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. Are there any other inator of this rational?” The reader-scribe re matching factors found in both the numer sponds in the affirmative, and the conversa ator and the denominator?” tion for the remainder of the simplification sounds something like this: Reader-scribe: “No.”

©2017 AFB, All Rights Reserved Journal of Visual Impairment & Blindness, September-October 2017 467 4x The interactive, bidirectional communica 2 tion also allows the student to render mathe matical formulas without ambiguity. We will Student: “Read what is there.” take, for example, the rendering of the qua dratic formula. In written form, the quadratic Reader-scribe: “A rational with a numera formula is: tor of 4x and a denominator of 2.” -b ± b2 - 4ac x = Student: “Reduce the 4 and the 2. Make 2a the 2 in the denominator intoa1and make the 4 in the numerator into a 2. The student demonstrates proﬁciency with the Read back what is left.” formula by making the following statements to a reader-scribe: Reader-scribe: “A rational with a 2x in the numerator anda1inthe Student: “The quadratic formula is x denominator.” equals a rational.”

2x numerator x = 1 denominator

Student: “My answer is 2x.” Student: “The numerator of the rational is term plus or minus term.” The role of the reader-scribe must be filled by an individual who has mastery over the term ± term x = course content, and training is critically im denominator portant. A reader-scribe who mistakenly says + “the square root of x 1 divided by 2” will Student: “The first term is negative b, put the student at a disadvantage because that and the second term is a square root.” statement can be interpreted five different ways: -b ± x = denominator 1 x + 1 x + 1 x + or or 2 2 2 Student: “The inside of the square root is term minus term.” 1 x + 1 or x + or 2 2 -b ± term - term x = The Process-Driven Math reader-scribe is denominator trained to hide mathematic syntax by break ing the expression into pieces, a method that Student: “The first term in the square ensures a consistent and unambiguous render root is b squared, and the second term in ing of the equation. Likewise, the student the square root is 4ac.” must also be trained to give an accurate audio rendering of the mathematic manipulations -b ± b2 - 4ac x = being executed. denominator

468 Journal of Visual Impairment & Blindness, September-October 2017 ©2017 AFB, All Rights Reserved Student: “The denominator of the ratio ing. Braille and the Nemeth code will con nal is the factor 2a.” tinue to be important and necessary tools for individuals who are visually impaired, but -b ± b2 - 4ac additional tools are needed for people who x = 2a have tactile limitations and for others who have not been taught to read braille or Nemeth code. Process-Driven Math was developed to Without using Process-Driven Math, a person help these students so they, too, would have speaking the quadratic formula might say “x the opportunity to succeed in mathematics equals negative b plus or minus the square and persist in college. root of b squared minus 4ac divided by 2a,” Auburn University at Montgomery is now a statement that could be transcribed in many employing Logan to train tutors to become different ways. The student using the Process- reader-scribes for Process-Driven Math. Driven Math approach to verbally render Members of the Logan Project have since mathematical formulas can be graded using worked with a second student who was se the same standard for accuracy that is used for verely visually impaired. This student had sighted students. Thus, that student is com never been taught braille or Nemeth code, but peting on a more equal footing with sighted instead used enlarged text, screen readers, peers. speech to text, and typing to access her cur Logan successfully solved well over one ricula. These tools had been sufficient for all thousand mathematics problem using Process- of her courses except mathematics. She strug Driven Math. He fully drove all of the intel gled to properly discern exponents, even lectual processes and proved his competency when materials were enlarged. More signifi in the mathematics classes required by the cantly, she did not find that typing adequately university. Logan always had the ability, but supported her efforts to demonstrate a series what he lacked, and what threatened his per of complex transformations when simplifying sistence in college, was access to appropriate algebra expressions. She could write by hand, tools. Logan’s role has been far more than but the process caused significant fatigue as that of a student using a method created for she spent long periods of time bent over the him. He is one of the developers of the desk to keep her eyes extremely close to the Process-Driven Math method. His insights paper while writing. She was given the op into the experiences of a person doing math portunity to use any of the tools available to ematics solely by listening and speaking pro her, including Process-Driven Math, through vided crucial guidance in the creation of the out her college algebra course. She consis method. His success led to the establishment tently chose to use the Process-Driven Math of a university-funded research program approach because it significantly reduced the called the Logan Project. Researchers sought fatigue she experienced when writing by and received IRB approval from Auburn Uni hand. She successfully completed her college versity at Montgomery for the preliminary algebra course and is continuing to move for collection of data, and participants signed in ward in her education. formed consent forms for their participation One limitation of the Process-Driven Math in research activities. method is the subjectivity that exists between Braille is an essential literacy tool for peo the student and the reader-scribe during the ple who are visually impaired, and its use is communication of the mathematics. In addi directly related to educational attainment, em tion, Process-Driven Math is a highly labor- ployment opportunities, and independent liv intensive process in which the student is

©2017 AFB, All Rights Reserved Journal of Visual Impairment & Blindness, September-October 2017 469 dependent on the interaction with a trained disabilities. (NLTS2 Report). Washington, reader-scribe to work through lengthy prob DC: U.S. Department of Education. Re lems. In response to these limitations, mem trieved from http://nlts2.sri.com/reports/ bers of the Logan Project are seeking to de 2003_11/nlts2_report_2003_11_ch4.pdf velop Process-Driven Math into a software Cowan, N. (2010). The magical mystery four: How is working memory capacity limited, application. Software would eliminate the and why? Current Directions in Psycholog person who is in the role of the reader-scribe ical Science, 19(1), 51–57. and improve the autonomy of students using Frankel, L., Brownstein, B., Soiffer, N., & the method. Currently, the role of the reader Hansen, E. (2016). Development and initial can be met with software tools that produce evaluation of the ClearSpeak style for au unambiguous synthetic mathematic speech. tomated speaking of algebra. ETS Research One option is the use of software that imple Report Series, 2016(2), 1– 43. ments MathSpeak, a set of speech rules cre gh. (2004 –2006). MathSpeak core speciﬁca ated by Abraham Nemeth that map directly to tion grammar rules. Retrieved from http:// Nemeth code symbols (gh, 2004–2006; Nem www.gh-mathspeak.com/examples/grammar eth, 2013). Another option is software that rules integrates ClearSpeak, a set of speech rules Kapperman, G., & Sticken, J. (2003). A case for increased training in the Nemeth code and preferences that allows synthetic speech of braille mathematics for teachers of stu to closely approximate the way mathematics dents who are visually impaired. Journal are spoken in the classroom (Frankel, Brown- of Visual Impairment & Blindness, 97, stein, Soiffer, & Hansen, 2016). A Process- 110 –112. Driven Math software tool would combine Miller, G. A. (1956). The magical number existing synthetic mathematic speech capabil seven, plus or minus two: Some limits on ities with a component to automate the role of our capacity for processing information. the scribe. Thus, students could indepen Psychological Review, 63, 81–97. dently make real-time transformations to the Mullen, E. A. (1990). Decreased braille literacy: mathematics in discrete, user-controlled A symptom of a system in need of reassess steps. The goal of the project is to help many ment. RE:view, 22(3), 164–169. National Federation of the Blind Jernigan In more people who might be facing hurdles stitute. (2009). The braille literacy crisis in similar to those that Logan and other students America: Facing the truth, reversing the have faced so that they can be successful in trend, empowering the blind. Retrieved from college mathematics. https://nfb.org/images/nfb/documents/pdf/ braille_literacy_report_web.pdf REFERENCES Nemeth, A. (2013). MathSpeak (a talk on Amato, S. (2002). Standards for competence verbalizing math by Dr. Abraham Nemeth, in braille literacy skills in teacher prepara creator of the Nemeth math braille code). tion programs. Journal of Visual Impair Retrieved from http://accessinghigherground. ment & Blindness, 96(3), 143–153. org/handouts2013/HTCTU%20Alt% American Printing House for the Blind. 20Format%20Manuals/Math%20 (2014). Annual report 2014: Distribution Accommodations/07%20MATHSPEAK. of eligible students based on the federal pdf quota census of January 7, 2013 (ﬁscal Papadopoulos, K., & Koutsoklenis, A. (2009). year 2014). Retrieved from http://www. Reading media used by higher-education aph.org/federal-quota/distribution-2014 students and graduates with visual im Blackorby, J., Chorost, M., Garza, N., & Guz pairments in Greece. Journal of Visual man, A. (2003). The academic perfor Impairment & Blindness, 103(11), 772– mance of secondary school students with 777.

470 Journal of Visual Impairment & Blindness, September-October 2017 ©2017 AFB, All Rights Reserved Rosenblum, L. P., & Amato, S. (2004). Prepa graphemes for each character of the English ration in and use of the Nemeth braille code alphabet, which consumes a lot of space, re for mathematics by teachers of students with sulting in books that are very long. Contracted visual impairments. Journal of Visual Im braille, on the other hand, uses 189 contrac pairment & Blindness, 98(8), 484–498. tions, many of which represent whole words Schroeder, F. (1989). Literacy: The key to with one grapheme (the, can, and people, for opportunity. Journal of Visual Impairment example), which has led to shorter publications. & Blindness, 83(6), 290 –293. The creation and utilization of contracted braille has had an effect on the education of visually Ann P. Gulley, B.S., math and sciences student impaired students. As a result, there has been services coordinator, doctoral student, Learning Center, Auburn University at Montgomery, P.O. extensive research in this area. Box 244023, Montgomery, AL 36124; e-mail: Lee and Hock (2014) investigated the ef [email protected]. Luke A. Smith, Ph.D., assistant fect of using contracted braille on the spelling professor, College of Education, Auburn University proﬁciency of visually impaired students in at Montgomery, 7400 East Drive, Montgomery, AL 36117; e-mail: [email protected]. Jordan A. Price, a bilingual setting. Through doing quasi- B.S., graduate student, Samuel Ginn College of En experimental research, they concluded that gineering, Auburn University, 1301 Shelby Center, the most frequent errors were grapheme sub Auburn, AL 36849; e-mail: [email protected]. Lo stitution and direct translation of syllables of gan C. Prickett, undergraduate student, Learning Center, Auburn University at Montgomery; e-mail: the ﬁrst language. Troughton (1992) stated [email protected]. Matthew F. Ragland, Ph.D., as that the students who learn contracted braille sociate provost and professor, Auburn University at later in school show a better performance in Montgomery; e-mail: [email protected]. reading skills than those who learned contrac tions earlier. Wall Emerson, Holbrook, and D’Andrea (2009), on the other hand, con Practice Report cluded that if visually impaired students learn contractions at an earlier time, they will show better performance in their reading skills. A Comparative Analysis of In the educational system of Iran, in which Contracted Versus Alphabetical English is a foreign language, the duration of English Braille and Attitudes of primary school is six years. After that, these English as a Foreign Language English as a foreign language students enter Learners: A Case Study of a high school and start to learn English during Farsi-Speaking Visually an additional six-year period. In years seven Impaired Student and eight, English textbooks are embossed in Mohsen Mobaraki, Saber Atash Nazarloo, alphabetic English braille; but, for the next and Elaheh Toosheh four years, these books are prepared in the Education is an absolute right of all human contracted form. Visually impaired students beings, especially in today’s world, and there in Iran use alphabetic braille to read and write is a huge responsibility on those teachers who in their own mother tongue (Farsi). Con work with learners with disabilities. A very tracted braille is, therefore, a foreign concept special kind of education is needed for people to these students. who are visually impaired, and braille is one In Iran, the exposure of students who are method that is utilized by individuals to ac learning English to contracted braille in high cess educational materials. school has affected the reading skills of these There are two kinds of braille, alphabetic students. The present researchers interviewed and contracted. In alphabetic braille, there are four teachers regarding the teaching of