By Assad Ebrahim, on July 13th, 2016 (10,112 views) |
Topic: Education
Thoughts on Teaching Mathematics in an Exploratory, Dialectical, Topical format.
(2nd ed. July 13th 2016; 1st ed. Jan 31, 2010)
Mathematics is a richly spun tapestry, threaded with interconnections from a multiplicity of endeavors, perspectives, and disciplines, both theoretical and applied. Yet contrary to this “non-linear” reality, the typical pattern of school and even university mathematics is both linear and restricted.
This article takes a look at what lies behind the way things are, and what could bring positive change.
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If you haven’t done so already, you may want to start by reading the Preface to the Computing Series: Software as a Force Multiplier, Sections 1-3.
Introducing the LaTeX typsetting platform
If symbols, formulas, and equations comprise a large portion of your professional communication, then you will gain significantly by becoming proficient with the LaTeX (pronounced “lay-tech”) document preparation platform. With the right tools and a little practice, the relative ease of creating beautiful mathematical documents with LaTeX will likely mean that you leave Office in favor of LaTeX for your technical writing.
This article introduces the LaTeX platform (short for Lamport-TeX, after the mathematician Leslie Lamport), illustrates its capabilities, and highlights the key differences between using LaTeX or WYSIWYG “what you see is what you get” word processing systems such as Office.
For those that like to know the human side of the tools they use, we provide a brief history of the legendary TeX (pronounced “tech”) platform, which underpins all variations of which LaTeX is one, looks at the philosophy motivating the development of TeX, and something about its legendary creator Donald Knuth.
 Don Knuth, Leslie Lamport, and an illustration of of why writing mathematics in LaTeX is easier than in Word.
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By Assad Ebrahim, on April 15th, 2010 (12,298 views) |
Topic: Education, Maths--Philosophy
An examination of mathematical methods and the search for mathematical meaning.
This article curates a reading list (most sources available freely online) organized into a set of encounters that lie outside the standard mathematics curriculum. They are intended to enrich the reader’s understanding of mathematics and its place in scientific inquiry, increase her/his connection to the historical and philosophical questions behind the mathematics of the past and present, and gain greater satisfaction from further mathematical study. The reader should come away with a better understanding of the culture of mathematics: what mathematics is, mathematical method and meaning, and the relation of mathematics to the empirical world and to science.
We look at seven topics. These may be covered in any order, to suit your particular interests.
- What is Mathematics? (Its Nature and Characteristics)
- Reality, Truth, and the Nature of Mathematical Knowledge
- What is Proof? and the Problem of Certainty
- Some Readings in the History of Mathematics and the Evolution of Its Ideas
- The Search for Foundations in Mathematics
- Mathematics and Science
- Thoughts on Mathematical Practice and Mathematical Style
There is no core body of technical material to master in this course; the important thing is a feel for how, why, and in what context the core ideas of mathematics evolved, getting to the essence of their motivation, and understanding the fruits of these efforts. The course such as the below should appeal to all those who have an itch to scratch beneath the surface of mathematics, who find themselves asking “but why?”. It could be useful in all three tiers of education: secondary, post-secondary (undergraduate), and graduate, appropriately restructured.
- Secondary school elective: to encourage bright students in mathematics, science and technology to enter the university with a broader perspective on the mathematics they will be rapidly learning there.
- University elective course: offered as a writing-intensive seminar, intended primarily for students in the sciences and engineer: mathematics, physics, engineering.
- Graduate level course: offered in the first year of graduate school in mathematics or applied mathematics as a supplementary seminar.
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By Assad Ebrahim, on January 31st, 2010 (8,104 views) |
Topic: Education, Maths--Philosophy
The mathematician Alfred North Whitehead observed that “[Advancement occurs] by extending the number of important operations which we can perform without thinking of them.” (Introduction to Mathematics, 1911 ) This is certainly true in mathematics where the development of judicious notation, accompanied by good mathematical technique, extends the capability to perform chains of complex reasoning accurately and efficiently. Through proper problem formulation (tractable yet generalizable), one can sometimes pass from a single insight to the solution of a family of problems, and in some cases, to the solution to the general question itself.
Here, mathematical computing can provide a useful benefit: helping to efficiently explore conjectures, dispatch with false directions, and save time during the development, error-checking and validation stages of obtaining general results. In industry, where specific or semi-general results are needed fast, such tools allow rapidly working up the required material and providing the necessary certainty before the fully general results or complete proof are ready.
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