From my memories of studying mathematics in middle school and then high school, it was never entirely evident to me that mathematics was the study of objects. This realignment of my perception of mathematics took place only much later, when in the gradual process of traveling outwards and onwards I could see the structure of what it was that I was learning. The notion of structure and ‘object’ in mathematics is very real. I consider the work of a mathematician to be not too dissimilar to the efforts of a geologist studying a mineral formation. Like geologists, mathematicians often encounter these structures in the field, in the work of true scientists who collect data and in their observations seek to give rise to a deeper, theoretical understanding of the patterns at work underneath the observed phenomena. There is a sense that someone exploring the world of mathematics will run into similar concepts, whether that is because of our conscious effort to classify and abstract already existing concepts, or perhaps because our brain is structured in such a way that we naturally have only so much capacity to observe the fine-grained. The extent to which a mineral grown in a mathematician’s mind gives rise to a mineral found in some remote subterranean cave can be experimentally determined, because we are always operating under the overarching the notion of “features” as a means of classification. This connects to one of the qualms I have with early mathematics education: it often fails to encourage the discovery of mathematics. Given the basic definitions and assumptions, one would like, as in a chemistry lab, to carry out the reaction oneself, to grow the mineral and observe any unusual behavior. But because mathematics is a science in which one is capable to making claims that are entirely and completely true (with proof) under a particular logical framework, it has the capacity to abuse the learning process and to deny mathematical learning its truly experimental nature by bluntly forcing the facts. In addition, during my three semesters so far in college, I have already encountered numerous courses in mathematics which, although encouraging the cultivation of a deep understanding of the material, fails to do so both explicitly (in terms of in-class opportunities to discuss material) and implicitly (in regulating the quantity of out-of-class work to allow for enough time to process the material). I am sure that this is not a unique experience: everyone at one point or another discovers the way they learn best. The way mathematical concepts grow is shaped by these conditions, and it can make the difference between a fragmented mineral and one that is sturdy and stands the test of time.
