Combatting Math Anxiety: Abstractness of Mathematics

Abstract vs. Concrete

Wait, but doesn’t everyone learn numbers before even attending preschool? What do I mean by difficult?

 When children first encounter numbers, they do not learn them as abstract objects; the initial way of understanding numbers accompanies the idea of unit. Imagine kids learning to add for the first time: many of them do so by counting their fingers. Teachers may support their learning by providing explanations, “If you have one apple, and a friend gives you another apple, then you end up with two apples.” Such units of fingers and apples provide a concrete metaphor to describe an abstract operation “1+1 = 2”, through which students learn basic arithmetic between natural numbers. Here, notice students have yet to learn the full picture of what 2 means – they have only learned the metaphor which can only paint a partial picture (i.e., the duality of an identical object) of what the number can highlight.

Such metaphoric ways to learn mathematics can be powerful, as they project an abstract object into an object that one can directly perceive. As students progress in their journey of school math education, however, such concrete metaphors become less and less applicable to the concepts that they learn, failing to paint the full picture (In this article, I use concrete as the antonym of abstract, to mean a quality of an object that does have spatial or temporal location). One of the greatest barriers that students face in school mathematics, for example, is the idea of variables. Variables, represented through algebraic symbols, make mathematics even more abstract than numbers: not only do they have no spatial and temporal location, they do not even have a location on a number line. One may even say that variables have a higher level of abstractness than numbers! Two common ways to combat this abstractness in math classes are 1) to force students to memorize formulas and procedures to solve problems through rote practice or 2) by introducing variables as unknowns.

Though probably still common, most math educators would agree that (1) is not ideal. The reason for its prominence is because memorization is the easiest way to combat abstractness. It is clear what they need to do, and the complexity of abstractness of variables need not be addressed. However, this also means that students would not learn anything mathematical. Such a way of teaching may exacerbate students’ misbelief against math. Again, one reason why students find mathematics difficult is because there’s too much to memorize. Speaking as a mathematics major (and many mathematicians would agree), memorization is not a crucial part of mathematics. What a surprise!

I could go on about memorization, but the issue more relevant to the topic of abstractness of mathematics is (2), introducing variables as unknowns. An unknown, in contrast to a variable, does have a location on a number line: a location that is unknown. By defining variables as unknowns, the concept becomes more concrete and less abstract. However, that does not depict the full definition of variables. Usiskin (1988), while providing a framework for students’ conception of Algebra, categorizes this way of understanding variables (i.e., defining variables as unknowns) as viewing Algebra as “a study of procedures for solving certain kinds of problems” (p. 9). In other words, viewing variables as unknowns pertains only to procedural solving of certain problems, failing to highlight other important aspects of variables (e.g., values of a variable may vary!) outside of a limited context of school math problems. Knowing that a majority of students interpret letters in mathematics as specific unknowns (Kieran, 2007, p. 1), it is understandable why many students feel frustrated seeing letters in mathematics: it is only useful in a school setting, being completely irrelevant to their lives!

With the case of variables, we observed a conflict that occurs when only concrete metaphor is used to scaffold understanding of an abstract object. Since the level of abstractness only increases as students proceed in their journey of school mathematics education (e.g., functions, irrational numbers and their notations, calculus, sets, constructing definitions, proofs), I believe it is crucial for students to understand abstract objects as abstract, without relying on concrete metaphors.

But how could one understand abstract objects as abstract? I believe utilization/combination of different types of reasoning (e.g., visualization, symbolic manipulation, explaining with words) is necessary. The effectiveness of a type of reasoning varies depending on the individual. I, for instance, love using symbolic languages of mathematics to describe any abstract concept. Using everyday languages such as Korean or English has never really been helpful for me, and visualization often would not make sense because the visuals can be restrictive of what I see and imagine. However, I acknowledge that symbolic language is just one of the many ways of understanding mathematics. Visualizations and colloquial language may manifest novel insights that symbolic language may not immediately show. With various types of reasoning, different facets of the abstractness may be manifested, providing a step closer for students to understand mathematical objects as abstract.

Combatting Anxiety

For that matter, in my tutoring sessions, I endeavored to provide opportunities for students to explore different types of reasoning by spending most of the time solving various problems in depth. However, solving problems was a big stress for students. Although I would explain that I ask the question (“How did you get the answer?”) regardless of the correctness of their answers and that mistakes are more beneficial than getting correct answers, students with high anxiety would continuously avoid elaborating on their ideas by saying “I don’t know.” This observation may be explained by Interpretation Account that Ramirez et al. (2018) proposed, where they argue that students’ development of math anxiety is “largely determined by how they interpret previous math experiences and outcomes” (p. 151). No matter how many times I reassure students that I won’t be judging their mistakes, math anxiety might arise if they interpret their mistakes as something negative, and my demand for their reasoning would exacerbate their anxiety if they perceive the experience as embarrassing.

Therefore, I used problem-solving as a medium for endorsing abstract reasoning, not as a goal. I did so by 1) questioning what is already known to students, and 2) designing impromptu problems that align with student mistakes that relate to the original problem. For example, my former student, Kate, had a belief that the volume of any solid is length times width times height. She would utilize her own idea of length and width (e.g., two legs of a right triangle), and simply multiply the measures to find the volume. With her, I created a problem about the volume of a rectangular prism and had her compare the answer with the original problem (i.e., finding the volume of a triangular prism with the same height). Notice I was challenging what is already known to the student (i.e., volume = l*w*h), and I provided her a visual reference for her to make sense of the abstract idea of volume. Realizing that the triangular prism clearly had less volume and with some symbolic manipulation, she came to understand that the triangular prism had half of the volume as its rectangular counterpart. Explicitly stating that she was the one who came up with the answer, she seemed to have gained more confidence in explaining her answers even when it is wrong, looking much less anxious.

Brown and Walter (2004) discuss the benefits of doubting what is already known: “it is challenging the given that frequently opens up new vistas in the way we think” (p.17). In other words, just like the case of 1 as one apple, variables as unknowns, and volume as l*w*h, students may only possess a partial – metaphoric – understanding of abstract concepts of mathematics. In such situations, challenging students to face the gaps in their knowledge through personalized impromptu problems can elicit novel ways of thinking and understanding mathematical concepts. And when they do come up with their own ways of thinking and understanding, the experience would benefit how they perceive their own mathematical competence, encouraging them to face abstractness without relying on concrete metaphors.

Now I wonder: what if we teach students so that they can challenge their own knowledge themselves, exploiting their mistakes as a way to explore the world of mathematics, especially the world of abstractness? Would it allow students to come up with their own way of understanding abstract objects as abstract? Would it help them combat math anxiety? I believe so. With such an educational setting, I hope students could divorce from the anxiety of finding correct answers and focus on the aesthetic experience that the mathematical journey renders. Well, of course, to claim that this to be true, the field requires a lot more research on this topic.