If calculus equalled torture in high school, then you may be skeptical of Maria Droujkova's argument that a kindergartner can handle fractals – and what's more, that even the most math-anxious parent can teach them. The U.S.-based curriculum developer and math education consultant is the co-author of a creative, new book called Moebius Noodles, a collection of free-play activities for young children that expose them to higher math concepts. (For parents who don't know the math terms, there are a lot of colourful pictures (and Lego!) In an e-mail interview with the Globe and Mail, Droujkova shared how she thinks the ability for children to be creative in math is underestimated by schools, and why bedtime should include math stories.
You make the case that even five-year-olds can do calculus. Can you explain what you mean?
I want to acknowledge these struggles, and place them apart from what we actually do with kids. There are degrees of abstraction and mastery in every human endeavour, including calculus. For example, little kids make up stories, but we don't expect them to write War and Peace.
Why it is so hard, as you put it, to differentiate between the "math skills" and a "mathematical state of mind?"
The same reasons why space flight was hard, and took the humanity a long time to develop: research and infrastructure. We need more basic research to understand the mechanisms of learning, and then we need the tools to implement that research. It's a very complex field of study.
Do you think we underestimate how much children can absorb about higher math at young ages?
I think we need to continue rethinking what children do in math, not only how much. To do so, we may want to focus on what mathematics is: the study of pattern and structure. For example, it is traditional to teach young kids to name triangles, circles, and squares. You can become ambitious and make this list of named shapes longer, for example, add a couple of dozens pretty names of regular polygons (icosagon, anyone?). Children can be more creative in math at young ages.
You mention the "grief stories" you often hear from parents who hated math. Are we creating another generation with their own grief stories?
This is one of the biggest fears of parents – of anyone who thinks about children. Yet each generation has its own struggles and difficulties. People, including children, can do hard work and face scary things if the overall enterprise is meaningful to them. That's one reason I like to talk about math adventures. When kids find meaning in what they do, it can still be difficult, or scary, but it feels very different from meaningless suffering.
You write in your book, that the way we think of math "is as tragic as if kids had to learn the "Isty-Bitsy" spider by heart before they can hear any more complex music." But do you respond to mathematicians who argue that early math students need to have strong – as in rapid recall – of simple addition and times tables so they can move on to higher concepts?
I need more data from neuroscience and cognitive science before I can claim that kids need (or do not need) particular facts before they can tackle particular concepts. Mathematics is very nonlinear. For example, to understand what the number 8,374 means, you need powers and multiplication (8 times the third power of 10). But to understand powers and multiplication, you need to use (large enough) numbers – unless you approach all these ideas in their child-friendly, visual forms that don't require prerequisites. This is one of my pedagogical quests: helping kids learn complex ideas with minimal prerequisites, or none at all, to ground mathematics in immediate, embodied experience. And there is good news for those mathematicians you mention! Patterns and properties kids notice within such conceptual activities can help with fluent computation and memory.
You suggest parents try not to ask math questions to which they already know the answers. What do you mean by that?
This is an exercise that, again, helps kids (and parents!) to make mathematics their own. There are tasks with infinitely many answers, for example, "Make a 3-D shape out of grapes and toothpicks." Other times, the number of answers is finite, but there is an open choice, for example, "Will you choose triangles, squares, or hexagons for your tessellation?" Asking children questions to which you don't know answers simply means children have choices. It's about agency and autonomy in mathematics.
Some parents and teachers will argue it's important that young students learn basic addition and multiplication even if it means they have to do some rote memorization. Isn't that covering all the bases, mathematically speaking? What's the harm?
The harm is not in rote memorization at all, but in the absence of meaning, of the reason why, of understanding what it's all about. As with other activities, rote memorization must be meaningful to the person doing it. One child in my math circle memorized 100 digits of pi because he loved the idea of pi and the process made him feel good. If children need a body of facts for a problem-solving or an art project, they often want to memorize. There are countless situations where rote memorization is meaningful.
Try this at home
Watch video about math together. The TedEd videos include a collection on math topics. Droujkova, who says parents often develop their own math understanding by playing with their kids, recommend this video.
Explore math based art online, or do crafts steeped in math, such as making snowflakes or creating patterns. Here's a favourite from Moebius Noodles.
Read math-based stories. Droujkova points to the collection of math stories. Her personal favourites: the calculus story of Hotel Infinity, and the algebra story of King Shiram and his chessboard.
This interview has been condensed and edited.
Erin Anderssen