Groups Rule!

Maybe I'm slow, and you've all seen this sort of thing in action already, but my afternoon precalc class was a blast to watch in action today. I have 9 groups of 4, and gave each group a different word problem from the book. These were problems that needed the use of Law of Sines or Law of Cosines.

They were to solve the problem and be ready to be the experts on it when others asked them for help. I helped the groups who needed help, and as groups were getting close to done, I told each group their next problem, and who their personal experts were. I loved seeing one person walk over to another group to get help for their group. I loved how hard they all worked. It was great.

My morning class doesn't seem to get as engaged as this class, though. I'm trying to dream up a way to pull them in.

Teaching Question: Students Overusing Proportional Reasoning

I have heard, from a colleague who works with prospective elementary teachers, that many of them are not good at proportional thinking. My students (in pre-calc) seem to be fine at it, but ... they're using it even when it doesn't apply. My question for you is how to help them see why proportional reasoning is not always a sensible choice.

Problem #54. Determining a Distance: A woman standing on a hill sees a flagpole she knows is 60 feet tall [yeah, right]. The angle of depression to the bottom of the pole is 14 degrees, and the angle of elevation to the top of the pole is 18 degrees. Find her distance x from the pole.

One student wanted to average the two angles at 16 degrees each. Another said the observer could stand on a stool to be a little higher, so the angles would be 16 degrees each. Their answers were very close. There were other good (but wrong) methods that all came down to assuming this relationship was linear in a way that it's not. Since their answers were very close, it was hard to help them see what was wrong with their reasoning.

Can anyone help me here?

Math and Children's Literature: My Favorite Mathy Picture Books

I love kids' books and I love math. So I've gathered together quite a collection. My son, who has gone to free schools where he isn't required to do math lessons, has probably gotten more math out of reading these books than he has from any formal math lessons.

His favorites are probably a few from the I Love Math series, published back in the early 90's. Although they are out of print, inexpensive copies of most of them are available online. My son and I especially enjoy the stories (in every volume, I believe) about Professor Guesser, a cat detective who solves mysteries using mathematical reasoning. She's featured in the title story of The Case of the Missing Zebra Stripes: Zoo Math. Some of the zebras are missing their stripes, and Professor Guesser figures out what's really going on. These twelve books feel like math magazines, even though they're hardcover, because they have so many different sorts of content - they're full of stories, games, mazes, riddles, and lots of math.  (I think this series is good for ages 4 to 12. On all of my age ranges, I have just used my own judgment.)

Here are the other picture books you'll find on my Math Books page (tab above):

The Opposites, by Monique Felix (ages 2 to 6)

One of the earliest math skills, more basic perhaps than counting, is noticing attributes. This book has no words, and yet it tells dozens of stories, each about opposites. Noticing the one attribute that shows opposites in the detail-filled pictures is a math game your child will want to play again and again.

Quack and Count, by Keith Baker (ages 2 to 7)

This is a board book, so it's good for the youngest child who will sit and listen to a story. And it stays good because it's so luscious. Great illustrations, fun rhythm and rhyme, cute story, and good mathematics. 7 ducklings are enjoying themselves in every combination. “Slipping, sliding, having fun, 7 ducklings, 6 plus 1.” (And then 5 plus 2, 4 plus 3, 3 plus 4, and so on.) It would be great to have a book like this for each number, showing all the number pairs that make it. If I ever get to teach math for elementary teachers again, I'd love to get my students to make books like this one.

Anno's Counting House, by Mitsumasa Anno (ages 2 to 7)

Everything I've seen by Mitsumasa Anno is delightful. There is so much to see in his books, many of which have no words. In this book, ten people are moving from one house to another. In each two-page spread you can see one more person who's moved from the left house to the right, along with lots of furniture and other small items. 

Anno's Mysterious Multiplying Jar will appeal to older readers. There is one island with two counties, which have three mountains each ..., until we get to ten jars within each box - a lovely, very visual representation of factorials. Anno's Magic Seeds does have words, and tells a fascinating story, of a plant whose seed, when baked, will keep you from being hungry for a full year. The plant grows two seeds in a year, and one needs to be used to grow a new plant... You may also enjoy Anno's Math Games. Anno has written over 40 books, most available in English.

Two of Everything, by Lily Toy Hong (ages 3 to 7)
A poor old farming couple in China find a mysterious pot. When a hairpin drops in, they scoop two out.The math isn't discussed in the story, but it's pretty easy to add your own thoughts to this delightful tale of doubling.






How Hungry Are You? by Donna Jo Napoli and Richard Tchen (ages 3 to 12)

There are lots of great of great books on sharing equally. My favorite used to be The Doorbell Rang, by Pat Hutchins, but this one is even more delightful. The picnic starts with just two friends, rabbit is bringing 12 sandwiches and frog is bringing the bug juice. Monkey wants to come, "My mom just made cookies. I could take a dozen." They figure out how much of each goody each friend will get. In the end, there are 13 of them, and the sharing becomes more complicated. One of the delights of this book is the little icons showing who’s talking. Those would help kids to create a delightful impromptu play.

One Grain of Rice, by Demi (ages 5 to 12)

The greedy raja is gently outsmarted by a wise village girl named Rani. This is a very sweet take on the story of grains of rice put on a chessboard. (One grain on the first square, two on the next, then 4, 8, 16, …, until the board is filled. How much rice is that, anyway?)

 The Cat in Numberland, by Ivar Ekeland (ages 5 to adult)

The story starts when Zero knocks on the door of the Hotel Infinity. He’d like a room, but they’re all full (with the number One in Room One, and so on). Turns out that’s no problem. The cat who lives in the lobby gets confused - if the hotel is full, how can the numbers make room for zero just by all moving up one room? Things get worse when the fractions come to visit. This story is charming enough to entertain young children, and deep enough to intrigue anyone. Are you ready to learn about infinity with your 5 year-old?

You Can Count on Monsters, by Richard Evan Schwartz (any age)

Each number from 1 to 100 is a monster, and each one gets its picture on its own page. All of the numbers (except poor 1) are made up from their prime parts. The pictures are colorful, full of intriguing detail, and amusing. The pages in the front and back that explain prime factorization are unassuming, waiting for the reader to decide it’s time to find out more. This and Powers of Ten would both make great coffee table books, to peruse over and over.

Go to my Math Books page for reviews of chapter books for older kids (and books suitable for adults). There are lots of other good mathy kids' books, but these are my favorites.

I also love the idea of creating math lessons from good children's literature even when it wasn't intended for the purpose, but I've never done that myself. (I did find a good math lesson for my adult calculus students in the book Holes, by Louis Sachar.) Julie Brennan does wonders with this genre. Here's an excerpt from one of her chapters in my soon-to-be-published book, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers:

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I recall my daughter, Hannah*, running across the old question, “How do you split two things evenly among three people?” She has two very memorable experiences to draw on. One is a PBS Cyberchase episode she watched when she was five or six, where the kids had to split two apples exactly evenly among the heads of the three-headed dog or they were in trouble. She never forgot that each apple was split into thirds, and each dog got two thirds. Second, when she was around seven or eight, we were reading a Laura Ingalls Wilder book aloud, and we ran into the story of Laura and Mary getting two cookies from someone. On the way home, they agonized over wanting to eat the cookies themselves, but knowing they needed to share with their sister Carrie, and not knowing how to evenly divide the cookies. In the end, they erred on the side of caution, split one cookie between themselves, and gave the other whole cookie to Carrie. My daughters found it so funny that they didn’t think to divide the cookies up into thirds! Between these anchors, this idea of dividing and sharing proportionately is very real, and it gives a real sense of what 2/3 can be - two wholes divided three ways.

However you do it, I hope you'll enjoy finding math in great children's books!

[This post was written to share in the Math Monday Blog Hop. Thanks, Cindy.]




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* pseudonym

Online Course: WOW! Multiplication

Maria Droujkova is very fun to work with. I have only recently met Yelena McManaman - I suspect she will wow you too.  This is very different from most online courses. First, it's very short - two weeks, two hours a week. Second, it's very participatory - you learn about some interesting activities the first week, then try them out with your kids (or anyone, really) the second week. Third, you actually contribute to research in math education when you do this.

The course begins tomorrow, so if you're interested, today is your last chance to sign up.

Estimation and Orders of Magnitude

I just read a blog post by Jonathan Claydon titled Year of Estimation. Like me, he's loving the site estimation180.com. I hadn't paid attention to how the items on the site build up from earlier to later, going from tissues in a travel pack to a box to a bigger package. I may use that feature if I can get myself using the site more often. (There's so much I'd like to do, and not enough time for it all.)

I make my own estimation activities too. This past week in pre-calc I brought in a jar of little origami stars and asked my students to estimate how many there were. This worked into Kate's absolute value lesson. After I announced the correct number of stars, I asked them each to write down how far off they were, their error. Of course some of them had to do Actual Number - Guess, or A-G, and others had to do G-A (leading us to absolute value). I put all the guesses on a spreadsheet, showed my students how I wrote a formula to have excel figure the error, and then graphed guess versus error, giving an absolute value graph. We decided that the good guesses were the ones with an error less than 20, giving us a reason to solve | 225 - x | < 20 (225 was the actual number of stars and x was the value of a guess).

Jonathan pointed to an online quiz that I found intriguing, on the relative sizes of things. I also found it frustrating, because with one wrong answer you had to start over. The quiz has some hard comparisons. Here are some I got wrong:  Which is bigger, ...

• the Eiffel Tower or the Great Pyramid of Gaza?
• the width of Uluru Rock (in Australia) or the height of Angel Falls in Venezuela?
• the Milky Way or the Crab Nebula?
• the moon or Pluto?
• Russia's east to west length or the moon's diameter?

I wanted an easier version - with only items my students would know about - that they could use to think about orders of magnitude. So I created one myself. I alphabetized so it would be easier to search the list, and I lettered them for easier reference. Are any on my list still hard? Could you add in any and keep it relatively easy? Where am I jumping the most between orders of magnitude?


Quiz Yourself on Estimation
Put these in order from smallest to largest.
a. blue whale
b. California
c. carbon atom
d. dog
e. Earth
f. egg
g. eiffel tower
h. electron
i. giraffe
j. human
k. Jupiter
l. Milky Way galaxy
m. moon
n. Mount Rushmore
o. Niagara Falls
p. Oregon
q. Pacific Ocean
r. proton
s. red blood cell
t. soccer ball
u. sun
v. sunflower seed
w. tennis ball
x. United States
y. water molecule
z. white house



If you are working on estimation or orders of magnitude, you may also like some of these resources:

• On Being the Right Size, by J.B.S. Haldane, originally published in The World of Mathematics, by James Newman
• Powers of Ten, by Philip and Phylis Morrison (youtube video here and the similar Universcale by Nikon here)
• Mathsemantics: Making Numbers Talk Sense, by Edward MacNeal (for the one great chapter on estimation)

Week 3 of a Great Semester

I am still trying to squeeze out time to work on the book (Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers), so I seldom have the time to blog these days. But I just now had so much fun in my Pre-Calculus class, I have to write about it.

I've been noticing that I'm enjoying all four of my classes this semester. I usually have a favorite, and I am often struggling with a few disengaged students in at least one class. Somehow that hasn't materialized this semester. (One high school student, R, was being goofy the first day, and I called him on it in a puzzled sort of way. Turns out he is a great math student. Yay!)


Pre-Calculus
In pre-calc, the first unit I do comes from the parts of the review chapter that I thought were worth focusing on: Lines, Circles, and Inequalities. (I have them look in the first six sections of the text for problems that would get them stuck, and we work a bit on those, but I don't lecture on all those details.) Although my three topics seem unrelated, I find small ways in which they connect.

We are starting on inequalities, and I was explaining interval notation: "For x ≥ 4, we write [4, ∞). The bracket means we include the 4, and the parenthesis means we do not include infinity, which we never do, because infinity is not a number." R felt that infinity is a number, and explained why, using the phrase 'infinity principle'. I'm not sure what he meant by that, but it actually helped another student think about infinity as "a principle of numbers, not a number itself". I lent R The Cat in Numberland after class.

Yesterday I had used part of Kate's lesson to help them see absolute value as distance. Today I described | x - 5 | < 3 as meaning the distance between our number and 5 is less than 3. A student asked if that was the same as |x| - 5 < 3. I said "Great question. What do you think?" The class was divided. I asked if anyone could give a reason for why it might be the same or different. Someone said that absolute value is a grouping symbol. I agreed and asked how that made the two inequalities different. No one had an answer to that. S then said that the first one is always positive (on the left side), and the second one can be negative (if x=2, for example). I told her after class that that's called a counterexample, and is used often when we're trying to prove something isn't true.

At that point I said I was falling in love with this class, and called them mathematicians. I'm sure they think I'm a bit nuts, but hopefully "in a good way".

Getting back to the task at hand, we picked numbers on the number line for which our first inequality was true, and I got them to tell me I could make it solid (coloring in all the points between 2 and 8). After we did this very concrete process, I showed them the algebraic way to "solve it". I told them we read the solution, 2 < x < 8 as 2 is less than x, which is less than 8.

We then looked at | x - 5 | ≥ 3, and I got them to tell me points that worked first, and then walked them through the steps to get the solution of x ≤ 2 or x ≥ 8. Someone asked if it was ok to write 2 ≥ x ≥ 8. I replied that this says 2 ≥ 8, so it doesn't work.

I won't know until the next quiz how much of this is really making sense to them. It seems great right now, but I am often terribly disappointed once test time comes. The downfall of a good lecture is that it looks and sounds better than it really is.


Linear Algebra
Today I was 'covering' linear independence. The book gives a definition, and I wanted the students to see a need for the definition before I put it up. We had previously seen an example of a vector that was a linear combination of two other vectors. I used a similar example - two of the vectors would make a plane, while the third vector (a linear combination of the first two) would contribute nothing new. So we call this a linearly dependent set of vectors. (And, by our textbook's definition, this happens when there is at least one non-zero c_i in the equation c₁a₁+c₂a₂+...+c_na_n = 0.) Naturally, linear independence is defined to be the opposite situation. If c₁a₁+c₂a₂+...+c_na_n = 0 has only the trivial solution (all the c's = 0), then the set {a₁, a₂, ... a_n } is linearly independent.

That may not sound exciting, but I love how the various concepts in linear algebra all weave together. I couldn't stop myself from mentioning dimension today, even though the book doesn't get to that until the next chapter.


Calculus
Last night, we figured out a few derivatives (which I like to also call the slope function, to help the students keep their eyes on the meaning) using the definition. I keep asking and they keep telling me - it's just change in y over change in x, but I'll only know whether or not they really see that after the first test.

During the first two weeks, they were very confused. It's beginning to come together for them, I think.


I feel very lucky to be teaching students who are willing to play around with math. I also am seeing how my work with math circles, my work on the book, and my blogging have all contributed to my enthusiasm and my steadily increasing skills, even after 25 years of teaching.