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Process-Driven Math

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Process-Driven Math denied access to many career paths, especially those in science, technology, engineering, and Practice Perspective mathematics. Process-Driven Math is a fully audio Process-Driven Math: method of mathematics instruction and as­ An Auditory Method sessment that was created at Auburn Univer­ of Mathematics Instruction sity at Montgomery, Alabama, to meet the and Assessment for Students needs of one particular student, Logan. He Who Are Blind or Have Low Vision was blind, mobility impaired, and he could Ann P. Gulley, Luke A. Smith, not speak above a whisper. Logan was not Jordan A. Price, Logan C. Prickett, able to use traditional low vision tools like and Matthew F. Ragland braille and Nemeth code because he lacked the sensitivity in his fingers required to read Mathematics can be a formidable roadblock the raised dots. Logan’s need for tools that for students who are visually impaired (that would enable him to perform the rigorous is, those who are blind or have low vision). algebraic manipulations that are common in Grade equivalency comparisons between stu­ mathematics courses led to the development dents who are visually impaired and those of Process-Driven Math. With this method, he who are sighted are staggering. According to was able to succeed in both college algebra the U.S. Department of Education, 75% of all visually impaired students are more than a and pre-calculus with trigonometry. These full grade level behind their sighted peers in tools may help other visually impaired stu­ mathematics, and 20% of them are five or dents who are not succeeding in mathematics more grade levels behind their sighted peers because they lack access to, or knowledge of, in the subject (Blackorby, Chorost, Garza, & the Nemeth code. Guzman, 2003). Overall low achievement in Within many algebraic problems, there mathematics is directly related to the crisis in are processes (algorithms) that must be per­ braille literacy that experts in the field have formed to simplify expressions and solve noted with alarm over the last 25 years for variables. Process-Driven Math frees up (Amato, 2002; Mullen, 1990; National Fed­ working memory during the delivery of the eration of the Blind [NFB], 2009; Papadopou­ mathematic content to the student, and during los & Koutsoklenis, 2009; Schroeder, 1989). the student’s subsequent manipulation of the Over 80% of students who are visually im­ equation while working toward a solution. As paired are educated in mainstream, state-run a result, the student who cannot use or access schools (American Printing House for the Nemeth code can better focus on the required Blind [APH], 2014). These students are not algorithms because the cognitive load on receiving adequate instruction in the Nemeth working memory is greatly reduced. When Braille Code for Mathematics and Science using this method, a person who functions as Notation (hereafter, Nemeth code) and are not the student’s reader and scribe systematically given adequate access to Nemeth code mate­ reveals the algebraic expression in layers. The rials (Amato, 2002; Kapperman & Sticken, process is highly interactive and places con­ 2003; Rosenblum & Amato, 2004). As a re­ trol over the flow of information into the sult, visually impaired students often cannot hands of the student. During the simplifica­ persist in post-secondary education and are tion processes, the student himself systemat­ ically transforms the equation in discrete, We thank Steve Noble for sharing his knowledge of controlled steps that are carefully recorded so synthetic voicing and speech rules for mathematics. that every decision made is available to the ©2017 AFB, All Rights Reserved Journal of Visual Impairment & Blindness, September-October 2017 465 student for review. Take, for example, the logical and consistent with the landscape pre­ following algebraic expression: sented. The student can often make decisions that advance the simplification process with­ 4x2y + 4xy 2xy + 8y out having to hear the actual numbers and · x2 - x - 2 x2 + 2x - 8 symbols in a particular part of the expression. For example, the second rational in the ex­ From the outset, the visually impaired stu­ pression above must be inverted and the di­ dent who either cannot use or access Nemeth vision sign changed to a multiplication sign. code is at a severe disadvantage compared to The student says, “Take the inverse of the her sighted peers. The sighted student can second rational and change the operator that quickly assess the landscape of the expression preceded it from division to multiplication.” and develop a plan of action based on its The student is able to focus on the mathemat­ structure. If a student who is visually im­ ical process without having to hear the vol­ paired is given an audio rendering of the same ume of syntax in the second rational. After the expression in which each number and symbol student directs the change, the reader-scribe are read outright, he will have to recall all the then tells the student that the expression is details from working memory, analyze the “rational times rational” and rewrites the ex­ relationships of the component parts, and then pression to reflect that change: synthesize the landscape of the expression. 2 2 The cognitive load created by this type of 4x y + 4xy x + 2x - 8 X audio rendering will overload working memory x2 - x - 2 2xy + 8y (Cowan, 2010; Miller, 1956). When working memory is overloaded, that particular content At some point in the process, the student within the subject becomes inaccessible to the will ask to hear the numerator of the second student. rational. Instead of saying “x2 + 2x - 8,” The Process-Driven Math method begins the reader-scribe says “term + term – con­ with an audio rendering of the algebraic ex­ stant.” The use of proper vocabulary rein­ pression that significantly reduces the spoken forces the student’s understanding of the ex­ math syntax (that is, numbers, variables, sym­ pression. Terms are defined as elements that bols, and operators) found in the expression. are being added or subtracted. The word con­ It delivers the overall landscape of the expres­ stant improves the student’s understanding of sion without overwhelming the student’s this particular term because the student knows working memory. Numbers and symbols are that a constant is a number that stands on its temporarily hidden behind layers of appropri­ own without a variable. After hearing “term ate mathematic vocabulary. For example, in + term – constant” the student will ask to the initial Process-Driven Math audio ren­ hear what is inside each of the terms, in­ dering of the expression above, the student cluding the constant. The student will then will hear “rational divided by rational.” have the information needed to identify the (We chose the vocabulary term “rational” numerator of the second rational as a qua­ over the more common term “fraction” be­ dratic that should be factored. cause it builds a strong foundation for the With precision of language, the student will mathematic vocabulary that is often used in direct the reader-scribe to establish two sets subsequent college mathematics courses.) of parentheses, each with space for two terms, The student who hears “rational divided by and to place an operator of a plus sign in one rational” is now in control, ready to explore parenthesis and an operator of a minus sign in the pieces of the problem in a manner that is the other. 466 Journal of Visual Impairment & Blindness, September-October 2017 ©2017 AFB, All Rights Reserved ( + )( - ) Reader-scribe: “Yes, I see a factor of the quantity x + 1 in the numerator and the The student directs the scribe to put an x in the denominator.” first slot of each parenthesis, a 4 in the second slot of the first parenthesis, and a 2 in the 4xy( x + 1)(x + 4)(x - 2) second slot of the second parenthesis. ( x - 2)(x + 1)2y( x + 4) ( x + 4)(x - 2) Student: “Cancel them. Are there any other matching factors found in both the numer­ Without ambiguity, and without overloading ator and the denominator?” working memory, the quadratic is simplified and the student moves on to work in another Reader-scribe: “Yes, I see a factor of the part of the problem. quantity x + 4 in the numerator and the The student continues directing the reader- denominator.” scribe to read each section of the expression and to record the changes the student makes 4xy( x + 4)(x - 2) in each of those areas. After the student ( x - 2)2y( x + 4) simplifies both of the numerators and both of the denominators properly, the expres­ Student: “Cancel them. Are there any other sion becomes: matching factors found in both the numer­ ator and the denominator?” 4xy( x + 1)( x + 4)(x - 2) X ( - )( + ) ( + ) x 2 x 1 2y x 4 Reader-scribe: “Yes, I see a factor of the quantity x – 2 in the numerator and the The reader-scribe tells the student that the denominator.” expression is “rational times rational.” The student directs the reader-scribe to 4xy( x - 2) multiply across the numerators and multi­ ( - ) ply across the denominators to create one x 2 2y rational. The reader-scribe follows the stu­ dent’s instructions and makes the following Student: “Cancel them. Are there any other transformation: matching factors found in both the numer­ ator and the denominator?” 4xy( x + 1)(x + 4)(x - 2) ( x - 2)(x + 1)2y( x + 4) Reader-scribe: “Yes. I see a y in the numerator and the denominator.” The reader-scribe tells the student that there is one rational. The student is then close to fin­ 4xy ishing the simplification of this expression 2y and asks, “Are there any matching factors found both in the numerator and the denom­ Student: “Cancel them.
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