Professional Mathematicians Research Articles

Interplay between music and mathematics in the eyes of the beholder: focusing on differing types of expertise

This study explored the unique connections between music and mathematics as perceived by four groups of experts: professional mathematicians and musicians, as well as teacher educators in these two fields. Using 2 × 2 study design, we studied four groups of participants, comprising theorists and educators from various Israeli universities. During semi-structured interviews, the study participants were asked about their views on the connections between mathematics and music. This study proposes a model of experts’ conceptions of the connection between mathematics and music, which is of descriptive and explanatory power. that reveals differences between the four groups of experts. Theoreticians in both disciplines highlighted Mathematics as a key tool for music analysis and creation. Musical educators emphasized the role of music as a tool for learning mathematics. All the study participants, independently of the field of their expertise, value structure, beauty, sense of wonder, freedom and creative thinking as characteristics of both fields. Additionally, all the experts hold conceptions of the importance of integrating music and mathematics into various discipline. This study opened new doors for future research wherein utilization of experts’ insights to craft integrated study modules of music and mathematics can be explored, a pursuit that carries substantial significance.

Modeling of Some Classes of Extended Oscillators: Simulations, Algorithms, Generating Chaos, and Open Problems

In this article, we propose some extended oscillator models. Various experiments are performed. The models are studied using the Melnikov approach. We show some integral units for researching the behavior of these hypothetical oscillators. These will be implemented as add-on sections of a thoughtful main web-based application for researching computations. One of the main goals of the study is to share the difficulties that researchers (who are not necessarily professional mathematicians) encounter in using contemporary computer algebraic systems (CASs) for scientific research to examine in detail the dynamics of modifications of classical and newer models that are emerging in the literature (for the large values of the parameters of the models). The present article is a natural continuation of the research in the direction that has been indicated and discussed in our previous investigations. One possible application that the Melnikov function may find in the modeling of a radiating antenna diagram is also discussed. Some probability-based constructions are also presented. We hope that some of these notes will be reflected in upcoming registered rectifications of the CAS. The aim of studying the design realization (scheme, manufacture, output, etc.) of the explored differential models can be viewed as not yet being met.

Ramanujan Summation for Pascal’s Triangle

The concept of Pascal’s triangle has fascinated not only professional mathematicians but also everyone interested in exploring science. Similarly, the idea of Ramanujan summation has made a revolution in mathematical research after it was introduced by Srinivasa Ramanujan. In this paper, we will provide the Ramanujan summation methods for numbers located in slant diagonals of Pascal’s triangle and derive a generalized formula for such summations.

Fermat's Two Page Claim of his Last Problem Verified

In the 1650s Fermat claimed a short two-page proof of his last theorem but could not fit it into the margin. This article vindicates Fermat’s claim that has never been verified. The proof of Fermat’s claim has held up for over 360 years. The solution to the problem given in this article shows that the problem can be solved in two pages and clarifies why the problem resisted solution by thousands of mathematicians both professional and amateur. The author solved the problem in 1998 but opted not to publish it until recent years. However, the author featured the proof publicly in his website challenging anyone to find a flaw in the proof. No objections ever occurred. The proof uses only theorems and proof techniques available to Fermat. It is a devious proof that can be understood by a gifted high school student because it only uses theorems taught in high school along with an interesting algorithm devised by the author for the purpose of addressing infinity issues obstructing the finding of a simple two-page solution. The algorithm invented by the author ended the quest to solve the theorem and allows the viewer of the proof to understand why this problem has resisted so many attempts. The author believes this solution is the only one possible. The author Dr. Joseph E. L. Brierly is skeptical about the many-page solution claimed by Sir Andrew Wiles some years ago. Even professional mathematicians avoid attempting to verify pages of subtle reasoning. There is a strong likelihood of errors in proofs of that magnitude. As everyone knows a computer operating system invariably has multiple bugs due to the large number of lines of coding. In essence, there is no reason for an impractical-to-verify solution taking pages of subtle reasoning when there exists an easily reviewed two-page solution given in this research article. There exist few important theorems in mathematics and physics that require more than 5 pages of reasoning.

Design in the service of topology and the classification theorem

Topology is considered an advanced field in mathematics, and it might seem off-putting to people with no previous experience in mathematics. The classification theorem, which lies within the field of algebraic topology, is fascinating, but understanding it requires extensive mathematical knowledge. In this manuscript, we present a modular object that provides a glimpse into the construction of surfaces as defined by the classification theorem. This allows any user to explore different options and discover possible surfaces with three holes or fewer. This tactile experience of exploration grants the user intuition about the meaning of the theorem, without needing the rigorous proof; thus, it simplifies the depth of the classification theorem and its respective proof for non-mathematical audiences. We strive to accurately illustrate these esoteric concepts by giving them a form and to describe the modeling process of creating a tangible, touchable sculpture designed to intrigue and invite simple understanding. We believe that this playful object will be exciting for a variety of communities: university students, graduate students, teachers, professional mathematicians, and especially designers, who can use it to produce new tools.

Sums of the first n odd integers

There is a long tradition of mathematicians valuing collections of proofs, e.g. [1] is a famous recent collection. In the past, professional mathematicians have also written many different proofs of single important theorems, e.g. [2]. Proofs of the Pythagorean Theorem were collected by [3], and [4] discusses mathematical style through proofs of the irrationality of . The recent work of [5] is a discussion of mathematical style via a comparison of 99 different ‘proofs’ the following ‘theorem’.

Mathematical Creativity: Student Geometrical Figure Apprehension in Geometry Problem-Solving Using New Auxiliary Elements

<p style="text-align:justify">The definition of creativity among professional mathematicians and the definition of mathematical creativity in the classroom context are significantly different. The purpose of this study was to investigate the relationship between students’ mathematical creativity (i.e., cognitive flexibility) and figure apprehension when solving geometric problems with novel auxiliary features such as straight lines and curved lines. In other words, this study determined if geometry knowledge influenced mathematical creativity (cognitive flexibility) in problem-solving. Grade-12 students participated in the intervention. The high school that is the research topic attempts to equip students with academic abilities and is, except for vocational schools, the most popular form of high school among all other types. Such a school was chosen for the study so that a significant proportion of students in Makassar could be represented. In this study, we discovered a relationship between cognitive flexibility and the geometric ability of pupils while solving problems involving auxiliary lines. This indicates that the usage of auxiliary lines as a reference for developing pupils’ creative thinking skills must be advocated. In addition, good geometric abilities (e.g., visual thinking, geometrical reasoning) will encourage pupils to generate various problem-solving concepts. This finding contributes significantly to future research by focusing on auxiliary lines.</p>

Broadening views of mathematical creativity: Inclusion of the undergraduate student perspective

Numerous conceptions of creativity exist in the literature; yet these are commonly based on the perspectives of professional mathematicians. Including students’ perspectives in creativity is crucial not only for a more robust picture of what it means to be creative but also to combat damaging dominant narratives about who can be creative. We examined calculus students’ views of mathematical creativity, a group not often considered in the creativity literature, to broaden future considerations of creativity. Interviews with N=55 calculus students across various institutions were conducted. Results show six emergent wide-ranging themes of these students' creativity views: actions and attitudes, application, different ways, originality, outside authority, and understanding. Of these six themes, understanding was striking due to a clear distinction between students who felt understanding was required first to be creative and students who felt creativity could lead to better understanding. Our themes provide insight into what may resonate for some students, which may serve as coding parameters in qualitative and quantitative studies for researchers conducting future work about mathematical creativity.

Paradigm crisis in the step from tertiary to secondary mathematics education

We consider the following hypothesis: whereas, the institutions in charge of the teaching of mathematics for future professional mathematicians (IPM, in short) share a didactic paradigm (strongly based on logical arguments), there does not seem to be such a thing in secondary mathematics education. In this work, we analyse the notion of didactic paradigm and its possible uses in research, we sketch a description of the didactic paradigm shared in IPM, we indicate the problems that it would have in being adopted in secondary education, and we consider the failure of consecutive reforms in the teaching of secondary mathematics as an evidence of the difficulties in finding a suitable didactic paradigm at that level. Part of our study uses the paradigm of mathematical modelling as a reference paradigm.

Sums of Powers Via Matrices

Power sums of integers have always been fascinating for professional mathematicians and amateurs alike. There is something intriguing about these sums, a mixture of complexity and simplicity that h...

Professional mathematicians do not differ from others in the symbolic numerical distance and size effects

The numerical distance effect (it is easier to compare numbers that are further apart) and size effect (for a constant distance, it is easier to compare smaller numbers) characterize symbolic number processing. However, evidence for a relationship between these two basic phenomena and more complex mathematical skills is mixed. Previously this relationship has only been studied in participants with normal or poor mathematical skills, not in mathematicians. Furthermore, the prevalence of these effects at the individual level is not known. Here we compared professional mathematicians, engineers, social scientists, and a reference group using the symbolic magnitude classification task with single-digit Arabic numbers. The groups did not differ with respect to symbolic numerical distance and size effects in either frequentist or Bayesian analyses. Moreover, we looked at their prevalence at the individual level using the bootstrapping method: while a reliable numerical distance effect was present in almost all participants, the prevalence of a reliable numerical size effect was much lower. Again, prevalence did not differ between groups. In summary, the phenomena were neither more pronounced nor more prevalent in mathematicians, suggesting that extremely high mathematical skills neither rely on nor have special consequences for analogue processing of symbolic numerical magnitudes.

The Aesthetic Psychology of Mathematics

Mathematicians often describe arguments as “beautiful” or “dull,” and famous scientists have claimed that mathematical beauty is a guide toward the truth. Do laypeople, like mathematicians and scientists, perceive mathematics through an aesthetic lens? We show here that they do. Two studies ask people to rate the similarity of simple mathematical arguments to pieces of classical piano music (Study 1) or to landscape paintings (Study 2). In both cases, there is internal consensus about the pairings of arguments and artworks at greater than chance levels, particularly for visual art. There was also some evidence for correspondence to the aesthetic ratings of undergraduate mathematics students (Study 1) and of professional mathematicians (Studies 1 and 2).

Divisibility Patterns within Pascal Divisibility Networks

The Pascal triangle is so simple and rich that it has always attracted the interest of professional and amateur mathematicians. Their coefficients satisfy a myriad of properties. Inspired by the work of Shekatkar et al., we study the divisibility patterns within the elements of the Pascal triangle, through its decomposition into Pascal’s matrices, from the perspective of network science. Applying Kolmogorov–Smirnov test, we determine that the degree distribution of the resulting network follows a power-law distribution. We also study degrees, global and local clustering coefficients, stretching graph, averaged path length and the mixing assortative.

A distinct cortical network for mathematical knowledge in the human brain

How does the brain represent and manipulate abstract mathematical concepts? Recent evidence suggests that mathematical processing relies on specific brain areas and dissociates from language. Here, we investigate this dissociation in two fMRI experiments in which professional mathematicians had to judge the truth value of mathematical and nonmathematical spoken statements. Sentences with mathematical content systematically activated bilateral intraparietal sulci and inferior temporal regions, regardless of math domain, problem difficulty, and strategy for judging truth value (memory retrieval, calculation or mental imagery). Second, classical language areas were only involved in the parsing of both nonmathematical and mathematical statements, and their activation correlated with syntactic complexity, not mathematical content. Third, the mere presence, within a sentence, of elementary logical operators such as quantifiers or negation did not suffice to activate math-responsive areas. Instead, quantifiers and negation impacted on activity in right angular gyrus and left inferior frontal gyrus, respectively. Overall, these results support the existence of a distinct, non-linguistic cortical network for mathematical knowledge in the human brain.

The brain-structural correlates of mathematical expertise

Studies in several domains of expertise have established that experience-dependent plasticity brings about both functional and anatomical changes. However, little is known about how such changes come to shape the brain in the case of expertise acquired by professional mathematicians. Here, we aimed to identify cognitive and brain-structural (grey and white matter) characteristics of mathematicians as compared to non-mathematicians. Mathematicians and non-mathematician academics from the University of Oxford underwent structural and diffusion MRI scans, and were tested on a cognitive battery assessing working memory, attention, IQ, numerical and social skills. At the behavioural level, mathematical expertise was associated with better performance in domain-general and domain-specific dimensions. At the grey matter level, in a whole-brain analysis, behavioural performance correlated with grey matter density in left superior frontal gyrus – positively for mathematicians but negatively for non-mathematicians; in a region of interest analysis, we found in mathematicians higher grey matter density in the right superior parietal lobule, but lower grey matter density in the right intraparietal sulcus and in the left inferior frontal gyrus. In terms of white matter, there were no significant group differences in fractional anisotropy or mean diffusivity. These results reveal new insights into the relationship between mathematical expertise and grey matter metrics in brain regions previously implicated in numerical cognition, as well as in regions that have so far received less attention in this field. Further studies, based on longitudinal designs and cognitive training, could examine the conjecture that such cross-sectional findings arise from a bidirectional link between experience and structural brain changes that is itself subject to change across the lifespan.

Researching how professional mathematicians construct new mathematical definitions: a case study

ABSTRACTIn this work, we study the mathematical practice of defining by mathematics researchers. Since research is an important part of many professional mathematicians, understanding how they do research is a necessary step before thinking about future researchers’ undergraduate and postgraduate education. We focus on the defining process associated with the generalization of existing definitions as a way of constructing new ones. Data of this qualitative study come from a case study whose subject is a mathematics researcher in the area of differential geometry. We have interviewed this researcher and collected her research documents. From our analysis of the data, we have identified four phases in the defining process (Finding an opportunity to generalize an existing concept, Proposing a new definition, Justifying that the new definition is valid and Continuing the chain of definitions), which we will describe in detail in Section 4.

Cortical circuits for mathematical knowledge: evidence for a major subdivision within the brain's semantic networks.

Is mathematical language similar to natural language? Are language areas used by mathematicians when they do mathematics? And does the brain comprise a generic semantic system that stores mathematical knowledge alongside knowledge of history, geography or famous people? Here, we refute those views by reviewing three functional MRI studies of the representation and manipulation of high-level mathematical knowledge in professional mathematicians. The results reveal that brain activity during professional mathematical reflection spares perisylvian language-related brain regions as well as temporal lobe areas classically involved in general semantic knowledge. Instead, mathematical reflection recycles bilateral intraparietal and ventral temporal regions involved in elementary number sense. Even simple fact retrieval, such as remembering that 'the sine function is periodical' or that 'London buses are red', activates dissociated areas for math versus non-math knowledge. Together with other fMRI and recent intracranial studies, our results indicated a major separation between two brain networks for mathematical and non-mathematical semantics, which goes a long way to explain a variety of facts in neuroimaging, neuropsychology and developmental disorders.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Research mathematicians as teacher educators: focusing on mathematics for secondary mathematics teachers

We believe that professional mathematicians who teach undergraduate mathematics courses to prospective teachers play an important role in the education of secondary school mathematics teachers. Thus, we explored the views of research mathematicians on the mathematics that should be taught to prospective mathematics teachers, on how the courses they teach can serve teachers in their work with school students, and on the changes they would implement if their courses were designed specifically for prospective teachers. We constructed profiles of the four mathematicians based on their responses to a clinical interview. We employed the construct of mathematics teacher-educators’ triad in the reflective analysis of our findings and extended the construct based on the results of this study. In conclusion, we commented on potential ways to draw stronger connections between university mathematics and the mathematics taught in schools.

Is it mathematics or is it school mathematics?

In my professional life, I have repeatedly had cause to return to the question of‘What relationship does the “mathematics” experienced in classrooms have to the mathematics studied in universities, or used outside academia? What relationship should it, and could it, have, and how do we get there?’It's time I began to frame some responses! In doing so, I am very aware that membership of The Mathematical Association is broad church, ranging from those with an amateur interest in mathematics (and mathematics education), through non-specialist teachers, to professional mathematicians. As such it is a rich community, but one in which it is challenging to address such questions in a way which is meaningful to all. However, I shall attempt to do so and I hope that the balance will in a small way also prove illuminating to the reader. In particular, I have a hope that it will point to the breadth and depth of what we expect from our teachers of mathematics, at all levels, and to a valuing of that.

On the role of visual experience in mathematical development: Evidence from blind mathematicians

Advanced mathematical reasoning, regardless of domain or difficulty, activates a reproducible set of bilateral brain areas including intraparietal, inferior temporal and dorsal prefrontal cortex. The respective roles of genetics, experience and education in the development of this math-responsive network, however, remain unresolved. Here, we investigate the role of visual experience by studying the exceptional case of three professional mathematicians who were blind from birth (n=1) or became blind during childhood (n=2). Subjects were scanned with fMRI while they judged the truth value of spoken mathematical and nonmathematical statements. Blind mathematicians activated the classical network of math-related areas during mathematical reflection, similar to that found in a group of sighted professional mathematicians. Thus, brain networks for advanced mathematical reasoning can develop in the absence of visual experience. Additional activations were found in occipital cortex, even in individuals who became blind during childhood, suggesting that either mental imagery or a more radical repurposing of visual cortex may occur in blind mathematicians.