March 28, 2009
Writing about web page http://www.mathematica.gr/forum/viewtopic.php?f=57&t=262
Trying to make my own clock, using a circular canvas, the first thing I needed to do was to find its centre. How can one do this: Pick any point of the circumference (call it P) and draw an arc from circumference to circumference. Use the two intersection points of the arc with the circumference to draw 2 new circles (or arcs), with the same radius as the first one. The 2 new circles meet at the centre of the first circle as well as at a second point, Ã. Use Ã as the centre of a new circle with radius ÃP. This new circle intersects with the very first arc drawn at 2 points, call the Ä and E. Use the 2 points as centres of circles of radii ÄÑ and ÄÅ. These 2 final (!!) circles meet at the centre of the circular canvas started with. There is an image of this construction at the website above (instructions of the website are in Greek nevertheless!)
February 22, 2009
Motivation and afftect in mathematics education are areas I haven't substantially explored yet, any ideas are welcome! I plan to related historical and cultural issues in mathematics education with motivation issues and how are they interrelated, has anyone looked at that?
February 11, 2009
January 25, 2009
November 04, 2008
Discussing on the fairness of mathematics the other day, brought me face to face with a new question. How fair is it for maths teachers? They get to handle the most difficult not to mention hated lesson of all, at great costs. Increasing demands of mathematics education have brought maths teachers to a very critical position: holding that uni position in their own hands.
Is this true? It could be, maybe a little. But ending up chasing students to do their work is not a solution. who is going to chase them later, when they are on their own? What difference is one year older going to make? These thoughts trouble me as I stand on both sides the researcher and the practitioner.
October 08, 2008
Öáßíåôáé ðùò ôá ìáèçìáôéêÜ êáôÜ ôïí Clawson åßíáé áôáîéêÜ. Áõôü ôï äéêáßùìá ßóùò ôïõò ôï ðáñåß÷å ç äýíáìç ôçò áðüäåéîçò, ìéá éäÝá ðïõ ãåííëçèçêå óôçí áñ÷áßá ÅëëÜäá êáé Þôáí ç ðñþôç öïñÜ ðïõ óçìåéùíüôáí êÜôé ôÝôïéï óôçí éóôïñßá ôçò óêÝøçò. Ç áðüäåéîç ìå ôç äýíáìç ôïõ ëïãéêïý óõëëïãéóìïý ðïõ êñýâåé ìÝóá ôçò, ôçí åëåõèåñþíåé þóôå íá ôçí êáôáíïÞóåé ï êáèÝíáò ðïõ åíäéáöÝñåôáé íá ôçí êáôáíïÞóåé' äåí ðñüêåéôáé ðéá ïýôå ãéá äïãìáôéêÞ ðßóôç ïýôå ãéá ìõóôéêÜ êáëÜ öõëáãìÝíá áðü êëåéóôÝò êïéíüôçôåò Þ ïìÜäåò. Ç áðüäåéîç áðïôåëåß êáé ôï êñßóéìï óçìåßï ðïõ äéá÷ùñßæåé ôá ìáèçìáôéêÜ áðü üëåò ôéò Üëëåò åðéóôÞìåò. ÊÜèå ôé ðïõ Ý÷åé áðïäåé÷èåß èá ðáñáìåßíåé áäéáìöéóâÞôçôï ãéá üëïõò ôïõò áéþíåò ðïõ èá Ýñèïõí.
Ðáñ'üëç ôçí ïñèüôçôá ôçò ðáñáðÜíù áíÜëõóçò, ïé Shan êáé Bailey âñßóêïõí üôé óôéò ìÝñåò ìáò åéó÷ùñïýí êáé ðÜëé ôáîéêÜ ÷áñáêôçñéóôéêÜ óôçí ðñïóâáóéìüôçôá ôçò ìÜèçóçò ôùí ìáèçìáôéêþí. ÓõóôÞìáôá üðùò áõôÜ ôïõ setting êáé ôïõ streaming, ìå êáôçãïñéïðïßçóç ôùí ìáèçôþí óýìöùíá ìå ôéò éêáíüôçôÝò ôïõò, ç ïðïßá óõ÷íÜ êáôáëÞãåé óå êáôçãïñéïðïßçóç ôùí ìáèçôþí óýìöùíá ìå ôçí êïéíùíéêÞ ôïõò ôÜîç Þ ôçí ïéêïíïìéêÞ äõíáôüôçôá ôùí ãïíéþí ôïõ, åðéôñÝðïõí íá ìåôáôñÝðïõí ôá ìáèçìáôéêÜ óå ìÜèçìá ðïõ åðéôñÝðåé ôçí ïõóéáóôéêÞ åðéôõ÷ßá ìüíï óôçí åëßô ôùí ìáèçôþí. Ðüôå ìðïñïýìå íá ãíùñßæïõìå ùò åêðáéäåõôéêïß óå ðïéá "êáôçãïñßá" èá ðñÝðåé íá ìðåé ï êÜèå ìáèçôÞò; ÐïôÝ. Ìüíï ïé ßäéïé ï ßäéïò ìáèçôÞò ìðïñåß íá áðïöáóßóåé áöïý ôïõ ãßíïõí äéáèÝóéìåò êáé ãíùóôÝò üëåò ïé åðéëïãÝò: ïé ìáèçôÝò áðéëÝãïíôáò ìáèçìáôéêÜ êáôåýèõíóçò Þ êïñìïý, êáôáöåýãïõí ìüíïé ôïõò óå Ýíá öõóéêü streaming, ÷ùñßò íá Ý÷ïõí áðïêëåéóôåß áðü ïðïéáäÞðïôå äéáäéêáóßá åê ôùí ðñïôÝñùí Þ åñÞìçí ôïõò.
November 25, 2007
Writing about web page http://www.thalesandfriends.org
Last Friday, 23rd of November 2007, I attended an event that included speaches about Mathematical Literature. Thales and Friends is an organisation that promotes the idea of reading mathematical novels and even use them for educational purposes. I love literature myself; I had first met Mathematical literature through "The parrot's theorem", then "Uncle Petros and Goldbach's Conjecture", more recently (although it's the oldest by far) "Flatland" and many more. The website held manypleasant surprises; it has a large amount of information about mathematical literature as well as some great ideas and ready-made material to work some of the proposed books with the students. It is a wonderful tool for the maths teachers. It also has a Greek version, which appears to be somewhat richer, since the base of this organisation is in Athens and the first actions are taken there (such as seminars and other events). I think it is a website really worth to visit.
November 14, 2007
Thomaidis and Tzanakis (2007, Educ Stud Math 66:165-183) conducted research about "historical evolutions and students' conception of the order relation on the number line". They take this as the opportunity to discuss the "controversial relation between the historical evolution of mathematical concepts and the process of their teaching and learning". Is it possible then to follow this "genetic" approach instead of the formal axiomatic one? They cite Harper (1987) who appears to believe that the first approach is possible since the learning route is parallel to the historic one. They also cite Sierpinska (1994), who, on the other hand, believes that the similarities (Harper's parallel roads) are just that (similarities); there is nothing beyond some coincidences that emerge because of the use of language or because of specific mechanisms used when learning. Brousseau (1983) does not argue against using historical approach, but is completely against reproducing whole historic situations; this could be, not only confusing, but also uneconomic (timewise). Herscovics (1989) also finds common aspects between the procedure of learning and the historic evolution but since the modern learning enviroments are so different this "parallelism" should not be taken too literally, as it is asserted.
After presenting their research results, Thomaidis and Tzanakis elaborate the previous opinions a bit further and suggest that, one could exploit these similarities between the learning procedure and the historical development "to foresee possible persistent difficulties of the students; and to make teachers more tolerant towards their students' errors, by increasing their awareness that these errors and difficulties do not simply mean that the student hasn't studied enough, but may have deeper epistemological roots which should be explored and understood thoroughly".
What I see through this article, is once again the need for the maths teachers to be educated about the history of the discipline that they teach. In this case, being aware of difficulties encountered in the past, they may actually get to understand students' weakenessess and be more able to help with it. If they are completely unaware of how a notion evolved, they will have their own experience alone to guide them throught the teaching of this notion. Students, moreover, need to know that they do have someone to rely upon, when things just make no sense.
November 11, 2007
Jones (1969) in Historical Topics for the Mathematics Classroom, of the National Council of Teachers of Mathematics claims that History and up-to-date information about the development of mathematics could a a useful tool for the teacher who wishes to teach "why". Jone's "whys" are of three kinds: chronological, logical and pedagogical. Briefly, chronological whys are about looking at specific facts, (such as the fact that there are 360 degrees in a full turn) or more generally at definitions or axiomatic systems and their development. Logical whys serve into building an understanding of the structure of the axiomatic system through its development and not through its finalised form, which possibly contradicts the historical development as well as "the way perceptions grow in the minds of many of our students" (p.2). Pedagogical whys "are yhe processes and devices that are not dictated by well-established arbitrary definitions and do not have a logical uniqueness", where history may serve in helping the teacher identify a process during which students use a pedagogical sequence to guide their thoughts and finally achieve an in-depth understanding.
What is very interesting is the fact that Jones often implies that all these tools and pedagogical reccomendations can function in the hands of skillful teachers, who, in their turn have very clear purposes in using these tools and make a very detailed planning for their employment. The author ends up suggesting that "the ingenuity of the teacher" is one of the factors that will determine the approach that may be used. Moreover, knowing history on the behalf of the teacher, may help in clearly distinguishing between old mathematics, newer concepts and end up with an informed perspective on what "modern mathematics" is.
Taking into consideration the fact that, as Jones mentions, history should be an important component in teachers' education programs, I have come to think that before discussing about employing students' syllabuses based upon history (and culture), the first step shuld be to make sure that the teachers have a good understanding of what they teach, not just as a tool, but mainly as a human procedure that assists in moving the world forward. As the article begins with the words of Bazun, a maths teacher; "algebra is made repellent by the unwillingness or inability of teachers to explain why...", I come to think that he must have a large proportion of rightness in his sayings. Is the modern maths teacher in position to answer to questions such as "why didn't we divide the circle into 100 degrees in the first place?" Probably not. Students perhaps receive this inability and they are therefore lead to the dislike towards something that come to be meaningless in its gist, even for the person who is in charge to teach them that very discipline.