also like that we seldom use up a whole class period for testing.

I don't really have a sense yet of whether students are doing better with this system. I think there are students who would have had to drop who are sticking with it. That seems to be the biggest improvement.


Less Textbook
The required textbook costs about $140. It's the 5th edition, and there are very minor changes from the 4th edition. On my syllabus I told students they could get the required book, or they could get any Beginning Algebra textbook. Few opted to get a book by a different author, and I realized I like having them all getting their homework from basically the same book. I have sheets with suggested homework problems for both 4th and 5th editions of our text. It turns out, there are plenty of used copies of the 4th edition, for 3 or 4 dollars each! So next semester I'm going to require our official text, 4th or 5th edition. (This semester there were a bunch of people who never got a text, and I eventually bought 8 copies of the 4th edition and sold them to students. One of them said he felt like he was at a chop shop.)

I like them having the book for the homework. It's easier to remind them what they ought to do. (And students in a class like this need some help getting themselves on track.) But I'm having fun avoiding the book in my decisions about what to do with our classtime. After 20 years of teaching this course, I would have thought I knew it pretty well. But it was only this term, because of avoiding using the book, that I noticed that I don't like the organization of the chapter on polynomials.

• 5.1 Exponents
• 5.2 Adding and Subtracting Polynomials (does not need a section, I knew that already)
• 5.3 Multiplying Polynomials
• 5.4 Special Factors (using FOIL, and multiplying (a+b)(a-b)...)
• 5.5 Negative Exponents and Scientific Notation (should be two sections)
• 5.6 Dividing Polynomials (I've always skipped dividing by a binomial - they aren't ready for it, and dividing by a monomial is like work we've done earlier, so it's quick)

I think the negative exponents belong after 5.1, and scientific notation can easily follow that. The adding, subtracting, and dividing are just footnotes for the next main topic, which is multiplying polynomials. I keep reminding them that we're doing this in preparation for factoring, which will help us solve problems having to do with gravity (for example).

I don't know if this is helpful for anyone else, but I think I'll be happy later that I wrote this now. After I've gotten used to this new system, I'll start trying to do something about video lectures, so we can invert the class. (Lecture as homework, problem-solving in class.) That might take me another year...

Questions for SBG Advocates and Practitioners

My main question is how many students you have, but I'd also like to know what grade level you teach, and what courses.

I teach community college math. This semester I'm teaching 3 sections of beginning algebra. I started with over 40 students in each section.

I'm asking because I can't imagine using some of the systems I've seen described, when I have this many students. But even if you only have 25 students, I know high school teachers usually teach 5 or more classes, and that would be about the same number of students I have.

I'm adjusting my systems as the term progresses, and I'll report soon on what I'm doing. I don't think I'll know until the end of the semester how well it's working, though.

Today Is the Day for "REBEL Education Blogs"

I heard it from Cooperative Catalyst. The idea is to post your own alternative thoughts on educational reform, and then post a link at wallwisher.

How has 'education reform' become such a mean-spirited and small-minded pursuit? Teacher-bashing, ignoring the realities of students' lives, judging education by standardized tests, competing for funding, etc. (I guess the answer is that someone needs a scapegoat, and teachers are the current candidate. Perhaps they want to get rid of (some of?) the last strong unions?)

Let's cherish our young people, and honor the amazing people who dedicate their professional lives to working with them. Let's fund all the schools adequately. Let's remember that we are a democracy, and educating for democracy requires an environment respectful of children's needs.

My vision is of children freely following their own interests, but perhaps that only works when they've gotten the same sort of freedom in their families. Deborah Meier, in The Power of Their Ideas, shows a school where educating for democracy is really happening. It's in New York City, and is proof that open education works in urban areas, with diverse groups of students, not just with the privileged. It's not as free-form as my vision, but it's really working and it's beautiful. (The book was written in the nineties, but the schools are still going strong.)

I found Ira Socol's blog on wallwisher*. In his current post he links to a post he wrote about his amazing high school in New Rochelle in (I think) the seventies. Here are some of the founding thoughts:

The following quotation from [Thoreau's] Walden expresses compactly the major beliefs which generate the form of the new program:

Students should not play life, or study it merely while the community supports them at this expensive game, but earnestly live it from beginning to end. How could youths better learn to live than by at once trying the experiment of living?

In other words, we are assuming (1) that learning takes places best not when conceived as a preparation for life but when it occurs in the context of actually living, (2) that each learner ultimately must organize his own learning in his own way, (3) that "problems" and personal interests rather than "subjects" are a more realistic structure by which to organize learning experiences, (4) that students are capable of directly and authentically participating in the intellectual and social life of their community, (5) that they should do so, and (6) that the community badly needs them.

This set of beliefs is sometimes referred to as the "judo" principle of education. Instead of trying to forestall, resist, or neutralize the natural curiosity, intelligence, energy, and idealism of youth, one uses it in a context which permits both them and their community to change. Thus, the experimental program reduces the reliance on classrooms and school buildings; it transforms the relevant problems of the community and the special interests of individual students into the students' "curriculum"; it looks toward the creation of a sense of community in both The Program students and adults.

Unfortunately, this lovely program doesn't exist any more. I hope the rough times we're going through push people to experiment with programs like this again.


Visions of Math
I want to get more specific, and think about math. In my Why Math? Why School? post, I replied to Deborah Meier's disappointinly shallow conception of math with a paraphrase of what Diane Ravitch had said about some other subjects:

We will teach mathematics because it is important and beautiful. We will teach it not because it will save our society, not because we "must" know particular techniques, but because we simply do not have it in our hearts to do otherwise.

In the comments, Ben Blum-Smith wrote:

I think there's something really deeply empowering about mathematics. I believe the rich deep study of mathematics cultivates curiosity, profound resourcefulness, tolerance of frustration, persistence, and an amazing trust of your own mind. I think these are some of the really big reasons why it's an important part of education.

Thinking about why we teach math will help us think about how we might teach it, if we could change the world, and offer students a fulfilling, mind-nourishing set of experiences in school.

 Suburban Lion (also found on wallwisher) dreamed up A Rebel Math Curriculum:

In this Rebel Education, gone are the days of Algebra, Geometry, More Algebra, Trigonometry, and Calculus. Gone are the days of lengthy multiple choice tests. Teachers assess students by analyzing the products they create and encourage the students themselves to critically reflect on their own creations. Students are not pressured to meet Imperial standards, but instead are responsible for setting their own goals for improvement each semester. The students don’t feel like they are competing to score higher than their classmates, but instead learn to recognize that each of their classmates has a different set of skills and that by cooperating they can achieve things that they could not do alone. While the Empire is pumping out clone after clone, the Rebels are producing a diverse array of students with varying sets of knowledge and skills.

His vision is filled with games and computer programming. I think games are one great way to pull students into thinking about math, and computer programming works great alongside that. I'd add:

• Cooking (for elementary math)
• Building 
• Puzzles
• Science
• History (which adds so much context to math)
• "Living Math" (stories that bring math concepts to life, like The Cat In Numberland)

Finding ways to bring together all of the rich ideas we hope students will learn, instead of separating them into 'subjects' will make math so much more accessible.




Let's all think together about all this, and blog together on November 22.







_____
*I don't like that wallwisher hides the url of the site it's linking to. Perhaps there's an easier way I'm not seeing, but I've been googling the blog name to get a proper link.

Systems of Equations Gets Better Without the Textbook

All I have in my classroom is a blackboard, so I can't do cool video clips easily. But I can bring in problems that get students to see more meaning in whatever they're learning. Right now they're embarking on learning how to solve systems of equations. Every algebra textbook I've used leaves the story problems for last in this chapter, but my students this semester seemed to ask much better questions than previous groups had, when I started out with a story.

I checked out my bookmarks last week, and found these ideas over at Dan Greene's blog (The Exponential Curve). I decided to start with the race between the Tortoise and the Hare (download the first .doc file for this and the following problem).

On Monday I wrote this on the board:

WarmUp

Since he's so slow, the Tortoise gets a headstart of 16 feet; he "runs" at 1/2 foot per second. The Hare cheats and starts at the 2-foot line; he runs at 4 feet per second. 

I started by asking if they knew the fable. Many of them told me 'slow and steady wins the race'. We joked a bit, and I said that sometimes the Hare did win, and that's why the Tortoise was getting a headstart. Then I suggested we find out when the Hare would catch up with the Tortoise.

I made a table of values for both of them at once, and many students found that confusing. So in the second class, I started by writing out full sentences. "After 1 second, the Tortoise is at ___ and the Hare is at ___." We figured that out, and did it for 2 seconds, and then put it all into a table. They found the animals' positions at 10 and 20 seconds in small groups, and saw that at 10 seconds the Hare had gotten ahead. So then they looked for the time at which the two were even.

I told them this method (Guess and Check with a Table of Values) was not in the book, and didn't really require algebra, but might get tedious with a real problem with uglier numbers. We kept exploring this problem by creating equations and graphing. The students had taken a test last week on graphing applications, but many are still struggling with the meaning of rate of change and interpreting the y-intercept. We used those ideas to go from the scenario to the equations, and I think some students started to see more connections between the pieces.

After we had the equations, we used the substitution method to solve (even though we had the answer already). I really liked how it went, and am so happy I'm not basing my lessons on the textbook. They're expected to do lots of homework to solidify the concepts, and I wish I would have given them more clarity about that at the beginning of the semester. I think I'll be able to do this much more smoothly next semester.

Today we did another problem of Dan's that felt similar to me, but probably not to them. I changed the names from Goofus and Gallant to Richie Rich and ... my first class suggested Tiny Tim, my second class went with Charlie Bucket.

WarmUp

Richie Rich got $170 on his birthday, and then spent $15 a day on Starbucks and Hot Cheetos. Charlie Bucket only got $10 on his birthday, but he saved it and earned $5 a day with his newspaper route.

I asked what questions they had. The two morning classes came up with different questions, but both (eventually) asked when they'd have the same amount of money. (Thank you!) They still struggled, but I think more and more of them are getting it.

I'm mostly ignoring systems with no solution and dependent systems. I used to teach everything that was in the book, but these notions don't seem necessary on a first pass through systems of equations. I want them to start out grounded in real problems. (I'm hoping the cartoon characters keep it from feeling like pseudo-context...) As they get better at algebra, they can move toward considering the 'what-if' questions. (What if the equations represent parallel lines?)

Today was a good day.

[I still feel like I'm pulling them through the material, and they aren't getting much deep thinking. But I need to recognize that even that is a big accomplishment for many of them.]

Casting Out Nines

Did you know that the digits of any multiple of 9 add up to a multiple of 9?* For example, 9 * 125 = 1125, and 1+1+2+5 = 9.  And that's the basis for lots of number tricks.

Here's a version of an online trick using a crystal ball to predict your number.

But some of the tricksters aren't careful enough about their math, and can mess up. Shecky, at Math Frolic, pointed to this one (get your calculator out), that tells you your original number, and the digits you chose later in the process. But it messed up for me. I think it will mess up about one-tenth of the time, actually.

Here's how it goes:

On a PIECE of PAPER, write a number between 10 and 10,000.

I chose 13.

Multiply it by 4.

52

Now add 5 to the result.

57

Now multiply the result by 75.

4275

Now choose any TWO digits from the result and add them to the result.
Example: 591673 + 69

I chose the 5 and 2, and added 52.

Now multiply the result by 3.
Lets multiply the result by 3 again to make it more difficult.

I multiplied by 9 to get 38943.

Now substitute one digit with an "X" and enter your result below.

I chose the 9.

The result was this:

 

The first number was: 12
Then you choose: 52
And you substituted the number 0 with an "X"

 

If I went back one step and put the x in for the 8 instead, it got everything right. 

 

What went wrong? 

 

By the way, casting out nines refers to a way to check your addition (important back when we didn't use calculators and computers). It depends on the fact that the remainder after division by nine is always the same for a number and the sum of its digits. 

______
*Why?

Ahh... The Purposes of Our Schools

It is good to see someone speaking out against the ugly recent portrayals of teachers as somehow the problem of our schools. I'm not sure when I subscribed to the Journal of Educational Controversy blog, but I've enjoyed a number of their posts. When I went to their website, I found this interesting math education article before I found the blog I've been reading.

William Ayers is a professor of education at the University of Illinois. He's also the radical friend the right tried to use against Obama. Obama unfortunately felt he had to distance himself from this passionate, well-spoken man.

I've been gathering links to good pieces reviewing Waiting for Superman. I'll post those after I've seen it.

Dots On a Circle

Here's the problem:

Draw a circle, put a few points on it, connect them all, and count how many regions. If you have no points you get no lines, which gives 1 region - the whole interior of the circle. One point still gets you no lines and one region. Two points gives one lines and splits the interior of the circle into 2 regions. What happens for 3, 4, or 5 points?

It looks like a very familiar pattern, doesn't it?* Now check to see if the pattern holds for 6 points. Hmm...

The goal with this problem is to find an expression (formula) for the number of regions when there are n points. After many years of playing with this problem on and off, I came up with a formula, but I didn't understand why it worked, and so I couldn't be sure if it would always work. I was running an online study group of people working through Harold Jacobs book, Mathematics: A Human Endeavor. This problem appears in the book, and I wanted to be able to lead any discussion that might come up about it. So I needed to understand why my formula worked. I turned to Google, and found lots of answers. They made sense, and I thought I understood.

But my understanding was shallow, and my memory is terrible, so I forgot. That was perfect. I knew I was capable of understanding it, and the next time around, I refused to look it up. With lots of work and my fair share of false starts, I finally figured it out. At first I felt bad about my solution method - I felt like I'd hit the problem with a big hammer, instead of delicately teasing it apart. I'm proud of it now, and I've written a paper describing my solution. I won't post it here, because I think this problem is so worth playing with, I don't want to make finding a solution online any easier than it is. But you're welcome to email me to request it.

I started writing this post over a year ago, and abandoned it, because I couldn't figure out how to say enough to pull people in, without giving too much away. James Tanton has just posted a video that offers a new twist on this problem, which got me thinking about it again, and I'll leave you with that.






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* This picture comes from James' video.

Thinking Mathematics 1: Arithmetic = Gateway to All, by James Tanton

I wrote about Math Without Words, one of James Tanton's textbook alternatives, about a month ago. He had kindly sent me a box of seven of these books back in the spring, and I was too overwhelmed by the magnitude of it all to manage to review them, until I realized it would be much better to do it one by one. Although I haven't yet delved into Thinking Mathematics 1: Arithmetic = Gateway to All as thoroughly as I'd like, I've seen enough to be delighted.

He's just posted a video about his favorite puzzle, in which he describes the Victorian ceiling in his childhood bedroom.  I loved that story when I read it in this book, and shared it once before. Here's an excerpt from his print version of the story:

My career as a mathematician began at age ten. I didn't realize this at the time, of course, but in retrospect it is clear to me that my journey into the rich world of mathematical play - and I use the word play with serious intent - was opened to me thanks to a pressed-tin ceiling in an old Victorian-style house.

I grew up in Adelaide, Australia, in a house built in the early 1900s. The ceiling of each room had its own geometric design and each night in my bedroom I fell asleep staring at a 5x5 grid of squares above me, lined with vines and flowers.

I counted squares and rectangles in the design. I traced paths through its cells and along its edges. I tried to fit non-square shapes onto the vertices of the design. In short, I played a myriad of self-invented games and puzzles on that grid of squares as I fell asleep.

The puzzles he gave himself when he was a child have found a home in Math Without Words. That book was out of print for a while, but it's now back in print, and I hope one day it will be considered a classic.

The story above goes on in his six-page introduction to the Thinking Mathematics series. He had a high school teacher who had the students each draw 3 right triangles and measure the sides to verify the Pythagorean Theorem (sounds good so far, but...). When James asked "How do we know this isn't just coincidence?", the teacher just said "Go back and draw another three right triangles." He knew at that point that he was on his own!

James, on the goal of the Thinking Mathematics series:

... to simply re-examine the standard K-12 mathematics curriculum, starting with matters of arithmetic and algebra, moving on from there, to revel in the delight of intellectual play and not knowing. Is zero even or odd? Is negative zero (if that makes sense) the same as zero? Why is 3⁰=1 and 0!=1? Why does the divisibility rule for 3 work? What's a divisibility rule for 7? Why is negative times negative positive? What does Pascal's triangle really tell us? Should we trust patterns? What is infinity? What is this thing called synthetic division and what is it really doing? Is there such a thing as base one-and-a-half? How many prime numbers are there? Why are primes interesting?

Chapter one, on the counting numbers, includes history, activities, and explorations related to the basic properties. My favorite activity is:

Chapter two, on figurate numbers (square, triangular, ...) includes this challenge:

As I turned to chapter three, on factors and primes, I thought he might include the locker problem. He does, but he adds some new twists:

Anyone up for these research questions?


And chapter four is what decided me to buy a download version of the book, so I can offer some of the delights in this chapter to students of mine who struggle with negative numbers. James' favorite model for negative numbers is holes and piles in sand.

Alas, he hasn't addressed subtracting a negative, which may be my students' biggest challenge. Let's see if I can improve on my 'taking away a debt makes you richer' rule with his piles and holes in the sand...

^-5-^-3 means that we have 5 holes and we are taking away 3 holes. Well, that clearly leaves us with 2 holes. James says there's no such thing as subtraction - we can add the opposite instead. Yes, taking away 3 holes is the same as adding 3 piles (into the holes!), we still have 2 holes left.

Now let's try a harder one: ^-3-^-5 means that we have 3 holes and we are taking away 5 holes. Hmm... Well, we just saw that the way to take away a hole is to add a pile, so that taking away 5 holes amounts to adding 5 piles. And we know what to do with 3 holes and 5 piles already. Now I'll go test this out with my students.

This book has 15 chapters, so I can't go through it all in this one review. But I trust that you'll find gems in every chapter. The table of contents is here. The print version costs $27.50 and the download version costs $20.

Enjoy!