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.

Once Again, Day One

We start classes tomorrow. No, I'm not ready yet. I've been learning how to use our online site, which comes from "Desire 2 Learn." (I hate that name. I feel like I'm spouting propaganda every time I say it. I'll be calling it d2l, and mentioning the problem of the name in each class. Gag.) I want students to be able to see material early, get copies of material when they miss class or lose their copies, etc. I'm having trouble uploading one particular file, and can't figure it out. Sigh.

I'll be teaching two sections of pre-calculus, one of calculus, and one of linear algebra. In each of my classes, I'll move the desks into groups of four, and hand out the syllabus and a sheet summarizing unit one and listing the homework. I'll also pass around a phone list for them to put their name and info on, which I'll copy and hand out the next day. And they will put their name on a 3x5 card, which I'll use to call on people randomly. Here are my planned day one activities for each class:


Pre-Calculus

Estimating 
We'll be using estimation180.com. (Thank you, Andrew Stadel.)
Our first problem: Breaths in a day (my guess: 8000, google, 17,000 to 28,000) I was 9000 off, 9000/17000 = 53% low

Visual Patterns
We'll be using visualpatterns.org. (Thank you, Fawn Nguyen.)
Our first problems: #2 (easier) and #1, pretty hard

Equations for Graphs
We'll be using Daily Desmos. (Thank you, Team Desmos and all the contributors.)
Our first problem: 110a2 (use two eqns, or challenge yourself to find just one)

Graphing Stories
We'll be using graphingstories.com and what I downloaded from Dan Meyer's blog. (Thank you, Dan Meyer.)

I have had endless trouble with the projection systems in my classrooms, so I plan to test it all out in both of them today. I plan to do one of these activities (or a quiz) with my students each day at the start of class. I think this will fill my hour up quite nicely.




Linear Algebra

Calculus (starts Tuesday evening)

On Screen:
Screen 1: What is the meaning of acceleration? (Write what you think it is on your paper.)
Screen 2: A rock is thrown upward. It reaches 11 feet, and falls back down.
What is the acceleration of the rock at the instant it reaches the very top of its motion?
Think about it on your own, without discussing it yet. We’ll vote, then discuss, then vote again.
Screen 3: A rock is thrown upward. It reaches 11 feet, and falls back down.
What is the acceleration of the rock at the instant it reaches the very top of its motion?
A. up B. down C. none D. not enough information

I don't want to bother with clickers, so I need to stop by the copy shop to get my vote cards made up. (I got this idea from Kate. Mine will be a bit different. I'll post more if it works out, and maybe even if it doesn't. Kate has a good calculus question at that post.)


Tangent Task
For the past two semesters, on day one, I had students carefully graph y=x², and then show a line tangent to the graph at x=2. After that they were supposed to estimate the slope of the tangent line. It worked great in the fall. But in the spring, a bunch of them knew the derivative 'rule' and that destroyed the activity. I'm going to use a circle and a few graphs I've drawn this time, so they can't use 'rules'.

This class meets for 2 1/2 hours, so I'll have lots more planned, but I get to work on that tomorrow and Tuesday.



On the first day or two I also:

• Explain how Donut Points work (Every time someone in class catches me in a math mistake, that's a donut point. When the class has gotten 30 donut points, I bring in donuts.);
• Talk about the difference between what people think math is and what it really is;
• Talk about mindsets, stereotype threat, and how neurons grow when we learn new things  (I like talking about myelin growth);
• Explain how to get cheap textbooks (online, used, I allow older editions);
• Explain that I will stamp homework each day;
• On day one I ask them to find interesting things on the syllabus, on day two I ask them to share, giving me the opportunity to explain test retakes.

You may find other helpful ideas at my previous Day One posts, here, here, and here.

Book Review: String, Striaghtedge, and Shadow: The Story of Geometry, by Julia Diggins

This book was recommended on Living Math Forum. I can see why. The storyline is very engaging overall, and gets you thinking about the history as if it were happening while you watch. But...

My biggest issue is that it is so gendered:
"A string can usually be found in a boy's pocket..."
"Ancient men discovered the ideas and constructions of elementary geometry ..."
"Through the ages, men have searched to find the secrets of the universe."
"A long, long time ago primitive men observed the lines and curves and other forms of nature."
"It was from this inner sense - man's sensitivity to the order and harmony of the universe - that geometry really began."

I don't think writers do that so much these days. (This book was written in 1965.) Writers sometimes still say 'he' when they mean all of us, and still sometimes say 'man' to mean people, but not often. Reading this book made me think modern writers must be avoiding this construction, even if unconsciously.

Why does it matter? Research shows that we need to feel a part of  a community in order to do our best thinking. Women and girls are shut out by this sort of writing. As much as I might like the content of this book, it sets me up as an outsider (even though the author is a woman!), and that's part of how stereotype threat happens.

I haven't read the whole book, but I did discover one error, I believe. The discovery of the fact that the square root of two is irrational seems to be described incorrectly:

"Then was it a ratio of whole numbers between 1 and 2? ... They tried every possible ratio, multiplying it by itself, to see if the answer would be 2. There was no such ratio.
After long and fruitless work, the Pythagoreans had to give up. They simply could not find any number for the square root of 2."

There are two problems here. One, they couldn't have tried 'every possible ratio', because there are an infinite number of possibilities. More importantly, it wasn't about giving up. If I understand the history correctly, they actually proved that no such ratio can exist. This notion of proof is a very important foundation - it's part of what mathematics is. So her version of this story takes away some of its drama.

The Pythagoreans believed, as she says, "that the universe was ruled by whole numbers." So to prove that a length exists which cannot be described by a ratio of whole numbers was extremely unsettling to them.

How do we prove that the square root of two cannot be a ratio of whole numbers? There is more than one way to do it. You might like Kate and Justin's way more than the one I usually use. It's less dependent on being comfortable thinking with variables. Here's the way I think of it:

If the square root of 2 could be represented by a fraction, we could writes that fraction in simplest terms as a/b. Then we'd have (a/b)²=2, or a² = 2*b² . Since the right side of this equation is even, the left side must be, too. If a² is even, a must itself be even. Let's call it 2c. Then our equation becomes (2c)² = 2*b² , or 4*c² = 2*b² , or 2*c² = b² .  Now the left side of this new equation is even, so the right side must be too. And that means we can write b as 2d. But if both a and b are even numbers, then the fraction can be simplified. We started out with what we thought was a fraction in simplest terms, and found out that it could be simplified. This is a contradiction. It happened because we tried to write the square root of 2 as a fraction - it can't be done, and this proves it.

Proof by contradiction is a bit weird. I think I might like Kate and Justin's proof better myself.

Well, you might like String, Straightedge, and Shadow, even with its flaws. I might, myself. But I have decided not to include it in my Book Picks section.

Book Review: How to Count Like a Martian, by Glory St.John

Once again, my work on the last bits of Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is getting me to write things that belong here.

Today I'm working on the Book Picks section, one of the resources you'll find at the back of the book. Much of it comes from what I've already written for the Math Books page of this blog. (You can see the tab for it above.) But there are some great books I hadn't written up yet.

Right now, I'm writing a description of How to Count Like a Martian, by Glory St. John. It's running too long for the Book Picks section, so I'm posting it here. I'll pare it down afterwards.

A really good way to understand place value is to work with other number bases. How to Count Like a Martian is a detective story in which the history of other number systems plays a starring role.

 “Out of the depths of the dark and starry night come the first of the faint and mysterious sounds … At your radio telescope, you are expertly tuning the dials.” You have just received a message from Mars. “You know that this is not a message in words. Martians and Earthlings would have too much trouble trying to find the same words to succeed that way. But there is another kind of language that both Martians and Earthlings understand.”

Numbers… And so you research the number systems that have been used on Earth, hoping that will help you decipher this message. The book proceeds to explain eight different counting systems, including the abacus, and computers. 

In the process, the concepts of place value (she just calls it place), base, and zero are explored. By the end of the book, you can see that the beeps and bee-beeps of the message you received are just the counting numbers, Martian style.

How to Count Like a Martian was written in 1975, when there were still dials and tape recorders. those two items may be the only evidence of its age. I wonder if any young kids will like it as much as I do. Please let me know if your kid loves this book.

KenKen: A Simple Puzzle That Goes Deep

In the conclusions I wrote for Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, I mentioned KenKen*. Our copy editor asked what it was. As I searched for a good reference to put in a footnote, a memory began to surface of an article I read long ago. Memory is funny. It turns out I had read the article over four years ago, and I still remembered a particular word from it - midnight. That led me to the article I had read, but it's unfortunately behind a paywall. Luckily, I had downloaded it the first time I read it, and finally found it on my own computer.

I had saved it back then because I was using KenKen with the kids at Wildcat Community FreeSchool. The puzzles I was sharing with the kids seem pretty easy, but to solve them requires holding addition facts in your head while also thinking logically about the relationships. This is a good way to deepen your hold on those facts. I wanted the parents to understand how valuable this simple puzzle was, so I copied the article.

'Midnight' was in the first paragraph, in the dramatic opening of the story written by Leo Lewis for The Times of London:

At one minute to midnight every September 30, the decrepit, cluttered schoolroom of Tetsuya Miyamoto stands frozen in time. Breaking the sepulchral silence of the Yokohama side street, the clock ticks over into the first day of October and a fax machine in the corner shudders to life. 

Throughout the rest of the night, page after page spews out of the machine, each one representing a different seven-year-old child, each one an application form pregnant with parental hopes and fears. 

The class these parents so desperately wanted their kids in consisted of puzzle-solving sessions. Tetsuya Miyamoto provided the KenKen Puzzles he had invented, and the children would then work alone for 40 minutes on up to three puzzles. The first is a 4 by 4 grid, the next is a harder 5 by 5 grid, and the third one is harder yet. After they've worked on the puzzles alone, the group works together on a puzzle Tetsuya Miyamoto puts on the board. He calls on a student for a number, then says right or wrong. That's it (according to Leo).

Before I go any further, I'd better share a KenKen puzzle with you.  Here's today's puzzle from the New York Times. If you like it, go there for more puzzles. Since this puzzle is 4 by 4, each row and column will have the numbers 1 to 4 in it. The clue on the middle top, 6X, means that the two numbers for that outlined box must multiply to 6. We know we can't use 1x6, because 6 won't be used in this puzzle. Is that enough to get you started?

Tetsuya Miyamoto designed his KenKen puzzles to draw students in and get them sweating:

Every puzzle, says Mr Miyamoto, contains a “trick, a discovery – a story”. The puzzle works in his classroom, he says, only because the children want to root out the clues and persevere with the discovery process. “As the feeling of achievement increases, so too does the level of concentration,” he says.
...
By combining the four main mathematical functions of addition, subtraction, multiplication and division, the brain is forced to dart between competing theories. The puzzle, he says, is impossible to solve without the scientific process of trial and error. 
...
The puzzle, Mr Miyamoto says, draws out the primal, self-starting learning instinct of human beings – an instinct that is notoriously suppressed by the fact-cramming teaching methods of the Japanese education system, but which he says needs to be encouraged in people of all backgrounds.

I think U.S. schools are headed toward that same sort of fact-cramming. It was never a good idea, but it seems clear to me that we need particular facts less than ever with the internet at our side. What we need are understandings of how it all fits together.

Mr Miyamoto’s theory is that the brain – of a child or adult – is failed by conventional teaching. By concentrating on a “third way” of problem-solving, he believes that the mind becomes a more potent tool for dealing with the rest of life...

I wish I knew what second way is implied here. I'm assuming fact-cramming is the first way.

For both children and adults, runs Mr Miyamoto’s theory, the brain feeds on what it has worked out for itself rather than what it has been told to focus on.

This important idea has been stated many ways by many excellent teachers. I am reminded of the quote the Kaplan's use to define their math circle philosophy:

"What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises."    --G. C. Lichtenberg

There are other ways to make arithmetic challenging and appealing, some of which you'll find in Playing With Math. But KenKen is a particularly easy one to bring into your life. Enjoy!




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*The name KenKen is trademarked. Because of this, you can also find these puzzles under other names. Calcudoku seems to be the most common.