Why are we so lousy at teaching math? Why can't it be taught so kids love it? Lukas WinklerPrins thinks there is a better way. As a 21 year-old mathematician studying metrics, dynamic systems, involved in STEAM and a major Lego lover, his advice is first-hand, based on recent experience and worthy of experimentation.
When you think of math, where does your mind go? Tiresome sets of problems and difficult-to-understand literature? If so, you are not uncommon—there are not many who stay curious in math past grade school.
Sentiments of sadness at the public state of mathematics are well articulated in Paul Lockhart’s A Mathematician’s Lament and P.R. Halmos’ Mathematics as a Creative Art. Both authors espouse mathematics as a creative, thriving field, but bemoan its opaqueness and terrible methods of teaching, with the former being largely a result of the latter. The articles articulate how American math curricula virtually prohibit the viewing of beauty in mathematical ideas, while simultaneously failing to provide meaningful everyday examples. The first point has to do with the chronology of math education.
When educators shield students from the terrors of “higher mathematics” (proof writing and analysis), they are inculcating a fear of it. A beginning violinist can gain much from hearing a masterpiece performed, just as a budding painter can learn by observing and emulating the masters in art museums—even if neither of them can fully comprehend the thoughts and meaning that went into creating the pieces. The deep understanding comes through material practice, knowledge of context, and an ability to self-discover. Almost all three of these steps are scrapped in math education (only bits of the first, through rote memorization, remain).
To be fair, this might be a problem at multiple levels. Students can be complacent or passive, teachers not well-trained in their field, and university professors doing arcane research instead of bringing their knowledge to the public sphere. But we have to start somewhere.
I propose a three-pronged approach to tackle this at the K-8 teaching level.
Many approaches to early-age math education start with physical play. The most famous set of math toys, perhaps, are Cuisenaire rods* (used for ratios in Montessori-style schools). This teaching method comes out of necessity: it is the most directly relevant to children of a young age. Tangible objects are immediate and visual, giving students spatial and relational understanding of numbers as objects with weight, color, and shape—attributes the human body is adept at measuring and understanding. Alan Kay referenced a letter Einstein wrote to the French Mathematician Jacques Hadamard: "'I have sensations of a kinesthetic or muscular type.' Einstein could feel the abstract spaces he was dealing with, in the muscles of his arms and his fingers…”. Hands-on projects in math education give students the opportunity to form this sensibility through experience and introduction to some high-level concepts early on. The scope of tactile representation is limited, however, and ultimately students must learn to work with symbols.
Symbols & Language
Among the most difficult things in the study of mathematics for me personally is notation. Mathematics notation is capricious and context-sensitive, and as such the language is difficult to read and unintuitive. I again implore math educators to focus on the feel, or intuitive understanding of an idea. Language is important: it allows mathematicians to find unintuitive conclusions through intuitive use of syntax. How can we merge these two lines of thought?
As a transitional period of mathematics study, we can still use sensory means of explanation alongside the corresponding equations and symbols. In this sense Math can learn much from Language instruction—the idea of a thing must be known before the word can be learned. Groundwork on concepts can be made through intuitive means, but the notation can come as a simultaneous layer on top; thus, notation will be taught while accommodating for its arbitrariness. As students progress, they will carry their intuition into further symbolic manipulation.
Beyond the issues of understanding and semantics, students must _care_ about what they study. Common rhetoric encourages focusing on “applications” of mathematics—word problems. I warn against this. Teaching by problems is constraining; elegant theories and patterns get squeezed into templates of problems, and the student will find it difficult to pull ideas from diverse fields to solve new and unseen challenges. Modern students need to know how to navigate ambiguous and unknown problems.
Instead, I advocate for “play”. Play should be a bit messy, aimless, and bored, because these are ripe environments for creative action. But the classroom can be a gently guiding force. A community of students studying what they enjoy (through self-directed play) is a more effective learning environment than forced classroom material. Allow the student to guide herself through issues and questions that arise naturally. The key component is making sure the student can justify their choices and explain their thinking, pushing the student to become meta-cognitive and envision alternative possibilities.
All together, this means that a lesson should:
- Introduce concepts through visual & tactile means for a more direct connection to the student.
- Focus on use, meaning, and relations of an idea before enforcing a certain terminology or symbolism.
- Allow for students to play with ideas themselves, nudging them towards correct use through communal experimentation.
Starting here, I hope we can help set the foundations for a generation of students who feel more comfortable, creative, and insightful in the field of mathematics.
BIO: Lukas WinklerPrins is a mathematician and apprentice at Atelier Boris Bally. His work on metrics and dynamic systems has taken him to Thicket, a social design lab, Community Systems Foundation, and NSF grant work at Brown. He helped start Brown STEAM, a diverse team dedicated to innovation between the disciplines at Brown University, and serves as a STEAM advisor for three independent schools throughout the country. Lukas has also served as an organizer for Brickworld Chicago, the largest LEGO fan convention in North America. Contact him through LTWP.NET.
*Personal Note: I had Cuisinaire rods as a kid and loved them!! Math was fun and playful...so I used them with my kids! I don't know if it's related but they both love math!