The problem of applied mathematics is to account for the ’unreasonable effectiveness’ of mathematics in empirical science. A related question is, are there mathematical explanations of scientific facts, in the same way there are empirical explanations of scientific facts? Philosophers are interested in the problem of applied mathematics for two main reasons. They are interested in whether the use of mathematics in empirical science is sufficient to motivate ontological conclusions. The indispensability argument suggests that the widespread application of mathematics obligates us to accept mathematical entities into our ontology. The second primary philosophical question concerns the details of the applications of mathematics. Philosophers are interested in what sort of relationship between mathematics and the physical world allows mathematics to play the role that it does. In this thesis, I examine both areas of literature in detail. I begin by examining the details of the indispensability argument as well as some significant critiques of the argument and the methodological conclusions that it gives rise to. I then examine the work of those philosophers who debate whether the widespread application of mathematics in science motivates accepting mathematical entities into our ontology. This debate centers on whether there are mathematical explanations of scientific facts, which is to say, scientific explanations which have an essential mathematical component. Both sides agree that the existence of mathematical explanations would motivate realism, and they debate the acceptability of various examples to this end. I conclude that there is a strong case that there are mathematical explanations. Next I examine the work of the philosophers who focus on the formal relationship between mathematics and the physical world. Some philosophers argue that mathematical explanations obtain because of a structure preserving ’mapping’ between mathematical structures and the physical world. Others argue that mathematics can play its role without such a relationship. I conclude that the mapping view is correct at its core, but needs to be expanded to account for some contravening examples. In the end, I conclude that this second area of literature represents a much more fruitful and interesting approach to the problem of applied mathematics.

This project is aimed at generating new knowledge and improving our understanding of how Modified for Language Support-Worked Example Pairs (MLS-WEPs) contribute to effective mathematics learning and teaching in an ESOL (English to Speakers of Other Languages) context. The current study investigated a novel instructional approach to help English Language Learners (ELLs) develop better understanding in mathematical reasoning, problem solving, and literacy skills (listening, reading, writing, and speaking) while they are still developing their English language proficiency. The current study followed a wait-list control design, with both the treatment and control groups receiving intervention materials. The intervention materials were administered multiple times with different topics (units) throughout the study. The lessons were audio-recorded when the selected topics were taught. Pretest and posttest were given each time when the selected topics were taught. The data analysis for this study included both qualitative and quantitative analyses. The present study revealed the following results: (1) MLS-WEPs not only enhanced ELLs’ ability to solve mathematical problems, but also improved their written explanation skills and enabled them to transfer such skills to different mathematical concepts; (2) when controlling ELLs’ prior knowledge, the effectiveness of the MLS-WEPs intervention did not vary by their English language proficiency; (3) the MLS-WEPs intervention materials facilitated teachers to provide ELLs with more opportunities to read, write, and speak in mathematics and enabled teachers to ask more and deeper questions. However, worked example comparisons did not appear to motivate the participant teachers to promote equitable participation in mathematics classrooms. These findings provide direct empirical support for the need to reform mathematics teaching and learning in the ESOL context. / Math & Science Education