BackgroundThe COVID-19 pandemic has created ethical challenges for intensive care unit (ICU) professionals, potentially causing moral distress. This study explored the levels and causes of moral distress and the ethical climate in Dutch ICUs during COVID-19.MethodsAn extended version of the Measurement of Moral Distress for Healthcare Professionals (MMD-HP) and Ethical Decision Making Climate Questionnaire (EDMCQ) were online distributed among all 84 ICUs. Moral distress scores in nurses and intensivists were compared with the historical control group one year before COVID-19. ResultsThree (...) hundred forty-five nurses (70.7%), 40 intensivists (8.2%), and 103 supporting staff (21.1%) completed the survey. Moral distress levels were higher for nurses than supporting staff. Moral distress levels in intensivists did not differ significantly from those of nurses and supporting staff. “Inadequate emotional support for patients and their families” was the highest-ranked cause of moral distress for all groups of professionals. Of all factors, all professions rated the ethical climate most positively regarding the culture of mutual respect, ethical awareness and support. “Culture of not avoiding end-of-life-decisions” and “Self-reflective and empowering leadership” received the lowest mean scores. Moral distress scores during COVID-19 were significantly lower for ICU nurses (p < 0.001) and intensivists (p < 0.05) compared to one year prior.ConclusionLevels and causes of moral distress vary between ICU professionals and differ from the historical control group. Targeted interventions that address moral distress during a crisis are desirable to improve the mental health and retention of ICU professionals and the quality of patient care. (shrink)

There are tensions within formal education between imparting knowledge and the development of skills for handling that knowledge. In the primary school sector, the latter can also be squeezed out of the curriculum by a focus on basic skills such as literacy and numeracy. What happens when an explicit attempt is made to develop young children's reasoning—both in terms of their apparent cognitive abilities and their basic skills? This paper reports an independent evaluation of an in-class intervention called ‘Philosophy for (...) Children’, after just over one year of schooling. The intervention aims to help children become more willing and able to question, reason, construct arguments and collaborate with others. A group of 48 volunteer schools were randomised to receive P4C or act as a control for one year. This paper reports the CAT results for all pupils in years 4 and 5 initially, and the Key Stage 2 attainment in English and Maths for those starting in year 5. There was no school dropout. Individual attrition from a total of 3,159 pupils was around 11 percent—roughly equal between groups. There were small positive ‘effect’ sizes in favour of the P4C group in progress in reading and maths, and even smaller perhaps negligible improvements in CAT scores and writing. The results for the most disadvantaged pupils were larger for attainment, but not for CATs. Observations and interviews suggest that the intervention was generally enjoyable and thought to be beneficial for pupil confidence. Our conclusion is that, for those wishing to improve attainment outcomes in the short term, an emphasis on developing reasoning is promising, especially for the poorest students, but perhaps not the most effective way forward. However, for those who value reasoning for its own sake, this evaluation demonstrates that using curriculum time in this way does not damage attainment, and so suggests that something like P4C is an appropriate educational approach. (shrink)

The goals of the present study were to test whether (and which) basic numerical abilities can be improved with training and whether training effects transfer to improvement in children’s math achievement. The literature is mixed with evidence that does or does not substantiate the efficacy of training basic numerical ability. In the present study, we developed a child-friendly software named ‘123 Bakery’ which includes four training modules; non-symbolic numerosity comparison, non-symbolic numerosity estimation, approximate arithmetic and symbol-to-numerosity mapping. Fifty-six first (...) graders were randomly assigned to either the training or control group. The training group participated in 6 weeks of training (5 times a week, 30 minutes per day). All participants underwent pre- and post- training assessment of their basic numerical processing ability (including numerosity discrimination acuity, symbolic/non-symbolic magnitude estimation, approximate arithmetic, symbol-to-numerosity mapping), overall math achievement and intelligence, 6 weeks apart. The acuity for numerosity discrimination (approximate number sense acuity; ANS acuity) significantly improved after training, but this training effect did not transfer to improvement in symbolic, exact calculation, or any other math ability. We conclude that basic numerical cognition training leads to improvement in ANS acuity, but whether this effect transfers to symbolic math ability remains to be further tested. (shrink)

This study aimed to examine the relationship between parents' intelligence mindset and children's math anxiety and the mediating role of parents' failure mindset and evaluations of their child's math performance. A total of 419 Chinese students and their parents were recruited to complete a series of questionnaires on topics such as math anxiety, parent's failure mindset, parent's intelligence mindset, and parents' evaluations of their child's mathematical performance. The results revealed that parents' intelligence mindset was not correlated with (...) children's math anxiety. However, parents' intelligence mindset indirectly predicted children's math anxiety through the chain-mediated role of parents' failure beliefs and parents' evaluations of their child's math performance. Further, sex differences were found through a multigroup analysis, which showed a chain-mediated effect between parents' intelligence mindset and girls' math anxiety. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Rembrandt’s Art: A Paradigm for Critical Thinking and AestheticsMark S. Conn (bio)IntroductionThe purpose of art is to lay bare the questions, which have been hidden by the answers.—James BaldwinPhilosophers have asked, How do we know the world? Over centuries, many visual artists have responded to this question by provoking us to see the world differently—through their own eyes. Rembrandt, by no small measure, is one of those artists. While (...) many people have attempted to understand and interpret the art of this quintessential master, few have used his art as a tool to teach critical thinking. The challenge, then, is to set forth a paradigm, using Rembrandt’s art as the vehicle, to teach critical thinking skills. These skills can be applied across the curriculum without losing an appreciation for the aesthetic characteristics of his art.Several sections in this article begin with a foundational discussion of the connection between art and the general curriculum, including how the effectiveness of that curriculum may then be measured. Continuing with a working definition of critical thinking, I demonstrate how Rembrandt’s work relates particularly well to the social studies curriculum. Definitions of critical thinking skills are discussed just prior to introducing the paradigm. A discussion of the paradigm’s relationship to aesthetics and necessary concepts is followed by the conclusion, which contains the graphic illustration of the paradigm. [End Page 68]Foundational DiscussionIn order to create a paradigm that encompasses Rembrandt’s art, critical thinking, and aesthetics, it is important to understand the connection that art has to the curriculum. By examining the way in which this relationship has evolved and is measured, we discover its foundation. It also produces a rationale for the paradigm.For decades the educational community has debated the validity of including arts in the curriculum. There are those who believe that in order to justify inclusion in the curriculum, there needs to be evidence that art has a quantifiable impact upon learning. Others believe that there is a need to teach art for art’s sake.1 Elliot Eisner stated, “More often than we would like, arts educators receive requests to justify their professional existence or existence of the arts in our schools on the basis of their contribution to non-art outcomes.”2 According to Eisner’s research, there have been no studies that show a statistically significant relationship between the arts and academic success.3 Additionally, in their final report evaluators from Transforming Education Through the Arts Challenge (TETAC) concluded that “Analysis of site-specific data found no significant relationship between program implementation and performance on reading and math tests.”4 There are many sources that can be cited that report the same findings. In assessing these kinds of findings we find the conclusions to be approximately the same.If we look at the U.S. Department of Education’s National Assessment of Educational Progress, which each year releases “The Nation’s Report Card,” we find a tremendous amount of data.5 However, most of what we find is data that is analyzed scientifically, with a long but incomplete list of incomplete variables. Therefore, it is difficult to come to any consensus about which programs, in which states, are successful and why. All we actually see is who is supposedly doing better in various curriculum areas based upon standardized tests that vary from state to state. Consider an example of current thinking regarding curriculum and testing from an article by Diane Ravitch, former assistant U.S. secretary of education, and Deborah Meir, senior scholar at New York University’s Steinhardt School of Education:Both of us acknowledged that our choices involve risks. A national curriculum might be unwieldy and superficial (“a mile wide and an inch deep—ironically the charge directed at our current incoherent and fragmented curriculum”) as well as politically compromised, while a local one might reflect the low expectations of the local community as local foolishness and local biases (some schools, for example, might teach intelligent design). We agreed that the measurement of “results”—what constitutes intellectual achievement—has been badly distorted by current local and state tests, which undermine high-quality tasks and make... (shrink)

In J.M. Coetzee’s novel The Schooldays of Jesus (2016), the question of finding the ‘right education’ for a young child is a central and recurring theme. In particular, the novel presents us with t...

We prove Fraïssé’s conjecture within the system of Π11-comprehension. Furthermore, we prove that Fraïssé’s conjecture follows from the Δ20-bqo-ness of 3 over the system of Arithmetic Transfinite Recursion, and that the Δ20-bqo-ness of 3 is a Π21-statement strictly weaker than Π11-comprehension.

Children’s math self-concepts—their beliefs about themselves and math—are important for teachers, parents, and students, because they are linked to academic motivation, choices, and outcomes. There have been several attempts at improving math achievement based on the training of math skills. Here we took a complementary approach and conducted an intervention study to boost children’s math self-concepts. Our primary objective was to assess the feasibility of whether a novel multicomponent intervention—one that combines explicit and implicit approaches (...) to help children form more positive beliefs linking themselves and math—can be administered in an authentic school setting. The intervention was conducted in Spain, a country in which math achievement is below the average of other OECD countries. We tested third grade students (N= 180;Mage= 8.79 years; 96 girls), using treatment and comparison groups and pre- and posttest assessments. A novelty of this study is that we used both implicit and explicit measures of children’s math self-concepts. For a subsample of students, we also obtained an assessment of year-end math achievement. Math self-concepts in the treatment and comparison groups did not significantly differ at pretest. Students in the treatment group demonstrated a significant increase in math self-concepts from pretest to posttest; students in the comparison group did not. In the treatment group, implicit math self-concepts at posttest were associated with higher year-end math achievement, assessed approximately 3 months after the completion of the intervention. Taken together, the results suggest that math self-concepts are malleable and that social–cognitive interventions can boost children’s beliefs about themselves and math. Based on the favorable results of this feasibility study, it is appropriate to formally test this novel multicomponent approach for improving math self-concepts using randomized controlled trial (RCT) design. (shrink)

Numbers. Natural numbers -- Integers -- Real numbers -- Irrational and transcendental numbers -- Funny numbers. Zero -- e : the unnatural natural number -- [Phi] : the golden ratio -- i : the imaginary number -- Writing numbers. Roman numerals -- Egyptian fractions -- Continued fractions -- Logic. Mr. Spock is not logical -- Proofs, truth, and trees : oh my! -- Programming with logic -- Temporal reasoning -- Sets. Cantor's diagonalization : infinity isn't just infinity -- Axiomatic set (...) theory : keep the good, dump the bad -- Models : using sets as the LEGOs of the math world -- Transfinite numbers : counting and ordering infinite sets -- Group theory : finding symmetries with sets -- Mechanical math. Finite state machines : simplicity goes far -- The turing machine -- Pathology and the heart of computing -- Calculus : no, not that calculus- [lambda] calculus -- Numbers, booleans, and recursion -- Types, types, types : modeling [lambda] calculus -- The halting problem. (shrink)

If is a random sequence, then the sequence is clearly not random; however, seems to be “about half random”. L. Staiger [Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 103 159–194 and A tight upper bound on Kolmogorov complexity and uniformly optimal prediction, Theory Comput. Syst. 31 215–229] and K. Tadaki [A generalisation of Chaitin’s halting probability Ω and halting self-similar sets, Hokkaido Math. J. 31 219–253] have studied the degree of randomness of sequences or reals by measuring their (...) “degree of compression”. This line of study leads to various definitions of partial randomness. In this paper we explore some relations between these definitions. Among other results we obtain a characterisation of Σ1-dimension in terms of strong Martin-Löf ε-tests , and we show that ε-randomness for ε is different than the classical 1-randomness. (shrink)

Long ago, when Alexander the Great asked the mathematician Menaechmus for a crash course in geometry, he got the famous reply ``There is no royal road to mathematics.'' Where there was no shortcut for Alexander, there is no shortcut for us. Still, the fact that we have access to computers and mature programming languages means that there are avenues for us that were denied to the kings and emperors of yore. The purpose of this book is to teach logic and (...) mathematical reasoning in practice, and to connect logical reasoning with computer programming in Haskell. Haskell emerged in the 1990s as a standard for lazy functional programming, a programming style where arguments are evaluated only when the value is actually needed. Haskell is a marvelous demonstration tool for logic and maths because its functional character allows implementations to remain very close to the concepts that get implemented, while the laziness permits smooth handling of infinite data structures. This book does not assume the reader to have previous experience with either programming or construction of formal proofs, but acquaintance with mathematical notation, at the level of secondary school mathematics is presumed. Everything one needs to know about mathematical reasoning or programming is explained as we go along. After proper digestion of the material in this book, the reader will be able to write interesting programs, reason about their correctness, and document them in a clear fashion. The reader will also have learned how to set up mathematical proofs in a structured way, and how to read and digest mathematical proofs written by others. This is the updated, expanded, and corrected second edition of a much-acclaimed textbook. Praise for the first edition: 'Doets and van Eijck's ``The Haskell Road to Logic, Maths and Programming'' is an astonishingly extensive and accessible textbook on logic, maths, and Haskell.' Ralf Laemmel, Professor of Computer Science, University of Koblenz-Landau. (shrink)

_'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.'_ _– The Guardian_ In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from maths (...) to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel’s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. A gripping combination of science and accessibility, _Godel’s Proof_ by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity. _Kurt Godel_ Born in Brunn, he was a colleague of physicist Albert Einstein and professor at the Institute for Advanced Study in Princeton, N.J. (shrink)

Sabine Hossenfelder’s recent book “Lost in Math” has attracted numerous responses, including by notable physicists such as Frank Wilczek. In this article we focus on Wilczek’s remark on that book, in particular on the perils of postempirical science. We also discuss shortly multiverse hypothesis from philosophical perspective. In last section, we offer a resolution from the perspective of Neutrosophic Logic on this problem of classical tension between mathematics and experience approach to physics, which seems to cause the stagnation of (...) modern physics. (shrink)

In a previous study we examined reading, spelling, and maths skills in an unselected group of 129 Italian children attending fifth grade by testing various cognitive predictors; results showed a high degree of predictors’ selectivity for each of these three behaviors. In the present study, we focused on the specificity of the predictors by performing cross-analyses on the same dataset; i.e., we predicted spelling and maths skills based on reading predictors, reading based on maths predictors and so on. Results indicated (...) that some predictors, such as the Orthographic Decision and the Arithmetic Facts tests, predicted reading, spelling and maths skills in similar ways, while others predicted different behaviors but only for a specific parameter, such as fluency but not accuracy, and still others were specific for a single behavior. To interpret these results, we propose a novel model of learning skills separately considering factors in terms of competence, performance and acquisition. Reading, spelling and calculation skills would depend on the development of discrete and different abstract competences. By contrast, overlap among behaviors would be accounted for by defective acquisition in automatized responses to individual “instances”; this latter skill is item specific but domain independent. Finally, performance factors implied in task’s characteristics may contribute to the partial association among learning skills. It is proposed that this new model may provide a useful base for interpreting the diffuse presence of comorbidities among learning disorders. (shrink)

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. Yanofsky describes (...) simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there. (shrink)

In these days, there is an increasing technological development in intelligent tutoring systems. This field has become interesting to many researchers. In this paper, we present an intelligent tutoring system for teaching mathematics that help students understand the basics of math and that helps a lot of students of all ages to understand the topic because it's important for students of adding and subtracting. Through which the student will be able to study the course and solve related problems. An (...) evaluation of the intelligent tutoring systems was carried out and the results were encouraging. (shrink)

We discuss the nature of reality in the ontological context of Penrose’s math-matter-mind triangle. The triangle suggests the circularity of the widespread view that math arises from the mind, the mind arises out of matter, and that matter can be explained in terms of math. Non-physicists should be wary of any claim that modern physics leads us to any particular resolution of this circularity, since even the sample of three theoretical physicists writing this paper hold three divergent (...) views. Some physicists believe that current physics has already found the basic framework for a complete description of reality, and only has to fill in the details. Others suspect that no single framework, from physics or other sources, will ever capture reality. Yet others guess that reality might be approached arbitrarily closely by some form of future physics, but probably based on completely different frameworks. We will designate these three approaches as the fundamentalist, secular and mystic views of the world, as seen by practicing physicists. We present and contrast each of these views, which arguably form broad categories capturing most if not all interpretations of physics. We argue that this diversity in the physics community is more useful than an ontological monoculture, since it motivates physicists to tackle unsolved problems with a wide variety of approaches. (shrink)

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, (...) 1970) that if T is a Π 1 1 set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model. (shrink)

We attribute three major insights to Hegel: first, an understanding of the real numbers as the paradigmatic kind of number ; second, a recognition that a quantitative relation has three elements, which is embedded in his conception of measure; and third, a recognition of the phenomenon of divergence of measures such as in second-order or continuous phase transitions in which correlation length diverges. For ease of exposition, we will refer to these three insights as the R First Theory, Tripartite Relations, (...) and Divergence of Measures. Given the constraints of space, we emphasize the first and the third in this paper. (shrink)

_'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.'_ _– The Guardian_ In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from maths (...) to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel’s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. A gripping combination of science and accessibility, _Godel’s Proof_ by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity. _Kurt Godel_ Born in Brunn, he was a colleague of physicist Albert Einstein and professor at the Institute for Advanced Study in Princeton, N.J. (shrink)

Decades of research have focused on children's reasoning about math equivalence problems for both practical and theoretical insights. Not only are math equivalence problems foundational in arithmetic and algebra, they also represent a class of problems on which children's thinking is resistant to change. Feedback is one instructional tool that can serve as a key trigger of cognitive change. In this paper, we review all experimental studies on the effects of feedback on children's understanding of math equivalence. (...) Meta-analytic results indicate that feedback has positive effects for low-knowledge learners and negative effects for high-knowledge learners, and these effects are stronger for procedural outcomes than conceptual outcomes. Findings highlight the variable influences of feedback on math equivalence understanding and suggest that models of thinking and reasoning need to consider learner characteristics, learning outcomes, and learning materials, as well as the dynamic interactions among them. (shrink)

Descartes’s arguments about the certainty of mathematics in the Rules for the Direction of the Mind cannot be understood independently of his attack on the authority of ancient authors. The author maintains this view by reading Descartes’s claims about mathematics through the lens of status theory, a framework for disputation revived by Renaissance dialecticians. Within status theory, “certainty” was closely associated with consensus. The essay shows how Descartes used status to attack the authority of the ancient authors and elevate mathematics (...) as an alternative. Descartes’s arguments about the certainty of mathematics are colored by his effort to make mathematics satisfy the demands of status-based disputation. Specifically, in order to elevate the status of mathematics, Descartes portrayed it as more common—prone to consensus—than he really believed it was. (shrink)

The current study aimed to investigate the extent to which familial history of reading and math difficulties have an impact on children’s academic outcomes within a 3-year longitudinal study, which evaluated their core reading and math skills after first and second grades, as well as performance on complex academic tasks after second and third grades. At baseline, parents were asked to complete the Adult Reading History Questionnaire and its adaption, Adult Math History Questionnaire, to index familial history (...) of reading and math difficulties, respectively. Preliminary findings established the psychometric properties of the AMHQ, suggesting that it is a reliable and valid scale. Correlation analyses indicated that the ARHQ was negatively associated with children’s reading skills, whereas the AMHQ was negatively related to math outcomes. Path results revealed that the ARHQ predicted children’s performance on complex reading tasks indirectly via their core reading skills, and the AMHQ was linked to complex math outcomes indirectly via core math abilities. The ARHQ was also found to be negatively correlated with measures of children’s math performance, with path findings suggesting that these relations were indirectly explained by differences in their core reading skills. These results suggest that assessing familial risk for academic difficulties may be crucial to understanding comorbid etiological and developmental associations between reading and math differences. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Math Anxiety:Making Room to BreatheValerie Allen (bio) and Todd Stambaugh (bio)"Don't do that to me, Professor," the student said, and everybody laughed, for by this late in the semester, the atmosphere was relaxed. The instructor in question had just reached the point in a worked problem when they could move from reasoning about specific numbers to stating a general principle: x≤y≤z, meaning that y—the value we sought—was always (...) going to be greater than or equal to one value (x) and less than or equal to the other (z). The student who had spoken had earned everyone's affection over the weeks, open about their difficulties with math, ready to risk being wrong, and to ask obvious questions that others may have been reluctant to owning. They had hung in there, despite presenting a classic case of math anxiety, a syndrome long recognized in the scholarship of teaching and learning.Among math anxiety's many symptoms, breathing disruption ranks high: hyperventilation (rapid breathing) and hypoventilation (barely or not breathing). When the energy-conserving parasympathetic nervous system is overtaken by the energy-expending sympathetic nervous system, the long exhalations that decelerate the heart rate are replaced by short, panting puffs from the upper lungs. The timed test provides the perfect environment for math anxiety to flourish. Stress diverts cognitive resources needed for procedural execution, so working memory consequently shrinks.Self-rated metrics for math anxiety have been in place for decades (Richardson and Suinn). Relatively trivial interventions can yield nontrivial results: deep breathing exercises and positive affirmations have a beneficial impact, as does writing about one's nerves immediately prior to an exam (Brunyé et al.). More systemic changes require a reorientation in teaching whereby conceptual mistakes are positively rewarded as opportunities for learning; students compose questions instead of answers; students solve problems together rather than perform solo (Boaler).Return now to that opening scenario to watch it become stereotype incarnate: the student was mature, Black, and female; the teacher, younger, white, and male; her comment was triggered by his shift from a specific [End Page 217] set of numbers into symbolic notation. There is robust evidence to demonstrate that certain demographics are consistently underrepresented in the U.S. math community: women and "African-Americans, Hispanics, American Indians, Alaska Natives, and Pacific Islanders," according to the Mathematical Association of America (MAA) ("Resources for Minority"). Routinely feeling that they do not belong, such groups are particularly vulnerable to math anxiety.Several remedies for such disparity suggest themselves: appointing more women and people of color in the profession; one-on-one tutorial assistance; research projects about the students' personal environment; privileging pre-test, low-stakes work over timed tests; assigning group rather than individual work. When all is said and done, however, such strategies "have little to do with mathematics content" (Martin 304). Mathematicians still need to be able to conceptualize, describe, and manipulate abstract objects, and there seems to be a suspicion that, at heart, abstraction is gendered masculine and colored white.It is possible that the rhetorical immediacy and context-dependence of Black English has misguidedly reinforced such a perception. For poet June Jordan, Black English "devolves from a culture that abhors all abstraction" (Jordan 367). Psychiatrist Alvin Poussaint asks, "Is there a way to say metamorphosis in Ebonics?" (O'Malley). Yet, despite Black English's avoidance of nominalization and similar verbosities that afflict academic prose, abstract logical reasoning flourishes in vernacular idiom, as linguist William Labov demonstrated half a century ago. Our aim is not to explain the asymptote in Black English but rather to consider a feature characteristic of those very minority cultures historically underrepresented in math—storytelling—and to ask how it may be foregrounded in mathematics, where it is decisively sidelined. Storytelling has been claimed by every culture and ethnic group at some point as the repository of wisdom, practical skills, religious beliefs, and shared identity and values; but when it gets corralled into the category of "fiction," and when it gets opposed to "fact," a narrowed concept of truth dispatches storytelling to kindergarten and recreational math. Word problems may supply scenarios but offer none of a plot's satisfactions... (shrink)

_'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.'_ _– The Guardian_ In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from maths (...) to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel’s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. A gripping combination of science and accessibility, _Godel’s Proof_ by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity. _Kurt Godel_ Born in Brunn, he was a colleague of physicist Albert Einstein and professor at the Institute for Advanced Study in Princeton, N.J. (shrink)

In this paper a Knowledge-Based System (KBS) for determining the appropriate students major according to his/her preferences for sophomore student enrolled in the Faculty of Engineering and Information Technology in Al-Azhar University of Gaza was developed and tested. A set of predefined criterions that is taken into consideration before a sophomore student can select a major is outlined. Such criterion as high school score, score of subject such as Math I, Math II, Electrical Circuit I, and Electronics I (...) taken during the student freshman year, number of credits passed, student cumulative grade point average of freshman year, among others, were then used as input data to KBS. KBS was designed and developed using Simpler Level Five (SL5) Object expert system language. KBS was tested on three generation of sophomore students from the Faculty of Engineering and Information Technology of the Al-Azhar University, Gaza. The results of the evaluation show that the KBS is able to correctly determine the appropriate students major without errors. (shrink)

Explore how mathematics has shaped how we look at and understand the world. From figuring out Earth's circumference using geometry to developing load-bearing arches, young readers will be engaged as they learn about real world math applications.

This study aimed to explore the influence of the visuospatial active working memory sub-components on early math skills in young children, followed longitudinally along the first two years of primary school. We administered tests investigating visual active working memory (jigsaw puzzle), spatial active working memory (backward Corsi), and math tasks to 43 children at the beginning of first grade (T1), at the end of first grade (T2), and at the end of second grade (T3). Math tasks were (...) select according to the children’s age and their levels of formal education: the “Battery for the evaluation of numerical intelligence from 4 to 6 years of age” (BIN 4–6) at T1 to test early numerical competence and the “Test for the evaluation of calculating and problem-solving abilities” (AC-MT 6–11) to test math skills at T2 and T3. Three regression models, in which the predictors were identified through a forward selection based on the use of the Bayesian information criterion (BIC) index, were performed to study the relationship between visual and spatial working memory and math ability at the three points in time. The results show that spatial working memory influences early numerical performance at T1, while early numerical performance is the unique predic-tor of math performance at T2. At the end of the second grade, the regression model reveals a relationship between math performance and both visual and spatial working memory and the attenuation of the importance of domain-specific predictors. The study depicts the dif-ferent implications of visual and spatial working memory predictors over the children’s de-velopment periods and brings additional evidence to the debate on the relationship between visuospatial working memory and math ability in young children. (shrink)

A beneficial effect of gesture on learning has been demonstrated in multiple domains, including mathematics, science, and foreign language vocabulary. However, because gesture is known to co‐vary with other non‐verbal behaviors, including eye gaze and prosody along with face, lip, and body movements, it is possible the beneficial effect of gesture is instead attributable to these other behaviors. We used a computer‐generated animated pedagogical agent to control both verbal and non‐verbal behavior. Children viewed lessons on mathematical equivalence in which an (...) avatar either gestured or did not gesture, while eye gaze, head position, and lip movements remained identical across gesture conditions. Children who observed the gesturing avatar learned more, and they solved problems more quickly. Moreover, those children who learned were more likely to transfer and generalize their knowledge. These findings provide converging evidence that gesture facilitates math learning, and they reveal the potential for using technology to study non‐verbal behavior in controlled experiments. (shrink)

The purpose of this paper is to show that the mathematics of quantum mechanics is the mathematics of set partitions linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite (...) to more definite states is the partition join operation at the set level that prefigures at the quantum level projective measurement as well as the formation of maximally-definite state descriptions by Dirac’s Complete Sets of Commuting Operators. This development is measured quantitatively by logical entropy at the set level and by quantum logical entropy at the quantum level. This follow-the-math approach supports the Literal Interpretation of QM—as advocated by Abner Shimony among others which sees a reality of objective indefiniteness that is quite different from the common sense and classical view of reality as being “definite all the way down”. (shrink)

How math helps us solve the universe's deepest mysteries You must be able to test any physical theory in the real world. To most physicists, this is obvious. But since the 1980s, experimental physics has yielded vanishingly little insight into the fundamental physics of the universe. Meanwhile, some physicists have begun to probe the universe not with proton beams, but with pure math. They're less concerned with testable theories than with the drive to explain nature with mathematical beauty. (...) This approach is often pilloried by traditional scientists, who point out that such approaches have yet to make a correct prediction about the real world. But in The Universe Speaks in Numbers, Graham Farmelo offers a gripping tour of the history of math-based physics and explores why it may be the key to the next big breakthrough in our understanding of the nature of reality. (shrink)

Mathematics, even more than the other sciences, is often presented as essentially unique, as if it could not be any other way. And yet, prima facie alternative mathematics are all over the place, from non-Western mathematics to mathematics based on nonclassical logics. Taking inspiration from Robin Dembroff’s analysis of critical gender kinds, and from Andrew Aberdein and Stephen Read’s analysis of alternative logics, in this paper I will introduce a practice-centered framework for the study of alternative mathematics based on the (...) notion of critical math kind. After sketching a model of mainstream mathematics, I will provide examples of how deviation along several distinct dimensions can occur, and how deviations can vary in their gravity. I will then discuss how the framework can be used to think of questions concerning the alternativeness status and philosophical implications of alleged alternative mathematics, and help us in identifying alternatives that suit our purposes. (shrink)

Where does math come from? From a textbook? From rules? From deduction? From logic? Not really, Eugenia Cheng writes in Is Math Real?: it comes from curiosity, from instinctive human curiosity, "from people not being satisfied with answers and always wanting to understand more." And most importantly, she says, "it comes from questions": not from answering them, but from posing them. Nothing could seem more at odds from the way most of us were taught math: a rigid (...) and autocratic model which taught us to follow specific steps to reach specific answers. Instead of encouraging a child who asks why 1+1 is 2, our methods of education force them to accept it. Instead of exploring why we multiply before we add, a textbook says, just to get on with the order of operations. Indeed, the point is usually just about getting the right answer, and those that are good at that, become "good at math" while those who question, are not. And that's terrible: These very same questions, as Cheng shows, aren't simply annoying questions coming from people who just don't "get it" and so can't do math. Rather, they are what drives mathematical research and push the boundaries in our understanding of all things. Legitimizing those questions, she invites everyone in, whether they think they are good at math or not. And by highlighting the development of mathematics outside Europe, Cheng shows that-western chauvinism notwithstanding--that math can be for anyone who wishes to do it, and how much we gain when anyone can. (shrink)

Summary This paper reports on individual interviews with 175 children, from 33 inner London junior schools, at 11 years of age. The children were part of a large?scale longitudinal study of educational progress based at the Thomas Coram Research Unit, and were also interviewed at 7 years. Children were heavily aware of the importance of school work, which figured prominently in views on the best and worst things about school. Only 17% were not looking forward to secondary school. They were (...) looking forward to specialising in subjects not provided in depth at primary school. Maths was much the most popular subject, as it was at 7 years. There was no evidence that expressing a liking for a subject, or expressing a general interest in school, was associated with higher achievements. Success or not in academic work was perceived to be largely in children's own hands; external attributions were rare. Differences according to ethnic group and sex of children are discussed. (shrink)

Curiosity is widely acknowledged as a crucial aspect of children’s development and as an important part of the learning process, with prior research showing associations between curiosity and achievement. Despite this evidence, there is little research on the development of curiosity or on promoting curiosity in school settings, and measures of curiosity promotion in the classroom are absent from the published literature. This article introduces the Curiosity in Classrooms Framework coding protocol, a tool for observing and coding instructional practices that (...) support the promotion of curiosity. We describe the development of the framework and observation instrument and the results of a feasibility study using the protocol, which gives a descriptive overview of curiosity-promoting instruction in 35 elementary-level math lessons. Our discussion includes lessons learned from this work and suggestions for future research using the developed observation tool. (shrink)

In this paper, I discuss the social philosopher Pierre Bourdieu’s concept of habitus, and use it to locate and examine dispositions in a larger constellation of related concepts, exploring their dynamic relationship within the social context, and their construction, manifestation, and function in relation to classroom mathematics practices. I describe the main characteristics of habitus that account for its invisible effects: its embodiment, its deep and pre-reflective internalization as schemata, orientation, and taste that are learned and yet unthought, and are (...) subsumed by our practices, which we understand as something that “goes without saying.” I also propose that, similarly to Bourdieu’s concept of linguistic habitus, a math habitus is made up of a complex intertwining of collective and individual histories that turn into “nature,” which structure all individual and collective action and inform mathematical classroom practice. I suggest that individual math dispositions may be liable to reconstruction through the reconstruction of the collective math habitus, which follows from opening spaces for dialogue, problematization and reconstruction of the unthought categories of the doxa. This requires that students acquire new concrete and symbolic means with which to challenge their current sense of mathematics as a discipline, and mathematical practice tout court. Finally, I argue that community of inquiry, employed as a pedagogical model, provides an avenue for both: for opening those spaces for reflective dialogical inquiry into concepts and questions whose meanings and references have so far been taken for granted, and for acquiring critical thinking and dialogical skills and dispositions that are a necessary means for participating in such reflective inquiry that offers significant promise for reconstructing individual and collective habitus in school settings. (shrink)

The starting point of this paper is a statement that Benjamin makes in a group of notes he writes for his project of a detective novel. Benjamin writes here that «criminal and detective could be so friends [so befreundet sein] as Sherlock Holmes and Watson». We’ll try to understand the meaning of this statement through the investigation of the detective topic in two moments of its fore and afterlife: its fore life in Benjamin’s meditation on the baroque and its after (...) life in Sherlock Holmes’s most recent apparition, in the BBC series Sherlock. One of the most interesting elements of this series is in fact the relationship – which is barely sketched in Conan Doyle’s stories – between Sherlock Holmes and his antagonist, the maths professor Jim Moriarty. We’ll see that in Benjamin’s notes for a detective novel the criminal is not a maths professor but a psychoanalyst. What is the difference, anyway, between professor Moriarty’s knowledge and Sherlock’s knowledge? In fact, we will find out that criminal and detective are closer to one other than what we may be induced to think – close but not coincident. (shrink)

When learning a new math concept, should learners be first taught the concept and its associated procedures and then solve problems, or solve problems first even if it leads to failure and then be taught the concept and the procedures? Two randomized-controlled studies found that both methods lead to high levels of procedural knowledge. However, students who engaged in problem solving before being taught demonstrated significantly greater conceptual understanding and ability to transfer to novel problems than those who were (...) taught first. The second study further showed that when given an opportunity to learn from the failed problem-solving attempts of their peers, students outperformed those who were taught first, but not those who engaged in problem solving first. Process findings showed that the number of student-generated solutions significantly predicted learning outcomes. These results challenge the conventional practice of direct instruction to teach new math concepts and procedures, and propose the possibility of learning from one's own failed problem-solving attempts or those of others before receiving instruction as alternatives for better math learning. (shrink)

Learning disabilities are highly prevalent on college campuses, yet students with learning disabilities graduate at lower rates than those without disabilities. Academic and psychosocial supports are essential for overcoming challenges and for improving postsecondary educational opportunities for students with learning disabilities. A holistic, multi-level model of campus-based supports was established to facilitate culture and practice changes at the institutional level, while concurrently bolstering mentors’ abilities to provide learning disability-knowledgeable support, and simultaneously creating opportunities for students’ personal and interpersonal development. Mixed (...) methods were used to investigate implementation of coordinated personal, interpersonal, and institutional level supports for undergraduate science, technology, engineering, and math (STEM) students with learning disabilities. A one-group pre-test post-test strategy was used to examine undergraduate outcomes. Participants included 52 STEM undergraduates with learning disabilities, 57 STEM graduate student mentors, 34 STEM faculty mentors, and 34 university administrators and personnel as members of a university-wide council. Enrolled for two years, undergraduates were engaged in group meetings involving psychoeducation and reflective discussions, development of self-advocacy projects, and individual mentorship. Undergraduates reported improved self efficacy (p =.001), campus connection (p <.001), professional development (p ≤.002), and self advocacy (p <.001) after two academic years. Graduate student mentors increased their understanding about learning disabilities and used their understanding to support both their mentees and other students they worked with. Council members identified and created opportunities for delivering learning disability-related trainings to faculty, mentors and advisors on campus, and for enhancing coordination of student services related to learning and related disorders. Disability-focused activities became integrated in broader campus activities regarding diversity. This research explicates a role that college campuses can play in fostering the wellbeing and the academic and career development of its students with developmental learning and related disorders. It offers an empirically tested campus-based model that is multilevel, holistic, and strengths-based for supporting positive outcomes of young people with learning disabilities in STEM. Moreover, findings advance the knowledge of supports and skills that are important for self-regulating and navigating complex and multi-faceted disability-related challenges within both the post-secondary educational environment and the young adults’ sociocultural context. (shrink)