As a result of Cantor’s developments, one could divide the mathematical community into three sorts. There were the finitists, typified by the attitudes of Aristotle or Gauss, who would only speak of potential infinities, not of actual infinities. Then there were the intuitionists like Kronecker and Brouwer who denied that there was any meaningful content to the notion of quantities that are anything but finite. Infinities are just potentialities that can never be actually realised. To manipulate them and include them within the realm of mathematics would be like letting wolves into the sheepfold. Then there were the transfinitists like Cantor himself, who ascribe the same degree of reality to actual completed infinities as they did to finite quantities. In between, there existed a breed of manipulative transfinitists, typified by Hilbert, who felt no compunction or need to ascribe any ontological status to infinities but admitted them as useful ingredients of mathematical formalism whose presence was useful in simplifying and unifying other mathematical theories. “No one,” he predicted, “though he speak with the tongue of angels, will keep people from using the principle of the excluded middle.”
School classrooms can be exceedingly difficult places in which to learn mathematics well.
Teachers vary in passion and ability (as is natural); class behaviour and attitudes (and bullying) can alter the learning environment radically; learning in a large group at the same speed and with little opportunity to receive one-on-one help leads to boredom or frustration for many/most students; textbooks vary in quality and frequently lack stimulating material (because publishers restrict the size of the books); and sometimes students are negatively influenced by parents and peers who share their own bad experiences or attitudes.
Home educators, too, often feel that they lack sufficient understanding to teach their children mathematics … and the resources that they have access to can sometimes be bland, unstimulating, and even poorly explained or constructed.
My wife and I home educated our daughter and I have taught and tutored mathematics for many years. I am aware of the difficulties listed above and have often been asked by students and parents to create a website to share my understanding and resources with others. Crystal Clear Mathematics is the result.
Because this medium is not interactive (in the sense of allowing immediate interaction between you and me) I cannot provide the quality of tutition that I would like to offer you … but I hope you find the videos and the supporting files to be useful none-the-less.
This is more than what I have expected! The summary idea is brilliant! Thank you very much! Definitely worth sharing!!!
Kenzi B (on a CCM YouTube video about What to Do When You Sit Down to Study Mathematics)