Monday, February 8, 2016

Math Jokes!

For a long time, I thought "math joke" was an oxymoron.

Then I had kids.

At dinner, they say, "Dad! Give us a math joke!" Below are some that I've given them. Enjoy these denizens of the intersection (Dad jokes) ∩ (math jokes). I may update the list from time to time....



Q: When the Great White shark showed up, where was the octopus?

A: Octogone!







Q: If you cut up a general into 1,000 pieces, what is one of the pieces called?

A: A milli-leader!





Q: Where did the mathematician's family eat dinner every night?

A: At the multiplication table!





Q: Why were 10 and 11 mad after the race?

A: Because 20 won!





Q: What did 0 to say to 8? (I heard this one yesterday from Sophie, age 9.)

A: I like your belt!





Q: What did the pirate say to 5 × 16?

A: Ahoy, 80!





Q: What do da glasses help with?

A: Da vision!





Q: What do you call an angle after it gets in a car crash?

A: A wrecked-angle!





Q: What do you call a seagull who won't give up on her geometry homework?

A: A trying-gull!





And last, here is a joke/riddle. Feel free to enter your solution in the comments!

ipip  ipip  ipip
ipip  ipip  ipip
ipip  ipip  ipip
ipip  ipip  ipip


Tuesday, February 2, 2016

How We've Been Using Flashcards

Saturday School is never very long, but over time we still manage to do a range of things, including solving word problems, learning concepts, and doing exercises that build fluency and fact recall. Our materials include worksheets that I create, workbooks off the shelf such as Kyoiku Dojinsha and others, released test questions, homemade flashcards, and pennies or dice that we might use to play a math game.

Concerning flashcards in particular, a researcher in mathematics education saw this article of mine and later emailed me an important tip that I wanted to pass along:
[Flashcards] ... are a good fluency method [but] please stress that students should be spending most of their time on the cards they do not know yet or on those they know but are not fast on yet. Most drill uses many problems students know and thus is a big time waster.
A similar message appears in this document for educators (emphasis added):
Organizing practice so that it focuses most heavily on understood but not yet fluent products and unknown factors can speed learning. To achieve this by the end of Grade 3, students must begin working toward fluency for the easy numbers as early as possible.
Time is always scarce, and it is valuable to customize flashcard work to the student's current state of mastery. Some digital apps do allow you to adjust the settings so that students are focusing on the facts they need to focus on—maybe some of them even adjust themselves automatically over time. In any case, here's how I've tried to accomplish something similar using old-fashioned flashcards. I make no claim that my method is the best, or even better than others! But it has worked for us, and it's been fun.

Here's how it works:
  • The flashcards are kept in a "piggy-bank" made from an old tissue box.
  • We work at the dining table. My wife or I will draw a flashcard from the bank and show it.
  • If the answer comes back "lickety-split," then the card is set aside. Otherwise, the card goes back in the bank. If there is doubt, then the card goes back in the bank.
  • If the student is drawing a complete blank, then I prompt with a strategy, for example if the problem is 6×8 and the reaction is a blank stare, then I might prompt with "Do you know 5 × 8?" Then the student can say "Oh right, that's 40, so 8 more is 48." Of course, the card goes back in the bank, but it has been a good just-in-time learning opportunity.
We'll do anywhere from 5 to 15 minutes of this, and at the end, the student "owns" all of the cards that have been set aside. The student highlights the known facts on a map (images below) and puts the cards into a keepsake box.


This work proceeds in tandem with our other activities, including worksheet practice on facts (customized according to what the map says). Eventually, the day comes when there are no more cards left in the bank and the entire map has been highlighted. Time to celebrate! From then on we'll still do occasional maintenance practice to keep the facts secure (worksheets or a configurable app), and of course there are the worksheets that come home from school.

The way the system works is that the bank gets emptier and emptier over time, with known facts exiting the process as they become known. That's how the game exemplifies the advice I shared at the outset about customizing fact practice.

Although initially the bank contains many already-known facts like 2 + 1 or 3 + 0, this is intentional in order that the early sessions will feature easy, known facts and establish a foundation of confidence as we embark on the process. The rapid progress on Day 1 creates excitement. Pretty soon, those easy known facts exit the system and the bank becomes nicely focused on the student's individual horizon. Of course, this also means that the work is getting tougher over time, so I pay attention to motivation and emotions while we work.

***

Here are the two maps we've got going right now (addition and multiplication):



(The addition map is like the one I showed in this post.) We use magnets to pin the maps to metal shelves above the kids' desks. Each weekend, they chart new progress using a highlighter.

The maps are a great way to recognize accomplishment. The kids love adding to the maps and seeing their maps fill up over time. Completing the map brings a strong sense of satisfaction and achievement.

For my purposes, the map also suggests hypotheses about where prerequisite concepts might be lacking. I can address those directly in a separate line of work.

Here's what our flashcards look like. They are made out of ordinary 3x5 note cards. (To save paper, I write several problems on each note card and then cut them out with scissors.)


Now, the setup is actually a little different depending on whether we're talking about addition or multiplication. Let's consider addition first. By the time we're doing this, the student can mentally calculate all or almost all of the sums on the map, but some of the facts are very slow, and not too many of the facts are known from memory. So after I have written out all of the sums, I give the flashcards to the student and she goes through them one by one, writing each answer on the back of the card.

With the cards complete, we're ready for the funnest part: turning an empty tissue box into a bank. Here is a picture of last year's "piggy bank" of addition facts (piggy-posterior not shown).


Here are the containers they use to hold the facts they "own":



Multiplication differs from addition in the setup, because at the time when we first create the flashcards, the student doesn't really know how to calculate all of the products yet. Products only go into the bank if the student has seen them before and can calculate them mentally (perhaps slowly in some cases). I did a little probing at the outset to find out roughly where we were.

So that's how we've used flashcards in Saturday School. Readers might wonder why we only included sums and products, since students also have to be fluent with differences and quotients. Including differences and quotients is good, and I also like flashcards that show the entire fact family (you cover up two of the numbers, and the student tells you the third). In my case however, it happens that the kids are so secure with the relationships between operations that, presented with 14 − 9 or 24 ÷ 6, they just answer by consulting their mental "lookup table" of + or ×. They also get direct practice with differences and quotients in worksheets that I give them or that come home from school.


Posts on Saturday School:

 http://jzimba.blogspot.com/search/label/saturday%20school

Friday, January 29, 2016

On Certainty

Once it was part of my job to work alongside experts in Quality Assurance. Actually, many of my colleagues weren't experts, just go-getters willing to give QA their best shot. The experts weren't always the best at this work, either, because being really good at QA is partly a matter of temperament. Watching the QA team operate, I came to think that when it comes to checking a piece of work, there are two kinds of people: those who are trying to make sure the work is correct, and those who are trying to prove that it isn't. You want the second type of person on your QA team. A checker should revel in finding errors, not aim to show that there aren't any. 

If there are two mentalities, one corresponding to the jaded TSA employee and the other corresponding to the lovingly patient KGB interrogator, then I'm interrogatory by nature. But I do catch myself thinking the wrong way sometimes, such as when I'm finished putting away the pieces of a board game. 'Wouldn't it be nice, now (says my brain) to put on the box top and take the game back to the shelf!' Yes, it would be nice, but there is still another piece on the floor. Trust me, there is. One of my personal folk theorems is, 'There's always one more.' Need a paper clip or a rubber band? You're in luck, because there's one more in that drawer. Look long enough and you will find it. Nobody in history, to my knowledge, has ever run out of paper clips.

Another doctrine of mine is The Fundamental Theorem of Travel Delays, which I deduced circa 2010. This theorem says that The number of travel delays is not equal to 1. Corollary: If Delta Airlines, MTA, or Amtrak announces a delay, then start researching other plans, because they are going to announce another delay. Any other outcome would violate the theorem. Here is another example of the theorem in action: if you are entering a New York subway station, and if the person ahead of you swipes their MetroCard and gets an error message with a beep, then for the love of God, get yourself out from behind them. Where there is one beep, there will be another.

Even though I have the temperament for it, QA probably wouldn't be a good profession for me. I would spend too much time checking a piece of work when there were other pieces of work to be checked. There is such a thing as being overly perfectionist. Ignatius Reilly, the main character in the novel A Confederacy of Dunces, had a job pasting due-date slips into library books. "On some days," he said to his mother, "I could only paste in three or four slips and at the same time feel satisfied with the quality of my work." 

A paradox that made a strong impression on me as a child was the paradox of Caesar's dying breath. Every breath you take, so the saying goes, probably includes at least one air molecule from Caesar's dying breath. Amazing, isn't it? Although the probability is minuscule that any randomly selected air molecule boasts such a pedigree, nevertheless, a breath of air contains so many molecules that the chances of entirely avoiding the imperial ones are low. It's like playing Russian roulette with a gun that is nearly empty but pulling the trigger eighty sextillion times. There's no future in it. Similarly, in the design of population studies, sometimes you don't need your sample to be a large percentage of the population if the sample is large in absolute terms. Intuitions like these inform my neurotic approach to copy editing: while any given word is nearly certain to be correct, in a long enough run of words there must be errors.

Addiction therapists describe gamblers who think that if a game offers a player a one-in-four chance of winning, then the player is certain to win by playing the game four times. I created a much harder puzzle once to test the solver's sense of such things: if the probability of winning a game is the same as the probability of losing the game a million times in a row, then is the probability of winning the game less than, equal to, or greater than one-in-a-million? I find this puzzle challenging! But the belief about the game of one-in-four is so wrong, I cannot believe that anybody believes it. Somehow those therapists are tricking people into giving the wrong answer. That said, if you can easily be tricked into giving an answer that on second thought you realize is wrong, then the real problem is not your intuitions about probability—it's your neglect of the habit of giving second thoughts to things. To build this habit, it is necessary to err frequently.

Socrates was unsure of everything save his own power to sniff out error. To detect error, it helps to believe in it.  'Out of the crooked timber of humanity no straight thing was ever made.' Certainty is a state of mind normally denied us, but if there is one thing we can be sure of it's mistakes. Between us, I despair of proofreading this page.

Saturday, January 23, 2016

Book Review: The Complete Sherlock Holmes


The Complete Sherlock Holmes, Volume I and Volume II

Sir Arthur Conan Doyle

Introduction and Notes by Kyle Freeman

Softcover, 709 pp. and 709 pp.

Barnes & Noble Classics, 2003




These Barnes & Noble Classics editions of Sherlock Holmes are authoritative, affordable, and printed in good-quality format with few typographical errors. Mr. Freeman's editorial contributions include a couple of enthusiastic and informative introductions, a detailed timeline, and a number of textual notes; all this adds significantly to the book's value. If you want a complete edition of Sherlock Holmes, then these two volumes will serve you well.

Some of the tales collected here aren't worth reading nowadays (I'll give details below), but much of The Complete Sherlock Holmes is still first-rate detective fiction. As soon as I finished The Complete Sherlock Holmes, I added Conan Doyle and his works to my continually updated list of favorite genre fiction.

The Sherlock Holmes tales have certain conventions and repetitive features, which might be tedious for some readers. However, these Holmesian and Watsonian hallmarks are effected differently from story to story, and having all of the stories next to one another also reveals variety within the genre: there are examples here of the puzzle story, the diplomatic intrigue, the urban crime story, and the Gothic horror tale. Some of my favorites, in chronological order:

A Scandal in Bohemiathe story with Irene Adler
The Adventure of the Speckled Banda tale of the macabre
Silver Blaze
The Musgrave Ritual
The Reigate Puzzle
The Naval Treaty
The Final ProblemHolmes dies...or does he?
The Hound of the Baskervilles (novella)atmospheric adventure on the moors
The Adventure of the Empty HouseSherlock Holmes returns!
The Adventure of the Priory School
The Adventure of the Second Stain
The Adventure of the Bruce-Partington Plans
The Adventure of the Lion's Mane

On this website you can see how Holmes aficionados rank the stories. And here is Sir Arthur Conan Doyle naming his own favorites.

Skippable:
  • Most of the stories in the 1927 collection The Case Book of Sherlock Holmes
  • Study in Scarlet (novella) 
  • The Sign of Four (novella) 
  • Valley of Fear (novella)
Fans of Sherlock Holmes usually rate Study in Scarlet and The Sign of Four more highly, but I found the former work immature and the latter work convoluted and overlong. For that matter, I assume that a single Holmes novella is all the average reader really wants to invest in, and in that case there's no question that the novella you want is the Gothically delicious Hound of the Baskervilles.

Monday, January 18, 2016

Kid Cryptograms

As a Christmas present, my kids bought me The Complete Sherlock Holmes, Volume I and Volume II. Inspired by the stories, I created some cryptograms for my kids to play with over the weekend.

The first cryptogram was a book cipher, which was clearly a hit!


To decode a book cipher, you first have to know what book to use. So I included a cartoon showing two of our bookshelves, along with enough information to identify the book to be used for decoding (it was The Wind in the Willows). Now, with the right book in hand, turn to each indicated page number and write down the word that is reached by counting the indicated number of words from the top of the page.

The second cryptogram was based on hidden words. To set the stage for the game, I hid a toy monster somewhere in the house, along with two quarters. Then I created an encrypted message; when decoded, the message reads as if the monster were asking for help. To allow for two players, I divided the cryptogram into two parts, one consisting of the odd-numbered words and the other consisting of the even-numbered words. It looked like this:


To decode the message, replace each word with a word that hides inside it. (For example, replace the word STAVE with the word SAVE.) Then bring the two halves of the message together to reveal the monster's plea for help—and its location in the house.

Sunday, January 10, 2016

Freudenthal's Impossible Problem

I received the following email from a friend just as I was cramming myself into a middle seat for a cross-country flight:

I am thinking of two whole numbers greater than 1 whose sum is less than 100. I tell this to Jason and Brendan, and give Jason the product and Brendan the sum.

Jason: I don't know what the two numbers are.
Brendan: I knew Jason wouldn't know what the two numbers are.
Jason: Ah, now I know what the two numbers are.
Brendan: Ah, now I know what they are as well.

What are the two numbers? Assume Jason and Brendan are perfectly logical, honest, smart, etc.

Talk about good timing! The hours of the flight passed quickly as I scribbled figures in my notebook. (I had the answer by the time we landed, but it was not easy.)

This beautiful puzzle is justly famous; you can read about it online by searching 'Freudenthal's Impossible Problem.'

Favorite Children's Books

These are books to read to young children. Table 1 lists board books. Babies and toddlers can also enjoy picture books—see Table 2. Table 3 gives some good early nonfiction books. All three tables include books for a range of ages, so pick and choose.

The hyperlinks take you to a Google search on the ISBN. Also search online for Caldecott Medal and Newberry Award winners from any decade (older often means better). There are also some good books in this list (PDF).

People often ask me which math books to buy for little ones. There are a few counting-specific books in Table 3, but when the time comes, any book is a counting book. You can just ask the child to count the number of caps, or bumblebees, or flowers, etc., in whatever book you happen to be reading at the time.


Table 1. Favorite Board Books—Fiction


Sandra Boynton Moo, Baa, La La La! 067144901X
Sandra Boynton But Not the Hippopotamus 0671449044
Sandra Boynton The Going to Bed Book 0671449028
Sandra Boynton Blue Hat, Green Hat 0671493205
Jan Brett The Mitten 0399231099
Jan Brett The Hat 0399234616
Margaret Wise Brown Goodnight Moon 0694003611
Margaret Wise Brown The Big Red Barn 0694006246
Margaret Wise Brown The Runaway Bunny 0060775823
Eric Carle The Very Hungry Caterpillar 0399226907
Alexandra Day Good Dog, Carl 0689807481
Bruce Degen Jamberry 0694006513
Melanie Gerth Five Little Ladybugs 1581172184
Piers Harper, Sonia Black Snow Bear 0439544262
Beatrix Potter The Tale of Peter Rabbit 0723247706
Roger Priddy Fluffy Chick and Friends 0312494300
Roger Priddy Halloween Jack 0312500076
Roger Priddy Alien Al 0312498934
Roger Priddy Chirpy Chick 0312516258
Raffi Five Little Ducks 0517800578
H.A. Rey Curious George (many stories)
Esphyr Slobodkina Caps for Sale 0064431436
Rosemary Wells Max's Bath 0803701624
(many possible) Mother Goose (many possible)


Table 2. Favorite Picture Books—Fiction


Francesca Assirelli Chicken Licken 1846433274
Ludwig Bemelmans Madeline 0670445800
Bros. Grimm / Bernadette Watts Little Red Riding Hood 0735840083
Peter Brown Mr. Tiger Goes Wild 0316200638
Virginia Lee Burton The Little House 039525938X
Virginia Lee Burton Mike Mulligan and His Steam Shovel 0395169615
Rene Cloke My Big Book of Brer Rabbit Stories 0517228718
Wende and Harry Devlin Old Black Witch! 1930900627
Julia Donaldson The Gruffalo 0142403873
Marjorie Flack, Kurt Wiese The Story About Ping 0448482339
Emily Gravett Blue Chameleon 144241958X
Emily Gravett Orange, Pear, Apple, Bear 1416939997
Russel and Lillian Hoban Bread and Jam for Frances 0060838000
Russell and Lillian Hoban Bedtime for Frances 0064434516
Russell and Lillian Hoban A Bargain for Frances 006444001X
Russell and Lillian Hoban A Birthday for Frances 0064430073
Russell and Lillian Hoban A Baby Sister for Frances 0064430065
John Klassen This Is Not My Hat 0763655996
Ruth Krauss The Happy Day 0064431916
Munro Leaf, Robert Lawson The Story of Ferdinand 044845694X
Leo Lionni Inch by Inch 0688132839
Arnold Lobel Days with Frog and Toad 0064440583
Arnold Lobel Frog and Toad Are Friends 0064440206
Arnold Lobel Frog and Toad All Year 0064440591
Arnold Lobel Frog and Toad Together 0064440214
Robert McCloskey Blueberries for Sal 0670175919
Tomie de Paola Finn McCoul 0823403858
H.A. Rey Curious George (many books)
Suejean Rim Birdie's Big-Girl Shoes 0316044709
Maurice Sendak Where the Wild Things Are 0060254920
Dr. Seuss Various titles (many books)
David Shannon No, David! 0590930028
William Steig Sylvester and the Magic Pebble 1442435607
William Steig Dr. De Soto 0312611897
William Steig Amos and Boris 031253566X
David Ezra Stein Dinosaur Kisses 076366104X
John Steptoe Mufaro's Beautiful Daughters 0688040454
James Stevenson That Terrible Halloween Night 068884281X
Judith Viorst Alexander and the No-Good (etc.) 0689711735
Mo Willems There Is a Bird On Your Head! 1423106865
Mo Willems Don't Let the Pigeon Drive the Bus! 078681988X
Mo Willems We Are In A Book! 1423133080
Gita Wolf, Joydeb Chitrakar The Enduring Ark 9380340184
Jane Yolen, John Schoenherr Owl Moon 0399214577


Table 3. Favorite Early Nonfiction


(several) My First Little House Books (series) (many books)
Laurence Anholt Degas and the Little Dancer 0764138529
Mitsumaso Anno Anno's Counting Book 069001287X
Eric Carle From Head to Toe 0694013013
Eric Carle Brown Bear, Brown Bear What Do You See? 0805047905
Eric Carle Polar Bear, Polar Bear, What Do You Hear? 0805053883
Eric Carle A House for Hermit Crab 0887081681
Henry Cole Unspoken: A Story from the Underground Railroad 0545399971
Nicola Davies Big Blue Whale 0763610801
Richard Ferguson Deep Blue Sea 1405321504
Flensted Counting Chickens 1609050339
Gyo Fujikawa Baby Animal Families 1402757026
Taro Gomi Spring Is Here 0811823318
Karen Katz Where Is Baby's Belly Button? 0689835604
Susan Korman Box Turtle at Silver Pond Lane 1568999356
Martin it et al. Chicka Chicka Boom Boom 1442450703
Wendy Pfeffer From Seed to Pumpkin 0064451909
Roger Priddy Happy Baby Colors 031249047X
Roger Priddy Happy Baby ABC 0312491697
Roger Priddy Happy Baby Words 0312490097
Roger Priddy Happy Baby 123 0312490232
Roger Priddy My Big Truck Book 031251106X
Peter & Connie Roop Keep the Lights Burning, Abbie 9780876142752
Paul Showers You Can't Make a Move Without Your Muscles 0690041845
Alastair Smith Dinosaurs: Lift-the-Flap 0794504183
Joan Sweeney Suzette & the Puppy: A Story About Mary Cassatt 0764138529
Matthew van Fleet Tails 0152167730
Matthew van Fleet Moo 1442435038
Wilson and Bloom Elephants: A Book for Children 0500543445