I have had considerable success helping students learn the multiplication tables by finding familiar melodies and replacing the words with the numbers in a table. We sing the 3s to “Frere Jacques,” the 4’s and 6s to “This Old Man,” the 7s to “Head, Shoulders, Knees and Toes,” the 8s to “Happy Birthday,” and the 9s and 12s to the “Fee Fi Fiddle E-I-O” part of “I’ve Been Workin’ On the Railroad.” The 6s can also be sung to “Baby Beluga” if you know that; it works with both the verse and the chorus. W didn’t use songs for the 2s, 5s, 10s and 11s because these are easy enough to learn without songs.

Once students can sing any table, for example the 3s, they can solve problems like 8 x 3, 4 x 3, 9x 3, and also division problems like 18 divided by 3, 33 divided by 3, and 21 divided by 3. Granted, they probably need to use their fingers to count the numbers as they sing them to get the answers, but in my opinion that is fine at this stage. Once they are comfortable with these, you can teach them how we carry in multiplication. Then they can solve problems like 46 x 3 and 59 x 3. As soon as they are consistently solving these accurately, you can challenge them further by multiplying three and then four-digit numbers by three: 637 x 3, 428 x 3, 5294 x 3, 8137 x 3. As you can see, all of these can be solved just using the song for the 3s (“Frere Jacques”) and carrying. And for me, seeing a 2nd grader solving problems like this with a big smile is just great.

Perhaps it goes without saying that as soon as students know more than one table, including those without songs, you can be mixing them on a practice page. This does make it more challenging, so if your student gets confused, you could simplify to just one table being practiced at a time. However, the ability to shift between tables should be practiced as soon as possible.

We also need to work to have students free themselves from these songs and from the activity of counting up. The songs are a tool to help them solve certain problems, and solving them helps them feel capable and makes math enjoyable. But the songs can also become a crutch. As their teacher, you can find that balance between time using the songs and time working to learn the number facts just by recognition.

This morning I was sitting behind our house with a good view of one of our hummingbird feeders and saw what I think was a mourning dove come flying over our house. Immediately a hummingbird darted after it, as if to drive it away from the area. But then a wasp went to the feeder and repeatedly drove the hummingbird away. The hummingbird kept trying to get to the feeder, and at one point was able to get on the opposite side from the wasp and get a drink. I could see it watching the wasp through the feeder. Soon the wasp left. When another hummingbird came, one chased the other away – I couldn’t tell which one.

Later I was watching the bees drinking from the pond. I have a hose going from the pump to the top of a stack of stones. As the water flows down the rocks, bees have plenty of places to drink. Sometimes bees fall into the pond. Usually they swim around until they reach a plant, a stone or the side of the pond and then they climb out. Today I saw water striders attack bees as they fell into the water. They were able to kill the bees in just a few seconds. In the past I had seen water striders eating bees that were in the water. I had thought that they were just scavengers and were eating bees that had drowned, but now I know they are predators.

We also have a small snake that has been visiting the ponds regularly. I call him Sneaky Pete. He swims around in the pond a grabs the fish food when he can, but the fish don’t seem to be bothered by him.

Now that the internet has put a vast library at our fingertips, homeschooling high school students may be easier than ever. But taking in information is only one part of learning. The importance of continually inspiring teenagers and engaging them to work with and transform ideas is overlooked in many schools, which is why some parents choose to homeschool. And with all of the "newness" that has flooded the web, students need help to find those age-old treasures that have guided humanity through its gradual awakening.

One of these treasures is a 2500 year old book, “The History of the Peloponnesian War.” It was written by a Greek man named Thucydides, who wrote it as a lesson to future generations. He saw that war was inevitable and set out to document it before it started! He foresaw that the forces in play would continue to work far into the future, and he wanted people to have “an exact knowledge of the past as an aid to the understanding of the future.”

When I used this book as a text with a class of 10th grade students at Austin Waldorf School, they were amazed at how the dynamics between various powers and the perspectives presented in speeches, could, with a few changes, have been taken from current events. (I have turned my notes from this class into a homeschooling course.)

Or how many students have studied geometry without learning how to use a compass and straight-edge to accurately divide a circle into four, five, six and seven equal parts? This is a skill that cannot be learned at the computer screen. Even if the instructions are read online, the hands must take up these other tools and practice. And why is this relevant? Because every healthy teenager is aware that she or he is a mystery, and to learn about other mysteries both honors the mystery of their individuality and provides assurance that they can, with study and practice, master the mysteries of their own existence.

These and many other treasures are available to students today, either to supplement traditional education or to replace it with a homeschooling option. I invite adults and students to contact me to learn about other inspiring resources or to inform me of resources that you have found valuable.

An Overview of the Elementary Math Curriculum

Please note: This is my own interpretation of the Waldorf math curriculum, based on my training at Rudolf Steiner College and my work with students. Some variation from other representations of the Waldorf curriculum is to be expected. It is just an overview, and many details have not been included. Also, some aspects have been listed in one year, but they could begin sooner and continue after.

1st Grade: Introduce the qualities of the numbers 1 to 12. Introduce the four arithmetical processes by working from the whole to the parts. (Work with multiple addends (e.g. 3+5+2+4+6) to prepare for multiplication, which is a “code” way of writing statements such as 3+3+3+3+3.) Act out number stories using stones or other manipulatives to explore numbers less than 20. Write and read simple number statements. Introduce number families (e.g. 8=6+2, 8=2+6, 2=8-6, 6=8-2). Practice counting by ones, twos, tens, hundreds and fives. Practice counting backwards by ones from ten (and later from 20). Explore the ways we can make 10 by adding two numbers together. Practice math facts using dice.

2nd Grade: Rhythmic counting, artistic representations of the multiplication tables (such as number stars), place value, writing and reading numbers into the millions, carrying in addition (introduced as “bundling” groups of ten, or I use bagging groups of ten apples), borrowing in subtraction (“unbundling” or emptying bags as needed), the long division way of writing division, more number families, expanding numbers (e.g. 347= 300+40+7), counting by tens from any number, mental math, square numbers (introduced showing how 4, 9, and 16 objects naturally form a square when pushed together).

3rd Grade: Solidify previous work, addition and subtraction facts for sums less than 20, the first 12 multiplication tables up to 12 times each number. Learn how we measure time, weight, distance and volume. Introduce and work with division with remainders, carrying in multiplication and simple long division. More mental math.

4th Grade: Expand and solidify previous work. Introduce and work with fractions, including adding, subtracting, multiplying and dividing fractions. More long division and mental math.

5th Grade: Expand and solidify previous work. Decimals, formulas for area and perimeter of squares, rectangles and triangles, charts and graphs and negative numbers.

6th Grade: Algebra, business math, geometric formulas for perimeter, area, circumference, and volume.

7th Grade: Exponents, powers and roots, linear equations, the geometry of angles, the Pythagorean Theorem, prime factoring.

8th Grade: Expand and solidify all previous work. Number bases and quadratic equations.

If you read the letters from the Founding Fathers, or journals such as those of Lewis and Clark, you will see that at that time spelling was still rather free. In many ways, the development of individual consciousness follows the same progression as the development of human consciousness, which means that many students will be in 7th or 8th grade before they really are ready to focus on spelling. Because spelling is easy to test, teachers love to “teach” it, often beginning in first grade. Is it any wonder that many children at this age are simply not interested, or don’t have the memory forces to retrieve this information at will? Rather than making these students feel inadequate, we can make the search for unusual spellings an exciting journey, an exploration into the labyrinth of English spelling.

I have found that one good way to work with the apparent lawlessness of our spelling is to encourage students to keep a book of “treasure words” that show how intriguing spelling can be. In the same way that children will find many interesting shells at a beach, or stones in a riverbed, they will find many words that, when they look and listen closely, are treasures because they are unusual.

Some words are interesting in themselves. Examples that come to mind are “iron,” “skein” and “choir.” However, in many cases, the unusual qualities of a word only become apparent in contrast to another word with a similar sound but different spelling, or similar spelling but different sound. One of my favorite examples is the combination “you” and “ewe.” Look at the word “most” and compare it to “cost” and “toast.” Look at “many” and compare it to “zany” and “penny.” Look at way, weigh and whey, and at father, lather and bather, and at eat, grate and great. If we present words like these as exciting treasure rather than as dangerous villains waiting to trip us up, children are more likely to become curious and to look at words more closely.

In addition to writing treasure words in a special book, students should have regular times when they review these words, reading them, using them in speech and writing them. Some students enjoy using these words in poems since they may have already found words that rhyme with them. A delightful poem based on unusual spellings can be found if you type “Poem: I take it you already know…” into a search engine.

When we take this approach to spelling, we may also relax our demands for correct spelling from children, so that they can enjoy the activity of writing and the quirkiness of our spelling. I can assure you that any ways that you can increase your child’s enjoyment of writing will be of far more lasting value than a high score on a third grade spelling test.