# Applications

## Carencias transpositivas de la noción de derivada en bachillerato (Teaching Derivatives)

El siguiente trabajo fue escrito por Diego Duque como parte de los requerimientos para obtener el título de Máster Universitario en Estudios Avanzados en Pedagogía en Facultad de Educación de la Universidad Complutense de Madrid. Puedes descargar el PDF aquí.

RESUMEN:

La derivada ha sido un contenido matemático con una amplia dificultad para la mayoría de los estudiantes de Bachillerato y universitarios. En su didáctica pueden emplearse representaciones gráficas evitando así su estudio únicamente desde un enfoque analítico. Sin embargo, se ha demostrado que existen carencias en su aprendizaje debido a la falta de ilustraciones que representen su significado.

Por tanto, para innovar y poner en práctica técnicas de aprendizaje de la derivada, se utiliza un modelo epistemológico de referencia haciendo uso de la Teoría de la Transposición Didáctica. Una vez finalizado el modelo, este se compara con currículos y libros de texto de la LOE y de la LOMCE. De esta forma se evalúan las carencias didácticas que existen hoy en día en España acerca la derivada. Tras su subsanación se podría fomentar un aprendizaje de la derivada de mayor calidad en el alumnado de Bachillerato.

Palabras clave: derivada, didáctica, modelo epistemológico de referencia, praxeologías, transposición didáctica

The following paper was written by Diego Duque as part of his thesis to obtain the Degree Máster Universitario en Estudios Avanzados en Pedagogía at Facultad de Educación in Universidad Complutense de Madrid. You can download his work as a PDF file (In Spanish).

ABSTRACT:

The derivative has been a mathematical content with a large difficulty for most high school and university students. Its didactic can use graphical representations avoiding its study uniquely from an analytical approach. However, it has been shown that there are gaps in its learning due to the lack of illustrations representing its meaning.

Therefore, to innovate and implement the practice of derivative’s learning techniques, an epistemological reference model is used by using the Theory of Didactic Transposition. Once the model is finished, it is compared to syllabuses and textbooks from the LOE and the LOMCE. Thus, it evaluates didactic gaps that exist today in Spain about the derivative. The improvements could foster a better quality learning of derivative in high school students.

Keywords: derivative, didactic, epistemological reference model, praxeology, didactic transposition

## Introduction To Analysis I (Honors)

This is a graduate level course being offered at Indiana University by the Math Department. I took this course on Fall 2013 with Prof. Alberto Torchinsky. The book we used was Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (International Series in Pure & Applied Mathematics) by Rudin. All assignments are from this book. I do not like this book very much and instead prefer other like: Elementary Analysis: The Theory of Calculus by Ross and The Way of Analysis, Revised Edition (Jones and Bartlett Books in Mathematics) by Strichartz. The following is a compilation of my solutions to assignments given in class.

## Assignments

- Assignment 1 (with solutions)
- Assignment 2 (with solutions)
- Assignment 3 (with solutions)
- Assignment 4 (with solutions)
- Assignment 5 (with solutions)
- Assignment 6 (with solutions)
- Assignment 7 (with solutions)
- Assignment 8 (with solutions)
- Assignment 9 (with solutions)
- Assignment 10 (with solutions)

## Exams

- MidTerm 1 - [Solutions]
- MidTerm 2 - [Solutions]
- Final Exam (with solutions)

## Resources

- Notes
- Complement Notes to Rudin (very useful! taken from George Bergman website, check it out)
- Cantor Set
- Cauchy Criterion
- Infinite Series, Convergence Tests and Leibniz Theorem
- Guide to Open set Proofs (taken from Professor Sormani's website)

## Video Lectures

- The University of Nottingham - U-Now Open Courseware - Mathematical analysis

(This is a great resource as a complement to your lectures. I highly recommend it)

## Math

In the following list you can find resources, i.e., practice problems, solutions, study guides, or miscellaneous, of the various math courses at the undergraduate and graduate level I have taken. Please feel free to contact me at with any questions or suggestions. For resources of courses I have assited or teach look at my teaching section. For resources on the Computer Science side, take a look at my Computer Science section.

- M311 Calculus III
- M312 Calculus IV
- M343 Introduction To Differential Equations I
- M403 Introduction to Modern Algebra I
- M403 Introduction to Modern Algebra I (Honors)
- M409 Linear Transformations
- M413 Introduction To Analysis I (Honors)
- M436 Introduction To Geometries
- M447 Mathematical Models and Applications
- M451 Mathematical Finance
- M463 Introduction To Probability I
- M464 Introduction To Probability II
- M595 Seminar in Teaching College Math I
- M800 Reading and Research
- S520 Introduction to Statistics
- S620 Introduction to Statistical Theory
- E520 Algebra Throughout the Curriculum
- CO5412 Nonlinear Optimization I (Spanish)

## Introduction To Geometries

This is a ungraduate/graduate level course being offered at Indiana University by the Math Department. I took this course on Fall 2014 with Prof. Matthias Weber. In this course we did not use any textbook but rather follow a class diary written by Prof. Weber after each class. The following is a compilation of my solutions to assignments given in class.

## Assignments

- Assignment 1 - [Solutions]
- Assignment 2 - [Solutions] - [Code]
- Assignment 3 - [Solutions]
- Assignment 4 - [Solutions]
- Assignment 5 - [Solutions]
- Assignment 6 - [Solutions]
- Assignment 7 - [Solutions]
- Assignment 8 - [Solutions]
- Assignment 9 - [Solutions]
- Assignment 10 - [Solutions] - [Code]

## Exams

## Resources

- Stereographic Projection and Riemann Sphere
- Hyperbolic Geometry (taken from Prof. Charles Walkden's webpage, check it out)
- Taxicab Geometry
- Mobius Transformations and Circles (taken from Prof. Rich Schwartz's webpages)

## Mathematical Finance

This is a ungraduate/graduate level course being offered at Indiana University by the Math Department. I took this course on Spring 2015 with Prof. Chris Connell. The textbook we used was An Elementary Introduction to Mathematical Finance by Ross. This is an ok book but at times a bit confusing. The following is a compilation of my solutions to assignments given in class, note that all assignments and quizzes are from the book.

## Assignments

- Assignment 1 (with solutions)
- Assignment 2 (with solutions)
- Assignment 3 (with solutions)
- Assignment 4 (with solutions)
- Assignment 5 (with solutions)
- Assignment 6 (with solutions)
- Assignment 7 (with solutions)
- Assignment 8 (with solutions)
- Assignment 9 (with solutions)

## Quizzes

- Quiz 1 (with solutions)
- Quiz 2 (with solutions)
- Quiz 3 (with solutions)
- Quiz 4 (with solutions)
- Quiz 5 (with solutions)
- Quiz 6 (with solutions)
- Quiz 7 (with solutions)
- Quiz 8 (with solutions)
- Quiz 9 (with solutions)