- By meril
- 25 April, 2016
- Comments Off on Discovering Algebra
It has been the plaintive cry of generations of high school students – What do I even need algebra for anyway? Or trigonometry? And it’s a legitimate question: Before educators talk about how best to teach or learn algebra, we should talk about if and why we need it in the first place. Because this should be something we can do. If, as we say, mathematics is an essential forum for the development of rational and abstract thinking, can’t we turn these rational tools inward, reflexively, to try and make some sense of why we approach mathematics the way that we do?
Amazingly, the traditional focus of school mathematics on functions, trigonometry, logarithms, and such can be traced back to the 18th century and the great Swiss mathematician, Leonhard Euler’s formulation of what he saw as a manageable chunk of introductory algebra and geometry for students. Intentionally or not, Euler inadvertently inspired a schism that can still be felt today. Though Euler was, himself, a pioneer of Infinitesimal Analysis, or what we now refer to as Differential Calculus, he felt that beginner students should focus on learning finite mathematics before embarking on the then-contentious explorations of the infinitely small numbers used in Calculus to find limits. In the 19th century, educators seized upon Euler’s delineation and designed an intricate, interrelated, self-contained system of finite mathematics to function as a gymnasium for training young minds. This is the world of trigonometry and logarithmic functions that has traumatized high school students ever since.
In creating this self-contained, finite gymnasium, school mathematics in the 19th century cut itself off from the vital momentum of mathematical exploration and discovery. School mathematics today remains anchored to these 18th century ideas embedded in a 19th century paradigm. What students learn in school today has little to do with the ideas and questions that engage contemporary mathematicians.
Perhaps educators starting as far back as the 19th century have misinterpreted Euler’s original insight.
Clearly, the cutting edge of contemporary mathematics is a bit much to teach to school-level students. As Euler did, we need to plan backwards, starting with articulating where we hope they will get to, and we need to also focus on teaching the core concepts as we now understand them, as well as the techniques and tools of the trade. But what if the project of school mathematics got completely sidetracked more than two centuries ago?
From a 21st century perspective, what might we now identify as the new fundamentals of contemporary mathematics? Might is be more useful to learn mathematics through explorations of modeling and transformation, rather than than through trigonometry and logarithms? Any effort to make mathematics more engaging for students needs as a first step to shake off the confines of these centuries-old conceptions about what mathematics actually is. Euler was a brilliant thinker who pushed the boundaries of mathematics. I suspect, if he were alive today, he would not be as stuck on his own 18th century notions as we in our modern classrooms still seem to be.
As part of a group project, I recently outlined a new approach for grade 9 algebra. Exploring Data and Discovering Algebra in a Technology-Rich Environment starts with real-world data, and uses simple spreadsheets and graphing software to discover patterns and to eventually create in students the need for algebra, and then guides them through its discovery. This is not at all how algebra is usually taught. Algebra is most often presented as an essential tool to be mastered, even though students may never have felt the need for this tool in their lives before they encounter it in school.
Our perspective is that algebra is meta-arithmetic thinking. If approached the right way, it can be experienced as a powerful tool for generalizing about quantitative relationships. But before students begin to feel any need to create a generalized formula, they need to be immersed in data and exposed to repetitive examples where patterns can begin to emerge.
In Exploring Data and Discovering Algebra, our strategy is to anchor mathematical thinking in present day reality. Mathematics is approached as a mode of communication where a student’s ability to articulate and present their ideas develops together with their understanding. The building and presentation of mathematical models, the visualization of patterns, and playful exploration combine in a guided discovery of algebra.
In our approach, the tools of mathematics are key. (Essentially, algebra itself is a conceptual tool, but there are other preliminary and less-abstract tools we teach students to use along the way.) A knowledge of dynamic spreadsheet software, graphing software, and algebraic software, as well as multimedia presentation software are core tools in our course. In particular, GeoGebra, with its ready, multifaceted, simultaneous visualizations of data through tables, graphs and algebraic expressions, offers an exciting approach to this subject matter.
Young people today have proven themselves ready and willing to immerse themselves in the virtual worlds of gaming culture. Similarly, we invite them to immerse themselves in the virtual networks of the real world communities who work with data in their professional lives. In so doing, we meaningfully situating our classroom activities in real-world contexts such as sports, entrepreneurism, ecology, and art.
Ultimately, understanding algebra is less important than understanding the world we share. The best way to learn how to use a tool is to build something that requires that tool. Rather than teaching mathematics and then showing how it is useful, we aim to engineer a situation where mathematical understanding is required and then we let students learn on the fly.
Biermann, H. R., & Jahnke, H. N. (2014). How eighteenth-century mathematics was transformed into nineteenth-century school curricula. In S. Rezat, M. Hattermann & A. Peter-Koop (Eds.),Transformation – A fundamental idea of mathematics education. (pp. 5-28). New York: Springer. doi:10.1007/978-1-4614-3489-4