Throughout the centuries there has been many variations of writing numerals and counting. An example of something completely different, and outdated, Yan, Tan, Tethera method which was most commonly used to count sheep, although has also featured in a Scottish dance. Today, in Britain, we use the decimal system for counting which is based on ten; the number of fingers and thumbs we have. This system has derived from an ancient Hindu system which was developed by Arabs during the ninth and tenth centuries before quickly spreading throughout Europe.

When teaching, to explain the concept of place value, we write out the names of columns to demonstrate the value of each number:

This number is then 3452. We can describe this number as consisting of three thousands, four hundreds, five tens and two units. When written out in words it would read; three thousand, four hundred and fifty-two. Zero is an important feature in that it is described as a place holder; if there were no tens in three thousand, four hundred and fifty-two, it would become three thousand four hundred and two which would be written as 3402.

Thus, in our decimal system the concept of place value can be explained as; the place of a digit in a number tells you the value of that digit.

A very significant feature of the place value system is the principle of exchange; it is described in Haylock as being “the principle at the heart of our place value system”. It is the idea that a ten in one place can be exchanged for one in the next place to the left, for example 10 hundreds could be exchanged for 1 thousand.

It is this principle of exchange which makes the place value system fundamental to the way we complete calculations, which is how I can introduce the element of Connectedness (Ma, 2010, page122). My understanding of Connectedness is that there are a lot of areas of mathematics which overlap, such as a concept being shared within different areas. So when I was reading about place value it reminded me of the element of Connectedness, and this is due to the principle of exchange. Within addition calculations you ‘carry’ numbers, this is the process of replacing ten in one column by one in the column to the left. The importance of place value here is imperative as without an understanding of it, you would be unable to complete the calculation.

An understanding of place value is also vital to complete subtraction sums consisting of ore than 1 digit. It is needed to complete column method of subtractions using the principle of exchange to overcome the difficulty caused when the digit to be subtracted in a particular column is less than the one it is being subtracted from. This demonstrates the element of Connectedness.

I have taught a few lessons on place value before moving onto multi-digit addition and subtraction calculations. From giving those lessons, I understand how difficult it is to understand sums using ‘carrying’ and ‘borrowing’ if you do not have a knowledge of place value. I found that the best way to teach place value was by using base ten blocks, which most schools have access too. They consist of units, tens and hundreds, and makes it a lot easier to demonstrate how you can exchange 10 units for 1 ten and 10 tens for 1 hundred. It is also possible to use these blocks to practice addition and subtraction with exchanging. The physical aspect to the lesson definitely helps the child understand what is happening.

I love music and I always have. I was never able to sing so I turned to instruments instead, just like my dad and my granddad. From a young age I played the trumpet in jazz bands, brass bands and windbands and have never doubted it wasn’t for me. Since moving to university, I have stopped playing as much but this input reminded me just how much I miss it.

Within music, there are a lot of scales and the whole tone scale is  particular favourite of mine. After this input I started researching further  the relationship between maths and music and discovered there was a link between the whole tone scale and Pythagoras. The whole tone scale is demonstrated at 1:01.

The whole tone scale was developed my Pythagoras by using the ratio of the frequency of the sound waves of two tones. Octaves have a ratio of 1:2, and so Pythagoras used smaller intervals to develop the whole tone scale.

Also mentioned is the symmetrical nature of the scale which can be seen here:

Symmetry in music is common. A well-known composer who used it was Bach, and it can be seen in his Goldberg Variations at 3:11.

Another Mathematician, who’s work featured in music is Fibonacci. His work features in the design of the piano keyboard as well as in pieces of music. You can read about this in my blog about Fibonacci.

I never thought that one day my love for music would increase my enjoyment for maths, but it definitely has! I now know longer view maths as a scientific and analytical subject, instead, I can see the art and emotion and beauty of it.

Named after Leonardo of Pisa, the Fibonacci sequence is a sequence of numbers defined by the fact that each term is the sum of the previous two terms. Thus, the first fifteen terms are as follow:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377

 Leonardo of Pisa, nick named Fibonacci, was an Italian Mathematician who was a major figure in spreading the Hindu-Arabic numerals to the rest of the world. These numerals are the ones we use today and replaced the previously used Roman numerals.

The Fibonacci sequence was first presented as a solution to a rabbit problem:

Fibonacci’s sequence also frequently appears in nature in the form of a spiral. It can be seen in flowers, pine cones and pineapple fruitlets, giving the seeds the most efficient and even distribution in the space available. The most common example is in a sunflower; the seeds appear in a spiral with the seeds coming out of the centre which usually grow in formations of 55 clockwise and 89 anticlockwise; both Fibonacci numbers.

Within a lecture focusing on Fibonacci and art we drew our own spirals using Fibonacci’s numbers:

We also investigated the mystery of Phi and the Golden Ratio, it relates to the spiral also, as with each 90 degree turn it gets 1.6180 times wider. From Fibonacci numbers, you can get the ratio by taking any two successive numbers and putting them into this formula:

The Golden ratio has been commonly used in art and these are only a few examples:

My main interest, however, is in music rather than art and after the past few years of playing piano, I was amazed to find out Fibonacci played such a large role in music. Fibonacci’s sequence is a framework which appeals to many composers, possibly because the golden ratio. The Golden Ratio often features to generate rhythmic change or develop a melody line.

Fibonacci’s numbers feature in the piano keyboard:

Examining the scale of C to C, you can identify thirteen keys; eight white and five black. This can be further related to Fibonacci using the layout of the keys, with the black keys in groupings of twos and threes.

Fibonacci’s numbers also feature in an octave. Within a notes octave there is a span of thirteen notes and a scale is made up of eight notes. If you investigate a chord, the finding is that the chord’s foundation is the third and fifth notes which are based on a tone made up of the two steps, and one step from the root tone, as demonstrated below: