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Calculus for kindergarteners

Redesigning the way we teach math

Many of us raised in Canada and the U.S. were exposed to the traditional sequence of math instruction as children, beginning with counting and working our way through the different and increasingly complicated tasks. We assume that the math we learn follows some sort of natural order so that when we finally are introduced to calculus we assume we have reached the pinnacle of high school math. I’d hazard a guess that most of us can all think back upon countless childhood hours bent over math drill sheets, stuffing our growing brains with mathematical factoids. These practice sheets are tantamount to torture for many kids, and although we’d like to believe this challenging time spent is justified, a growing number of curriculum reformers are claiming that this strategy is inefficient – and they may be right.

Natural Math
Maria Droujkova, math educator and curriculum designer, is a vocal part of this mathematical revolution. Droujkova says, in various talks such as at SPARKcon and the Computer-Based Math Education Summit, that the current method of instruction for mathematics “has nothing to do with how people think, how children grow and learn, or how mathematics is built.” The entrenched system of forcing children to execute repetitive and boring computations over and over doesn’t teach them anything about why these numbers actually matter and what they can do. In fact, Droujkova points out that these activities, despite their relative “ease,” are actually harder for children to do because they drain cognitive resources like working memory, attention span, and accuracy. These exercises stress finicky manipulations of numbers as opposed to understanding grand underlying patterns. This misplaced focus can distract students from the true purpose of math – resolving and analyzing patterns – and potentially discourage many potential engineers, statisticians, designers, et cetera, from following that path.

Instead, Droujkova argues in an interview with The Atlantic that mathematics education should reflect the “playful universe” of advanced mathematics. Mathematics is a diverse and creative macrocosm encompassing over sixty disciplines that influence the way that we think about and execute almost everything in life. In order to restore children’s interest in math, Droujkova advocates an alternative method she calls “Natural Math.” Natural Math harnesses a child’s productive and creative instincts and channels them into different games and free play exercises. These games are designed to teach fundamentals of a variety of mathematical principles. Droujkova claims that once children understand underlying concepts foundational to many upper level math disciplines, they begin noticing changes and can engage with mathematical patterns in complex ways. Following this format, math educators should strive to create exercises that are rich and complex (can be interpreted in different ways) but easy (conducive to immediate play). Some example activities Droujkova provides in her talks include constructing with Lego, making origami, and using imaginary tools like a “function box” (which manipulate inputed variables according to an unstated rule that the child must figure out).

“[The current method of instruction for mathematics] has nothing to do with how people think, how children grow and learn, or how mathematics is built.”

A hierarchy of learning

Droujkova states that there are many complex yet easy aspects to all branches of math, but in particular she has zeroed in on calculus and algebra. She emphasizes the importance of calculus and algebra as pattern-drafting tools that can be used in designing and creating. This creative and productive element is thought to allow kids to engage in free play while simultaneously learning. In her book, co-authored with Yelana McManaman, Moebius Noodles: Adventurous Math for the Playground Crowd, she explains the principles of teaching math using her application-based approach and outlines different activities that can be used to introduce complex mathematical concepts to young children. Such activities include making fractals which touches on notions of recursion and infinitesimals, and using “mirror books,” where mirrors reflect upon other mirrors, allowing children to assess concepts of infinity and transformation. Clearly this resembles less of the calculus you did in high school, or are doing here at McGill, but Droujkova argues that it provides the grounded, hands-on fundamentals of the discipline that can be built upon in future years once students move onto the use of abstract words, graphs, and formulas.

Perhaps Droujkova and her like-minded associates are striving toward higher levels of learning in children, using a hierarchy of learning outcomes to measure educaional value as postulated by Bloom’s taxonomy. At the very bottom of the taxonomy is “remembering” – recalling facts and concepts – moving up toward more complex and generative types of learning such as “analyzing” – drawing connections between ideas – “evaluating” – justifying one’s stand – and “creating” – producing new or original work at the pinnacle. Creating teaching material that accesses the top tiers (that is, evaluation and creation) of Bloom’s pyramid is difficult to do at all educational levels. It is even a pervasive issue in college and university, where students learn to memorize and regurgitate facts without truly grasping any of the deeper conceptual or applicable understanding (recall that multiple choice midterm you did last week). In the case of elementary math, it is easy to see how those painful addition and multiplication worksheets we all encountered as children access only the bottom tier of this hierarchy, whereas Droujkova’s alternative may challenge children to analyze and create.

A new door for children

An entertained student is an engaged student, and an engaged student is more likely to stay in school. As it stands, mathematics can turn a lot of children off from schooling, as they find it difficult and boring. Difficulties at school are even greater for disenfranchised populations who may not have the kind of support network and constant supervision that those with higher socioeconomic status, who are largely white, benefit from. Perhaps anticipating this problem, Droujkova, and colleagues have made their material and courses available for free under Creative Commons, and have designed activities that require only easily accessible materials. Furthermore, Droujkova emphasizes the importance of math circles in creating learning communities for students who may otherwise lack this supportive network. Their goal is to empower local school systems to begin teaching in a way that breaks down as many mathematical and financial barriers as possible.

Of course, the Natural Math camp has met resistance from different camps who claim that Natural Math is not a tenable option. Some critics state that this type of math may put undue pressure on children to learn complex concepts at ever younger ages, somehow justifying the behaviour of overly strict parenting (“My 3 year old can design fractals, can yours?”). Others argue that the focus on play will prevent children from learning traditional calculation skills, ultimately setting them behind.

These concerns are justified and important to take into consideration when dealing with the important issue of mathematical literacy. It is optimistic to think that this more generative and pattern-oriented type of math education will result in a more math-literate population. Curriculum reform is indeed a slow-going process, and simply changing math curriculum itself does not address systemic issues such as poverty or social constructs that dictate who can excel at and love math. However, as Droujkova argues in her interiew with The Atlantic, the emphasis in our current education system on precision and replication does not have any true relation to the real world and that instead the logic puzzles and open projects that Natural Math advocates allow kids to explore, innovate, and interact with their world in new and signifiant ways. Thus, different approaches that attempt to fundamentally shift the way our children think and learn will no doubt have a signifiant impact on how they respond to both the challenges of today and tomorrow.