math

Parallelograms generalise Pythagoras to any triangle

There’s always space on the inter-web for another proof of Pythagoras’s Theorem.  Here’s one that uses the following equal areas property of parallelograms. This kind of area chopping and shape translation is a feature of Euclidean geometry and our senses support it’s veracity at the order of size of the classroom. The squares on the …

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The Ghosts of Departed Quantities and the difficulties of teaching and learning caculus

Bishop Berkeley writes this attack on the apparent supernatural reasoning involved in calculus.  The infidel was probably Halley (of comet fame) or Newton. If pupils find the subject difficult to understand at school, and teachers find it difficult to teach, then the reason may be articulated in this book by the great man. Quotes include: …

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depression near Greenland – logarithmic spirals in nature

Logarithmic spirals frequently occur in nature.  Is this such a manifestation? The general equation for a logarithmic spiral is as follows. Changing the variables and produces spirals of different qualities.  The is really an enlargement scale factor but the controls how the spiral grows per revolution. If is more representative of an ammonite then seems …

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third century wisdom of Diophantus gives us the essential truth about teaching and learning mathematics

The Google facsimile of the 1621 edition of Diophantus’s Arithmetica lays out, in original Greek and Latin translation, the essentials for success in teaching and learning mathematics.  Also included is the translation of the second half by Heath. When I first studied and learned to love mathematics I was beset by friends and family who stridently claimed that …

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