Are mathematical symbols merely symbolic notations that exist separately from the abstract mathematical truths that underlie their actions? Some would say yes, that mathematical cognition occurs after we have converted the symbols into meaning, a meaning that exists outside of space-time, and a meaning that is as equally contained in an ancient mathematical manuscript as it is in a recent explication. Others think that perhaps mathematical symbols do something more than simply express mathematical concepts: that they “enable us to perform mathematical operations that we would not be able to do in the mind alone.” By assuming part of the cognitive load, mathematical symbols become “epistemic actions.” From the perspective of a mathematics philosophy, a causal account of how we obtain math knowledge is complex. Elementary numerical knowledge represents our ability to perceive natural numbers. From a young age we can see when there is more or less of something. Likewise we can “match the number of voices [to] the number of people speaking,” or the number of items in a hand to the number of items on a screen. However, many mathematical truths don’t commonly originate from direct sensory observation. Rather, they are attained from cultural sources. The mechanisms taught require cognitive reconstruction in order to understand the abstract structures underlying the concept. In many instances, mathematical mechanisms are not closely matched to our intuitions. For this reason we must build an entire mental framework around math concepts in order to make sense of them. An obvious example can be found in the number zero which equals both nothing and something at the same time. On its own it equals nothing. As a placeholder it means much more. All the properties of zero--for instance that it is the neutral element in addition (1 + 0 = 1), or that any number elevated to zero potency equals one, or 0! = 1! = 1--are not inferable. One of the ways we support mathematical cognition is through external devices such as calculators and slide rulers. Any device outside of the human that assumes some of the cognitive load is an epistemic action in that its “primary goal is to obtain information about the world.” When coins are sorted by value and stacked in tens, for instance, it is much easier to count them than if they are in a disorganized pile. By designating part of the task to the environment in this way, performance improves. In comparison, a notebook serves as an external memory device for someone with short-term memory loss. The extended mind thesis presents the notion of active externalism wherein objects within the environment function as part of the mind. The mind and the environment are a coupled system that function together with the same purpose. According to De Cruz and De Smedt, “[a] way to interpret the extended mind without contributing cognition and agency to artifacts is to argue that not all concepts are mental representations. One must then suppose that not all concepts can be entertained by human minds, due to intrinsic limitations of human cognition.” Using the example of ultraviolet light, which humans cannot perceive due to our lack of receptor cells for ultraviolet light, De Cruz and De Smedt show that we can entertain the concept of ultraviolet light without actually having a mental representation of it because of the special instruments we’ve developed to capture it. Similarly, “mathematical symbols can represent objects that are not representable with our internal cognitive resources alone [and this implies] that not all mathematical concepts are mental representations.” Through their research De Cruz and De Smedt found that the use of symbols is cognitively demanding on students despite their familiarity with symbolic notation systems. They prefer to use lengthy descriptions that imagine particular situations to solve problems such as “how to obtain the number of girls in a class when the number of boys is known, and you know that boys outnumber girls by four.” They also discovered difficulty for students shifting from one symbolic representation to another. Students consistently failed to represent 1+7 in blocks; rather than laying them out in two separate groups one of one block and the other with seven blocks, they arrange the blocks such that they spell 1+7 by their configuration. Students require explicit instruction to use the symbolic objects successfully. To understand symbols requires that we “decouple meaning from materiality.” While this is a process that occurs early on in development—by the age of two most children can discriminate between representational objects (e.g. a photo of a bottle of milk) and the actual object—the parts of the brain occupied with this endeavor differs from task to task and culture to culture based on the way the symbolic systems were learned to begin with. Chinese-speaking students who learned math with an abacus rely more on motor-related brain areas when performing mathematical tasks than English natives whose language-related brain areas are more active during the same task. Likewise, there are differences in cognitive organization between literate and illiterate people. Brains of illiterate people show more white matter, the function of which is “to connect different functional areas of the nervous system [indicating] that literate people have less connections in their brain.” De Cruz and De Smedt explain that literate people can afford to forget since they can use daybooks, notes and the like to store memory. Mathematical symbols refer both to procedures (the value of pi is the ratio of a circle’s circumference to its diameter) and to objects (structurally π represents 3.14159265…). Studies indicate that the process of moving from procedure to mathematical object occurs in students as they learn new math skills. In a study of adolescents’ working with negative numbers, many mistakes were made due to “a studious application of rules, rather than an intrinsic understanding of negative numerosities; [thus] the best predictor of success in individual students was their ability to use the minus sign correctly, not their conceptual understanding. Adolescents aren’t the only ones struggling with negative numbers. As it happens, negative numbers stumped mathematicians for eons and were largely rejected until well into the 1800s. De Cruz suggests that negative numbers are non-intuitive because they lack evolutionary import; “there seems to be no compelling reason why natural selection should have equipped [us] to deal with negative quantities.” Thoughts such as these just wouldn’t endure without mathematical symbols to carry them, for the demand on our cognitive resources and attention is great in the course of the problem solving and cultural conditioning that occurs moment to moment in life. Could it be that students find math difficult for the simple reason that we weren't designed for it? Perhaps we didn't evolve to intuit higher mathematics equations, but when we look how far we've come along in our understanding of the universe through mathematics we cannot deny that something has evolved: perhaps it is our capacity for evoking mental representations from symbols. One thing is for certain, we didn't come to this kind of understanding overnight. Understanding mathematics requires skillful integration of a variety of innate and acquired numerical processes beginning at the simplest level. Therefore, the next step in understanding why students perceive math as difficult involves looking at the ways in which math is taught. References De Cruz, H. H., & De Smedt, J. (2013). Mathematical symbols as epistemic actions. Synthese, 190(1), 3-19.
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AuthorNatalie Nickerson; that's me. Archives
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