Quality of Numbers

This post originally appeared as a guest post for the blog Homemade Math.

I love math. I always have. I think partly because I was able to see the magic of the patterns through the humdrum of algorithms and worksheets that were mundanely fed to us in school. I was lucky in that. So many cannot, and math becomes a tedious bog to drudge through each step heavy with misunderstanding from the last step. I knew early on that I wanted my children’s education with math to be different than mine. I wanted them to see the wonder, the light, the beauty in it. With my second child, I came across that beauty quite literally in the Waldorf approach to math, especially their beautiful circles. Waldorf has a whole child approach where art, movement, and play constitute as great a role in education as academics.

I was first drawn to its approach because of the amount of play that it has with small children. Play is imperative for young minds. After reading Active Arithmetic by Henning Andersen and Making Math Meaningful by Jamie York, I could see the approach had so much movement and play before anything tangible hits the page – skip counting games, hopscotch, drawing in the air, drawing in the sand, collecting, sorting and playing with natural objects. I decided to incorporate a lot of Waldorf Math with my youngest. I pull from a lot of different sources, but I like how they go through the first 12 numbers one by one and explore their qualities which impart number sense. For this post I’m going to use the number “8” as an example.

One of the characteristics that Waldorf is known for is the chalk drawings. I had an epiphany one day to use the drawing for our number units. I don’t have a chalk board that we keep out, so I used crayons and paper to create the drawing that I leave up while we are exploring our number. It has several different pieces to it that we do one at a time. At the top is the number “8” as a digit, spelled out, and as a Roman Numeral. In the center of this page is the circle with dots. To the right and left are the groups for eight, and at the bottom is a bunny skip counting eights. These things get integrated piece by piece in different ways and at the end of our journey of eight, they are incorporated into her notebook.

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To begin we may start out with using different things to make the number eight. We may make it with Cuisinaire rods, scarfs, books. With sixes, she used six of each color rods to make all her sixes. This is a fairly simple task and so we many also talk about another aspect, perhaps grouping. After a few days, she will practice writing the number eight both in our numeric character and as a Roman Numeral in her notebook.

Grouping can be done so many ways. Really you just need is eight of something. It can be acorns, rocks, leaves. We often use our math rods. I ask, “What is eight?” At first I had to show her what I meant by this. I might say, “six and two are eight.” This is similar to educationunboxed.com’s “Who can come to 8’s party?” game. This is addition, but more importantly this is grouping eight into different groups. Being able to regroup numbers is the basis for understanding how numbers work, and thus understanding later concepts of 2-digit addition, subtraction, multiplication, and long division. For my chalk drawing done with crayon on paper, the groupings are in two groups each, but when we are playing with rods or other objects, any number of groups works. How the question is asked is important as well. When asking “What is eight minus two?” or “What is six and two?” a child is limited to only one answer. Asking “What is eight?” illustrates the idea that eight could be grouped in all kinds of ways and leaves room for several answers. Also note that when using just two groups with the math rods, the numbers make a staircase and their corresponding groups make the inverse of the same staircase. This observation is used in the Gattegno approach to math. These exercises can be done once or a couple of different times before showing the correlation with numeric characters and then placing into her notebook as dots. Sometimes, it doesn’t even make it to the notebook. What’s important to me is exposure and understanding.

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The skip counting using dots on the circle is one of my favorites. This is from the book mentioned earlier Active Arithmetic. This activity corresponds with using a train of the same color Cuisinaire rods to make 8, such as four red 2’s to make 8 or two purple 4’s to make 8. This can be done so many ways with hoola hoops or circles on paper, but ours started with a circle drawn in our sandy roads. Then she picked eight acorns and places them on the circle. We drew a line from one acorn to another skipping ever other one to make a square. Then we go to the next dot and skip every other one to make another square. This begins to show the relationship between the numbers, and it reinforced on a numberline. For instance the 8-pointed star created by placing eight dots on the circle and connecting every other dot show a pattern of two fours. Notice that 4 dots on the circle make a square, and so those two 4’s make an 8. This natural progression of circles is a great way to illustrate factors long before they know the word or even fully understand multiplication.

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This skip counting is reinforced on the number line. Here we first skip by 2’s and then by 4’s and lastly by 8’s. When we look at each 8 jump, we see it contains four 2’s and two 4’s. You can also use this numberline for common multiples. My daughter loves having little jumping animals to the side to jump the numbers: bunnies, foxes, grasshoppers.

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In addition to these activities we also may do a good deal of movement and games. I will draw a number line on a sidewalk with chalk for her to physically jump to each number. I have even seen one of these number lines made on a wood floor with painter’s tape and chalk. We will use rhythm and movement to practice skip counting aloud as well. This kind of art, drawing, play, and movement incorporates the whole child. This is why I decided to pull from Waldorf math to mix into our homeschooling.

In the original post, Becky included my instagram post found here.

Della Parker-Hanson is a scientist turned homeschooling mother. She has two children, 13 and 7, that she had been schooling from the beginning. She has a blog called, The Beauty of Play at https://thebeautyofhomeschooling.wordpress.com/ on which she shares homeschooling ideas and experiences. She can also be found on Facebook at https://www.facebook.com/beautyofplay/ and on Instagram at https://www.instagram.com/beauty_of_play/. She is currently developing an online self-paced course for parents to teach the quality of numbers as described above. If you are interested in the class, you sign up on her email list on her blog.

3 Comments Add yours

  1. Karen Rivera says:

    Excelent post!! Love it Della!!

    Like

  2. Sarah Love says:

    I love this approach! I’m curious what age group this is pertaining too? My son is 5.5 and loves numbers.

    Thanks!

    Like

    1. Your son is the perfect age. I would say quality of number is great from anywhere from 4 to 7, varying the activities for the age. Waldorf pedagogy explores quality of numbers even in the older middle school grades, but in geometry. There is so much that can be done here.

      Like

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