September 6, 2022: My Post to Coursera's Constructivism in STEM Class (I pulled in my previous blog on mathematical notation for inspiration)
Standstills in Mathematics Education: Getting Past the Intimidation of Mathematical Notation
In this paper, I will discuss Two Conceptions of Mathematical Notation by Anna Sfard at the University of Jerusalem. The environment in which the use of operational versus structural understandings of mathematical notation were used was a secondary school environment. Specifically for this paper, this secondary school constituted 96 students, ages 14-17. They were asked to “translate four simple word problems into an equation”. They were also “asked to come up with verbal prescriptions (algorithms) for calculating the solutions of the same problem”, respectively. Finally, the environment is one where the notion of concept is just developing. Operational proceduralism, or the use of the program, is the earlier stage of development that is installed as an avoidance mechanism according to the hypothesis of Sfard as a substitute for the nonverbal, spatial crystallized concepts behind mathematics. Computer programs often supplant symbolic algebraic notation for the students, who have trouble understanding the full static picture of “operations of the named sort occurring in all pasts, presents, and futures” represented by the mathematical notation. Therefore, the only further characteristic is an avoidance of the next developmental stage, a stage where many students often get stuck (including myself).
The learning environment for the student must be avoidant for a reason. From my interaction with students in my extensive experience as a math educator, I know that avoidance in mathematics often comes from insufficient understanding. As these students are clearly still in the procedural explication/operational stage of understanding mathematics (they are still in the maze running it out), they have not yet themselves devised the full concept of the complete map of the system of many of the algebraic symbols. Therefore, though the symbols may be used, they are still trying to conceive of them verbally. We hear such understandings when math is compared to a language. However, we can see that math is explicitly stated to be nonverbal in the following; “Italians and Arabs in the 1600’s used ‘syncopated’ or verbal descriptions of functions up to that point, at which point a transition to symbolic algebra [which therefore is specifically and definitively nonverbal] began.” Slowly the transition from word to function, function and its product to operation, and operation to atemporal spatial representation of a “crystallized” function begins. What is often left behind in teaching, according to Sfard–being in particular an opinion with which I professionally concur–is the constructivity of the situation. Notations are presented erroneously in two ways, as 1) frozen, perfect facts and 2) as verbal statements. Both of these are in error. Notations are often still heated “under the hood”, where the name of the representation is still having its analytic “gerrymandered constituencies” voting self-elements in and out in the halls of mathematics according to their comprehensive powers when put into rigorous practice.
Additionally, verbal statements usually are these very analytical definitions that are verbal and therefore do not challenge the student to engage in the nonverbal, spatial reasoning that makes symbolic reasoning so difficult. Essentially, the issue here is students do not know there is a sort of “code switching” going on in learning types, and therefore think their verbal comprehensive mechanisms are inadequate, when instead teachers should be teaching nonverbal/spatial through visuals like the unit circle, physical puzzles such as Hanayama lock puzzles, and other methods that introduce this type of reasoning. Therefore, to improve the environments for conceptual, representative, and symbolic understanding, procedural/algorithmic understandings should slowly be transitioned out of support and replaced with symbolic/spatial/visual reasoning. This means creating environments that are visually and symbolically rich, emphasizing maps as third order representation, like concept maps, electoral maps, or even watching auctions or real estate price changes in real time. In addition, physical puzzles like Hanayama brand lock-and-key type puzzles are very much encouraged.
By emphasizing the development instead of the “immediate perfect comprehension” of mathematical symbols, such immediate comprehension of words can be expected of words but not of “named crystallized functions”. Once this is established as being the nature of a symbol in a developmentally patient and educated manner, students can begin to form the correct understanding and mode of approaching mathematical notation. They will stop being so frightened, scared, and without tools when they know their verbal reasoning is not insufficient, but simply the wrong mental position for the task. Specifically, classrooms must be designed to acknowledge the verbal shared ground, incubate from within the operational dance, and fly up with a freed a novel conception into the symbolic view constitutive of the mathematical heights.
Sfard, Anna. (1987). Two conceptions of mathematical notions: Operational and structural. Proceedings of the Eleventh International Conference for the Psychology of Mathematics Education. 3.
