Archive for May, 2008

Enter stage right . . . the math gnomes

Monday, May 26th, 2008

OK, so I’ve come to the realization that there is no way I am going to “catch up” in my documentation of our first-grade year. Time is at such a premium right now, and what I really want to be able to do is blog about what is going on in our lives currently, not what we did months ago. So I’m giving up on my initial idea to record all of our main lessons from this year. Let’s face it, it’s just not going to happen in a timely manner.

But, before I turn over this new leaf, I do want to record our follow-up math lesson to the Quality of Numbers, mainly because I think it might be useful to somebody else out there looking for an inspirational launch pad. I had a great time preparing for this main lesson block. It was definitely a highlight in our year.

The first math main lesson, the Quality of Numbers, was followed by a main lesson block on learning the lower case letters, which we did by creating a 2008 calendar (For more on this see the Christopherus First Grade Syllabus). This was a great project to do in January and provided the opportunity for a lot of writing and copying from the board, as well as practicing the numbers and learning the months of the year. As a side note, I really like the advice offered by Live Education! that recommends allowing the child to copy from the board letter by letter; each letter first written by the parent-teacher, then by the student. Once we started doing this there was noticeable improvement in Starling’s written work. Her sentences were much neater, with fewer mistakes, and she was able to relax into the process of copying from the board.

After the calendar main lesson block we returned to math, and a continuation of our story involving the characters of Phaedra and Adira. Starling and Chickadee were very excited to hear about the Land of Numeria, and were eager to find out who the little men were in the picture.

[A word about my decision to use the ubiquitous Waldorf math gnomes to introduce the four math processes. . . At first I resisted the idea. Waldorf education often gets criticized for being overrun with references to elemental creatures, and the math gnomes seemed to me to be a bit too clichè. There were plenty of other options. Donna Simmons uses squirrels. Live Education! introduces a whole host of different characters. But after further thought I came to recognize that there is something special about introducing the math concepts via the personalities of these little men (or women, if one chooses), and I knew that Starling would immediately identify with these endearing characters.]

Once again, I looked to Marsha Johnson for inspiration. I love Marsha’s ability to create pictures with words. Shamelessly, I lifted some of her sentences and phrases for my own stories. Below is the tale that I told on the first day of our main lesson block on the Four Processes. It tells of Phaedra and Adira’s introduction to Gnome Share (division). Unlike the customary introduction of the Four Processes by way of addition, Waldorf education advocates introducing the process of division first, emphasizing the concept of “from the whole to the parts”. During the course of the story, Starling and Chickadee also got to meet Gnome Share, a needle-felted fellow who emerged from a basket at the appropriate moment. Gnome Share quickly took on a life of his own, as the girls incorporated him into our daily lives. Here he is with Gnome Times:

And here’s the story:

Phaedra and Adira did it! They answered the twelve ancient questions, satisfied Cipher, solved the code on the golden scroll, and opened the padlock on the great wooden doors. When the doors swung open the two girls stood in amazement, their eyes as big as saucers, their hearts full of anticipation and joy. What was this new world? Cipher called it the Land of Numeria. She said it was the key to all understanding about the magical world of numbers. But what did it all mean?

As Phaedra and Adira stepped forward through the threshold of the door they found themselves standing on a beautiful rainbow path that sparkled in the sunlight. Just near to the path, the girls saw a large hill, rising up with many rocks and gravel slides, and little narrow paths leading in and out of big stones, places only goats or dwarves could tread, perhaps little gnomes could squeeze by those big rocks and narrow passageways . . . and indeed do you know, there in the damp dirt of one of the paths, the children could see what certainly looked like small footprints!

Though the girls did not know it, the hill was filled with little caves and big caves, and inside those caves were tunnels leading to and fro, and open ones that led in and out of rocky hills, some way up high, overlooking the land, and others down below, just near the bubbling streams that flowed downwards into the green valleys. Built into one of the rocky outcroppings was a small wooden door, similar to the one the girls had just entered, but much smaller. The girls immediately walked up to it and listened. Standing before it they heard a tap, tap, tapping going on deep inside the hill. They couldn’t have yet known, but indeed inside the hill, the gnomes were very busy, as always, working day and night to dig beautiful colorful jewels and precious metals out of their rocky beds. Tap, tap, tap, crack, crack, crack, went the small hammers of the busy gnomes.

As the girls were standing, listening, the doors suddenly opened and there before them stood a gnome dressed all in red. He was a stout and plump fellow with a white beard, kindly face and sparkling blue eyes. He wore a red shirt, red trousers, a red hat, and big boots with thick soles, and on his wide belt he wore a lovely embroidered pouch, with a curious symbol on it. “Hello” he said, as he made a low bow to Phaedra and Adira. “Welcome to the Land of Numeria. My name is Gnome Share and I am happy to make your acquaintance! Do come in and I’ll show you around.” The girls bent down and entered through the small door. Inside, lanterns glowed and lit the way as Gnome Share led the children down a winding path. The girls looked all around as they walked. Piles of colored gems lay in large heaps on the rocky ground. Ponies with sacks tied over their backs trotted merrily up the path toward the girls and gnomes everywhere were busy at work. “I’m one of the Gnomes in charge here” said Gnome Share “There are four of us. I’m sure you’ll meet the other three during your stay.”

After a short distance the girls came to a large room, lit by lanterns and candles. It was all aglow in sparkling colored light. The girls had never seen so many precious jewels–rubies, sapphires, diamonds and millions of colored gems that reflected the lamp light. “It is my job,” Gnome Share said “to divide the gems, jewels and precious metals so that they are equally shared among the people of Numeria. I make sure that everything that is found is shared. There are so many gemstones to go around. Here have some yourselves!” and Gnome Share picked up a sackful of colored stones. “These are for the two of you. But do be sure to share them.” “Oh, yes . . . share them. . . of course we will. But how can we be sure to share them equally between us?” Phaedra asked. “Oh,” the kindly Gnome said “That is easily accomplished. Watch. . . . There are six gems in the sack . . . .here is one for you, Phaedra, and one for you Adira, another one for you Phaedra, and another one for you Adira, one more for you Phaedra, and one more for you Adira . . . as you see we now have two piles of gemstones with 3 in each. It’s as simple as that!” The girls smiled and looked at each other. What a wonderful little man! But how busy he was! “Could we help you in your work?” the two girls asked enthusiastically. “Of course!” came the reply. For the rest of the day, Phaedra and Adira busily counted and divided the gems. What a lovely man he was and how happy they were to be in his presence.

After telling the story I drew a chalk drawing of Gnome Share on the board for Grace to use as a guide for her own drawing. We then wrote the following verse adapted from Dorothy Harrer’s Math Lessons for Elementary Grades:

Gnome Share is a kind old gnome
With each one he will share.
The jewels which he gathers.
He divides with greatest care.

The following day, I presented Gnome Share’s symbol and Starling drew a large division sign in her main lesson book. We then did some dividing (sharing) with twenty-four glass gemstones that were stored in a little red flannel sack. Gnome Share wanted to share the gemstones between himself, Starling, Chickadee, and me. Starling divided the twenty-four gems between the four of us and was able to determine the number of stones in each pile. She then illustrated what we did by drawing four piles of six gemstones in her main lesson book. Under her illustration she wrote the “number sentence”: 24÷4=6

Next came Gnome Minus. Here’s his introductory story:
One day as Phaedra and Adira were counting and separating the jewels with Gnome Share a funny little gnome dressed all in blue entered the Treasure Room. He was more slender than Gnome Share, with a tall blue cap and the same merry blue eyes. Instead of a white beard, his was the color of coal.

“Minus!” said Gnome Share “How very good to see you! Come and meet my new friends, Phaedra and Adira. They have been helping me to divide the jewels.” The girls moved closer to the slight little man and shook his hand. “Oh,” he said “How delightful! What a joy! My name is Gnome Minus Takeway. How wonderful to make your acquaintance! I was just thinking the other day how much Gnome Share needs some helpers. He has so very much to do.” And pausing for a moment, Gnome Minus turned to Gnome Share and said, “Gnome Share I have collected some jewels for the Treasure Room. Twelve jewels to be exact. Now, where are they?” he said, and the small gnome began to search his pockets. “Now, I know they are in here somewhere. Yes, here are a few. One, two, three, four, five . . . . five? Well, for heaven’s sake where are the others?” The girls looked at Gnome Minus and then at Gnome Share. “Oh dear, it’s my awful pockets again! The gems have fallen out of my pockets, so that I only have five left!” And at that Gnome Minus pulled his pockets out of his pants to reveal several large holes. “I know that I should mend my pants, but I am so very busy and I never seem to find the time.” “Phaedra, Adira, would you be so kind to help me find the other gemstones?”

“Now, if I started with twelve stones and I only have five remaining. How many did I lose?” The girls looked at the little gnome. “Well, let’s use some of the jewels here to figure it out. I know that I started with twelve gems and now I have five. Let’s take five jewels and move them over here.” And the little gnome moved five gems into a small pile. “Now how many remain?” he said. Phaedra counted the remaining gemstones. “Seven!” she said. “That’s right seven. Can you help me to find the seven gems that I lost?” the funny man said with an imploringly look on his face. “I feel awfully foolish for losing them and I know that Gnome Share would like to have them. “Of course, we can.” Phaedra and Adira said in unison, and they began to search for the gems.

And here is lovable Gnome Minus himself (with Gnome Add): He quickly became our favorite. Starling and Chickadee particularly loved sliding gemstones out of the holes in his pockets.

His verse which I changed slightly from Harrer’s version is:

Minus is such a silly gnome.
He loses everything.
His ragged pockets are empty
For he subtracts everything.

The next day, and last day of our four day rhythm, I introduced Starling to Gnome Minus’ symbol and we did some subtraction scenarios with the glass stones. The most effective one was the removal of a certain number of stones from a sack containing twenty four gems. Starling then had to determine how many stones were remaining in the sack. We choose one of the problems to illustrate in her main lesson book accompanied by the subtraction equation.

And so our lessons went, for three weeks in total. Gnome Add made his appearance next, followed by Gnome Times. During week three we reviewed the four processes and the gnomes bid goodbye to Phaedra and Adira. The stories from weeks two and three can be found here.