Sunday, October 31, 2010

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!

Friday, October 29, 2010

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.

Wednesday, October 27, 2010

[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...

Friday, October 22, 2010

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.

Sunday, October 17, 2010

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.

Tuesday, October 12, 2010

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:

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.

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.]

Monday, October 11, 2010

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.
Now add 5 to the result.
Now multiply the result by 75.
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. 


Tuesday, October 5, 2010

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.

Monday, October 4, 2010

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.

* This picture comes from James' video.

Sunday, October 3, 2010

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.


Saturday, October 2, 2010

Math for Love and Love for Math

Dan Finkel loves math. His blog, Math for Love, is one of my favorites. Here's a short video of him talking about the perennial question, "Why are we doing this?"

I've slept better the past two nights, but I was still awake for about an hour in the middle of the night last night, thinking about my difficult class. I might get some videos of people in the fun class, talking about what's starting to change for them, and then tell my difficult class that they can become a community of learners, too, if they choose it.

Here's are some snapshots in words...

A has taken this course 4 times before this, and has always dropped when they got into graphing, because she just wasn't getting it. She's starting to get it now. I asked her to help B, who has been an amazing community organizer, emailing other students, suggesting times to study together, encouraging them to stick with it. A promised to do that.

B came by my house this afternoon for some tutoring. When I was teaching at a community college in Michigan, I had long lines of students waiting for their turn with me during my office hours. Here in California, they don't come for help as much (at first). Perhaps I've changed, but I think it's a difference in the students here, they don't expect as much help from their Professors, and think they have to go to tutors for help. (Is this really true?)

C walked out once, angry with me. She gets pretty angry at herself too. Each test that comes back with a 'Redo' at the top makes her mad. But then she comes in, and aces her retake. She's doing great in this class, but she would have gotten C's if I used a 'normal' grading system. [I do not call what I do Standards Based Grading, even though I was inspired by those folks to expand my Mastery Tests to be most of the grade.  Mastery Tests seems like a simpler name for it. They get to retake until they've got a score of 85% or better.] Won't it be great when grading systems like this are the new normal?!

D complained at the beginning of the term about all the pictures I use (trying to get them to really understand fractions, instead of memorizing steps). She also complained about me asking them to do things before I explained how. "Most teachers explain first, and then let us work on it." She struggles with math, and got 100% on her retake yesterday. I asked if she liked my method better now. She hesitated, and I think she's not really believing in the pictures and the trying things out before getting fed the steps. But she's happy about her progress, and that's what counts.

This class starts at 10am two days a week, and at 9:40 the other two days. Many of the students are coming in around 8:15 to study together. We are very lucky that our classroom is empty before our class meets - that's rare. They're making great use of this lucky break. One of them, E, often goes up to the board during this time to explain things to the others. He got a chance to do that in class on Monday or Tuesday, and he did a great job. (He made a mistake at one point, and I waited, on edge, hoping the others would correct him. It got fixed, but never got pointed to directly.) They were all shouting things out, as they moved together through the problem. I told E after class that I knew he'd make a great teacher some day, and his eyes lit up. Watching this beautiful scene during class, I thought this group would do great on the test. When they didn't, I was really discouraged. (What I said to them in class the next day is that I know they're working hard, and working together, which can make a big difference; we need to figure out together what they need to do to work more effectively.) But they are keeping their spirits up.

Most of the students I've described are older students, and have made a commitment to themselves that they're going to do it - they're going to pass this class. My commitment to them is to push them to do more than pass. I love their dedication.
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