(October 29, 2012) Keith Devlin concludes the course by discussing the development of mathematical cognition in humans as well as the millennium problems
How the ancient Greeks shaped modern mathematics – video animation
Testing spaces in order to choose the best option for me to study calculus and its history
Take a look and contribute if you like
My presentation at the Research in Progress day organised by the British Society for the History of Mathematics took place the 22 of February at Queens College in Oxford.
It was in front of a small but very knowledgeable audience. Many of them mathematics professors interested in the use of the history of mathematics for mathematics education and others historians of mathematics interested in the dissemination of historical material regarding the development of mathematics. Many of them are authors of interesting books like Jacqueline Stedall, Peter Neumann, Jan van Maanen, Steve Russ among others. Some of them new researcher in their 2 or 3rd year all of them researching in the history of mathematics with no further application.
One talk : The Jesuits in the Polish-Lithuanian Commonwealth and their treatment of the mathematical sciences. Dagmar Mrozic, interested me because I saw again how education is really a powerful tool. The other interesting talk was done by an undergraduate student of Exeter University, Ryan Stanley, who won the prize for the best essay. He wrote about Cantor, Dedekind and the rigor of calculus. One of the things I liked most was his attitude in the talk. Really captured my attention. Interesting material although with some misconceptions but as he said, he just looked at what someone said about what someone wrote about what he thought and so on…so the chances of understanding something wrong are high. Or even to understand right what others (2nd sources) understood wrong. He was very humble but very secure. Enjoying the adventure of knowing and exploring the life and ideas of two great mathematicians. He had a voice and assumed a position that really amused me for half on hour.
The public engaged very much with students designing a PLE for the learning of mathematics and the idea to develop some topics for A-level related with the calculus. For me the power is the issue that they are going to be the designers of that space. Radical constructivist maybe? There are some initiatives already in integrating the history into the classroom but have not been executed. The overall comment is that it is not easy to do. I talked to skeptic professors that had maybe tried and they found it difficult. There is a difficulty I know, but the challenge is to see how can this be done. Look at what things that did not work in their cases and see how to change them.
One aspect that is still not resolved is the how to concretise, materialise the technology that is going to bring alive the PLES of students. Are they going to have just a bundle of tools loose and spread in the web (Siemens and Downes) or are they working with a pre-made space where some tools are available that are fixed and interoperable (Mash-up) or are they having a common platform (Moodle or desire2learn) and loose tools spread in the web. There is a factor of rapid change that I need to take into account if this is going to be sustainable within time. If the technology is to close it won’t work. I have to think in the structure and integration of RME-DS and A Universe of Knowledge. A universe of knowledge could be a platform where students can put in their intelectual artifacts and teachers their inputs for the particular learning experience. But the PLE I do not know how to work it out so it is able to last and can navigates the changes to come.
There is work to do regarding the development of the idea of the calculus flower. Learning more in depth about the calculus and its component. Analysing each of them in depth and understanding in which ways they are connected and intertwined. Calculus could be seen as fabric where the threads are carefully interwoven. Or maybe better as a flower which petals are connected and -sometimes they overlap. Look for that structure among the components.
Task: Look at the syllabus of the Vol. 1 of Apostol. Think about the order. Discuss.
Here my abstract for the talk:
Our way of thinking and feeding our mind has changed with humans’ intellectual tools throughout history. For example, the technologies of the map and the clock advanced the evolution of abstract thinking. A more conceptual example would be the calculus, an intellectual tool that from its very beginning has not only kept on changing our way of thinking and feeding our mind, but also revolutionised humankind, allowing it to ‘see’ the world from a different perspective. The world we live in —the earth— could now be seen from an unexpected place such as the moon. This fact changed radically the perspective we had of our own planet. The calculus, in the words of Morris Klein, is definitely a landmark in human thought.
Bringing this subject to life, integrating Newton’s original notes (available from the Newton Project) and some relevant passages from history, I aim to enrich the understanding of the roles played by cultural and mathematical context in the invention of new branches of mathematics; hence making mathematics more human, one of the fundamental ideas Hans Freudenthal had 30 years ago. The particular examples of what this integration will look like in practice are still open for research.
Just as the calculus has changed humans’ way of thinking, so are digital tools and the Internet changing how young people approach knowledge and therefore the way they learn. As the calculus gave a new perspective to the study of space, digital tools combined with the Internet are changing the perspective of ‘space’ and in particular of ‘learning space’. Colliding virtual and physical spaces into one that I will call ‘Dynamic Space’.My research interest is precisely in how the learning of mathematics develops in such a space.
To summarize, the aim of my research project is to design a learning intervention using cultural context and historical material as a tool to foster connections among seemingly fragmented bits of inorganic mathematical knowledge, and to promote in 17–18-year-old students the learning of mathematics through knowledge re-invention, making mathematics a human activity. This is an idea that lies in the foundations of Realistic Mathematics Education theory, a way of learning introduced by Hans Freudenthal, for which, following van Maanen’s and Lawrence’s ideas, history will serve as a guide throughout the process, using technology as a workbench for the crafting of intellectual artifacts by young students in the learning process. If there are any contributions, advice or resources you wish to share with me, please visit http://MatHistory.wordpress.com, a collaborative place I have created for this purpose. It will grow organically with my work and your contributions. Thank you!
Chapter 1: The language of the universe
Chapter 2: The genius of the east
Chapter 3: The forntiers of space
Chapter 4: To infinity and beyond
Imagine if we would introduce calculus with such a text:
“The sense of intellectual discomfort by which the calculus was provoked into consciousness in the seventeenth century lies deep within memory. It arises from an unsetting contrast, a division of experience. Words and numbers are, like the human beings that employ them, isolated and discrete; but the slow and measured movement of the stars across the night sky, the rising and the setting of the sun, the great ball that arise at the far end of consciousness, linger for moments or for months, and then, like barges moving on some sullen river, silently disappear -these are all of them, continuos and smoothly flowing processes. Their parts are inseparable. How can language account for what is not discrete, and numbers for what is not divisible? ” (p. xi. A tour of the calculus)
And as the index of the course this text:
The overall structure of the calculus is simple. The subject is defined by a
- Fantastic leading idea
- One basic axiom
- A calm and profound intellectual invention
- A deep property
- Two crucial definitions
- One ancillary definition
- One major theorem
- The fundamental theorem of the calculus
The fantastic leading idea: The real world may be understood in terms of the real numbers
The basic axiom: Brings the real numbers into existence
The calm and profound invention: The mathematical function
The deep property: The continuity
The crucial definitions: Instantaneous speed and the area underneath a curve
The ancillary definition: A limit
One major theorem: The mean value theorem
The fundamental theorem of the calculus is the fundamental theorem of the calculus
Imagine that this would be the index or syllabus of the course and the instructions were: Pick one of these topics, the one that makes you wonder more, that makes you awe, develop it until you can integrate it into a meaningful whole with your course.
They have to choose which group takes which topic, that will make them start to get involved in this aspects of the whole in order to see why they want to work on a particular topic. And if there are any groups coinciding in their choice they would have to negotiate (which is a very important skill to learn). They will need to figure out -a little bit- what calculus is about but as they have some mathematical background they will find their way out
Despite my idea, I will name a few studies that had brought to the field -didactics of mathematics- evidence of the advantages of introducing history and cultural context into the teaching of mathematics.
Doorman, M. and van Maanen, J. (2008). A historical perspective on teaching and learning calculus. Australian Senior Mathematics Teacher. Vol. 22(2) pp. 4-14.
Ramon Llull (Catalan: [rəˈmon ˈʎuʎ]; c. 1232 – c. 1315), T.O.S.F. (AnglicisedRaymond Lully, Raymond Lull; in Latin Raimundus or Raymundus Lullus orLullius) was a Majorcan writer and philosopher, logician and a Franciscan tertiary. He is credited with writing the first major work of Catalan literature. Recently surfaced manuscripts show him to have anticipated by several centuries prominent work on elections theory. He is also considered a pioneer of computation theory, especially given his influence on Gottfried Leibniz.Llull is well known also as a glossator of Roman Law.
In 2010 Taschen republished the work in a facsimile edition. (Wikipedia)
See more here