LONG DIVISION, MAPPINGS, AND METHODS: A COMPREHENSIVE RESOURCE OF ELEMENTARY DIVISION ALGORITHMS FOR MATHEMATICAL EDUCATORS By Jason J. Passey A project submitted in partial fulfillment of the requirements for the degree of MASTER OF EDUCATION IN CURRICULUM AND INSTRUCTION WEBER STATE UNIVERSITY Ogden, Utah December 16, 2021 Approved ______________________________ Sheryl J. Rushton, Ph.D. ______________________________ Louise Richards Moulding, Ph.D. ______________________________ Katarina Pantic, Ph.D.2 Acknowledgments I would like to thank Weber State for reinstating my enrollment and for Dr. Louise Moulding and her mediation efforts. For my wife Tiffany and her belief in my abilities and encouragement and for making the necessary communications and innumerable sacrifices that got my MEd off the ground. For Dr. Sheryl Rushton’s untiring patience during editing and the cordial manner in which she conducted our conversations. And for Dr. Katarina Pantic and her technical support and about this, and that, which it should be. I am also grateful for the encouraging words of Tom Erekson and friends and students who urged me onward in this great endeavor. All of whom were answers to my prayers. May division never be the same. 3 Table of Contents NATURE OF THE PROBLEM……………………………………………….….………5 Literature Review ………………………………………………………………...7 People Like Options……………………………….……………………...8 The Written Notation……………………………...………….………….11 The Language of Mathematics……………………………….………….14 History of Division Algorithms and Their Notations …………………...20 Comprehensive Resource……………………………………….….…….29 Summary…………………………………………………………………...…...32 PURPOSE…………………………………………………………………………….….34 METHOD…………………………………………………………………………….….36 DISCUSSION……….…………………………………………………………….….….40 Conclusion…………………………………………………………………..….43 REFERENCES……………………………………………………………………….….45 APPENDICES Appendix A: Division Algorithms –Methods and Written notations…..…....…51 Appendix B: Comedic Division Links…….………………………….……..…55 4 List of Figures Figure 1. Gerbert’s Divisio Ferrea or Iron Division……………………….…….….…..22 Figure 2. Method of Maximus Planudes…………………………………….…………..24 Figure 3. Addition & Subtraction Method scale…………………………….…………..31 5 NATURE OF THE PROBLEM People like protocol options, from grandma recipes, and bookstore self-help books, to DIY (do-it-yourself) YouTubers, humans hunt for and enjoy knowing options of how to do things. Many how-to guides are practical principles, like ways to clean crayon marks off walls, while others showcase historical and cumbersome practices compared to modern equivalents, which are often more formal; consider blacksmithing demonstrations to make a nail. Further, formal protocols share the harnessing of mathematical principles to bring about consistent results, such as a proprietary chicken recipe to lifesaving heart surgery. Using mathematics combined with this drive for new options instigated a reliable way to invent rather than relying solely on the process of chance. It is recognized as an integration of mathematics and science. It is taught as mathematical procedures such as addition, subtraction, multiplication, and not the least of all four topics, division. These operations are introduced at the foundational level of education which is followed by higher-level mathematics and science, and consequently this technological world (Klein & Milgram, 2003). Society has directed many to mathematics and science coursework, to maintain and advance this modern world. Those directed are incentivized further with the knowledge that such pursuits likely lead to computational skills, higher-paying STEM careers, and more lifestyle options (Alexander, Graham, & Harris, 1998). However, many students struggle with division protocol (Watson, 2012). From confusing language, competing notation symbols, and complementary methods used for scaffolding. This critical division-learning struggle correlates to when the supermajority of children lose interest to pursue STEM fields, despite efforts nationally to reverse such 6 feelings (Kim, 2018; Potvin & Hasni, 2014). Further, mathematical education in the United States still lags well behind the lead nations, ranking 38th of 71 countries (Desilver, 2017). Many teachers do not know any division-algorithm(s) (DA) beyond the basic algorithm(s) they learned in their youth; despite society's awareness of many of the issues involving division or the potential rewards of teachers knowing more than one algorithm holds for their pupils (Klein & Milgram, 2003). The above DA refers to all possible combinations of the two aspects of division which are methods and written notations. The former is the way the division problem is thought of and calculated while the latter is the arrangement once recorded. For some, what they learned about division methods and written notations, was not the assured optimal DA. Rather, in each region of the world, before science methods informed nearly every aspect of their lives, an influential person may have used certain written notations and methods that gained favor at the time and remain in use in that region today (Smith, 1923; Smith, 1925). Studies concerning division options, originating from these regions and comparison rankings for strengths and weaknesses of the various options, appear to be lacking. Further, understanding which methods and written notations are the most accurate, efficient, teachable, and why, is unclear. Recent efforts have increased knowledge of division methods to aid the way the written notation of the traditional standard-algorithm (SA) is scaffolded. The SA refers to the following written notation symbol ( ⟌ ) which places quotients over the dividend with the divisor to the left used in conjunction with the method found below the dividend of multiply-subtract-bring down-divide. Explicit knowledge of lead nations’ methods and written notations might 7 also be beneficial to know among other iterations in existence. Teachers equipped with such knowledge would be able to introduce students to other options and verify a variety of methods and written notations. Better access to a comprehensive resource of DA could help expand teachers’ repertoire to help address the needs of struggling students. More readily available DA would also help administrators and other stakeholders better accommodate those who struggle. It may be the SA with its myriad of scaffolding methods to uphold its written notation is not the most comprehendible and mathematically effective for many people. And having DA outside those usually taught may allow for better understanding between individuals (Orey, 2001). If teachers are going to engage students in problem-solving using a variety of DA, then there is a need for a resource that provides a comprehensive collection of DA with explanations modeling how to use them. This resource would need to be accessible in its format, such as a website with links to video demonstrations for each DA or other information such as its background if known; ordered in a way that would anticipate the more useful DA first or in convenient groupings. For comprehensive purposes and more rigorous research, regardless of popularity and educational efficacy, all known DA should be included. Furthermore, a teacher should be able to validate how a student achieved a certain result with such a resource. This assessment in turn may help student-teacher relations as communication increases and bridges the critical disconnect which too often takes place at this age. Literature Review 8 This literature review will highlight the many nuances regarding the mathematical operation known as division. Also, how tradition and the perception that assumes such topics are long-settled and elementary yet, have caused division’s nuances to be overlooked. Then, it will inspect the knowledge gap of students who struggle with division and the language used to communicate division; to give insight into possible significant solutions to prevent this gap. Next, the review will take a look at the historical context to examine the division knowledge gap which hinders teachers from assisting students better at this critical age, when they make long-lasting decisions. Finally, it will explore the need for a resource, available to teachers, with comprehensive information regarding division options. People Like Options Considering all the options for division, they can fittingly be divided into two aspects: written notations and methods. Written notations are concerned with the symbol employed and the arrangement or mapping of the quotient, divisor, and dividend. While methods refer to the thinking behind a solution or the idea of “show your work”. Together they are DA. For example, the twelfth-century al-Hassar suggested one written notation option, now preferred by some for performing division —the fractional notation common in the USA. While al-Hassar’s name is not well known, he likely influenced Leonardo of Pisa, aka Fibonacci. Fibonacci is known for introducing the Hindu-Arabic numeral system to Europe in his 1202 Liber Abbaci (Hoover, 1972; Schwartz, 2005). Cajori (1928) referred to another prominent mathematician, Gottfried Wilhelm Leibniz, who is the same man who has been credited as the creator of the most mathematical written notations still in use today and who at one point experimented with 9 three different written notations for division. The first mentioned was a C symbol that opened upwards no other description was included, the second was the colon ( : ) symbol as division which he is said to have introduced, the third was the bar (─) symbol as in no other demonstration was shown. He further quoted Leibniz from a letter to Johann Bernoulli to gain his support, “As regards signs, I see it clearly that it is to the interest of…students, that [society] should reach agreement on signs” (Cajori, 1925, p. 415). To aid student efficiency. Beyond Leibniz’s efforts to consider multiple written notation options, efforts since appear to have revolved around methods. Ralston (2003) addressed an ongoing cyclic debate about using calculators in place of the SA. Despite digitization, the arguments read like a study John (1930) conducted using fifth graders on methods for long and short division. In John’s study short was considered faster and long more accurate. The recommendation was that the more fluent short division be withheld until after the more accurate long division was understood. The SA has arguments for and against. Star (2005) conducted a study of students’ use of the SA. He found that students use what they are comfortable with, if given that flexibility, yet were not given other options to choose from. When encouraged to invent their own, higher use of an invented method was also witnessed. While the study was unclear what percentage of students discovered or came to understand the SA, those who did used it almost exclusively. The post-test scores of SA users were some 30% higher than those who did not, and students who used the SA were likely to finish more problems. So, in this study accuracy and fluency were found to be higher among those students who adopted the SA. 10 Similarly, Realistic Mathematics Education (RME) and Everyday Mathematics (EM) have been significant forces in the educational world and sought to shift the notion that arithmetic’s priority must alter from progressive complexity to progressive schematization through the use of alternative methods. They both distill to the same thing —modification of the interaction with the written notation in common use. Specifically, RME translates into the day-to-day experience in this way; the mathematics problems given to students are static, meaning they have similar difficulty from the beginning of the year to the end of the year. They are often the same problem or very comparable while the method used to work with it refines over time. Thus, at the beginning of the year they are presented with a problem, and all year long they work with that same type of problem using a variety of methods to solve it from simple concrete manipulations to algorithmic. Differentiation was touted as a major characteristic of RME, as well as the opportunity for the creation of shortcuts and tricks (Anghileri, Beishuizen. & Van Putten, 2002; Randolph & Sherman, 2001; Treffers, 1987; Van den Heuvel-Panhuizen, 2001; Zulkardi, 1999) According to Ralston (2003) in the USA, EM has been misunderstood by many educators as the idea that algorithm strategies are not needed just the use of calculators and number sense. Mathematicians wanted the program's intent to be clear that EM uses many methods and guides the learner to what is called a focus-algorithm which is the SA particularly the written notation (Isaac, 2001; McConnell et al., 1993; Ralston, 2004; Treffers, 1987). 11 To clarify, realistic in RME refers to imaginable or attainable for a child rather than real-world as it is often confused. Additionally, Van den Heuvel-Panhuizen (2001) said some RME characteristics are the ideas of human interaction or social activity. The long-term benchmark framework and mental arithmetic goals are considered the backbone of harder mathematics. When this idea comes to drill and practice, they are not ostracized but rather seen differently, as more student-driven. Van den Heuvel-Panhuizen continued by evaluating various studies, international tests, and concluded that those tests do not test the true strength of RME or are inconclusive. Alluding to the fact that the Dutch at the time were the leaders of Trends in International Mathematics and Science Study (TIMSS) for the western nations. The only other significant distinction mentioned was related to methods which indicated similarities concerning problem sets, i.e., the use of static problem sets which students work with for long periods. For example, 81 people at six people per table, how many tables are needed? Modified slightly to 81 people at seven cups per pot, how many pots are needed? (Van den Heuvel-Panhuizen, 2001, pp. 14-16). While RME and EM sought to improve options of how to teach the SA they have not addressed possible written notation concerns. The Written Notation The previous section made known the two domains, methods and written notations for DA, which could be studied and summarized some historical efforts concerning methods. Despite efforts involving methods, many feel insufficient progress has been made (Alexander et al., 1998; Desilver, 2017). This perception of progress is also evidenced by the overwhelming amount of literature seeking to remedy the division struggles, which does not include related entrepreneurial “Mathnasiums” or bulletin 12 board tutor advertisements. This section highlights the concerns of division performance and recent attempts to address the performance in studies, or lack thereof, surrounding the other domain of written notations. And ultimately to bring into focus the big picture that many students struggle with division. This lack of progress is not unexpected when considering that common mathematical reform efforts, that focus on parsing methods, are not new, but instead are repackaged from an earlier time. Expecting a different outcome seems unreasonable. Ralston (2003) expressed this struggle as a simple calculator vs. skill debate. Where the calculator aids utilization of a breadth of mathematics concepts much quicker. But the user has limited knowledge of how the calculator actually does the computing. While the skillful user may not perform as many concepts –they can do so, albeit slower, without the calculator. which is an example of depth in the long-standing breadth vs. depth argument. But is more accurately thought of as curriculum efficiency vs. concept mastery because it may be more helpful when considering how to teach mathematics to see the struggle as to how fast students can perform a set of operations vs. knowing the relation of operation a to operation b or c. Whitehead in 1916 articulated the arguments of many historical figures concerning the above dichotomies saying that “specialization is needed to advance, contribute, and be successful in today’s world and to keep the attention of pupils” (1929). All this seems to mean the struggle depends on the subject and the individual. In the end, if individuals struggle with division and consequently do not feel confident about mathematics, current societal concerns, that not enough people pursue its disciplines namely mathematics and science, will be ignored. 13 Due to misconceptions about the long division tradition, many mathematical texts and most in this literature review intended to address the problems of struggling students, rarely recognize one issue regarding the written notation; especially in connection with the method used. Their authors and instructors chose instead to present tricks and gimmicks, so the student will use the SA; and often in such cases, as previously mentioned, this model is referred to as the focus-algorithm or another name with similar connotations (aka attainment-target). To ensure the use of this attainment-target or standard procedure, method accommodations are made to the student in numerous ways, often only giving verbiage that other written notations and DA exist, rather than actually demonstrating any visual (Isaac, 2001; McConnell et al., 1993; Van den Heuvel-Panhuizen, 2001). One exception from Bley and Thornton (2001) recommended in a note for students with severe reversal tendencies [switching divisor and dividend], was to use the division sign in reverse like this: ¯¯¯(. This suggestion assumes that the SA was universally accepted all over the world, has no drawbacks, and is “better” except in severe reversal cases. Fortunately, they did recognize the written notation problem this section seeks to highlight and to assuage readers they gave three points why, saying “the [method] is the same, but… the [written notation] is more consistent with the needed language, eye tracking, and perceptual organization.” (p. 264). Mullis, Martin and Foy (2008) reported on a collaboration that existed around the world, to study mathematics outcomes at various ages which may give insight into which DA and particularly written notations result in higher mathematical achievement. The International Association for the Evaluation of Educational Achievement (IEA) 14 headquartered in Amsterdam and directed through the International Study Center at Boston College administered the 2007 TIMSS. From a press release, the results of the 425,000 students overall and at least 4500 from each country showed a shift somewhat from the previous 2003 tests. The Netherlands previously led the western nations in several categories, but the results of this round of international tests showed that the Netherlands only topped the USA in grade 4 mathematics. Two Asian countries repeatedly ranked in the top three of the four years assessed: Singapore and Taiwan. Comparatively, the USA ranked tenth once, seventh once, and ninth twice. Other factors noted were reported by students or connected with high student achievement via their teacher and principal taken from data collected in connection with the test of participating countries such as positive outlook, feeling of preparedness, relevance for the future, satisfaction, classroom expectations, parental expectations, and properly equipped facilities (Martin & Mullis, 2008). Unfortunately, no mention was found or time given to detail different nations’ written notations and thereby possible effects. This information is important in order to better address written notation issues. Additionally, if the USA ranks number one that leaves little room for improvement. If that is not the case then significant improvement is still feasible. The Language of Mathematics The efforts of Anghileri (2004) highlighted through a study of the USA, England, and the Netherlands revealed that even though USA students scored higher on tests of accuracy, their understanding of division is brought into question. The USA placed success on the ability to formally calculate not on the ability to problem-solve. On that 15 note, the Dutch are said to be the winners, due to their flexibility and balance of accuracy and number sense. Devlin (2000) introduced his work saying that language and mathematics are enabled by the same features in the brain i.e., to communicate and to understand. Extrapolating on Devlin’s idea, if it is a desire to learn or teach mathematics, then reasoning that mathematics is like other spoken languages suggests a reasonable manner how we might learn mathematics. These foregoing ideas coupled with Maslow (1943), and synthesized, suggests to learn English, the following chronology occurs and may be used to help learn mathematics the easiest way. The first is assuring the comfort and trust or the basic needs of the learner are met. Such as hunger or a safe and secure learning environment. This environment helps set the stage for the second step or an atmosphere for active listening. Minimizing noise and other distractions helps create an environment for focusing on the remaining sounds. The third step then becomes hearing. Hearing is a crucial step for fluency in any language. Evidenced by historians and archeologists the world over who are blocked in their efforts to understand civilizations whose spoken languages are extinct (Preston, 2014). This also supports that hearing is the dominant communication input into the brain. And if hearing is a prerequisite to link the fourth step of speech, evidenced by those born deaf (Temurova, 2020). Then the fourth step of speech is a prerequisite to link to the fifth step of reading and reading links to the sixth step of writing. From this last step of writing; mathematics previously done mentally can now expand from its simple roots to self-actualizing according to Maslow with more complex operations and uses. 16 Moreover, if a student’s long division language or method is uncertain then their written notation should likewise struggle. Orey (2008) adds to this thinking by saying “This [arithmetical basis] is learned while students are learning their native or formal school languages” (p. 6). Taking another note from a college course at Montana State University on the language of mathematics, “That language [mathematics], like other languages, has its own grammar, syntax, vocabulary, word order, synonyms, negations, conventions, abbreviations, sentence structure, and paragraph structure” (Esty, 2007, paragraph 10). Ma (1999) expressed throughout her study that language used by instructors was key and gave several examples compared with the USA counterpart, such as, elementary teachers in the USA conceptually misunderstand division leading to poor teaching, evidenced by the inability to formulate a scenario which correctly interprets 1¾ ÷ ½; which could be, a recipe calls for one and three-quarters cups of oats how many half-cup scoops of oats are needed? This content matter is then revisited during teacher trainings throughout their careers. As reasoned above, the easiest way to learn a new language first requires hearing it. If heard in context long enough, the average human mind can track the patterns and make connections, thereby learning the new language. This process can be quickened by the intentional study of the new language and further hastened with instructional help, such as a one-to-one translation for comparison or other programs which aids in the conversion of that language. Such interpreters only equate similar meanings, index a new relation, symbol, sign, or icon in the mind of the learner into a synonym for that idea. The same holds for mathematics. Orey’s (2001) wall matrix did just that, it indexed many 17 languages for the common mathematics vocabulary, such as the word or written notation for division. With this semiotic view as he called it (or to form meanings of the relationships of symbols, icons, and spoken language), mathematics for most students is merely a few years behind their native tongue (Orey, 2008). By kindergarten where mathematics is formally introduced, children have had roughly five years of hearing and three of speaking, so if they start hearing mathematics and creating those synonyms, by age ten they should be able to rigorously start learning mathematics structure or all the aspects of that language regardless of their native tongue. But contrary to what many believe, mathematics is not a universal language as the following will show (Orey, 2001). Professor Daniel Orey (2001) at California State University, Sacramento studied how mathematics has its own language variance. He defined mathematics as being more like dialects; the bulk may be similar but nuances persist which can confound those who have this experience. Orey (2001) coordinated his ethnomathematics research under the name of, Algorithm Collection Project (ACP) which gathered algorithm strategies of students as they enter the California school system from other countries and others willing to participate. Part of his research looked at the language ability of the students from mono to multilingual, the algorithm strategy they use, and their performance as they transitioned to life in America. He further suggested that USA students struggle in mathematics for the same reason as some of his students; they are not multilingual. The study mentioned earlier by Star (2005) of student use of the SA also effectively highlighted that a DA that communicates is beneficial. When learning a 18 subject such as mathematics, understanding key terms and ideas to avoid confusion, hinge on the definition-wording, the proper context, and accurate demonstrations. One study found that intense intervention with what was called demonstration-imitation-keywords worked well to teach students with learning disabilities. Rivera and Smith (1988) would model a method while vocalizing keywords such as divide, multiply, subtract, bring down, repeat then have the student imitate with access to notes or an anchor chart, and would repeat the demonstration if imitation was unsuccessful. In addition, Rosa (2004) suggested, how we verbalize the problem has as much to do with understanding as the written notation or method. For instance, consider the Franco-Brazilian DA language, in Portuguese, division is verbalized “23 divided by 7”, where 23 is the dividend and 7 is the divisor, whereas in North America there are several versions which are less uniform and can be regularly found as is said in Portuguese but also as “7 into 23” or “7 dividing 23” or “7 goes into 23” (Rosa, 2004, p. 14). Additionally, one could say “23 shared among 7” or “7 divided into 23”. Calls for division can be stated in other ways as well, which may be the source of some of the USA’s problems. Since people often remember in a certain sequence, then saying, ”23 divided by 7” or reading 23÷7 may often result in 23 being placed before the ÷ (division) parenthesis as in 23⟌7, which is wrong for the North American way of doing standard long division. Contrary to the positive results Star (2005) observed, portions of this review and from Rosa (2004) above introduced or alluded to an issue with the SA option, which in 2001 Bley and Thornton had already acknowledged existed and spoke of when transitioning from concrete teaching aids, such as apples or money, to abstract symbols such as the “… [written notation or] division sign [ ⟌ ] tends to confuse students with 19 reversal tendencies, whether visual or auditory. Yet the other [written notation] (÷) is not particularly useful…” (p. 264), but offered no substitute. Lampert (1992) investigated successful instructional strategies of how to teach long division and made note that about half of 12-year-old students made the aforementioned reversal mistake and did not put the divisor in front of the long division symbol, and the dividend inside instead, students interchanged the language as equal or “When… asked to interpret the division of a smaller number by a larger one, they often simply inverted the numbers” (p. 232). This study was one of the few sources that recognized the written notation ordering and verbalizing gaps. Another study was conducted by Anghileri, Beishuizen, and Van Putten (2002) who noted a similar mistake with a study of Dutch and English students attributing it to quotitive contrasted to partitive. Consider eight divided by two. The partitive division model refers to sharing among a known number of groups or how many are contained in each group; i.e., Fran has 8 candies to share with a friend how many candies do each receive? The result is two groups of four. While quotitive division calls for measured subtraction to establish the number of groups when the group size is known; i.e., A family of 8 wants to ride bumper cars. Each car fits two people. How many cars will they use? Resulting in four groups of two. A major catalyst to present the problems highlighted in this literature review and any subsequent work was the confusion that resulted from the abovementioned spoken-order-relationship issues with the dividend & divisor for placement using the SA. Orey (2008) suggested these problems may present a handicap for visual learners and an additional problem for students from other countries where the USA standard method of 20 long division, or SA, may be interpreted as a square root. Semiotics would seek to study these phenomena (Orey, 2008). Closely connected to that would be the written notations. Keep the reversal tendency and written notations in mind when we talk of mathematics, as Lampert (1992) stated, “The important issue is how the operation is interpreted, not whether it is or is not taught” (p. 276). Ma’s (1999) book focused most heavily on this idea that how the concept is interpreted or understood has more to do with accuracy than knowledge of a method. Division-related issues presented up to this point and Ma (1999) seemed to suggest that to use the parenthesis or long division symbol correctly requires excellent number sense or as mentioned above, to be consistent in speech to achieve success. So that when using 7⟌23 saying “7 dividing 23” or “7 divided into 23” would be helpful to understand the written notation. Or acknowledging other options exist, that the SA strength may be its method not its written notation, and simply not using such confusing arrangements at all. A conjecture, that surfaces about the confusion caused when the numbers need to be switched in the SA from notations such as 23 ÷ 7, may have been introduced from the fact that Arabic and many other languages are read from right to left and mathematics documents studied by westerners after learning a few of their numbers may have been studied from left to right (Cajori, 1928). Thus, there are clear origins from which the wording and semiotics have taken shape. This background clarifies further how to instill number sense, such as, communicating algorithm strategies, is beneficial. On the other hand, minimizing mathematical confusion will require energy and time to instill better number sense via consistent speech sequencing or other mechanisms [or even a new written notation] (Anghileri, Beishuizen. & Van Putten, 2002). 21 History of Division Algorithms and Their Notations The student struggle with the SA, in light of the domain distinctions of methods and written notations, suggests a new perspective teachers could utilize. Except that many teachers do not know any DA beyond the SA they learned. This limited propagation problem came about in what appears as happenstance, with the slow devolution of particular tastes or needs of printers, popular mathematicians, and scientists from around the world. As printing became possible those who desired to publish, discussed the printer limitations with peers, along with other operational considerations which altered or ended the use of some DA (Cajori 1928, 1929; Karpinski, 1925; Smith 1923, 1925, 1958). The reason this option limitation came about among teachers requires a contextual history which will also help to illuminate and inform the other issues focused on in this review. Aspects of several DA will be mentioned in this section, for the full written notation and method example see Appendix A. Further, the origins of the various division methods and written notations are old and difficult to find which has contributed to their lack of dissemination. The following is the historical context. Of particular note, is the occasionally termed long division symbol, depicted as |¯¯¯ or ⟌ . Neither of these symbols nor the ideograms (written notations) which predated them have any known studies to back their use (Cajori 1928, 1929; Karpinski, 1925; Smith 1923, 1925). Smith (1925) said the earliest recorded method and written notation of division was by the Egyptians and used multiplication which focused on halving and doubling or other easily remembered tables or portions thereof, instead of multiplication tables for 1-22 12 encouraged today. In Cajori’s (1928) work, the Egyptian written notations are shown to use a solidus or slash-like stroke in their hieratic (the priestly shorthand) and demotic forms (the peoples’ shorthand version) when specifying certain fractions, some examples: / or ৴ for 2/3, and × for 1/4. This author also said the Babylonians had this ideogram (Igi-Gal), to express division. In some visuals, the symbol looked more like two perpendicular lines divided by the third leftmost impression or mark. The Greeks had no known written notation for division except with fractions they used a bar above the letter-number representing the whole part, or denominator, like we do today. One of the next novel instances we have access to, Cajori referred to as a manuscript (MS) from India where an abbreviated word was used –bhâ short for bhâga, meaning part. By 1150 A.D., the Hindus were using relative positioning with the numerator above the denominator and no bar, while the Arab included a bar. Elsewhere Smith (1925) stated that as division use increased the DA that were written by column, rule, or table, if the answer was not written directly from the head or simply oral, also increased. An early example of this kind of DA that at a glance seems to Figure 1. Gerbert’s divisio ferrea or iron division Example: 900 divided by 8 900 / 10 – 2 -( 900 – 180) 0 + 180=180 -(100 – 20) 80 + 20=100 -(100 – 20) 0 + 20=20 -(20 - 4) 0 + 4=4 as remainder r 4 8 90+10+10+2 = 112 Figure 1. At right Gerbert’s handwritten method with an interpretation at left. 23 share traits with the SA but is illustrative of the diversity which existed. This DA was commonly attributed as Gerbert’s from at least 980 A.D. and was also called divisio ferrea or iron division (see Figure 1). It uses complementary divisor differentia, for example, 6 becomes 10 - 4 as the new divisor and was tracked downward as in the example in Figure 1. The handwritten is the actual written notation used to track the method, while the other shown is to aid understanding; such as horizontal lines are intended to match the handwritten example. The first phase of 900/8 shown at the top may be thought of as 10 - 2. The first multiplicand quotient estimation used here is 90 and is tracked and tallied at the bottom of the slate, wax, or dust board it utilizes. The 10 is multiplied by 90 for 900 which cancels the 900 noted below the 8 or divisor section by scratching that value out. Followed by 2*90 for 180 noted next which is canceled by the second multiplicand quotient estimation of 10 by 10 – 2 for 100 with 20. The 100 cancels a portion of the 180. Then the remainders were added up, as the new dividend of 100, then the process repeats. With four as leftover thus 4/8 remaining but remainders were often neglected. Another DA Smith (1925) described that manipulates the divisor and was called per ripiego, or by factors, such as 441 / 63, 63 becomes 9 and 7 and flows 441 / 9 =quotient. Quotient / 7 = answer or 441/9 = 49 which is then divided e.g., 49/7 = 7. Therefore 441/63 = 7. Smith described several other DA, each employing different methods and or written notations. One DA separates the dividend, called by parts or “per il sca pezzo” meaning “division by cutting up”. This method was used when the divisor was a multiple of ten. Otherwise, the result is only approximate. Given a dividend of 65,284 and a 24 divisor of 600, the dividend is parted between the hundreds and tens place value because the divisor is hundreds. If the divisor were 60, it would be separated between the tens and ones place values. The divisor is also split and appears to be separated with the most significant digit by itself to divide the large dividend part of 652/6. The original divisor divides the smaller part thus 84/600. From there the DA may follow other methods like the SA but the quotients are added at the end. Another DA, used at least since 825 A.D., was the visually named Galley or Scratch DA due to the scratching of numbers already used (Smith, 1925). Or the Galley was said to resemble the sails of a boat the longer the dividend became, due to the compact manner in which this subtraction method was tracked. This method was rather popular until printing had become more common and printers resisted printing non-standard symbols like scratched out numbers. Printers charged extra for non-conforming symbols and arrangements or used what they had. Smith also referred to a 14th-century man named Maximus Planudes who used a DA that resembles a more modern written notation, but progresses with a different method (see Figure 2). Figure 2. Method of Maximus Planudes 25 In the 1400s the idea of bring down as in the SA was in use in a DA called a danda or by giving a value from the dividend to the remainder not shown here see Appendix A (Smith, 1925). However, for the written notation the dividend is located at the top with the divisor to its’ right, consistent with our current language to refer to division. Thus, between Maximus Planudes and this DA appears to be the origins of the method common in Europe today. Smith (1923,1925) refers to a 1460 “Italian MS” (unpublished manuscript) the earliest record found of the Austrian Method or shorthand of the a danda. However, the written notation is different, where “–“ represents the fraction bar, an example might be 400 – 20 equals 8. Smith also cited Filippo Calandri’s 1491 published work containing “the first printed [written notation] in long division” with the divisor on the right (p. 255). Yet, today the divisor is on the left in the USA. Few DAs have or will be detailed in this literature review. However, a quick tally of known historical DA written notations’ relative positioning in Appendix A reveals a few facts which may aid understanding. Relative to the dividend, five DAs placed the quotient to the right and two placed it to the left. While four DAs placed the quotient below the dividend and six placed it above. Interestingly, DAs with the quotients on the diagonal had only three that existed and all are placed down and to the right. There was only one DA that placed the divisor relative to the dividend on the diagonal down and to the right. There are four divisors placed to the left of the dividend (like the SA) and six to the right. Only one placed the divisor above but seven DAs placed it below, see Appendix A. 26 Smith (1925) stated that the division symbol ÷, (historically called obelus) “first appeared in print [as division] in the Teutsche Algebra, by Johann Heinrich Rahn…in 1659” (p. 406). While Cajori (1928) stated Adam Riese used it to indicate minus in 1524 While the colon (:), for much of the world, initially meant ratio but is now more associated with division thanks to Leibniz (Smith, 1925), particularly on the European continent (Cajori, 1928). By 1837, the French were using 500)22 written notation with the quotient below the divisor, the method reads as 500 divided by 22. By 1888, Joseph Ray used the aforementioned SA (Orey, 2001). Continuing, Garlick (1897) in the US, attempted in his book to order and set out how a school should run at the primary level. Specifically, regarding long division written notation, he dictated using the quotient below the dividend or the older parenthesis written notation shown in Figure 2 and as follows: divisor)dividend(quotient. This notation was originally from Michael Stifel’s 1544 Arithmetica Integra, which was later stated as the “lunar symbols” (Cajori, 1928). Smith (1900) wrote a work remarkably different than a later work he co-authored. Where he said of textbooks and arithmetic teaching of the previous fifty years “…not only is this old-fashioned rule-learning (unhappily not yet extinct) a sham; it is wholly unscientific” (p. 31). By 1915, Smith seemed to side with the dominant USA’s SA mechanistic movement the same that stands today, and by this time had already cemented itself in the minds of Americans. Again, the SA refers to the following written notation symbol ( )¯¯¯ ) which places quotients over the dividend with the divisor to the left used in conjunction with the method pattern found below the dividend of divide-multiply-subtract-bring down. 27 In 1915 Wentworth and Smith published a textbook. In their division section, long division is depicted the same as is common today and has been the case for over 100 years via records herein. In that span of time insufficient scientific research has been found to strengthen the efficacy of the SA method and its written notations, which agrees with Smith’s 1900 work that it is wholly unscientific. As the timeline approaches the current era, the variety of written notations existent gives way to the mire of history, and only the opinions of a few influential persons remain in the form of written notations, of which some are previously mentioned and study of and discussion on the matter seems to cease. To the degree that the most recent well-publicized argument concerning the matter of written notations only expressed a need for a standard symbol for the operation. This need was recommended by The Mathematical Association of America in 1923 which was that the world mathematical community should abandon both the obelus (÷) and the colon (:) and adopt the solidus or slash (/), and to leave political divisions in the past and unite around a common DA (Cajori, 1928). The existence of more recent treatments on the matter has not been found beyond what is included in this literature review. This sequence of events puts today's teachers in a state where they do not know of other DA because they have not been made easily available. Consequently, the problem with teachers not knowing other DA was set in place and hyped over 100 years ago. Many methods have been scrutinized, but their counterpart written notations have not. Either way, teachers still do not have at their disposal a comprehensive toolbox of DA to draw from. This deficiency is further complicated by a debate that Alexander, Graham, and Harris (1998) addressed where 28 they defined skill as methods made automatic whereas, prior to that point the strategy was considered intentional. But Ralston (2003) believed calculators changed everything: “Only if you believe the argument…that facility with the traditional long-division algorithm is necessary for the subsequent study of mathematics could you still advocate teaching it, since the skill [automaticity of method] itself has ceased to be useful.” (p. 1248) Alexander, Graham, and Harris's (1998) work previously addressed this debate. They felt facility with a specific DA perhaps has run its course, but the skill to, mentally at will, use a non-calculator strategy has not. The implication, according to several publications suggests the written notation used strongly influences the method used. Stated another way, one written notation could suggest a corresponding method. These issues are important to understand, because a calculator strategy is easier to learn, and use versus locating paper-and-pencil to use a more traditional strategy and implement it for a specific situation; but mastery of the skill, or making it second nature has not ceased to be useful and may be generalizable, and calculators are optional at best (Klein & Milgram, 2003; Ma, 1999; Van den Heuvel-Panhuizen, 2001). In addition, the immediate application of more traditional strategies to decimals and fractions is clear, and facility with DA strategies leads nicely to algebra, geometry, and beyond (Klein & Milgram, 2003). Smith (1925) quoted Pacioli from 1494 when speaking of division, “if a man can divide well, everything else is easy, for all the rest is involved therein” (p. 132). Help for teachers may come from an Orey’s (2001) suggestion to be better in mathematics, that we return to the old idea of the well-educated person (a renaissance 29 man); which knew many languages. He credits this by saying that the ability to draw from more algorithm strategies leads to greater understanding. Orey (2008) later suggested one avenue of study to form a cognitive foundation for added mathematics ability would be to combine a European and Asian tongue (thus the language aspect addressed earlier) with a DA other than the SA. Rivera and Smith (1988) said, “…methods [italics added for emphasis] and materials such as calculators, and microcomputers should be studied to determine techniques that promote student acquisition and mastery of arithmetic facts. Basic facts are a part of many mathematical skills…” (p. 81). This suggestion again implies a problem for USA teachers and their students which is, they do not know alternative DA. But without a comprehensive resource to draw from this suggestion is only rhetoric. Whitehead in 1911 refuted a myth, still commonly heard today in speeches about what success would ultimately look like, saying, “that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.” (p. 61) As mastery is gained, fewer task-specific strategies or skills are needed; only time and interest or motivation brought on by some need are required to bring about the behavior (Alexander et al., 1998). The problem then for teachers is access to DA so they can be recognized and harnessed. Comprehensive Resource Options for different DAs exist in various resources and much could be done to improve access for teachers, and other interested parties. Because existing resources are 30 limited in scope, difficult to locate, and decipher, such an effort may further what Cajori (1929) wrote of in the concluding pages of his work “The individual designer of mathematical symbols…is doomed to disappointment” (p. 345). However, he goes on to explain the need for a uniform written notation and at the same time warns against it, saying that science revolves around innovation and that a uniform system might make science lethargic. But the dissemination of knowledge ultimately rests on a common language and notation. A resource is needed that includes the significant suggestion of Bley and Thornton (2001) that is tied to several concepts already considered, which is to reverse visually the SA written notation from )¯¯¯ to ¯¯¯( a sort of mirror image. This resource could include false methods such as, that used by Ma and Pa Kettle in the 1949 sitcom who invent an incorrect yet repeatable DA which overlooks the significance of place value or other micro-didactic examples invented by RME students. “More diversity in approach can lead to some good strategies but many students are unable to develop their own informal methods” (Anghileri, 2004). It should also include examples like the macro-didactic or big picture approaches Van den Heuvel-Panhuizen (2001) regarded as attainment targets. Similarly, Anghileri, Beishuizen, and Van Putten (2002) use a few examples to leave the reader with the idea that a standard, efficient, and intelligible spatial arrangement, for key information, is needed to cue memory and communicate recordings to others. Realistically to accommodate some DA methods in a resource the written notation may need to be modified, Orey (2004) related how he came up with an informal collection to accommodate doing long division which he placed in five categories, signs, 31 or patterns named North American, Franco-Brazilian, and Russo-Soviet. The other two are not named, but similar groupings could be used, see Appendix A. A decent collection of examples was found in a study comparing English and Dutch students using written DA: “counting, repeated addition, chunking (performed in different ways), reversed multiplication, dealing, estimate-adjust, repeated halving, repeated estimation, many of which will be incorporated into structured procedures for division calculations” (Anghileri, Beishuizen, & Van Putten, 2002, p. 151). Figure 3. Addition & Subtraction Method scale These lists and many available resources are typically intended as help for the SA method. but their scope is limited to only a few DA variations. Most DA methods are subtractive in practice, even if not subtractive, distinctions come from the amount which is added or subtracted, and how this amount is obtained (Thompson, 2008). These variations may be viewed from the efficiency perspective of an addition & subtraction method scale when placed as follows in Figure 3. At one end are choices less than the divisor, such as the common choice of halving to help with remainders, heading the other Exact from the head By the table Large Partial Quotients (Chunking) 10+ 3, 5-7, 9 Doubling By 1 as Repeated subtraction Small Partial quotients or Multiples of Divisor < divisor repeatedly subtracted (halving) 32 direction is doubling, times ten, and on to what is known as chunking or partial quotients, which a seasoned divider could do in a perfect estimation as is prescribed under the SA (Thompson, 2008). This addition & subtraction method scale suggests that any resource should include at least one example of each scale method. There are explicitly nine on the scale; Thompson’s (2005) DA are not listed. This suggests a collection of at least 10 or more DA which is simply not available. The effort to locate and decipher one of these lists for lesson suitability may quickly leave the sifter to conclude the cause is not worth the extra effort. To remedy this predicament for teachers and other interested parties, access to a comprehensive and uniformly comprehendible resource to expand teachers’ repertoire is needed. Summary When considering DA options, there are two aspects: methods and written notations. The former has have received the vast majority of the attention to date; and yet many students still struggle to make sense of division. While the latter written notations have been overlooked, despite statements that implicate this aspect does make a difference. Connected to either aspect is the need to be consistent in the language used to harness either aspect. History has neglected to formally study DA written notations. Leaving teachers to rely solely on the DA they learned, which for most, is the SA. Magnifying the topic of written notations may lead to some quantifiable improvement, whereas, past reform efforts tend to recycle ideas with unsatisfactory improvement. These reforms tended to ignore the real impact language has on mathematics to relate its methods and written 33 notations, while the literature indicates mathematical language could be researched further, as no exclusively written notation research was located or explicitly suggested. “In the end, we must realize that algorithms are fully conceptual cultural-historical products and should be taught as such” (Lee, 2007, p. 49). Associated with these realizations is the knowledge that there are at least a dozen written notations that could be studied, a few of which are realistically more feasible to bring about improvement. While efforts herein seek only to make DA options accessible for future study. To this end, an accessible comprehensive comprehendible resource is needed. 34 PURPOSE For the struggling students to overcome the often-overlooked problem of reversal tendencies as described by Bley and Thornton (2001), comparatively lower mathematics scores (Desilver, 2017), and the cyclic reform efforts that resorted to calculators (Ralston, 2003) this project was initiated. Perpetuated by the Long division Standard Algorithm (SA) or its academic tradition which had little scientific backing, and judged contemporary reform efforts such as EM, and RME, among others. Further, the SA appeared insufficient to the contempary demands that students perform mathematics at higher levels (Smith, 1900; Van den Heuvel-Panhuizen, 2001). The purpose of this project was to advance beyond the rhetoric of the past in a post-electronic age to organize a curriculum resource which distinguished a DA’s written notation from its method, where once scarce information was made more easily obtained by interested parties, such as researchers, which may have led to better DA solutions for those who struggled and their teachers. These efforts, coupled with Lampert’s (1992) knowledge that the mathematical vernacular had profound effects on student performance, have caused interested parties, reform efforts included, to seek out other methods that existed; and by association alternative written notations, all of which had yet to be formally studied. Of the readily available resources, the largest known collection of DA demonstrations was eight, with three more mentioned, but not explicitly demonstrated, found in Smith’s 1925 history of mathematics. The objective of this project was to serve concerned parties and to develop an organized comprehensive accumulation, and curriculum resource with significantly more DA available than in any of the previous collections. The project was made readily accessible, uniformly presented to improve 35 comprehension, a more comprehensive resource that better accommodated those who struggled to access, learn or teach division, and for those interested in conducting additional studies. To do so, a website was used to address accessibility. DA were organized into six sorted categories modeled using fairly consistent language and quantities, so that if one method was understood another may be discerned thereby; and every known DA was represented to some degree. 36 METHOD A comprehensive resource was needed to make DA strategies widely available to the public. Additionally, a new DA had been devised by the author prior to this project and was included as a model that encouraged users to systematically devise new DA by pairing different combinations of existing methods and written notations. Specifically, I attempted to resolve some of the SA problems –the key may be in this new DA or in another deliberate arrangement yet to be devised, which brings about a breakthrough in the reform patterns. Sources of written notations and methods in small or informal collections were located, identified, and extracted to accumulate a larger, more comprehensive resource. With the exception of three additions, that were previously unknown DAs that were happened upon while looking for something else. The efforts of locating and gathering the various collections, for eventual demonstrations of individual DA existent had largely been accomplished in the completion of this review. Further, this comprehensive collection of DA found a home in an accessible website format (see Appendix A), which utilized general feedback from three reviewers. The landing page titled Division Algorithms introduced the user to the purpose of the site and invites them to navigate to other portions in two ways from links located below the introductory passage or a menu located at the top-right corner called Sorted By/Home. From either location, the DA were sorted into the six categories detailed below. Also, on this page was a link to this published work. To sort the various DA, each of the six categories were intentionally selected and represented in several generally shared ways to think about the DA, which highlighted a 37 shared feature or ranked them. Thus, each category may or may not have all 30 known DA included, but all DA were accessed from multiple perspectives via links within the categories. Also, within some categories were subcategories or additional ways the DA were ranked or separated. The following were selected as the six categories and measures of rank: Method Name (alphabetical), Written Notation (relative to the dividend), Chronology (earliest record era), Geography (region of origin), Method Similarity (relative to the mechanism), Errors and Other Considerations (assorted unique features). The measures used to rank the DA lacked precision, to truly rank them would require significantly more data, which is beyond the scope of this project but could be the starting point for several other projects. Each DA was assigned its own page, except the USA standard algorithm, because doing so would have added nothing to the body of readily available information. Each page was organized with the same basic template. Which included a brief DA introduction, interest facts for further research, DA written procedure steps, clean before and after diagrams of each new mark, general observations and use tips (like feelings experienced after use and DA potential), if available a historical example picture, and space for a how-to video. One feature included was for users to submit new DA; therefore, the site had the potential to improve beyond the completion of the project. A consistent language and presentation were also settled on and used throughout any discussion and presentation of each DA particularly when a division was called for. The verbiage order used was the dividend “divided by” divisor but to be true to the historical manner method, I occasionally, had the steps call for division in the original verbiage and perhaps clarified in the explanatory notes. Division terms like Thompson’s 38 (2005) example were generally followed and I used words like divisor, remainder, chunking, and quotient. Scripting DA presentations was considered to maintain consistency but was abandoned due to time constraints and difficulty to execute properly. The presentation template consisted of method thinking in one column, written notations in the middle column, and explanatory notes in a third column. The flow of the DA was then represented using before and after diagrams with the thinking behind each new mark at left and any additional explanation or considerations at right. DA method steps were worded as close as possible to the original sources, if clarity could be maintained. Otherwise, new wording was used to bring about a better understanding for modern users. Dividend, divisor, quotient, and remainder were frequently used. Fractional vernacular, such as numerator and denominator, were avoided unless able to better explain the idea of a step. Other generally useful words needed to communicate the steps included multiply, subtract, bring down, add, tally, sum, total, complement, gap, carryovers, current, times table, stacked, place value, column, row, product, affix, prefix, subscript, superscript, portion, estimate, digit, figure, factors, standby, position, recent, as-is, recurring decimal, round, set, repeat. This convenient list is included here to highlight the range of ideas needed and that some steps repeat an operation, like addition, but were intermediary calculations so to distinguish one similar step from another, similar words were used such as “ current” and “as-is”. However, for many of the words, there was nothing special and could be substituted with another. Further, to show the extent of vocabulary needed to understand most of the DA, for wording research, and for purposes of lesson vocabulary preparation. 39 As this resource was developed, terminologies previously mentioned and visual aids such as color-coding were evaluated and synthesized to bring about a uniform and understandable presentation. I employed a color-coding system which relies mainly on setting newly written notations to bold and may have been simultaneously joined by the use of background shading, starting with grey and adding other colors as needed to distinguish one grouped area of focus from another or simply to add a splash of color. While there is room for improvement a uniform and consistent presentation was achieved and helped to highlight unique differences, which were often noted in the observations and explanatory notes sections and helped to prescribe or address particular issues for struggling students and their teachers. Exploration of extreme quantities was carried out with many DA to ascertain method capabilities, while with others, more elementary quantities were carried out, for easier recognition. This comprehensive collection of algorithm strategies was placed alphabetically in Appendix A. The usefulness of Appendix A was intended to be of greater value than the sum of the writings which contributed to its existence. 40 DISCUSSION With many teachers, struggling students, and others only aware of or struggling to understand the SA, there was a need to find, describe and make accessible other DA. Smaller lists of DA exist but are limited to only a sampling of those in existence and are often too difficult to access. Many of these limited lists only made mention of the existence of other DA and offered little in the way of guidance on how they were or could be conducted. A difference with the resource I have made available is to give explicit mark-by-mark or step-by-step guidance for each DA’s method and written notation. Including, as well, other information about the DA such as its origins, comparisons to other DA, and observations of the experience from someone familiar with all of the DA. This project was meant to be generalizable and accessible to the public-at-large. Only thirty DA were included at the time of this publishing, but I know there are more and I can think of at least five that will not be listed at this time. A reviewer of the project may have identified yet another. The project internet site developed for a readily accessible comprehensive resource was reviewed by three individuals. Each of the reviewers was tasked with reviewing a portion of the curriculum resource for thirty minutes and to attempt to learn a new DA then asked for open-ended feedback. They were tasked to relate their experience and the difficulty of navigating the site. Overall, the reviewers made no mention of the parent category of DA in relation to the two aspects of written notation and methods as such, therein. Reviewer A has taught for twenty-two years and is currently teaching 5th grade and has also been a comprehensive mathematics instructor 41 and a parent of four children, all of whom were past the years of long division introductions. Reviewer A was impressed with the amount of effort put into the internet site and reported viewing most of the DA. She mentioned several DA that stood out to her and why. One being Double Division as it reminded her of what she called double-digit division. She also said it used a mnemonic memory device which she called “weapons” like a two-digit number multiplied by 2, 4, 8… which become guides, but offered no further explanation. She also highlighted the Bidirectional DA which she said reminded her of a version called Lucky Seven, a partial quotient method, and how easy it was to make the connection to their background in partial product multiplication, which was a multiplication method she prefered for students to use. However, she said she loved the Austrian/short method for its use of single-digit divisors. She mentioned the Dots DA she would never use. Calling it “way too messy”. Overall, she mentioned “the layout and navigation are great! The categories seemed good”. Another, Reviewer B, was in his twenties, not married or teaching full time, but he did work as a mathematics tutor for students at various levels of mathematics. He described the website was “fun to look through”, and explained that he had not seen most of these methods. The website navigation worked well and he reported no issues. As he continued, he expressed that he could see himself using the resource more in the future to learn about a given method of division. Reviewer B even went so far as to thank me for organizing the website. 42 Reviewer C, who loved mathematics and has taught grades one, two, four, and six for more years than she cared to admit and had a minor in mathematics. She said the site was obviously for upper grades and up, but she would have added color and equation segments to the backgrounds to see, ” if it’s one they’d like to look further into”. This idea was actually considered and I had put in a different background for each category to get a sense of the idea, but I quickly found that it felt like I was at a different website for each click, which was very disagreeable to me and left the impression of shrinking the project. What I ended up doing was including a background on some pages behind the text rather than the title area to maintain the coherence of the project. However, the idea of a more fun common background could be implemented at a future update, but my target user was not the students themselves but more for the teacher or researcher. Review C assumed correctly that the spot-marked demo was intended for videos and she would put videos before the observation section as another way to quickly judge if they wanted to know more. No other reviewer expressed similar sentiments and may even have expressed favor for the state they experienced it in. Since I intended for the observations section to be the DA introduction, and was to get someone to watch a five-minute video or spend some time with its pictorial before-and-after-demonstration. I decided not to change anything on this point. She liked the color-coded parts of the final step. Reviewer C disliked the DA sorted alphabetically, she said it did DA a disservice, but rather, wanted to see them grouped by popularity, which I too wanted but such data simply did not yet exist and could be an added charge for someone going forward. She also wanted them organized by “regions that commonly use them”, which I thought was included under geography, and “great rare methods that need another look”. Otherwise, 43 she believed a user might get tired of the site. I had not considered this exact approach, this could be its own category or a sub category under one of the last two categories, but I included the idea under other considerations in the last category, for now. One feature she mentioned that I did a great job on was, that I did not believe I could take credit for, was the search box where she typed a method in and found it. Limitations I am potentially biased in the fact that one of the DA included is of my own making and was an attempt to correct erroneous perceptions experienced during my years of tutelage concerning the matter. The measures I took to ascertain the DA, while extensive, were nevertheless lacking a precision many would expect. In most instances, there was little to no overlap concerning particulars of each DA, as such gaps existed and the information was likely not corroborated with one or more sources. Other limitations that were recognized were that many such internet lists of DA have video demonstrations or specialized calculators to accompany the DA presentation. They also may have collaborative content teams, professional site builders, and as such their own domain name. Another possible limitation was the target audience. The resource was written more for adults which may limit its reach to the real target audience; the long division learner. Another limitation may be that I started this project many years ago with a different approach, as such citations may be dated and remnants of the other approach may be conflated. Part of the original plan was to formulate identical language phrases and vocabulary to communicate the steps of the DA. However, what was not clear until I initiated development was that if this plan was carried out the original thinking may be 44 lost in rephrasing, and this was not acceptable to me. So, let it be as-is, until such time as both can coincide. Conclusion From the efforts here it was intended with hope, that someone would take up the charge to identify measures to research, so that society would know better how the various DA compared to one another or at least for a given language. Meaning DA x was better for English speakers, and DA y was better for Chinese, etc. This information would end the era of uncertified division standards. And to have truly studied the language relationship of the written notation and common mistakes. And that large projects like TIMSS, would have identified methods and notations along with their ranked nations. And finally, to recommend a DA whether the SA or not, that had science backing its written notation, not just its method. And to have someone isolate the various methods and plausible written notations and intentionally combined them into new DA or having tweaked an existing DA. The best DA to teach to Elementary students was expected to be, the best DA that aligns dividend to divisor with the language, allowed for tracking or omissions depending on the needs of the teacher and pupil, and had an easily understood method. 45 REFERENCES Alexander, P., Graham, S., & Harris, K. (1998). A perspective on strategy research: Progress and prospects. Educational Psychology Review, 10, 129-154. Anghileri, J., Beishuizen, M., & Van Putten, K. (2002). 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How to design mathematics lessons based on the realistic approach. Retrieved 3 December 2021 from https://repository.unsri.ac.id/6362/1/rme.html. 51 APPENDIX A Division Algorithms Methods and Written notations 52 Taking a mathematical approach to the number of arrangements; how many permutation possibilities exist that could be studied? The daunting answer of where things are in relation to the dividend is 7000+ when restricted with a 3X3 grid, even more if adding linearity. These arrangements would include a divisor, dividend, remainder in decimal form or fraction, units for tracking what the number is, calculation space, and quotient. To organize a comprehensive collection of DA that do exist, a website has been built and can be accessed using the following URL address: https://sites.google.com/alpinedistrict.org/dividealot/sorted-byhome 53 The 30 DA included at the time of publishing are listed here alphabetically by name: 1. A danda 2. Argumental 3. Austrian/Short 4. Bidirectional 5. By Factors/Ripiego 6. Chinese, Old 7. Chinese Sticks 8. Chunking/Partial quotient 9. Comedic Division 10. Dots 11. Dot to Dot 12. Double Division 13. Egyptian 14. Ekadhikena Purvena 15. European 16. Exploration 1 17. Franco-Brazilian 18. Galley/Scratch 19. Gerbert/Iron Division 20. Golden Divison 21. Indo-Pakistani 22. Khanda/Partioning 23. Nikhilam 24. Paravartya Yojayet 54 25. Russo-Soviet 26. Sca pezzo 27. Straight 28. USA Standard Algorithm 29. Trachtenberg Fast 30. Trachtenberg Simple 55 APPENDIX B Comedic Division Ma and Pa Kettle video http://www.mathematics.harvard.edu/~knill/mathmovies/swf/maandpakettleaddition.html Retrieved 3 December 2021. The same method can also be viewed in this other well known series below. Abbot and Costello video https://people.math.harvard.edu/~knill/mathmovies/swf/inthenavy_28.html Retrieved 3 December 2021