Mathematics and Bowling

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As an international ten pin bowler I have experienced the sport to a great extent. The links between bowling and mathematics may not be obvious to an amateur but there are many underlying principles.

Firstly, the most complex mathematical connection with bowling is calculating and measuring a perfect fit to customise your own bowling balls. This calculation involves the degree of flexibility your fingers have, the length of your span and the width of your finger tips.

This is the information sheet used to drill a bowling ball. The measurements are written to the exact fraction. If the measurement is out, even by 1/32 then the ball can be extremely uncomfortable on the hand.

To workout your starting position in bowling you must use a calculation incorporating many different variables. There is ‘Laydown’ which is the distance between your ankle and where the ball passes your ankle and also ‘Drift’ is how much to the left or right you move when walking towards the approach. Here is an example of one of my starting positions:

A more complex equation for your starting position can incorporate the position and movement of your bowling ball on the lane. The more complex method includes the ‘Breakpoint Board’ which is the board that your ball begins to react to the lane condition. Here is an example of one of my complex starting positions:

Finally, after all the complex calculations used to perfect your technique there is then the score system that’s used which needs to be interpreted correctly. As an amateur who has never bowled other than for fun will not have even considered how the scores are calculated. Here is an example of one of my games scores:

In the bowling scoring system there are many rules used. Firstly, a strike (X) enables you to carry forward the next two shots played to add onto your total whereas a spare (/) enable you to carry forward the next one shot only. Therefore obviously if you do not spare or strike (for example the “8 1” ) your score stops there as you have missed the spare.

One of the main mathematical elements in ten pin bowling is average. When bowling a league or tournament it is whoever has the highest average over all their games who is the winner so you are always calculating your average of the games played to compare with other players. When bowling at an international level you are aiming to have a 200 average minimum throughout your games to be in with a chance of winning anything.

I hope this blog has widened your eyes on ten pin bowling as a sport as not many people consider bowling to be a sport. This is quite frustrating as it is such a complex sport that requires a lot of constant practice on your physical game as well your mental game and knowledge game.

Musical Maths

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As a musician myself, I can easily relate to mathematical connections with music.

Music is, and has been, my passion for many years. I see music as my method of relaxation. To sit back, listen and enjoy exactly what I am hearing.

As a teacher of music, when I am teaching the very basics of music I am also teaching the very basics of maths. Sometimes it can be difficult for children to understand music if they don’t have a knowledge of the basics of maths. On the other hand, sometimes children will understand and relate to maths better because of their musical development.

Obviously the basics of maths and music is the length that each note is worth. When teaching the children the not triangle this allows children to see how many notes would fit into the first long note, the semibreve. The basic structure of this can be easy to remember visually but to work out more difficult theory questions, it is a lot more beneficial to be able to use division. Here is an example of a basic note triangle.

When doing theory practice, I often use note sums to build competence with the value of notes. The children need to know confidently exactly how long each note lasts to allow them to calculate the sums. This is basically just simple mathematical sums but with notes instead of numbers.

These are a few basic examples.

On a more difficult level, musical scales have a very close link with the Fibonacci Sequence. The Fibonacci Sequence came from a theory on rabbits which gave out a set of very memorable numbers: 1, 3, 5, 8, 13 etc. A musical scale includes 8 notes all together and similarly with the Fibonacci Sequence the most important notes of the scales are the 1st, 3rd, 5th and 8th.

In musical scales we use all four of these notes to create an arpeggio of the scale. Arpeggio’s I initially learnt from the Disney film Aristocats. In this video Berlioz plays arpeggios throughout the song ‘Scales and Arpeggios’ as he plays the notes of the scale in this order: 1, 5, 3, 5: 1, 5, 3, 5 etc. Half way through when he jumps on the piano he plays the full scale as he runs up the keys from the 1st note to the 8th. Have a wee watch below.

Another interesting link with music and maths is that the Fibonacci Sequence is actually created the bass clef symbol which is used to determine the notes on the piano used. I have drawn the diagram below to show you:

I find it incredibly interesting how closely linked the two subjects are. Even just with thins as small as your fingers being numbered 1-5 to help with the order the music is played.

My passion for music has allowed me to enhance and progress with my mathematical competence.

my life has been invaded by mathematics

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The Discovering Mathematics module has enabled my brain to be extremely mathematically reflective on life. Over the course of this module I have had to keep notes of anything that has sprung to mind as I have had constant spur of the moment ideas.

I didn’t realise how mathematical my brain could actually be until I began to think of the relations to my every day life. I have really surprised myself!

I came across this picture last week and it made me laugh! Setting an alarm in the morning isn’t as easy as some people make out! When setting my alarm I have to think through my whole morning routine before I can do this. My method is to start with the time I need to be ready for then work backwards. For example: 8am is my leaving time so I subtract each routine; 15 mins for breakfast = 7:45am, 30 mins for showering/getting ready = 7:15am, 10 mins to get my things together = 7:05. This informs me I have to set my alarm for 7am (I need those extra 5 mins to moan about having to get up!)

Shampoo … Who would’ve though buying and using shampoo would be so difficult?

The shampoo my hairdresser recommended to use (Paul Mitchell) was rather expensive at £22.50 for one bottle therefore I had to use my mathematical skills to convince myself it was worth the money. To do this I had to take the price (£22.50) and to divide it by the amount of shampoo (1000ml) which gave me that it costs £0.02 per ml. On average I use 10ml per wash then I could calculate it was only going to cost me 20p per wash. That seemed reasonable enough to purchase. I forget the initial £22.50 and just think of the 20p!

Finally, on my 18th Birthday I was presented with a bottle of Grey Goose Vodka from my boyfriend. The only calculation to make was how many glasses can I have before I need to get another one! The bottle held 350ml of vodka and a single shot of vodka is 25ml. By dividing the two I discovered I could have 14 glasses before I would run dry … or should I say … run to the supermarket!

Health, Statistics, Data

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During our discovering mathematics module we had Dr Ellie Hothersall come to give us a presentation on the relationship between mathematics and the healthcare profession.

Ellie described that we expect everyone to have a basic understanding of mathematics but this isn’t always the case. This deems to be of huge concern as having wrong numerical calculations can be fatal within the healthcare profession.

I can relate very closely to the healthcare profession as my younger brother has spent the majority of the past four years in hospital after being diagnosed with leukaemia, cancer of the blood, when he was just five years old.

During this time in hospital a part of this was spent in the intensive care unit. This allows me to relate to the importance of data interpretation as my brother took very ill, very quickly, so the doctors took a blood sample. Within minutes the results arrived back as did a whole team of doctors. After analysing the data presented they immediately put him into a coma. Later the doctors told us at that point in time my brother had literally seconds to live.

This shows how important data interpretation and mathematical understanding is. Just being able to do quick calculations with the results saved my brothers life. Within seconds they had to insert numerous injections and wires to give the medication to save him.

This is what the doctors did, by correctly interpreting results, to save my brothers life!

Ellie emphasised that people involved with the health profession must be aware of the underlying mathematical principles of medicine when making medical decisions.

Another example of this importance is in March, 2010, an elderly lady died due to being given the wrong dose of the drug potassium chloride. The nurse calculated that the lady needed 50ml of the drug. The nurse then entered into the machine, the figure of the calculation she had previously done, which resulted in the lady’s potassium blood reading to be 7.4 instead of 4.5 (the norm) which led to her suffering a cardiac arrest. DAILY MAIL REPORTER, 23rd FEB 2011.

Both of these examples show an incredible importance to health professionals being able to do correct calculations and to interpret data and statistics accurately.

Teaching Mathematics to Children

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Mathematics in Scottish Primary Schools should be taught as a fun and creative subject which forms a core for other subjects. Children should enjoy learning mathematics and to be able to connect with mathematics by understanding the links to why they need to learn it.

Scottish Executive (2010) believe that teachers have the responsibility to promote the development of numeracy. To do this teacher are advised to use active learning strategies and to make relevant links to other subjects.

Next year for my second year placement I am attending an international school in Switzerland and their main aim is active learning. I am excited to see exactly how they use active learning and their pedagogical approaches which make this so successful.

Throughout my academic reading, linking maths with other subjects seems to be a very prominent subject. Winter et al (2009) say that to improve mathematical competence in children, teachers need to help the children make the necessary connections between learning. Liebeck (1984) agrees that teachers must provide the children with suitable experiences to perform successful matching.

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I believe this picture sums up exactly what everyone is trying to say. It includes active teaching strategies which I have already mentioned. Also real world applications as I’ve said children need to be able to make these appropriate connections before being able to learn fully. Teacher facilitation and learning environment I believe to be closely linked themselves as the teacher, by using suitable methods, need to create a fun and flexible learning environment.

I think this video has some great ideas for active learning in mathematics through the use of sport although I do have some criticism. I think the use of worksheets is unnessecary, I think as soon as the worksheets come out the children will disengage as for some children their gym time is the only time they enjoy because they don’t have to write!

There is verbal, fun methods of assessment rather than using worksheets. As long as the children are participating fully and enjoying what they are doing then there shouldn’t be a need for worksheets at that time. Children will being to relate physical education with writing and worksheets which isn’t the case.

4 ~ Longitudinal Coherance

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Longitudinal Coherance is the final of the four key aspects that Ma (2010) describes to be linked to having a profound understanding of mathematics.

Children need to be given the opportunities to develop confidence and coherence within mathematics. Having longitudinal coherence is when you can identify recurring themes and to see the relevance.

Wu (2002) describes longitudinal coherence as being able to see the interrelationships among topics. Seeing these relationships will allow for coherent development.

I believe longitudinal coherence to be when you have a fundamental understanding over mathematics as a whole. This allows you to explore and experience situations.

Ma (2010) describes Longitudinal Coherence to be using what has been learned during our mathematics journey to influence and support our current mathematical status.

3 ~ Basic Ideas

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Basic Ideas are the third of the four key aspects that Ma (2010) describes to be linked to having a profound understanding of mathematics.

Basic Ideas are ideas as they occur throughout mathematical learning which creates a solid foundation for future learning, Ma (2010).

When teaching mathematics the basic ideas should be highlighted to allow children to use these when approaching different situations. Children need to understand the basic ideas confidently to enable them to use these to help in more difficult questions/problems.

Ma (2010) believes that good mathematicians should be able to identify thee basic ideas prominent in topics to use them in future processes. Once the basic ideas have been learned they can then be use with knowledge to solve more complex calculations.

Bryce and Humes (2008) explain that having a ground knowledge of basic ideas means that you have a sound subject knowledge. We then need to use this knowledge to solve more complex calculations.

2 ~ Multiple Perspectives

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Multiple Perspectives is the second of the four key aspects that Ma (2010) describes to be linked to having a profound understanding of mathematics.

Multiple Perspectives is about knowing the base knowledge and concepts and understanding them to know when and where to apply them to new situations.

In schools, children should be taught different methods/approaches to use. Having single approach methods are no good to children, they need to be able to explore, Vale and Barbosa (2009). Children will learn more through trial and error as they are problem solving through exploration and discovery.

For teachers, teaching different methods will allow for equal learning opportunities. For students, this will cater for a variety of learning styles.

Multiple Perspectives allow for us to incorporate many skills such as; analysing, predicting, reflecting and evaluating. Having these skills will allow you to persevere with your situation until you have an appropriate answer.

Bryce and Humes (2008) both describe multiple perspectives to be having a problem, knowing what approach to use to solve it and why that approach would work best. It is about being flexible with your methods to adapt them accordingly to the situations.

1 ~ Connectedness

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Connectedness is the first of the four key aspects that Ma (2010) describes to be linked to having a profound understanding of mathematics. In this blog I will describe how I have developed to understand and interpret the meaning of connectedness.

Mathematics when taught at schools is not taught as an integrated subject but as a subject in isolation. When mathematics is not connected to real life children will find it difficult to link what they have learned to real life situations.

Children need taught mathematics through cross-curricular activities so they can experience mathematics as a whole. For example, STEM subjects (Science, Technologies, Economics and Mathematics) are so closely linked together therefore it should be obvious that these subjects are easily taught together with the natural link. These links should be highlighted to the children and used to their full potential.

MA (2010) describes connectedness to be having the ability to relate topics to one another. Having these connections allow you to use prior learning and knowledge and to apply these to new situations and contexts. Having the basic knowledge and understanding will allow you to work through new ideas and processes.

Harper (2015) described connectedness to be taking individual pieces of knowledge to make these into a unified body of knowledge. Tara was referring to having the basic understanding of the possible connections to link together with different situations.

Relational VS Instrumental

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Skemp (1989) demonstrates his passion for mathematics through his discovery between relational and instrumental mathematical understanding.

Skemp describes relational understanding to be useful for drawing upon in everyday life as it’s adaptable, easy, effective and is of great quality.

Relational understanding is described as being able to use the coherent knowledge of mathematical structures to make the appropriate connections, Lewis (1996).

On the other hand, there is instrumental understanding which is more time convenient. Skemp (1989) describes this to be habit learning. He says this as you can memorise the rules and methods to solve problems.

Here are some examples I have thought of:

Instrumental Understanding = Learning dance steps, order of movements or positions

Relational Understanding = Performing these dance steps more passionately.

Instrumental Understanding = Knowing the rules of driving a car

Relational Understanding = Knowing why these rules are in place