Lisa G. McCarville is a mother, educator and learning consultant who is working to help people shift their mindsets about what learning is. Lisa is a differently abled person and an advocate for all learners. Lisa works as a writer and an acting instructor/coach using theater games in social groups for people on the autism spectrum. She has a degree in Liberal Arts, 40 years of experience in the arts, and enjoys taking online open courseware classes at many universities. Lisa teaches herself about early education and extensively researches Waldorf, Montessori, and Multiple Intelligence methods of education. She lives with her husband and two big dogs in Arizona, where she homeschools her youngest son, who is on the spectrum. A version of this post originally appeared on Lisa’s blog.

Skip counting is the main underlying pattern of the multiplication tables. If you count by twos, fives, or tens, you are skip counting. Using a manipulative helps children to see the pattern in an abstract concept like counting by sevens. Why do my son and I use manipulatives? They work for us – it’s that simple. Memorizing multiplication tables is usually done through drills, worksheets and rote memorization. It takes a lot of effort and practice for spatial learners, such as my son. Cognitive research shows that not everyone learns the same way. During my own math education experience, I often had no concrete understanding of what I was memorizing. A manipulative that can be touched allows him to see the abstract connection to what he is memorizing. Creating a visual road map of the “big picture” is crucial to his motivation. He learns where math can take his brain and what he can do with basic math facts. My son is also learning that the effort of thinking is worth the work, that the process of math can be joyful. Montessori classrooms are filled with manipulatives as a part of the curriculum. This wooden manipulative is a Waldorf method used to teach children the relationships between patterns and math. The connection the learner makes with these manipulatives forms a bridge to more complex conceptual thinking skills and hopefully a passionate enjoyment of geometry. This very easily made manipulative for skip counting that I found on Pinterest has become a favorite with many parents on some of the Facebook pages I post on. This manipulative is taken from a traditional Waldorf math lesson done with colored chalk on a board, explaining the relationship of patterns to numeracy. My husband Mike did this project in about an hour, and since I had some of the materials on hand, like the dowels and number stickers, the cost was only the $1.75 used to buy the wooden circle piece at the craft store. I found self-adhesive number stickers at the scrapbooking section but you can use a sharpie, craft paint, or even a watercolor vegetable-based paint, polished down with beeswax to seal and finish it. You can do it on paper as well. I will share ideas about cheaper methods below.

The Manipulative being used to count by threes.

WHAT YOU NEED

• One round wooden plaque (size it for your learner’s fine motor skills)
• 10 wooden peg dowels with flat heads (such as axle pegs)
• Wooden craft glue
• A drill with a bit
• Number stickers 0 through 9, paint, or permanent marker
• Small balls of colored yarn
• Ruler/Compass/Protractor
• Pencil/Paper/Scissors
• Husband that will completely take over the project while you have a glass of wine (Optional)

Mike’s first step was to trace the wooden base onto paper to create a template. He divided the 360 degree circle by 10, which gave him 10 36 degree segments to measure and draw from the center point on the template. He scored through the paper onto the wood, using the pattern to mark where each dowel would go. One of my favorite exercises to do with young children is to imagine how the first humans discovered and recognized a circle. Was it an animal tied to a stake in the ground that made them see the circle shape? How would an animal make a circle like this? String and stick circle making exercises can be introduce how people solved math problems with simple tools. “How did they figure it out” discussions make math fun and engaging for young learners. A wonderful book on these subjects to read aloud to learners is“String, Straight-Edge & Shadow – The Story of Geometry” by Julia E Diggins After scoring the wood, Mike checked his work. This was as hard as it looks but he got it precisely right. An educator could easily turn this into a geometrical problem solving exercise for older children. If they can make a circle, find the center, and divide it into 10 equal fractions, you can even do the patterns with colored pencils on paper circles, leaving out the wood manipulative. Younger learners can use a 36 degree angle template to trace segments onto the circle.

After the math puzzle is solved and the 10 points are marked it’s time for drilling the dowel holes.

Mike measured the width of wooden dowel against his drill bits, until he found one that matched. He also put a piece of tape on the drill bit to mark how deep to drill, so that he could make each hole uniform. He then practiced on a piece of scrap wood.   After Mike drilled out the holes, he used a wood glue to secure the dowels. He also tapped the ends of dowels with a rubber ended hammer to make sure they were secure, and pressed on the adhesive number stickers. I like the multiples of three lesson as it shows a wonderful star pattern and the kids love to see how their answers form patterns with the manipulative. Make sure you loop the string around the peg dowels. Be aware of how the process works and show the child: if they skip count by 3 all the way up to 3×10 = 30, they can go counterclockwise, skip count by 7 and get the same star. It’s a fun “aha!” moment for kids to discover. I think it is easier to see these patterns with skip counting by 1 and then skip counting counterclockwise by 9. Most kids know number nine has some tricks and it may help them see the pattern. I keep a times tables chart on the table so that my son can reference it and self correct if he gets confused by the numerical language or sequence of what he is doing. This is empowering, and fosters abstract thinking: using rhythmical language, visual cues and hands-on activity across the midline all work together to master the process. Retention of these math facts through visual recognition of the patterns may help students who cannot memorize their multiplication facts in traditional ways. The best part is, it makes math fun while showing patterns in a simple, inexpensive way. Waldorf educators do this circled skip counting on chalk boards too and it is very beautifully rendered. My Pinterest link from above shows examples of this. If you use different colored yarn for different numbers you can layer patterns and see common denominators. In these picture, the green string marks threes and the orange marks sixes. My son is very serious here, but he does like this work. You can also use a knitting loom as a manipulative. My son said it was like a trampoline and bounced his hand on it!

Another variation of skip counting that my son created with pattern blocks.

The most important reason why I do what I do is these looks of happiness from my son. He loves learning this way. He uses what he learns in his own studies of art, drawing spaceships and futuristic communities using these shapes and patterns. He can synthesize what he learns and make it meaningful to his own interests. My goal is for him to be a lifelong learner and I think that making math enjoyable, practical and engaging leads to that goal.

Conor’s hexagon space station drawing.