Comparing Mathematical Explanations

2021 ◽  
Author(s):  
Isaac Wilhelm
Keyword(s):  
Mathematical Explanations

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

2021 ◽  
pp. 001440292110088
Author(s):  
Madhavi Jayanthi ◽  
Russell Gersten ◽  
Robin F. Schumacher ◽  
Joseph Dimino ◽  
Keith Smolkowski ◽  
...  
Keyword(s):  
Randomized Controlled Trial ◽  
Explicit Instruction ◽  
Controlled Trial ◽  
Number Line ◽  
Randomized Controlled ◽  
Fifth Grade Students ◽  
Linear Representations ◽  
Mathematical Explanations ◽  
Grade 5 ◽  
Number Line Estimation

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.

CRITICAL REVIEW OF EMULSION STABILITY AND CHARACTERIZATION TECHNIQUES IN OIL PROCESSING

2021 ◽  
pp. 1-36
Author(s):  
Vahideh Angardi ◽  
Ali Ettehadi ◽  
Özgün Yücel
Keyword(s):  
Separation Efficiency ◽  
Petroleum Industry ◽  
Cost Effective ◽  
Downstream Processing ◽  
Physical Factors ◽  
Comprehensive Review ◽  
Factors Affecting ◽  
Mathematical Explanations ◽  
The Stability ◽  
The Right

Abstract Effective separation of water and oil dispersions is considered a critical step in the determination of technical and economic success in the petroleum industry over the years. Moreover, a deeper understanding of the emulsification process and different affected parameters is essential for cost-effective oil production, transportation, and downstream processing. Numerous studies conducted on the concept of dispersion characterization indicate the importance of this concept, which deserves attention by the scientific community. Therefore, a comprehensive review study with critical analysis on significant concepts will help readers follow them easily. This study is a comprehensive review of the concept of dispersion characterization and conducted studies recently published. The main purposes of this review are to 1) Highlight flaws, 2) Outline gaps and weaknesses, 3) Address conflicts, 4) Prevent duplication of effort, 5) List factors affecting dispersion. It was found that the separation efficiency and stability of dispersions are affected by different chemical and physical factors. Factors affecting the stability of the emulsions have been studied in detail and will help to look for the right action to ensure stable emulsions. In addition, methods of ensuring stability, especially coalescence are highlighted, and coalescence mathematical explanations of phenomena are presented.

scholarly journals Calling differential DNA methylation at cell-type resolution: addressing misconceptions and best practices

2021 ◽  
Author(s):  
Elior Rahmani ◽  
Brandon Jew ◽  
Regev Schweiger ◽  
Brooke Rhead ◽  
Lindsey A. Criswell ◽  
...  
Keyword(s):  
Dna Methylation ◽  
Best Practices ◽  
Regression Model ◽  
Association Studies ◽  
Composition Analysis ◽  
Cell Type ◽  
Cell Type Specificity ◽  
Interaction Terms ◽  
Mathematical Explanations ◽  
Cell Type Specific

AbstractWe benchmarked two approaches for the detection of cell-type-specific differential DNA methylation: Tensor Composition Analysis (TCA) and a regression model with interaction terms (CellDMC). Our experiments alongside rigorous mathematical explanations show that TCA is superior over CellDMC, thus resolving recent criticisms suggested by Jing et al. Following misconceptions by Jing and colleagues with modelling cell-type-specificity and the application of TCA, we further discuss best practices for performing association studies at cell-type resolution. The scripts for reproducing all of our results and figures are publicly available at github.com/cozygene/CellTypeSpecificMethylationAnalysis.

Platonism and Mathematical Explanations

2021 ◽  
Vol 13 (2) ◽  
pp. 113-122
Author(s):  
Fabrice Pataut ◽  
Keyword(s):  
Mathematical Theory ◽  
Causal Explanation ◽  
Applied Mathematics ◽  
Propositional Attitude ◽  
Abstract Objects ◽  
Mathematical Explanations ◽  
Parsimony Principle ◽  
The Face ◽  
Numerical Formulation ◽  
The One

Ontological parsimony requires that if we can dispense with A when best explaining B, or when deducing a nominalistically statable conclusion B from nominalistically statable premises, we must indeed dispense with A. When A is a mathematical theory and it has been established that its conservativeness undermines the platonistic force of mathematical derivations (Field), or that a non numerical formulation of some explanans may be obtained so that the platonistic force of the best numerical-based account of the explanandum is also undermined (Rizza), the parsimony principle has been respected. Since derivations resorting to conservative mathematics and proofs involved in non numerical best explanations also require abstract objects, concepts, and principles under the usual reading of “abstract,” one might complain that such accounts turn out to be as metaphysically loaded as their platonistic counterparts. One might then urge that ontological parsimony is also required of these nominalistic accounts. It might, however, prove more fruitful to leave this particular worry to the side, to free oneself, as it were, from parsimony thus construed and to look at other important aspects of the defeating or undermining strategies that have been lavished on the disposal of platonism. Two aspects are worthy of our attention: epistemic cost and debunking claims. Our knowledge that applied mathematics is conservative is established at a cost, and so is our knowledge that nominalistic proofs play a genuine theoretical role in best explanations. I will suggest that the knowledge one must acquire to show that nominalistic deductions and explanations do indeed play their respective theoretical role involves some question-begging assumptions regarding the nature and validity of proofs. As for debunking, even if the face value content of either non numerical claims, or conservative mathematical claims, or platonistic mathematical claims didn’t figure in our causal explanation of why we hold the mathematical beliefs that we do, construed or understood as beliefs about such contents, or as beliefs held in either of these three ways, we could still be justified in holding them, so that the distinction between nominalistic deductions or non numerical explanations on the one hand and platonistic ones on the other turns out to be spurious with respect to the relevant propositional attitude, i.e., with respect to belief.

A Counterfactual Approach to Explanation in Mathematics

2019 ◽  
Vol 28 (1) ◽  
pp. 1-34 ◽  
Author(s):  
Sam Baron ◽  
Mark Colyvan ◽  
David Ripley
Keyword(s):  
Scientific Explanation ◽  
Mathematical Explanation ◽  
Test Case ◽  
Mathematical Explanations ◽  
Counterfactual Theory ◽  
Explanatory Structure ◽  
Counterfactual Approach

ABSTRACT Our goal in this paper is to extend counterfactual accounts of scientific explanation to mathematics. Our focus, in particular, is on intra-mathematical explanations: explanations of one mathematical fact in terms of another. We offer a basic counterfactual theory of intra-mathematical explanations, before modelling the explanatory structure of a test case using counterfactual machinery. We finish by considering the application of counterpossibles to mathematical explanation, and explore a second test case along these lines.

scholarly journals Counterfactual Scheming

Mind ◽  
2019 ◽  
Vol 129 (514) ◽  
pp. 535-562
Author(s):  
Sam Baron
Keyword(s):  
Mathematical Explanation ◽  
Explanatory Role ◽  
Unification Theory ◽  
Mathematical Explanations ◽  
Counterfactual Theory ◽  
Counterfactual Approach

Abstract Mathematics appears to play a genuine explanatory role in science. But how do mathematical explanations work? Recently, a counterfactual approach to mathematical explanation has been suggested. I argue that such a view fails to differentiate the explanatory uses of mathematics within science from the non-explanatory uses. I go on to offer a solution to this problem by combining elements of the counterfactual theory of explanation with elements of a unification theory of explanation. The result is a theory according to which a counterfactual is explanatory when it is an instance of a generalized counterfactual scheme.

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