# CTK Insights

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30 May

### Making Proof a Joke

TweetI just received a copy of Paul Lockhart's A Mathematician's Lament. He has much to say about the role of proof in mathematics and math education and their treatment in school curriculum. I have also written repeatedly about the rampant misconception exhibited by teachers of what a proof is and the process of arriving at […]

28 May

### Beware of Overconfidence

TweetIn one of the recent posts I quoted a retired teacher on the relevance of educational fads on success or failure of the teaching process. In his opinion there was none. And I believe this is indeed the case. All the educational theories, even when supported by the so-called research, remain what they actually are […]

21 May

### One word problem - many word problems

TweetAs every one knows, the word problems supplied by textbooks that attempt to appear realistic are mostly artificial, silly, and elicit from students, if not disgust, then a heartfelt jeer. Teachers who are obligated to follow a curriculum and a prescribed text find themselves in a bind: they often share students' perception of those problems. […]

13 May

### An acute triangle dissection for elementary school

TweetI have recently posted a simple result picked from a very early (1930s) Moscow Math Olympiad for the middle schoolers: In triangle ABC, AE and BD are the altitudes to sides BC and AC, respectively. M is the midpoint of AB. Prove that MD = ME. Vladimir Nikolin, an elementary school teacher from Serbia, noticed […]

06 May

### A curious variant of the Pythagorean theorem

TweetThe trigonometric form of the Pythagorean theorem is well known: sin²(α) + cos²(α) = 1. If α and β are two acute angles in a right triangle, then, since α = 90° - β and sin(90° - β) = cosβ, that identity can be rewritten as cos²(α) + cos²(β) = 1. Taking into account that […]

04 May

### Amazing iterations: parallelograms to ellipses

TweetSome results are made to be shared. Fabian Rothelius came up with a simple iterative process that generates parallelograms after parallelograms with all the vertices on just three ellipses. His explanation underscores the power of affine transforms. Start with a parallelogram ABCD and generate another parallelogram A'B'C'D': Repeat starting with A'B'C'D': And then go on: […]