Mathematical Explanation Research Articles

Conceptual (and Hence Mathematical) Explanation, Conceptual Grounding and Proof

This paper studies the notions of conceptual grounding and conceptual explanation (which includes the notion of mathematical explanation), with an aim of clarifying the links between them. On the one hand, it analyses complex examples of these two notions that bring to the fore features that are easily overlooked otherwise. On the other hand, it provides a formal framework for modeling both conceptual grounding and conceptual explanation, based on the concept of proof. Inspiration and analogies are drawn with the recent research in metaphysics on the pair metaphysical grounding–metaphysical explanation, and especially with the literature in philosophy of science on the pair causality-causal explanation.

Improving Struggling Fifth-Grade Students’ Understanding of Fractions: A Randomized Controlled Trial of an Intervention That Stresses Both Concepts and Procedures

Using a randomized controlled trial, we examined the effect of a fractions intervention for students experiencing mathematical difficulties in Grade 5. Students who were eligible for the study ( n = 205) were randomly assigned to intervention and comparison conditions, blocked by teacher. The intervention used systematic, explicit instruction and relied on linear representations (e.g., Cuisenaire Rods and number lines) to demonstrate key fractions concepts. Enhancing students’ mathematical explanations was also a focus. Results indicated that intervention students significantly outperformed students from the comparison condition on measures of fractions proficiency and understanding ( g = 0.66–0.78), number line estimation ( g = 0.80–1.08), fractions procedures ( g = 1.07), and explanation tasks ( g = 0.68–1.23). Findings suggest that interventions designed to include explicit instruction, along with consistent use of the number line and opportunities to explain reasoning, can promote students’ proficiency and understanding of fractions.

Unification and mathematical explanation

This paper provides a sorely-needed evaluation of the view that mathematical explanations in science explain by unifying. Illustrating with some novel examples, I argue that the view is misguided. For believers in mathematical explanations in science, my discussion rules out one way of spelling out how they work, bringing us one step closer to the right way. For non-believers, it contributes to a divide-and-conquer strategy for showing that there are no such explanations in science. My discussion also undermines the appeal to unifying power in support of the enhanced indispensability argument.

Does Volatility Harvesting Really Work?

A large number of academic papers claim that there is a consistent advantage to rebalancing an investment portfolio relative to a simple Buy&Hold strategy, while other academic papers claim the opposite. The papers on both sides are typically based on unclear mathematical arguments with only few real-world examples. This paper does not try to give a mathematical explanation but is entirely empirical. We use daily stock-returns for more than 900 U.S. stocks between the years 2007 and 2021, where we generate thousands of random portfolios of different sizes and start/end-dates, to compare the Rebalanced and Buy&Hold strategies on different performance metrics. Although the Rebalanced portfolios may seem to perform better than the Buy&Hold portfolios in terms of e.g. the arithmetic mean daily return, it actually depends on which period we consider. We also make preliminary experiments with intraday data, where it does appear that Rebalancing is consistently better than Buy&Hold, although these data-sets are so small that we cannot conclude this advantage holds in general.

Mathematical Explanation in Practice

The connection between understanding and explanation has recently been of interest to philosophers. Inglis and Mejia-Ramos (Synthese, 2019. https://doi-org.proxy.library.nd.edu/10.1007/s11229-019-02234-5 ) propose that within mathematics, we should accept a functional account of explanation that characterizes explanations as those things that produce understanding. In this paper, I start with the assumption that this view of mathematical explanation is correct and consider what we can consequently learn about mathematical explanation. I argue that this view of explanation suggests that we should shift the question of explanation away from why-questions and towards a “what’s going on here” question. Additionally, I argue that when we recognize the connection between understanding and explanation we naturally see how more than just proof can be explanatory. I expand this point by detailing how definitions and diagrams can be explanatory. In all, we see that when we take seriously the connection between understanding and explanation, we get a better sense of how explanation arises within mathematics.

Mathematical Modelling and Big-Data Analytics for Student Performance

Mathematical Modelling and Big-data Analytics are playing a vital role in educational databases. The result of integrating technology to predict student performance along with Mathematical Modelling and Big-Data Analytics helps us to make better decisions about teaching and learning. Modelling involves formulating real-life situations or to convert the problems in mathematical explanations to a real or believable situation. However, Mathematical modelling are an essential enabler in Big-data and Developments in Big-data analytics require not only more computing, but also new advanced mathematical approaches. In this paper, our main aim to see how Mathematical Modelling and Big-Data analytics help in students’ learning and how they relate to each other.

STUDENT EVALUATION MATHEMATICAL EXPLANATION IN DIFFERENTIAL CALCULUS CLASS

This study aimed to determine that the failure of students to evaluate mathematical explanations based on mathematics is influenced by sociomathematical norms, teaching authority, and classroom mathematics practice. The research method used is the case study method. The research data were obtained from inside and outside the research class. The data in the research class were in the form of field notes, video recordings of the class, video recordings of student group work, and student work. Data outside the research class is the result of interviews with three interview subjects. By studying the three evaluation methods students used in evaluating explanations, it was found that each student applied a different evaluation method at different times. The three evaluation methods contributed to some of the difficulties students experience in evaluating their mathematical descriptions. The results indicate that the failure of students in evaluating explanations is not solely due to errors in choosing the method, approach, or learning model used but can be caused by sociomathematical norms, authority, and classroom mathematics practices applied in the classroom.

Exploring Israeli high school graduates' explanations for the spread of the coronavirus.

The aim of this study is to explore Israeli high school graduates’ mathematical explanations for the spread of the coronavirus, given that the mathematics required to do so was part of their school curriculum. An online questionnaire consisting of two sections provided a variety of potential framings for explaining the phenomenon. The first section invited the participants to explain the spread of the coronavirus in terms of their school majors in general, with no specific reference to mathematics. The second section asked explicitly to explain the mathematical context underlying the phenomenon. In this section, the participants were asked to discuss the Prime Minister’s speech given in the media a few weeks earlier, in which he described the spread of the coronavirus as a geometric series. Data analysis of 87 participants’ responses to the questionnaire revealed 11 different mathematical ideas used to explain the spread of the coronavirus. These ideas included are as follows: doubling, sequence, exponential growth, using powers, tree diagram, recursion, fast-growing rate with covariation, probability, parabola and quadratic function, acceleration, and factorial. It was also found that the second section of the questionnaire elicited a wider range of mathematical ideas than the first one. We suggest possible explanations for the emergence of the mathematical ideas, which seem to reflect the graduates’ intuitive knowledge, influenced not only by their mathematics track level but also by their chosen high school majors. Possible implications are discussed.

There Are No Mathematical Explanations

If ontic dependence is the basis of explanation, there cannot be mathematical explanations. Accounting for the explanatory dependency between mathematical properties and empirical phenomena poses insurmountable metaphysical and epistemic difficulties, and the proposed amendments to the counterfactual theory of explanation invariably violate core commitments of the theory. Instead, mathematical explanations are either abstract mechanistic constitutive explanations or reconceptualizations of the explanandum phenomenon in which mathematics as such does not have an explanatory role. Explanation-like reasoning within mathematics, distinction between explanatory and nonexplanatory proofs, and comparative judgments of mathematical depth can be fully accounted for by a concept of formal understanding.

Unification and mathematical explanation in science

Mathematics clearly plays an important role in scientific explanation. Debate continues, however, over the kind of role that mathematics plays. I argue that if pure mathematical explananda and physical explananda are unified under a common explanation within science, then we have good reason to believe that mathematics is explanatory in its own right. The argument motivates the search for a new kind of scientific case study, a case in which pure mathematical facts and physical facts are explanatorily unified. I argue that it is possible for there to be such cases, and provide some toy examples to demonstrate this. I then identify a potential source of scientific case studies as a guide for future work.

Anharmonic effects on the dynamic behavior’s of Klein Gordon model’s

This work completes and extends the Ref. Tchakoutio Nguetcho et al. (2017), in which we have focused our attention only on the dynamic behavior of gap soliton solutions of the anharmonic Klein-Gordon model immersed in a parameterized on-site substrate potential. We expand our work now inside the permissible frequency band. These considerations have crucial effects on the response of nonlinear excitations that can propagate along this model. Moreover, working in the allowed frequency band is not only interesting from a physical point of view, it also provides an extraordinary mathematical model, a new class of differential equations possessing vital parameters and vertical singular straight lines. The dynamics around these singularities gives new informations of great interest, such as to better understand the break up (rupture) of a stretched polymer chain by pulling and its relation to soliton destruction and which until now had no mathematical explanations yet. The mathematical model of the differential equation we obtain corresponds, as an equivalent experimental model, to the nonlinear transmission electrical line introduced in 2009 in Ref. Yemélé and Kenmogné(2009), whose equations were corrected but not entirely resolved in 2016 in Ref. Yamgoué and Pelap(2016). Our dynamic study thus presents a theoretical prediction for their experimental set up. An extensive account of the bifurcation theory for modulated-wave propagation is given. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we not only make a total inventory of the solutions that can include such differential equations, but we derive a variety of exotic solutions corresponding to each phase trajectories as well as their different conditions of existence.

Mathematicians’ Assessments of the Explanatory Value of Proofs

The literature on mathematical explanation contains numerous examples of explanatory, and not so explanatory proofs. In this paper we report results of an empirical study aimed at investigating mathematicians’ notion of explanatoriness, and its relationship to accounts of mathematical explanation. Using a Comparative Judgement approach, we asked 38 mathematicians to assess the explanatory value of several proofs of the same proposition. We found an extremely high level of agreement among mathematicians, and some inconsistencies between their assessments and claims in the literature regarding the explanatoriness of certain types of proofs.

What could mathematics be for it to function in distinctively mathematical scientific explanations?

What could mathematics be for it to function in distinctively mathematical scientific explanations?

The Model Thinker by Scott E. Page

The 12th chapter of The Model Thinker begins with a quote by Claude Shannon:“Information is the resolution of uncertainty.”Making sense out of the deluge of information which virtually exists in today’s world requires much more than a unidimensional approach. With information comes complexity, and with complexity one needs to responsibly make sense out of it, be it to understand the world better or even to offer solutions to the problems at hand. Scott E. Page offers that solution through his book, The Model Thinker, which may very well serve as a manual to start learning about the ‘many-model approach’. The author bolsters his claim to harness the power of this many-model approach by quite rightly saying that, “we achieve wisdom through a multiplicity of lenses”. From politics, economics, market dynamics, social issues to interpreting disease epidemiology and contagions requires professionals and academics to think beyond the chaos of data, cohesively arranging them into information, deriving knowledge from it and applying what is learned in various aspects, thus forging the basis of wisdom. Not only does the author succeed in advocating this motion in the first few pages of his book, but he goes on to inform the reader of the 25 classes of modelling approaches as well. The book offers theoretical (or mathematical) explanations of each model, and the last few chapters are dedicated to how multiple models can be applied to solve real-life problems. A recommended read for anyone who is stepping into the world of dealing with data.

Optimized frame work for Reiner–Philippoff nanofluid with improved thermal sources and Cattaneo–Christov modifications: A numerical thermal analysis

This motivating analysis aims to present the thermal mechanism for mixed convection flow of Reiner–Philippoff nanofluid with assessment of entropy generation. The thermal performances of nanomaterials have been modified by utilizing the nonuniform heat source/sink, Ohmic dissipations and thermal radiation consequences. The assumed surface is assumed to be porous with non-Darcian porous medium. The modified Cattaneo–Christov relations are followed to modify the mass and heat equations. The invoking of similarity variables results in differential equations in nonlinear and coupled form. A MATLAB-based shooting algorithm is employed to access the numerical simulations. The physical aspect of thermal model is graphically addressed for endorsed flow parameters. The importance of entropy generation is visualized with associated mathematical relations and physical explanations. The numerical values are obtained for the assessment of heat and mass transfer phenomenon.

Distinctively mathematical explanation and the problem of directionality: A quasi-erotetic solution

The increasing preponderance of opinion that some natural phenomena can be explained mathematically has inspired a search for a viable account of distinctively mathematical explanation. Among the desiderata for an adequate account is that it should solve the problem of directionality —the reversals of distinctively mathematical explanations should not count as members among the explanatory fold but any solution must also avoid the exclusion of genuine explanations. In what follows, I introduce and defend what I refer to as a quasi-erotetic solution which provides a remedy to the problem in the form of an additional necessary condition on explanation.

Mathematical Analysis and Improvement of the Maximum Spatial Eigenfilter for Direction of Arrival Estimation

Abstract Maximum spatial eigenfiltering improves the accuracy of maximum likelihood direction-of-arrival estimators for closely-spaced signal sources but may interchangeably attenuate widely-spaced signal sources, producing a severe performance degradation. Although this behavior has been observed experimentally, it still lacks a mathematical explanation. In our previous work, we overcame these limitations using a differential spectrum-based spatial filter but this still caused a small degradation in the DOA estimate. In this paper, we develop a mathematical analysis of how the signal source separation and the Karhunen-Loeve expansion affect the passbands of the maximum spatial eigenfilter. The farther the sources, the less significant is the maximum eigenvalue of the spatial correlation matrix and its corresponding eigenvector. Then, the magnitude response of the maximum spatial eigenfilter no longer approximates the spatial power spectrum and is not guaranteed to place multiple passbands around the signal sources. Consequently, we propose a spatial filter built from the eigenvectors of the entire signal subspace. This filter showed an overall runtime smaller than that of our previous work. It also provides a significant reduction in the threshold signal-to-noise ratio for closely-spaced signal sources and does not hamper the estimation for widely-spaced signal sources.

How Can Microwave Observations at 23.8 GHz Help in Acquiring Water Vapor in the Atmosphere over Land?

Due to concerns about radio frequency interference from emerging telecommunications technology, there have been intensive discussions on the changes to the international Radio Regulations around 24 GHz at the recent World Radiocommunication Conference in 2019. Although the sensitivity to total precipitable water (TPW) at 23.8 GHz over land is small, and in some cases close to zero, state-of-the-art retrieval systems with no dependence on real-time ancillary data (e.g., numerical weather prediction (NWP) model forecasts), such as the National Oceanic and Atmospheric Administration (NOAA) operational Microwave Integrated Retrieval System (MiRS), have been producing reliable TPW products over both ocean and land, which implies that the microwave channel at 23.8 GHz is providing valuable information on TPW over land as well as over ocean. The contradiction between the zero or near-zero sensitivity and practical performance over land raises questions for the remote sensing community and public users of such data. In this study, we examine the underlying physics and include mathematical explanations, which address and clarify the apparent contradiction. The channel at 23.8 GHz is a direct measurement and indispensable for its combined use with microwave temperature and moisture sounding channels.

Regularity theory of elliptic systems in ε-scale flat domains

We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called ε-scale flatness condition, which could be arbitrarily rough below ε-scale. This particularly generalizes Kenig and Prange's work in [34] and [35] by a quantitative approach. Our result also provides a mathematical explanation on why the boundary regularity of the solutions of partial differential equations should be physically and experimentally expected even if the surfaces of mediums in real world may be arbitrarily rough at small scales.

The Metaphysics of Mathematical Explanation in Science

Debates between contemporary platonist and nominalist conceptions of the metaphysical status of mathematical objects have recently included discussions of explanations of physical phenomena in which mathematics plays an indispensable role, termed mathematical explanations in science (MES). I will argue that MES requires an ontology that can (1) ground claims about mathematical necessity as distinct from physical necessity and (2) explain how that mathematical necessity applies to the physical world. I contend that nominalism fails to meet the first criterion and platonism the second. I then articulate an alternative, Aristotelian approach to mathematical objects and defend such a view as meeting both criteria.