Swimming about in mathematics 13th February 2014

Swimming about in mathematics

Looking around the BBC website today I came across a piece by James Gallagher on the beauty that mathematicians routinely see in some of the more famous mathematical equations. Euler’s identity equation has been voted by many of them to be the most beautiful because it is simple to look at yet incredibly profound. It comprises the five most important mathematical constants (meaning they do not change) and it also comprises the three most basic arithmetic operations: addition, multiplication and exponentiation.

                                                                   iπ

               e + 1 = 0

e             (a transcendental number)

i              (fundamental imaginary number)

π             (another transcendental number)

1             (multiplicative identity)

0             (additive identity)

Here is a fun version of what the symbols denote when someone is learning to swim

e              (the aquatic learning potential we are born with)               

i               (the focus of our imagination)                                              

π              (the focus of our aquatic learning circle)                              

1              (the nature and focus of our aquatic guardianship)             

0              (Our openness to aquatic experiences)                                

This is a  little bit of fun and is not a formula of mathematical proof. The five constants (things that remain the same) have been exchanged for complex learn to swim variables (things that can change)

I enjoyed Prof David Percy waxing lyrical about Euler’s equation when he said “Given that e and i are incredibly complicated and seemingly unrelated numbers it is amazing that they are linked by this concise formula. At first you don’t realise the implications. It’s a gradual impact, perhaps as you would feel with a piece of music and then suddenly it becomes amazing as you realise its full potential”

He said beauty was a source of “inspiration and gives you the enthusiasm to find out about things”

Although I cannot pretend to understand complex mathematical formulae I do like to try and I like finding out about things, drinking in any beauty and taxing my brain with difficult ideas because they may bring me new understanding.

What else does the hidden beauty in mathematics have to say about learning to swim?

Marcus Du Sautoy mathematician and Prof for the public understanding of science sees beauty where Fermat proved a relationship exists between prime numbers and square numbers.

Any prime number that can be divided by 4 with a remainder of 1 is also the sum of two square numbers. eg 41 divided by 4 is 10 remainder 1 and is also the sum of 25 and 16 which are both square numbers.

“So if it has remainder 1 it can always be written as two square numbers and there is something beautiful about that.”

“It’s unexpected! Why should the two things (primes and squares) have anything to do with each other, but as the proof develops you start to see the two ideas become interwoven like two threads in a piece of music and you start to see them come together”

He said it was the journey not the final proof that was exciting “Like in a piece of music it is not enough to play the final chord”

Parallel thoughts ripple across the surface of my mind as I ponder how to explain to people that it is the journey that matters and it is not enough to mirror the stroke patterns of accomplished swimmers. Also beautiful truths hide in unexpected places and in the pool I glimpse this as a child takes itself aside to follow its own learning thread between group activities. This little piece of stolen time annoys many teachers but not me because it is often where that child finds some gold they need.

             

e   “we can learn to swim now because water shaped us in our ancestral past”


i    ”it matters where our focus of attention is when we are learning to swim”


π     “it matters what our kith and kin feel about being in water as it influences us”


1    ”we need to feel safe and be safe in water to say we can swim”


0   “Our perceptions about water can change”

Swimmer.......Who do you think you are? 17th January 2014

Swimmer......Who do you think you are?

                              What part has life on the land and life in the water played in our past?

Whenever I am near an inviting piece of open water I always contemplate what conditions were like for that first human ancestor when they chose to enter a local aquatic domain in their landscape. It must have happened at some point in our past when there was no one to teach free style or butterfly?
As earth provides two main elemental environments; terrestrial and aquatic, logic states that being able to use both of them confers a great advantage to a complex mobile creature. This simple piece of logic suggests to me that some of our ancestors became relatively more aquatic than we are today. To suggest otherwise removes ready and bountiful resources to our ancestors from the story of our past.
If we did evolve from tree dwelling apes it makes great sense that watery margins were a useful step.

If the logic of the statement above does not strike you then I need to try and explain why it makes such profound sense to me and also ask you to consider the possibility that the strength of your own personal relationship with water will have a strong bearing on how valid it feels.

The advantages that water could have provided for our semi-aquatic ancestor are:

1. Providing physical support, lessening the negative effects of gravity on the body to help us walk.
2. Access to new, varied, extensive, reliable and nutritious sources of food.
3. An extensive space that could provide a safe haven to rest, learn and play.
4. Disease reduction from dilution or removal of pathogens.
5. Aid to recuperation from physical injury and development of swim gait aids strength building.
6. Increased level of experience in individuals with a life span beyond reproductive years.
7. Grandparents helping to raise young, leading to experience transmission and cultural expansion.
8. An increased amount of shared survival skills lead to greater adaptability to overcome the challenges in numerous new environments and territories. Increased mobility and success worldwide.

The disadvantages that water could have presented to our ancestor are:

1. Predators and physical hazards were present in the water.
2. Local conditions changed suddenly through currents, tides, floods, squalls causing death.
3. Waterborne pathogens, disease and parasites caused disability, infections or death.
4. Drowning occurred due to a lack of positive experience or the example of an experienced other.

This leads me to make logical assumptions about what the first aquatic environment was like initially:

1. The water was clear enough for the floor and a lack of predators to be visible.
2. The water was warm enough to maintain a normal body core temperature relatively easily.
3. The environment was stable for long enough to provide local safe conditions on a reliable basis.
4. The water contained attractive resources which were relatively easy to collect.
5. The underwater substrate and water depth changes provided opportunities for gradual exploratory physical support in the form of banks, bars, spits, slopes, shelves, submerged trees, rocks and plants.
6. There was no initial pressure to enter the water other than to fulfil natural curiosity or retrieve food
7. The water could be entered and exited again relatively easily.

The reasoning I have used is based in part on observational experience over years of teaching swimming to people of all ages. I have also watched footage of how other animals engage with water and recognise patterns common to all. What matters most are the initial conditions which allow an animal capable of learning through play to be free of concern for their survival as they enter the water and move in it. If they are free of concern at this point their mobility is not hampered by an inescapable negative visceral response (panic) and they can feel how to travel freely. This state is crucial for animals to dive and swim successfully. Without it they hesitate, stiffly fearing the future. If their families are engaged in aquatic activities to make their living they learn fast no matter how cold the water is. When animals don't use it they lose connection with it. This is a natural process.

In my opinion the advantages provided by water exceed the disadvantages and every environment has its hazards which lead to positive behavioural changes (adaptation). Humans are arguably the most adaptable ape on the planet able to exist in almost every environment it has to offer and I feel that we became this way through an ancestor entering the water in a state of calm, living in the moment, deeply engaged with their environment and able to freely enjoy all of the benefits it had to offer. This phase did not have to last long as once trans generational experience was retained cultural learning had arrived and we became a supremely resourceful species. The rest was history.

Now we view our ancestral past through the hazy lens of a population largely disengaged from the natural environment. No wonder we cannot see the vital part that watery margins played in our past.