25 OCT 2017 by ideonexus

## Canons and Fugues

The idea of a canon is that one single theme is played against itself. This is done by having "copies" of the theme played by the various participating voices. But there are means' ways to do this. The most straightforward of all canons is the round, such as "Three Blind Mice", "Row, Row, Row Your Boat", or " Frere Jacques". Here, the theme enters in the first voice and, after a fixed time-delay, a "copy" of it enters, in precisely the same key. After the same fixed time-delay in the second v...Bach left his Musical Offering unfinished as puzzles for King Frederick to figure out.

29 SEP 2017 by ideonexus

## Roman Arithmetic

...basic Roman arithmetic is largely rather simple, even for those of us spoiled by Arabic notation. Addition is no sweat, because complex Roman numbers already use what math pros call additive notation, with numerals set beside one another to create a larger number. VI is just V plus I, after all. To add large numbers, simply pile all the letters together, arrange them in descending order, and thereâ€™s your sum. CLXVI plus CLXVI? CCLLXXVVII, or CCCXXXII. And one of the advantages of the Rom...07 AUG 2017 by ideonexus

## The Double Multiplicative Nature of Fraction or Ratio Equ...

Most real-world numbers arenâ€™t always so nice and neat, with wholenumber multiples. If, say, Plant A grew from 2 to 3 feet, and Plant B grew from 6 to 8 feet, then we would say that Plant A grew 1/2 of its original height, whereas Plant B only grew 1/3 of its original height. Such reasoning exemplifies multiplicative thinking and necessarily involves rational numbers. Consider a final example. If you ask a rising 6th grader to compare 13/15 and 14/ 16, chances are that the student will say...29 DEC 2016 by ideonexus

## Mathematics is Hard Work, Not Genius

What I fight against most in some sense, [when talking to the public,] is the kind of message, for example as put out by the film Good Will Hunting, that there is something you're born with and either you have it or you don't. That's really not the experience of mathematicians. We all find it difficult, it's not that we're any different from someone who struggles with maths problems in third grade. It's really the same process. We're just prepared to handle that struggle on a much larger scal...17 AUG 2016 by ideonexus

## Ways of Being "Good at Math"

Itâ€™s a common misconception that someone whoâ€™s good at math is someone who can compute quickly and accurately. But mathematics is a broad discipline, and there are many ways to be smart in math. Some students are good at seeing relationships among numbers, quantities, or objects. Others may be creative problem solvers, able to come up with nonroutine ways to approach an unfamiliar problem. Still others may be good at visually representing relationships or problems or translating from one ...17 MAR 2016 by ideonexus

## All the Ways of Intuiting 1729

Stanislas Dehaene brings up the Ramanujan-G.H.Hardy anecdote concerning the number 1729. The idea of running through the cubes of all integers from 1 to 12 in order to arrive at Ramanujan's spontaneous recognition of 1729 as the smallest positive integer that can be written in two distinct ways as the sum of two integral cubes is inappropriate and obscures the workings of the naive mathematical mind. To be sure, a computer-mind could come up with that list at a wink. But what would induce it ...05 FEB 2016 by ideonexus

## The Problem with How We Teach Math

Why do some children find Math hard to learn?Â Â I suspect that this is often caused by starting with the practice and drill of a bunch of skills called Arithmeticâ€”and instead of promoting inventiveness, we focus on preventing mistakes.Â Â I suspect that this negative emphasis leads many children not only to dislike Arithmetic, but also later to become averse to everything else that smells of technology.Â It might even lead to aÂ long-term distaste for the use of symbolic representations. Â...05 FEB 2016 by ideonexus

## 12 Tone Equal Temperament

On a standard piano keyboard, one octave is divided into 7 whole tones: A, B, C, D, E, F and G. In between these tones are 5 further notes which can be called either sharps or flats: A# (Bb), C# (Db), D# (Eb), F# (Gb), G# (Ab). (whether it's a sharp of a flat doesn't really matter, the note has the same frequency, just a different name). This gives us a grand total of 12 notes in one octave. If you were to measure the frequency of a note, then measure the frequency of a note exactly one octa...05 FEB 2016 by ideonexus

## The Mind on Music

For some reason that no one really understands, there is a psychological effect upon human listeners in regards to the musical scale. The tonic pitch, or tonal center is not only the mathematical center of the scale, but is the psychological center as well. Human perception of the tonic pitch in relation to the other notes of the scale gives each note of the scale, including the tonic pitch, a distinct "personality" or identity. If we were to label each note of the major scale with a number, ...27 APR 2015 by ideonexus