Daegan got a lot out of this video, and was inspired that night to practice drawing different numerals: Egyptian, Roman, and modern ‘Hindu-Arabic’:

And tonight the boys and I replicated the activity done during the documentary, in which mathematician Marcus du Sautoy explained binary numbers to Terry Jones using egg cups. (We used washed fruit cup containers). I started off by setting up our usual base 10 system (in blocks) beside binary, or base 2, cups:

Let’s begin where it is easiest, in the ‘tens’ place (second from the right in the blocks). Notice that as you move to the left (get larger), each place value increases by a multiple of 10. The hundreds are 10×10 (or 10-squared, or 10 to the power of 2); the thousands are 10x10x10 (or 10 to the power of 3), and so on. It is similarly with with binary, or base 2 system: each place value (‘cup’) to the left increases by a power of 2. 4 is 2×2, 8 is 2x2x2, 16 is 2x2x2x2, etc. Note too that we can express 10 as ‘10 to the power of 1’ and 2 as ‘2 to the power of 1’. The limiting case in both systems—the ‘ones’ or units as they are now more often called—is the base to the power of zero. Anything to the power of 0 is 1. (It’s one of those weird quirks of math; if this bit is fuzzy don’t sweat it).

Now, we’re all quite sure that every whole number we can think of can be expressed in our base 10 system, using the digits 1 through 9, with a 0 as a place value holder. But would you believe every number can also be expressed in binary, using nothing but ones (for a full cup) and zeroes (for an empty cup)? I know the boys didn’t believe it, and it is far from obvious: these cups are labelled 1, 2, 4, 8, and 16. How the heck do I make 22? Or 14? Or 3, for that matter?

So we made a chart, with the ‘normal’ base 10 numbers on the left, and the way we can express those numbers in binary notation—using nothing but ‘1’ and ‘0’—on the right. Here’s the completed chart (again, don’t worry if this doesn’t make sense yet—it will! Even my 6 year old was able to understand this!):

We then started building numbers by putting eggs in the appropriate cups. 1 was simple—an egg in the 1 cup—and looks the same written in binary as base ten: ‘1’. The picture below shows 2, which we’d write in binary as 1-0 (say ‘one-zero’, not ‘ten’); again, 1s show the ‘cup’ or place value filled, zeroes mean that place value is empty:

And 3, or 11 (‘one-one’) in binary; both the 2-cup and 1-cup are filled because 2+1=3.

And 4, or one-zero-zero (100), along with a hammy Gareth:

Can you figure out how to make 7? Yup—it is 4+2+1, so fill those cups for 111 (one-one-one):

So what number does 1010 represent in binary? If you said 10, congrats—you’ve got it!:

We continued on this way, and soon the boys felt confident enough to record the binary numbers in our chart, as well as create them:

I made a video of the boys building numbers 12 through 17 in binary (I’ll add it when I figure out how!) And yes, that’s the 6-year old rattling off (out of shot) the binary representation of 13. I am utterly amazed at Story of 1s ability to make binary numbers accessible to a child this age!

We ended the session by me asking the boys what the biggest binary number they could currently make (with 5 cups, or place values) was, and they quickly sorted it out. Gareth said it would be written “one-one-one-one-one” (11111) and Daegan added it up: 31. (16+8+4+2+1).

Who knew egg cups could be such a useful math manipulative, eh?

Ten Times Better turned out to be a far better book and better fit for our family that was apparent from the cover. I’d gotten it out of the library ages ago, and it had laid around, unread—until I got notice this week that it was due back. When I finally got around to reading it aloud to the boys as part of our math lessons that day, we all loved it! It is a series of playful poems (and playful artwork!) featuring different animals, with the first animal saying what he/she thinks the best number is, and why. And the second animal saying, in effect, “Oh, yeah? This ten-times-bigger number is better!” So the book starts with an elephant singing the praises of ONE (as it has one trunk), and a giant squid saying TEN (as in tentacles) is clearly better than one! And on it goes with very logical choices for the animals: a 3-toed sloth, a 9-banded armadillo, a giraffe with 70 spots on its neck. And my children’s favourite animal and line of poetry: “I have 60 teeth” says crocodile, “I’m a great  masticator. (That means I chew first, and ask questions later.).” A delightful book for kids learning multiplication and into animals.

The History of Counting was a book that caught Daegan’s attention; Gareth half-listened as he played on the floor nearby. It does exactly as its title suggests, outlining some of the major steps in humankind’s “invention of counting”, from using pebbles / physical markers (counting without numbers), to body counting (“1 is left little finger…11 is left neck…16 is right ear” up to a very limited number: 28), to using different words to represent the same number of different things (like we use “twins” to mean two babies who share a womb, but “duo” to mean two people in other contexts, like music), and so on. We visit the Sumerians (whose ‘base 60’ lives on today in our number system: 60 seconds in a minutes, 60 minutes in an hour, 360 degrees in a circle),  Egyptians, Phoenicians, Greeks, Romans, Arabs, Hindus, etc. up to modern day arabic numerals and the importance of zero and place value. Daegan was particularly engaged with the section on Roman Numerals, and the way the book showed their severe limitations when it came to computation. This is a very abstract concept for a 9-year-old, that’s for sure—understanding why place value was such a big leap over Roman Numerals—but attempting the addition problems in the book:

1084+1797

vs. the same numbers written in Roman Numerals:

MLXXXIV + MDCCXCVII

drove the point home. And even more so when I suggested he try multiplying those numbers instead of adding them! Happy reading!

So here’s the plan, (which you can just make out in the background of the pic) loosely starting around 8:30am and ending around 11:30, give or take:

Monday / Wednesday (and what we did today):

Language Arts Warm Up / All About Spelling / Language Smarts / Gareth reads to me; Daegan does Growing With Grammar solo / Gym and Snack / Logic (or Math) Warm Up / Mathematical Reasoning / Life of Fred

Tuesday / Thursday:

Logic (or Math) Warm Up / Math U See / Math Readers, Games, Investigations / Gym and Snack / Language Arts Warm Up / Word Family or Cut-and-Paste Story Sequencing (Gareth largely solo); Writing Paragraphs (Daegan with me) / Gareth reads to me; Daegan Reading Detective or Grammar and Punctuation solo.

I also have a whole variety of fun educational apps organized on the iPad if I need something to occupy one boy while I work with the other, or if one finishes earlier than the other.

The ‘Warm Ups’ are a collection of puzzle-based workbooks the boys enjoy, like Dr. Dooriddles, Analogy Challenges, Math Analogies, Mind Benders, Balance Math, Lollipop Logic, Logic Countdown, various Perplexor titles and so on. The boys pick what they like that day.  Both boys enjoy riddles and puzzles like these: they are our fun ‘warm up your brain’ activity.

We ended the day by watching a bit of the news coverage of Jack Layton’s death (the leader of the opposition in our parliament, who lost his battle with cancer this morning), along with some clips of his appearances on the Rick Mercer Report and 22 Minutes (you can find the latter on facebook). I also read the boys much of Jack’s final letter to Canadians, written 2 days before he died. I am glad to be back in our routines, but also glad our schedule allows rooms for days like this very sad day for Canada. I’ll leave you with Jack’s closing lines; the entire letter can be read here:

My friends, love is better than anger. Hope is better than fear. Optimism is better than despair. So let us be loving, hopeful and optimistic. And we’ll change the world.

This morning the boys and I read an intriguing book, The Token Gift  (Canada) (US).  It tells the fictional story of an old man from India, Rajrishi Mohan, discontent with what he has accomplished in life. If only he had been king, then he would have amounted to something! His wife reminds him to think back to his youth, and the game he created that is now played by so many—chaturanga, or, as it is known today, chess. Detailed, lush pictures take us back to his youth, and how the game of chess evolved from other games of chance using dice, to a 4-person game involving “4 divisions” (of the military: chariots, elephants, cavalry, and infantry) and a mix of strategy and chance, to the modern game of chess—pure strategy—as we know it today. The man is interrupted by a messenger sent from the king, who commands Mohan to the palace.

He makes a journey of several days, and meets with the king, who wishes to give Mohan a gift to thank him for inventing the game. “I have enjoyed played it since I was a boy—and the game’s emphasis on careful thinking and strategy has given our generals many advantages on the battlefield. I was surprised—and delighted—to learn that the inventor of this game lived in my kingdom.” Mohan refuses the offer, but the king insists, and Mohan requests what the king takes to be a small, or token, gift: for each of the 64 squares on the chessboard, he will accept some rice, with 1 grain for the first square, 2 grains for the second, 4 for the third, 8 for the fourth, and so on, doubling each time.

The king promises to provide this gift, and sets about meeting it. Servants start piling rice—1 grain, 2 grains, 4, 8, 16, 32, etc. but they soon run into difficulty, and the king sends for the Royal Mathematician. This part of the story is exceptionally well done, I think, as we learn that by square 16 there needs to be 32768 grains of rice—or about as much as in one sack of rice. Now the doubling begins anew, and by square 32 the king needs 32768 sacks of rice—more than the kingdom can grow in one year, but about equal to 1 shipment of rice.  And by square 48 he needs 32768 shipments of rice, or as much as the world can grow in a year, and by square 64 we learn it would take the world 32768 years to grow the amount of rice required! The king resigns in shame, for a man unable to keep his word is not a man of honour, not a man worthy of being king.

So now Mohan has his wish, and is king. But he wanders the halls of the palace feeling empty inside still—he realizes he is still the same person, and being king is not the answer. So his one and only decree is to abdicate his throne and reinstate the old king, pointing out that wisdom without honour—as he demonstrated in himself by tricking the king—is not desirable in a leader. “And,” he adds to the king, “Perhaps this will make you an even better leader, now that you know that things aren’t always as they first appear.” The story ends with a page about the possible origins of the game of chess, much of which was worked into this fictional account.

So there you go—a book with math, chess, history, a look into another culture AND a morality lesson! My two were quite taken with it; I’d recommend it for kids at least 6 or 8 years of age. Despite being a picture book, it is densely packed and with lots of text—not toddler or preschooler friendly, I’d think. Happy reading!

The series was good, and provided a fine overview of the Renaissance period, but it was definitely over Daegan’s head at points. (The suggested age range is grades 5 through 9). For example, there were many references to the church and religion (and its role in everyday life, the conflict with science a la Galileo, the break-up of the church into various Protestant factions after Luther) which I knew Daegs didn’t have sufficient background to fully understand. And it would definitely have been helpful to know more about ancient Greece and Rome. As the series stressed many times, the word ‘Renaissance’ literally means ‘rebirth’, as there was a rebirth in the science, art, philosophy, etc. of the ancients. Ancient Greece and Rome is our next stop in this grade 3 year where we seem to be doing non-stop social studies!

One thing I quite liked about the series was that they went to Europe and shot lots of video on site, allowing the viewer to see Renaissance architecture and art directly rather than a mock-up in studio. Similarly, they showed more real artefacts from the period (now in museums) than in the Life in The Middle Ages series. And it has further sparked Daegan’s curiosity about peoples of other times and places—my main goal.

So anyone out there have favourite resources (documentaries, movies, history books, novels, hands-on-art or science projects, whatever) for life in Ancient Greece and/or Ancient Rome?

The DVD set also includes a much shorter 3-part documentary (total, 78 minutes) called The Music of the Primes (your can read reviews from amazons UK site here),  which I enjoyed just as much, and was more accessible—in terms of math content—to our 8-year-old, who has some familiarity with prime numbers. (The 5-year-old enjoyed the “travel around the world” aspect of both DVDs, as well as the computer generated animations illustrating various mathematical ideas.) I do think older children, who have more background in both mathematical concepts and world history would get even more out of the DVDs, though, and we plan to borrow it again from the library in a few years.

One note of caution: this DVD is aimed at mature audiences and has a “European” sensibility. As du Sautoy travels around the world, we see lots of art from different areas, almost all of which celebrate the human form. There is explicit nudity, both male and female, in statues for example. And one of the examples given for the use of mathematics in Ancient China is how the Emperor could most fairly sleep with 121 women in his harem in 15 nights: his 1 queen, 3 wives, 9 mistresses, 27 concubines and 81 slaves. And many of the more modern day mathematical figures discussed were driven to insanity, suicide, or both—such as Alan Turing, the homosexual hero of second world war cryptography.

Given the paucity of video on the subject of mathematics—contrast all the science DVDs, and shows like Nova, Nature, Scientific American Frontiers and so on—I was glad to find this one and recommend it to older kids and adults (and youngers with adult guidance). Happy viewing!

I too have quite enjoyed them, as I missed much of this part of history in my own schooling, moving around as we so often did, and this before curriculum was standardized. I studied Japan 4 years in a row, but missed the Middle Ages entirely! Daegan and I were laughing at lots in The Doctor episode, learning about superstitious healing practices (“hang a dried toad around your neck to cure internal bleeding”), and that to get surgery done, you went to the barber! Doctors would diagnose you and recommend the surgery, of course, but manual labour was beneath them. The series is not all chuckles, however, with topics like feudalism, medieval warfare, the differences between serfdom and slavery, the Black Death, the role of the church, the role of women, etc. also being addressed. I find the style of presentation quite kid-friendly, with young adults dressed in period costume talking directly to the camera, at times arguing with the script, and simple historical re-enactments shown. I do have issues with the way some topics have been presented (sanitized, over-simplified, too much a pro-Western view, etc.) but, on the other hand, controversy is pretty much inevitable when discussing history in a short (20-30 mins per episode) DVD format, and makes for lively discussion points at the dinner table.

Gareth has also watched bits of various episodes but by and large doesn’t care for the series as yet. No surprise to me, as I don’t think 5-year-olds have enough context and understanding of their own culture to make sense of what they are watching. Whereas Daegan, at 8.5 years, really seems to connect with the show, particularly the “How Would You Like It?” segments where they explain how children were raised / educated at the time. The Knight sent his 12-year-old son off to live with another knight’s family as a squire for the next 8-ish years; the merchant’s son learned maths and literacy working in the family business, young boys worked as apprentices to masters in a variety of industries, required to do their masters bidding 7 days a week, etc..

If you are looking for an accessible, light (but not too light)  introduction to many aspects of life in the middle ages—not just a series of dates and battles and kings and queens—check the Life in the Middle Ages series out.

To give an example, Daegan and I most recently watched episode 4: Faith in Numbers, the episode Burke alludes to in his intro to the series as being about “how improvements in 14th century sheep farming by monks in France led to the modern computer” (via a series of connections, of course). We start at a decaying Roman aqueduct in France, learn about the use of water wheels for all kinds of purposes (milling grain, pressing grapes for wine, etc.) and are introduced to different systems of gears used in these inventions. We make various jumps: Cistercian monasteries, the Black Death, how the abundance of linen for “frilly knickers” led to an abundance of linen rag (when the undies wore out), which—in mills powered by gears—was mashed with water and gum to make linen paper (which was pressed to squeeze water out via gears again). From there we head to the invention of the printing press (more gears) to make use of the abundant paper, then to the jacquard loom and complicated patterns in clothing (like Chinese silks or English paisleys)—which again were automated using punch cards and gears. Finally we visit automated toys and amusements, the managing of US census data, to—finally—the modern (1970s) computer and its punch card. A fascinating show, with so many jumping off points to pursue further study or investigation (there were more connections than I wrote above)—it strews concepts, names, and places constantly. And yes, the boys were absolutely fascinated with how underwear became books!

Details about the series can be found here. Burke made two further Connections series in the 1990s, which were similar, but not as good as the original in my view. But a few notes of caution. The series is most definitely created for adults, not kids—the first episode is quite dark in tone, post-apocalyptic, with the aim to have us look at the way technology is so entwined in the fabric of our lives. “Look around the room you are in watching this show, and note how little would work without technology or a source or energy or power” ask Burke of the viewer. He talks about how this did happen during the massive blackout in NYC and other parts of the Northeast in 1965—and with the historical re-enactments, it may well disturb some younger viewers. In episode 3, there is brief nudity when a medieval woman is shown taking a bath as part of a town celebration, and there are other sexual references and innuendo in the series as well (done in dry British tone though, so may well go over your kids’ heads). Again, just a head’s up for your own family; my 8-year-old son loves the series and wants to watch more, and it has led to great discussions between us.

For older kids, or for your own further education, and especially for those who love science or history (this series has been a great way to get my science-loving kid curious about history, and I imagine it would work in the other direction as well), give Connections a try. Over the years as our family watches and re-watches the series, if the boys  understand more about each reference and connection (which from a wide variety of cultures, time periods and places), and use these connections to build up a “web” of knowledge, they are well on their way to being an educated person, in my view.

Local Calgary folks, the DVDs are at the library. As they are older you could probably find them online or on Netflix as well. Happy viewing!

We also read the poem In Flanders Fields, perhaps the most famous poem by a Canadian ever. Just today I found a wonderful book about the poem at the library, called In Flanders Fields: The Story of the Poem by John McCrae  (Canada) (US) by Linda Granfield.

It includes the entire text of the poem, one line per page, with moving artwork that helps interpret the poem for children, as well as factual details. Topics covered include John McCrae, WWI and its causes, a map of Europe showing where Flanders is, the role of various other countries in WWI, etc. I will be reading through all the material with Daegan (age 8 ) tomorrow; with Gareth (age 5) I think I will simply re-read the poem pages in the book, discussing the artwork as we go along.

We will also be watching the Canadian Heritage Minutes about John McCrae I found on youtube:

and perhaps a montage of some Heritage Minutes related to the Canadian Military. (BTW, if the face at the beginning of the 4th clip—about 4 minutes in—looks familiar, that’s because he now plays Spike on Flashpoint. He’s also an old friend of mine; Sergio and I worked together at a bookstore in Toronto for several years during our early 20s.)

And of course, there’s the option to attend a local Remembrance Day ceremony (a list of Calgary and area ceremonies can be found here; I believe the largest one takes place at the Military Museums), or watch one on television. Hope these links help you share Remembrance Day with your kids, and please share your favourite links on Remembrance Day, or similar days in your country, in the comments. I would love for my kids to learn more about other countries on this topic, and learn how WWI and other conflicts were truly WORLD wide in scope and impact.

Gareth (age 5) took this pic of me in pjs and robe, zoning on the couch. I even got a bit of a nap. Such a disappointment too, as we had planned a playdate with friends after their school on Friday, with the dad joining us after work and having a pizza night. And it was again crazy warm, hitting 20C (68F). Bah.

But the boys and I did read a few books, and watch some DVDs. We watched several of the The Way Things Work DVDs, which I wrote about here. And a few of a new series called Are We There Yet?: World Adventure.

Each DVD features real life siblings—young, I’d say under 10—visiting different locations around the world. We watched Hawaii, Egypt, and Sweden today. In Hawaii, a brother and sister learn to surf, go snorkelling, ride in a submarine, learn to hula, etc. In Egypt, two sisters visit museums, the pyramids, a bazaar, and make a mummy. In Sweden, a brother and sister team take part in a sailboat race, visit a museum with Sweden’s largest Viking ship, learn to be troubadours (busking for candy), and—my favourite bit—learn about Pippi Longstocking by watching a play and visiting her house/museum. In each location the kids meet up with experts (hula dancers, archaeologists, sailors, etc.) that show them around. Each DVD features 3 segments, each about 5-10 minutes in length. More details about the show, including a list of episodes and topics, can be found here, and pictures from the show here.

While my boys were engaged with these shows, a couple caveats. The show’s segments are short and fairly jumpy, providing a good overview of a place, but little detail. The shows are also focussed on the country from a tourist’s perspective, unlike the Families of the World series, in which you follow ‘a day in the life’ of children actually living in each location. But for strewing ‘rabbit trails’ to follow up on (Can we read Pippi Longstocking, Mom? Can we make a mummy? Who were the Vikings?) it is superb. And my kids particularly enjoyed watching kids their own age exploring the world.

Not sure where you can buy the DVDs; amazon doesn’t have them. I borrowed from the library. You might also check netflix, youtube, or treehousetv.com for clips.