Academic literature on the topic 'Word problems (Mathematics) – Juvenile literature'

In an elementary school classroom, as in real life, the lines between the content areas should be blurred, particularly between mathematical problem solving and mathematical situations contextualized in good literature. For that reason, I always look for interesting books about mathematical situations. Why use children's literature to teach mathematics? A good story often places mathematical problems in the context of familiar situations and is similar to, yet a much more elaborate version of, mathematical word problems. Assertions that children's inability to solve word problems results from their inability to read or to compute effectively simply are not true. The problem is that children do not know how to choose the correct operation or sequence of operations to solve the problem. To solve a problem situation presented in words, children need to be able to connect computational processes with appropriate calculations. Their difficulties lie in the fact that children simply do not understand the mathematics well enough conceptually to make the connection with the problem- solving situation. Using books with authentic problem situations may help children see that learning computation serves a real-life purpose.

AbstractResearch on mathematical problem solving has a long tradition: retracing its fascinating story sheds light on its intricacies and, therefore, on its needs. When we analyze this impressive literature, a critical issue emerges clearly, namely, the presence of words and expressions having many and sometimes opposite meanings. Significant examples are the terms ‘realistic’ and ‘modeling’ associated with word problems in school. Understanding how these terms are used is important in research, because this issue relates to the design of several studies and to the interpretation of a large number of phenomena, such as the well-known phenomenon of students’ suspension of sense making when they solve mathematical problems. In order to deepen our understanding of this phenomenon, we describe a large empirical and qualitative study focused on the effects of variations in the presentation (text, picture, format) of word problems on students’ approaches to these problems. The results of our study show that the phenomenon of suspension of sense making is more precisely a phenomenon of activation of alternative kinds of sense making: the different kinds of active sense making appear to be strongly affected by the presentation of the word problem.

Abstract. Introduction. The paper substantiates the relevance of studying set theory in mathematics and computer science lessons using modern technologies. The research aims to shed light on the capabilities of the computer algebra system Maple for specifying sets in various ways, demonstrating the validity of the laws (properties) of sets, and solving set theory word problems. Materials and methods. The methods applied in the research include analysis of school educational literature with a focus on identifying the types of tasks on set theory, analysis of methodological literature and the experience of teachers in teaching set theory, analysis of the functional and software capabilities of Maple, and synthesis of Maple capabilities to perform set theory assignments. Results. The paper has described possible directions of Maple application in teaching set theory. The examples of using the computer algebra system to specify sets, and demonstrate the validity of their laws (properties) are presented. The possibilities of visualizing educational material in Maple are illustrated by the example of solving set theory word problems of various degrees of complexity. Conclusion. The results of the study on the use of Maple in teaching set-theoretic transformations give an idea of ways to improve the learning process by visualizing educational material, getting time free from routine calculations, and imparting the study a research character. Keywords: computer algebra systems, Maple, set theory, membership tables.

Purpose The purpose of this tutorial is to discuss the use of curriculum-based language assessment (CBLA) with students who are English language learners and students who speak nonmainstream varieties of English, such as African American English. Method The article begins with a discussion of the discourse of mathematics and the role of the speech-language pathologist (SLP), followed by a review of studies that includes those that examined the performance of English language learner and nonmainstream dialect-speaking students on word-based math items. Results The literature review highlights the linguistic and content biases associated with word-based math problems. Useful strategies that SLPs and educators can incorporate in culturally and linguistically appropriate assessments are discussed. The tutorial ends with a discussion of CBLA as a viable assessment approach to use with culturally and linguistically diverse students. Conclusions Tests used at national, state, and school levels to assess students' math abilities have associated linguistic bias and content bias often leading to an inaccurate depiction of culturally and linguistically diverse students' math skills. CBLA as an assessment method can be used by school-based SLPs to gather valid and useful information about culturally and linguistically diverse students' language for learning math. By using CBLA, SLPs can help modify curricular tasks in broader contexts in an effort to make math, including high-level math, “accessible and achievable for all” students (American Speech-Language-Hearing Association, 2017).

The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where ‘pathetic’ lower bounds of the form n 1+ ϵ would suffice to derandomize interesting classes of probabilistic algorithms. We show the following: — If the word problem over S 5 requires constant-depth threshold circuits of size n 1+ ϵ for some ϵ >0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size). — If there are no constant-depth arithmetic circuits of size n 1+ ϵ for the problem of multiplying a sequence of n 3×3 matrices, then, for every constant d , black-box identity testing for depth- d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC 0 circuits of subexponential size).

Purpose – Word problems are still considered challenging for students when compared to other type of mathematics problems. Many emerging findings regarding this issue highlight that the challenges are predominately caused by linguistic aspects. This article aims to present a review and synthesis of literatures regarding the linguistic challenges of mathematics word problems and recommend solutions to address these challenges. Methodology – Systematic search was done and 35 articles from inside and outside Indonesia were selected. The linguistic challenges and recommended solutions found were analyzed using the main features constructing mathematics language: multiple semiotic system, particular features of vocabulary and grammar, and complex syntax. Findings – The review shows various difficulties shown by students in each feature of mathematics language. The review also recommends the practice of mathematics teaching and learning in which language aspects are discussed and exercised both among students and between the students and the teacher in order to help students face their linguistics challenges. It is also imperative for teachers to understand the structure and linguistic features involved in constructing word problems. Significance – This review breaks down the difficulties of mathematics word problems from the perspective of linguistic features constructing them. The findings of this review offer teachers different point of view to deal with teaching word problems, which is by understanding word problem as an entity of language rather than only as an entity of mathematics. This review also provides some solutions to help teachers address the difficulty for each linguistic feature.

Background: Workbooks were introduced by the South African Department of Basic Education (DBE) in 2011. Although the workbooks were designed as supplementary materials, in some schools they are used as the sole teaching text. Therefore, an analysis of the content coverage of the workbooks is warranted. This article provides such an analysis in terms of additive relation word problems.Aim: This article aims firstly to expound on the existing literature to propose a comprehensive additive relation word problem typology and secondly to analyse the prevalence of particular word problem types in the foundation phase Mathematics workbooks.Setting: This research was conducted in South Africa, focusing on additive relation word problems in foundation phase Mathematics workbooks.Methods: A comprehensive typology of additive relation word problem types was developed based on typologies used in previous studies. All the additive relation word problems in the 2017 Grades 1–3 foundation phase Mathematics workbooks were categorised according to this typology.Results: In total there were 61 single-step additive relation word problems with numerical answers across the three grades. This is a small number in comparison to other countries. There was also an uneven distribution of problem types, with more problems in the easier subcategories and fewer or no problems in the more difficult subcategories.Conclusion: This article provides evidence for the need to revise the word problems in the DBE workbooks. It also provides a theoretical framework to use in the revision of the workbooks and in any supplementary teaching material developed for teachers.

Mark Haddon’s The Curious Incident of the Dog in the Night-Time has achieved success as “the new Rain Man” or “the new definitive, popular account of the autistic condition” (Burks-Abbott 294). Integral to its favourable reception is the way it conflates the autistic main character, the fifteen-year-old narrator Christopher Boone, with the savant, or individual who exhibits both neurological problems and giftedness, thereby engaging with the way autism is presented in popular culture. In a variety of contemporary films and television series, autism has been transformed from a disability to a form of giftedness by relating it to abilities associated in contemporary media with a genius, in particular by invoking the metaphor of an autistic mind as a type of computer. As a result, the book engages with the current association of giftedness in mathematics and science with social awkwardness and isolation as constructed in popular culture: in idiomatic terms, the genius “nerd” figure characterised by an uncertain, adolescent approach to social contact (Kendall 353). The disablement of the character is, then, lessened so that the idea of being “special,” continually evoked throughout the text, has a transformative function that is related less to the special needs of those with a disability and more to the common element in adolescent fiction of longing for extraordinary power and control through being a special, gifted individual. The Curious Incident of the Dog in the Night-Time relates the protagonist, Christopher, to Sherlock Holmes and his methods of detection, specifically through the title being taken from a story by Conan Doyle, “Silver Blaze,” in which the “curious incident” referred to is that the dog did nothing in the night. In the original story, that the dog did not bark or react to an intruder was a clue that the person was known to the animal, so allowing Holmes to solve the crime by a process of deduction. Christopher copies these traditional methods of the classical detective to solve his personal mystery, that of who killed a neighbour’s dog, Wellington. The adoption of this title allows a double irony to emerge. Christopher’s attempts to emulate Holmes in his approach to crime are predicated on his assumption of his likeness to the model of the classical detective as he states, “I think that if I were a proper detective he is the kind of detective I would be,” pointing out the similarity of their powers of observation and his ability, like Holmes, to “detach his mind at will” as well as his capacity to find patterns in events (92). Through the novel, these attributes are aligned with his autism, constructing a trope of his disability conferring extraordinary abilities that are predicated on a computer-like detachment and precision in his method of thinking. The accessible narrative of the autistic Christopher gives the reader the impression of being able to understand the perspective of an individual with a spectrum disorder. In this way, the text not only engages with, but contributes to the construction of this disability in current popular culture as merely an extension of giftedness, especially in mathematics, and an associated unwillingness to communicate. Indeed, according to Raoul Eshelman, “one of its most engaging narrative devices is to make us identify with a mentally impaired narrator who is manifestly not interested in identifying either with us or anyone else” (1). The main character’s reference to mathematical and scientific ideas exploits an interest in giftedness already established by popular literature and film, and engages with a transformation effected in popular culture of the genius as autistic, and its corollary of an autistic person as potentially a genius. Such a construction ranges from fictional characters like Sheldon in The Big Bang Theory, Charlie and his physicist colleagues in Numb3rs, and Raymond Babbitt in Rain Man, to real life characters or representative figures in reality series and feature films such as x + y, The Imitation Game, The Big Short, and the television program Beauty and the Geek. While never referring specifically to autism, all the real or fictional representations contribute to the construction of a stereotype in which behaviours on the autistic spectrum are linked to a talent in mathematics and the sciences. In addition to this, detectives in the classical crime fiction alluded to in the novel typically exhibit traits of superhuman powers of deduction, pattern making, and problem solving that engage with the popular notion of genius in general and mathematics in particular by possessing a mind like a computer. Such detectives from current television series as Saga from The Bridge and Spencer Reid from Criminal Minds exhibit distance, coldness, and lack of social awareness or empathy with others, and this is presented as the basis of their extraordinary ability to discern patterns and solve crime. Spencer Reid, for example, has three PhDs in Science disciplines and Mathematics. Charlie in the television series Numb3rs is also a genius who uses his mathematical abilities to not only find the solution to crime but also explain the maths behind it to his FBI colleagues, and, in conjunction, the audience. But the character with the clearest association to Christopher is, naturally, Sherlock Holmes, both as constructed in Conan Doyle’s original text and the current adaptations and transformations of it. The television series Sherlock and Elementary, as well as the films Sherlock Holmes and Sherlock Holmes: A Game of Shadows all invoke a version of Holmes in which his powers of deduction are associated with symptoms to be found in a spectrum disorder.Like Christopher, the classical detective is characterised by being cold, emotionless, distant, socially inept, and isolated, but also keenly observant, analytical, and scientific; one who approaches the crime as a puzzle to be solved (Cawelti 43) with computer-like precision. In what is considered to be the original detective story, The Murders in the Rue Morgue, Poe included a “pseudo-mathematical logic in his literary scenario” (Platten 255). In Conan Doyle’s stories, Holmes, too, adopts a mathematical and scientific approach to construct patterns from clues that he alone can discern, and thereby solve the crime. The depiction of investigators in contemporary media such as Charlie in Numb3rs engages with these origins so that he is objective, dispassionate, and able to relate to real-world problems only through the filter of mathematical formulae. Christopher is presented similarly by engaging with the idea of the detective as implied savant and relying on an ability to discern patterns for successful crime solving.The book links the disabling behaviours of autism with the savant, so that the stereotype of the mystic displaying both disability and giftedness in fiction of earlier ages has been transformed in contemporary literature to a figure with extraordinary powers related both to autism and to the contemporary form of mysticism: innate mathematical ability and computer-style calculation. Allied with what Murray terms the “unknown and ambiguous nature” of autism, it is characterised as “the alien within the human, the mystical within the rational, the ultimate enigma” (25) in a way that is in keeping with the current fascination with the nature of genius and its association with being “special,” a term continually evoked and discussed throughout the book by the main character. The chapters on scientific ideas relate to Christopher’s world view, filtered through a mathematical and analytical approach to life and relationships with other people. Christopher examines beliefs such as the concept of humanity as superior to other animals, and the idea of religion and creationism, that is, the idea of humanity itself as special, with a cold and logical approach. He similarly discusses the idea of the individual person as special, linking this to a metaphor of the human mind being a computer (203, 148). Christopher’s narrow perspective as a result of his autism is not presented as disabling so much as protective, because the metaphorical connection of his viewpoint to a computer provides him with distance. Although initially Christopher fails to realise the significance of events, this allows him to be “switched off” (103) from events that he finds traumatising.The transformative metaphor of an autistic individual thinking like a computer is also invoked through Christopher’s explanation of “why people think that their brains are special, and different from computers” (147). Indeed, both in terms of his tendency to retreat or by “pressing CTRL + ALT + DEL and shutting down programs and turning the computer off and rebooting” (178) in times of stress, Christopher metaphorically views himself as a computer. Such a perspective invokes yet another popular cultural reference through the allusion to the human brain as “Captain Jean-Luc Picard in Star Trek: The Next Generation, sitting in his captain’s seat looking at a big screen” (147). But more importantly, the explanation refers to the basic premise of the book, that the text offers access to a condition that is inherently unknowable, but able to be understood by the reader through metaphor, often based on computers or technology as a result of a popular construction of autism that “the condition is the product of a brain in which the hard drive is incorrectly formatted” (Murray 25).Throughout the novel, the notion of “special” is presented as a trope for those with a disability, but as the protagonist, Christopher, points out, everyone is special in some way, so the whole idea of a disability as disabling is problematised throughout the text, while its associations of giftedness are upheld. Christopher’s disability, never actually designated as Asperger’s Syndrome or any type of spectrum disorder, is transformed into a protective mechanism that shields him from problematic social relationships of which he is unaware, but that the less naïve reader can well discern. In this way, rather than a limitation, the main character’s disorder protects him from a harsh reality. Even Christopher’s choice of Holmes as a role model is indicative of his desire to impose an eccentric order on his world, since this engages with a character in popular fiction who is famous not simply for his abilities, but for his eccentricity bordering on a form of autism. His aloof personality and cold logic not only fail to hamper him in his investigations, but these traits actually form the basis of them. The majority of recent adaptations of Conan Doyle’s stories, especially the BBC series Sherlock, depict Holmes with symptoms associated with spectrum disorder such as lack of empathy, difficulty in communication, and limited social skills, and these are clearly shown as contributing to his problem-solving ability. The trope of Christopher as detective also allows a parodic, postmodern comment on the classical detective form, because typically this fiction has a detective that knows more than the reader, and therefore the goal for the reader is to find the solution to the crime before it is revealed by the investigator in the final stages of the text (Rzepka 14). But the narrative works ironically in the novel since the non-autistic reader knows more than a narrator who is hampered by a limited worldview. From the beginning of the book, the narrative as focalised through Christopher’s narrow perspective allows a more profound view of events to be adopted by the reader, who is able to read clues that elude the protagonist. Christopher is well aware of this as he explains his attraction to the murder mystery novel, even though he has earlier stated he does not like novels since his inability to imagine or empathise means he is unable to relate to their fiction. For him, the genre of murder mystery is more akin to the books on maths and science that he finds comprehensible, because, like the classical detective, he views the crime as primarily a puzzle to be solved: as he states, “In a murder mystery novel someone has to work out who the murderer is and then catch them. It is a puzzle. If it is a good puzzle you can sometimes work out the answer before the end of the book” (5). But unlike Christopher, Holmes invariably knows more about the crime, can interpret the clues, and find the pattern, before other characters such as Watson, and especially the reader. In contrast, in The Curious Incident of the Dog in the Night-Time, the reader has more awareness of the probable context and significance of events than Christopher because, like a computer, he can calculate but not imagine. The reader can interpret clues within the plot of the story, such as the synchronous timing of the “death” of Christopher’s mother with the breakdown of the marriage of a neighbour, Mrs Shears. The astute reader is able to connect these events and realise that his mother has not died, but is living in a relationship with the neighbour’s husband. The construction of this pattern is denied Christopher, since he fails to determine their significance due to his limited imagination. Such a failure is related to Simon Baron-Cohen’s Theory of Mind, in which he proposes that autistic individuals have difficulty with social behaviour because they lack the capacity to comprehend that other people have individual mental states, or as Christopher terms it, “when I was little I didn’t understand about other people having minds” (145). Haddon utilises fictional licence when he allows Christopher to overcome such a limitation by a conscious shift in perspective, despite the specialist teacher within the text claiming that he would “always find this very difficult” (145). Christopher has here altered his view of events through his modelling both on the detective genre and on his affinity with mathematics, since he states, “I don’t find this difficult now. Because I decided that it was a kind of puzzle, and if something is a puzzle there is always a way of solving it” (145). In this way, the main character is shown as transcending symptoms of autism through the power of his giftedness in mathematics to ultimately discern a pattern in human relationships thereby adopting a computational approach to social problems.Haddon similarly explains the perspective of an individual with autism through a metaphor of Christopher’s memory being like a DVD recording. He is able to distance himself from his memories, choosing “Rewind” and then “Fast Forward” (96) to retrieve his recollection of events. This aspect of the precision of his memory relates to his machine-like coldness and lack of empathy for the feelings of others. But it also refers to the stereotype of the nerd figure in popular culture, where the nerd is able to relate more to a computer than to other people, exemplified in Sheldon from the television series The Big Bang Theory. Thus the presentation of Christopher’s autism relates to his giftedness in maths and science more than to areas that relate to his body. In general, descriptions of inappropriate or distressing bodily functions associated with disorders are mainly confined to other students at Christopher’s school. His references to his fellow students, such as Joseph eating his poo and playing in it (129) and his unsympathetic evaluation of Steve as not as clever or interesting as a dog because he “needs help to eat his food and could not even fetch a stick” (6), make a clear distinction between him and the other children, who despite being termed “special needs” are “special” in a different way from Christopher, because, according to him, “All the other children at my school are stupid” (56). While some reference is made to Christopher’s inappropriate behaviour in times of stress, such as punching a fellow student, wetting himself while on the train, and vomiting outside the school, in the main the emphasis is on his giftedness as a result of his autism, as displayed in the many chapters where he explains scientific and mathematical concepts. This is extrapolated into a further mathematical metaphor underlying the book, that he is like one of the prime numbers he finds so fascinating, because prime numbers do not fit neatly into the pattern of the number system, but they are essential and special nevertheless. Moreover, as James Berger suggests, prime numbers can “serve as figures for the autistic subject,” because like autistic individuals “they do not mix; they are singular, indivisible, unfactorable” yet “Mathematics could not exist without these singular entities that [. . .] are only apparent anomalies” (271).Haddon therefore offers a transformation by confounding autism with a computer-like ability to solve mathematical problems, so that the text is, as Haddon concedes, “as much about a gifted boy with behavior problems as it is about anyone on the autism spectrum” (qtd. in Burks-Abbott 291). Indeed, the word “autism” does not even appear in the book, while the terms “genius,” (140) “clever,” (32, 65, 252) and the like are continually being invoked in descriptions of Christopher, even if ironically. More importantly, the reader is constantly being shown his giftedness through the reiteration of his study of A Level Mathematics, and his explanation of scientific concepts. Throughout, Christopher explains aspects of mathematics, astrophysics, and other sciences, referring to such well-known puzzles in popular culture as the Monty Hall problem, as well as more obscure formulae and their proofs. They function to establish Christopher’s intuitive grasp of complex mathematical and scientific principles, as well as providing the reader with insight into both his perspective and the paradoxical nature of an individual who is at once able to solve quadratic equations in his head, yet is incapable of understanding the simple instruction, “Take the tube to Willesden Junction” (211).The presentation of Christopher is that of an individual who displays an extension of the social problems established in popular literature as connected to a talent for mathematics, therefore engaging with a depiction already existing in popular mythology: the isolated and analytical nerd or genius social introvert. Indeed, much of Christopher’s autistic behaviour functions to protect him from unsettling or traumatic information, since he fails to realise the significance of the information he collects or the clues he is given. His disability is therefore presented as not limiting so much as protective, and so the notion of disability is subsumed by the idea of the savant. The book, then, engages with a contemporary representation within popular culture that has transformed spectrum disability into mathematical giftedness, thereby metaphorically associating the autistic mind with the computer. ReferencesBaron-Cohen, Simon. Mindblindness: An Essay on Autism and Theory of Mind. Cambridge MA: MIT Press, 1995. Berger, James. “Alterity and Autism: Mark Haddon’s Curious Incident in the Neurological Spectrum.” Autism and Representation. Ed. Mark Osteen. Hoboken: Routledge, 2007. 271–88. Burks-Abbott, Gyasi. “Mark Haddon’s Popularity and Other Curious Incidents in My Life as an Autistic.” Autism and Representation. Ed. Mark Osteen. Hoboken: Routledge, 2007. 289–96. Cawelti, John G. Adventure, Mystery, and Romance: Formula Stories as Art and Popular Culture. Chicago: U of Chicago P, 1976. Eshelman, Raoul. “Transcendence and the Aesthetics of Disability: The Case of The Curious Incident of the Dog in the Night-Time.” Anthropoetics: The Journal of Generative Anthropology 15.1 (2009). Haddon, Mark. The Curious Incident of the Dog in the Night-Time. London: Random House Children’s Books, 2004. Kendall, Lori. “The Nerd Within: Mass Media and the Negotiation of Identity among Computer-Using Men.” Journal of Men’s Studies 3 (1999): 353–67. Murray, Stuart. “Autism and the Contemporary Sentimental: Fiction and the Narrative Fascination of the Present.” Literature and Medicine 25.1 (2006): 24–46. Platten, David. “Reading Glasses, Guns and Robots: A History of Science in French Crime Fiction.” French Cultural Studies 12 (2001): 253–70. Rzepka, Charles J. Detective Fiction. Cambridge, UK: Polity Press, 2005.