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Liping Ma’s Profound Understanding of Fundamental Mathematics

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As I write this blog, semester 1 of 2nd year is almost at its end. I get misty-eyed when I think about this model nearly being over as I have enjoyed it so much and learned things I would never have known about otherwise. Truly fascinating module. It’s nothing to do with complex equations or any horrible higher maths recaps. I have mentioned in a serious blog how i didn’t like maths in high school but ‘Discovering Mathematics’ has made me appreciate it again as i now have a deeper and broader understanding of fundamental mathematics and how this links with wider contexts.

Our first ever lecture in this module introduced us to PUFM (profound understanding of fundamental mathematics). At first I just brushed it over as I said to myself I can research into it later on in the module as I didn’t understand it and quite frankly found it confusing.

Once I came across Liping Ma’s book and read up about her theory then I was able to understand it better and see the links in all my future lectures. Ma wanted to understand why the U.S.A were in a much lower rank for test results than China was. Ma (2010) concluded that the reason the U.S.A were so behind was because teachers didn’t obtain an extensive understanding of elementary mathematics. She figured that during a teachers training they should be made aware and become habitual with basic (fundamental) mathematics as this is what the teachers in China have knowledge on from the start (Ma, 2010).

“By profound understanding I mean an understanding of the terrain of fundamental mathematics that is deep, broad and thorough. Although the term ‘profound’ is often considered to mean intellectual depth, it’s three connotations , deep, vast, and thorough, are interconnected.” (Ma, 2010, pp. 120).

To achieve the expected knowledge that Ma thought a teacher should have, she came up with 4 principles that would enable a teacher to have a profound understanding of fundamental mathematics:

Connectedness – “A teacher with PUFM has a general intention to make connections among mathematical concepts and procedures…” (Ma, 2010, pp. 122). This means being able to make links and see connections between mathematical concepts in a wide range of things in society. Also the importance of highlighting this to students when teaching so that they can discover and see these links. In the students learning this would mean that their knowledge learnt would not be fragmented but rather connected.

Multiple Perspectives – “Those who have achieved PUFM appreciate different facets of an idea and various approaches to a solution, as well as their advantages and disadvantages In addition, they are able to provide mathematical explanations of these various facets and approaches…” (Ma, 2010, pp. 122). This means the teacher should respect the multiple aspects of problems and solutions, moving away from there only being one answer. Together with allowing students to inspect these multiple aspects so that they have a flexible understanding of the topic.

Basic Ideas – “Teachers with PUFM display mathematical attitudes and are particularly aware of the “Simple but powerful basic concepts and principles of mathematics” (e.g. the idea of an equation)” (Ma, 2010, pp. 122). This simply means that the teacher should encourage children to explore the points relating to problems. Bringing thoughts back to the basics of mathematics to embolden the students understanding and make the subject less daunting. Students learning and understanding will therefore be more broad and in depth about the subject.

Longitudinal Coherence – “Teachers with PUFM are not limited to the knowledge that should be taught in a certain grade; rather they have achieved a fundamental understanding of the whole elementary mathematics curriculum.” (Ma, 2010, pp. 122). This means that the teacher needs to be able to see where the student is at in their studies and how to progress the student further or fix the problems they are having. The teacher should be of a mind to return and look at learning done in the past, but also able to plan in line with the direction of the classroom’s curriculum and accommodate the students needs within their studies.

“As a mathematics teacher one needs to know the location of each piece of knowledge in the whole mathematical system, its relation with previous knowledge.” (Ma, 2010, pp. 115).

The you look deeper into the 4 principles you can see why mathematics is so important for people to know about. In school, when thinking about the use of connectedness, we need to look across all the curricular areas to see where the links are. This is something a teacher can’t miss out on as mathematics, as I’ve discovered in this module, is in everything that we do in our lives and its important that from an early age we can see how mathematics links with the world around us. When we look into multiple perspectives a myth crops up in my mind that I have heard through out my education, that is that “there is only one process of finding an answer”. This is false. There are many ways in which a problem can be solved, it’s all about teaching the different roots and pathways. The principle of basic ideas is as fundamental to fundamental mathematics as you can get as fundamental is another word for ‘basic’. As professional teachers we need to understand that when teaching young children mathematics, we need to peel back and look deep into the roots so that children progress correctly and will enjoy maths. When we talk about longitudinal coherence, we talk about how we can progress a student further. Teachers need to recognise where a student is at with their learning and identify the correct steps in further educating the student and building on their previous knowledge.

So what have I learned fully for this module? Apart from Liping Ma’s theory about fundamental mathematics, I have learned that mathematics is precisely EVERYWHERE!! Connectedness in full power. I think it is important that when becoming a teacher you need to be aware of this so that you can educate your students on this so they can fully appreciate and enjoy mathematics to the full with the right rooted understanding. Furthermore I am a strong believer in if you’re an enthusiastic teacher while teaching your students will also be enthusiastic about the subjects you teach them. During this module my belief become more true to me as my lecturers were the most enthusiastic mathematics teachers I’d ever seen. This is contagious and made me become even more enthusiastic about the subject myself. Reflecting back on my thought of maths in high school, I’m a changed ‘learner’.

Overall, i think if we suppose a child to have a deep understanding of a specific subject, then so must we.

 

References:

Ma, L. (2010). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. 2nd ed. New York: Routledge.

The Winning Equation in Sport 🥇

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In Richard’s lecture today we looked at the fundamental mathematics within sport. Before the lecture I knew there was a lot of maths within sports but I had never looked deeply into each part of a sport and how it’s applied to mathematics.

Firstly in the lecture, Richard had us look at an old football score table for 1888-1889 and compare it to the ones we see today. We looked at the rankings, the wins, loses and draws and how many the won for and lost against. To try and make sense of the old table we redesigned it by firstly putting the football team names in order from rank 1-12 and followed that with their point beside them. By ordering the team we used the fundamental mathematics category, from Liping Ma, of basic mathematical principles (Ma, 2010). We used counting to help us order and matching to match up the points form the old table to then new order of teams.

Once we had created the new table and a few minutes of problem solving time we came to understand by using algebra we could figure out the total number of point by multiplying the amount of games won by 2 and adding the number of games they drew. We wrote this out as Wx2 + D = P.

After speaking a bit about maths in sport, we worked in small groups to come up with ways in redeveloping an existing sport based on its fundamental mathematics. My group chose to redevelop the sport of netball. In the redevelopment of our version of netball we considered the length of the pitch, where the players pass to/move to during the game and the point systems, considering the goals/basket heights.

By extending the court length, this increases the chance for changeovers and more passes to up the chances of conflict with the other team. Thus makes the game more exciting and unpredictable.

In normal netball the ball is passed to attack, who are closest to the goal. But we decided to make it less simple and create a new rule stating that the defence player now stand on goal side and the centre pass has to pass to either WA or GA (who are now of the other teams goal side). This means there will be a longer distance to pass the ball to get a goal.

Instead of having just 1 hoop to score, we added 2 more hoops creating 3 hoops for possible scoring (a little like the Harry Potter sport of quidditch) which all symbolise different scores. Highest hoop equals a higher score, the lowest hoop equaling a lower score. We also considered from where the player throws the ball to score. We thought by splitting the key in two would make a more fair goal as it is easier to score when you are closer to the hoop. Therefore the closer half on the key gives you a lower score than the further away half of the key.

In our new version of netball there are a lot more fundamental mathematics taking place that the player needs to be aware of. The player will have to use basic mathematical principles (Ma, 2010) when deciding what hoop to try to score in. The use of addition and subtraction would be used to put the hoop score along with the place they have shot from together. They would have to think about what angles would give them the best chance of scoring a goal. They would have to consider the distance they would have to throw the ball as the pitch is a longer length now, perhaps the consideration of an extra pass in place in the midfield?

In my own time I wanted to research about mathematics used in lacrosse as i play it at university. There are many basic mathematical principles relating to lacrosse:

  1. Perpendicular lines are used in lacrosse. To avoid your stick being check by an opponent you must keep it perpendicular to the ground as you can protect the whole stick with your body.
  2. Horizontal line or 180 degrees line are used when starting a match in the middle of the field, when two players from opposite teams throw the ball up in the air.
  3. The speed of the ball when it is passed. The speed can differ whether you are doing an air pass or a ground ball pass.
  4. The speed of the ball can also be effected by the force  you put into pulling the bottom of your stick back and pushing top of you stick forward to pass.
  5. The angle of the stick when you pass and receive the ball. Your arms should represent an acute angle when pulling back to pass the ball.
  6. The weight of the ball.
  7. Body weight distributed between your feet when passing, receiving a shooting.
  8. The motion and curve of the swing when throwing/passing the ball.
  9. The motion of cradling the stick.
  10. The diameter of the ball (approx. 25 inches).
  11. Segment bisectors, the line that cuts the pitch into two equal parts goes through the middle of the midfield third.
  12. Symmetry – the pitch is symmetrical when split directly down the middle.

In the future, during P.E lesson at school I would like to link the curricular areas and demonstrate to pupils how basic mathematical principles can be seen in the different types of sports they will be playing. I would encourage pupils to do sports outside of school as well and bring in their discoveries of maths within it. I remember in school we did a topic of DST (distance, speed and time). I could look at this with the pupils and apply sports to it. This would make maths a bit more enjoyable if they are passionate about a certain sport.

Overall, while playing a sport if you think about it mathematical and apply this is strategy it could improve your chances of winning for your team! The fundamental basic concepts that I spoke about (weight, motion, symmetry, angles, position, counting, adding, subtraction, multiplication, simple algebra, force, distance, speed and time (Ma, 2010, p.104). This therefore proves mathematics can be applied everywhere including playing a sport.

 

References:

Ma, L. (2010). Knowing and teaching elementary mathematics (Anniversary Ed.). New York: Routledge.

Edwards, A. (2012). [Website]. https://prezi.com/59smjt77b-sm/math-project-lacrosse/. (Accessed 09/11/17).

♪ Do Re Mi Fibonacci ♫

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My whole life I’ve been musical. My passion started when my mum bought me a harmonica and a second hand keyboard. I fell involve with music and have played an instrument ever since I was 8 years old. I started with the keyboard then progressed to cello, guitar, singing and saxophone. I did grades in saxophone and managed to get to grade 6 by the end of school. I was a key instrument in the school concert band which took me to 2 countries and many concerts around my home. I had a strict pattern of practising every night, much to my mum’s annoyance, and my sister also screeched her clarinet through the house most nights. My musicality and good rhythm combines with my other hobby of dancing, in particular tap which involves precise beats coming from your shoes.

Today in discovering mathematics, Paola talked about the links between maths and music. And not too surprisingly there’s quite a few.

  • Note values/rhythms
  • Beats in a bar
  • Tuning/Pitch
  • Chords
  • Counting songs
  • Fingering on music
  • Time signature
  • Figured bass
  • Scales
  • Musical Intervals
  • Fibonacci sequence

“Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry. As Stravinsky once said: “The musician should find in mathematics a study as useful to him as the learning of another language is to a poet. Mathematics swims seductively just below the surface.” (Sautoy, 2011).

As I have explored in a previous blog, the Fibonacci sequence (the golden ratio) exists in art and nature but did you know it’s also seen within music! If you’ve read my previous blogs you should be all clued up on it. The scales in music relate to the Fibonacci sequence, there are 13 notes in the span of any note within it’s octave. A scale has 8 notes in it, and within that the 3rd and 5th notes, along with the 1st note, create the simple foundations of any given chord. The scale is based on a tone, the tone is a weave of 2 steps and 1 step between notes (black and white) from the root tone (the 1st note of the scale) (Meisner, 2012).

The 5th note is the ruling note of the major scale. This note is also the 8th note of the 13 notes that are in an octave. This gives more proof to the theory of the Fibonacci sequence in music. What’s more, 8 ÷ 13 = 0.61538…, which resembles Phi (Meisner, 2012).

Compositions are frequently based on Phi. The timings in songs reflect the Fibonacci sequence in that when a song climaxes it often lands at 61.8% through the song. We can also find the golden ratio in the design of musical instruments. For example in the violin (Meisner, 2012).

I used to hate doing scales in music lesson. I would always make up rhymes to remember what notes are in which scales. But Paola taught us a mathematical process for know what every note is in every major scale! I wish i’d known this back in school. The pattern goes tone, tone, semitone, tone, tone, tone, semitone. A tone is when you skip a note in-between 2 other notes and a semitone is just one notes to the immediate next note. These all include the black notes (flats and sharps).

The pentatonic scale was a new concept that I hadn’t come across in my previous music knowledge. The scale is found all around the world is every country and is the foundations for a lot of classic hits. The pentatonic scale is made up of 5 key black notes. It has the same pattern as we discussed for the major scales, so a pentatonic C sharpe scale would go C#, D#, F, F#, G#, A#, C, C#.

In closing, interconnectedness is beaming in the subject of music and maths. Liping Ma’s theory is definitely becoming more and more accurate and clear. I can definitely see myself teaching music with connectedness in mind in the future to my classes, which will give them a more thorough understanding of it.

One more thing, did you know it’s impossible to tune a piano!

 

References:

Du Sautoy, M. (2011). ‘Listen by numbers: music and maths’ Guardian. Available at: http://theclassicalsuite.com/2011/06/listen-by-numbers-music-and-maths-via-guardian (Accessed: 08/11/17).

Meisner, G. (2012). [Website]. Available at: https://www.goldennumber.net/music/ (Accessed 08/11/17)

New Zealand to Hawaii in NO time at all 🌏

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I was excited about todays lecture as in Richard’s powerpoint title it contained the word ‘time’. I have always wanted to learn and find out more about time and timezones including the international date line.

Horology is the study of time and as humans we base our whole lives around this concept of ‘time’. Our life is a trickling hourglass, and when times up that’s our time up on this earth. We constantly are looking at clocks, updating calendars and setting countless alarms because as humans, many of us are useless as time keeping. It is the core to our daily schedules, without it we would be lost.

Throughout our childhood up to adolescence we become professionals at timing our every move. But this only comes from faults and mistakes. We’ve only not too rarely misjudged how long it will take to get up in the morning, ready to leave the house for school or work. You set your alarm the night before (3 or 4 alarms in my case) thinking in your head it will take you approximately x minutes to do each on your tasks in the morning, and by setting my alarm at 8:00am i should make it out the door at 8:45am to make it to school for 9:00am. BLEEEEP BLEEEP BLEEEP *snooze* … BLEEP BLEEP BLE. It’s 8:09am, you’ve slept in. (9 minutes off our time surly won’t make much of a difference). You’ve showered, gotten dressed, styled your hair and put some make-up on, eaten breakfast, packed your bag ready to go and your watch reads 8:51am. We know from experience that it takes 15 minutes to get to the school gates but you’ve only left yourself 9 minutes. You’re going to be late. You’ll know better for next time (hopefully). Now that I am 19 years old I am an expert at time keeping. I deliberately set my 3/4 alarms earlier than the time I need to get up at so I can even fit in a bit of 5 minute snooze time. I’ve dropped the hair na make-up and I know i can get ready in 30 minutes leaving the last 5 minutes for brushing my teeth and getting out the door perfectly on time.

This scenario is all too familiar with the majority of the nation and has a lot to do with fundamental mathematics. During the module we have spoken about Liping Ma. They categorised mathematics into 4 categories, these are: connectedness, multiple perspectives, basic ideas and longitudinal coherence (Ma, 2010). Within these 4 categories are what skills are needed for time management. You need to be good at the skills of organisation, estimation, planning, problem solving, sequencing events, etc.

Safe to say without our digital/analogical friend we would find it hard to tell the time of day or what day it is. Which brings me onto how we ever figured out time in the first place. Why did we need time? Did we always know it was time for dinner or time to get up out of bed? And what aided us to get to our 24 hour days? Richard asked us why was time important to humans in the first place? I answered that humans need to figure out time for farming seasons in order to survive. This lead onto, do other animals need to know time? We came up with the answer yes, they might not have busy schedules like us humans, but they are clever and migrate in the winter to somewhere warmer. So indeed they must have figured out that the cold weather relates to a certain time of year.

Before mechanical clocks were invented, we used inventions such as sundials and obelisks to help us tell the time. They tracked the movement of the sun through out the days so we could figure out that noon was when the sun is at its highest point in the sky and also figure out seasons, how long the sun waved over the sky in the day told us if it was summer (longer sky time) or winter (shorter sky time). The sun shines on the sundial and casts a shadow onto the face of the dial, alining with different hour-lines (sundial, 2017). Another interesting way that we think we used to attempt to first tell the time was with a manmade structure in Ireland called Bru na Boinne (or Newgrange). The building has a long, hollow passage way running all the way through the building end to end. Once a year during the winter solstice (the shortest day of the year) at 9:00am the sun shines right the way through the passage way with a bright beam for only a few minutes. Being able to understand the time and season was critical for survival, they need to know when to plant and harvest crops so when the sun shone through the building they would be able to understand its winter (Turtle, 2016).

In a day we have 24 hours on Earth. This isn’t the same on all planets. On Mercury, the year is shorter than it’s day…WHAT? This is because Mercury is closer to the sun so therefore it orbits the sun faster than we do on Earth. So because of this the planet Mercury turns slower than the time it takes to go around the sun once. So technically if you lived on Mercury you’d have a birthday EVERY SINGLE DAY!!! But why 12/24 hour time on Earth? One theory is that it is based on ancient Egyptian time. They said there was 10 hours of day, 10 hours of night, 2 hours of half-light day (dawn) and 2 hours of half-light night (dusk). They arranged these into 10 day groups, these were called decans and based on zodiac signs made of stars in the sky. The Egyptians must have understood the mathematical principle of ordinal sequencing, relative equidistance and that the Earth was spinning continuously, which they figured out from the stars.

Do you ever come across the situation where you’ve set your alarm for say 9:00am and you wake up at 8:59am. Spooky right!? Some may say that you therefore have an amazingly accurate body clock. Your body has an internal body clock which tells it what to do and when to change its behavioural, mental and physical states at certain times of the day. The are called Circadian Rhythms. The respond to the lightness and darkness of a living things environment, therefore being asleep and awake effects these rhythms. And it’s not just humans that respond to circadian rhythms, animals, plants, fungi and microbes do too!

Research shows that in the artic, the lack of daylight effects the inhabitant animal’s circadian rhythms. Circadian rhythms are only seen in the parts of the year that have daily sunrises and sunsets. One species of reindeer only has circadian rhythms in spring, autumn and winter but not in summer. Butterflies also have in interesting circadian rhythm which lies in their antenna! This clock in their antenna knows when to migrate during the autumn into winter. In plants their circadian clocks tell them when it is the right season for their flowers to bloom for the best chance of pollination (Circadian rhythm, 2017).

Your circadian rhythms can be disrupted in many ways. Once way is due to jet leg. When you have flying and you pass through the separate time zones, your body clock will be different from the local time zone clock. For example, if you fly east from Perth to Sydney in Australia, you “gain” 3 hours. When you land and wake up to it being 6:00am in Sydney, your biological clock is still running with the Perth local time, so your body is confused and thinks its the middle of the night at 3:00am. After a few days you body clocks with reset but this can make you more tired as your sleeping patterns are effected (Circadian rhythms, 2017).

In 1880 GMT (Greenwich Mean Time) was introduced as the central point for time zones across the world. Time zones split up the world, longitudinal ways, so that most countries fall into one time zone. This meant that each zone to the right of GMT increased by an hour and each zone to the left of GMT decreased by an hour until they meet around the other side of the Earth at the IDL (International Date Line). It is set up this way as each part of the world gets sunlight and night time at different times of the day as the Earth is spinning. Some argued that the world should have “practical time”. This means it would be the same time around the world, for example in the Uk we would be having our breakfast at 7:00am at sun rise and people in eastern Australia would be having their breakfast at 6:00pm at sun set. Seems a bit silly to me as then the Australians would be arriving at work in the dark…

During the year, in many countries, we have a concept called ‘daylight saving’. This is when we lose and gain an hour each year to help our society to continue to function and produce food when the day light decreases. In the spring time we gain an hour, we “spring” forward, and in autumn we lose an hour, we “fall” backwards. This mean in winter we will have lighter mornings as in winter the day light get shorter until the 21st of December (shortest day of the year).

Once you reach the other side of the world from the UK you will reach the International Date Line, about 180 degrees west or east of GMT. This is where time is either one day ahead or one day behind. The imaginary line from the north pole to the south pole is not straight though. It zigzags down the Earth’s surface to avoid political and country borders so that it doesn’t split any countries into two as that would be very confusing if your neighbour was in Tuesday and you were in Monday. When you cross the IDL from west to east (Australia to America) you take away a day, and if you cross east to west you add on a day. This is basically TIME TRAVEL, am I right!? Once a day for 1 hour and 59 minutes there are 3 dates present at the one time. Between 10:00 and 11:59 the 3 countries; American Samoa, New York and Kiritimati all have different dates. For example April 1st, April 2nd and April 3rd (What Is The International Date Line (IDL)?, 1995). So you can set off from New Zealand and arrive in Hawaii the day before you set off!

Overall, this is my favourite example of Liping Ma’s theory of connectedness (2010). Time is all about Mathematics and I feel Richard gave us a good insight into how they link and are used in our everyday lives. I have developed the understanding of connectedness throughout this topic and plan to deepen my understanding of this for my future students when connecting time in maths to wider contexts.

 

References:

Ma, Liping. (2010). Knowing and Teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States. New York: Routledge.

Sundial. (2017). [Website]. Available at: https://en.wikipedia.org/wiki/Sundial. (Accessed 06/11/17).

Turtle, M. (2016). [Website]. Available at: https://www.timetravelturtle.com/visit-newgrange-dublin-ireland/. (Accessed 07/11/17).

Circadian rhythm. (2017). [Website]. available at: https://en.wikipedia.org/wiki/Circadian_rhythm. (Accessed 07/11/17).

Circadian rhythms. (2017). [Website]. Available at: https://www.nigms.nih.gov/education/pages/Factsheet_CircadianRhythms.aspx. (Accessed 07/11/17).

What Is The International Date Line (IDL)? (1995).[Website]. Available at: https://www.timeanddate.com/time/dateline.html. (Accessed 07/11/17).

🌻 The Fractal Nature Of Reality 🍀

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I came out of this lecture completely mind blown at our world and what is produces naturally, we truly live in an extraordinary reality. Today we learnt about the Fibonacci sequence (the golden spiral) together with Phi (the golden ratio).

In 1509 there was an Italian mathematician called Luna Pacioli who published Divina Proportione, which was a treatise on a number that is known as the ‘Golden Ratio’. We symbolise this ratio by Phi (Φ). This ratio comes with fascinating frequency in nature all around us and mathematics (Pickover, 2009).

The Golden Spiral is made up of the Fibonacci sequence. The sequence is made by the fact that every number after the first two is the sum of the two preceding ones. The Rule is xn = xn-1 + xn-2. The sequence goes:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, … to infinity.

If you draw these numbers out in length x breadth boxes then it creates the golden spiral.

This spiral can be seen in a vast amount of nature from our galaxies to the shell of the nautilus. It is truly mind blowing to say the least, the fractal nature of reality.

A fractal is a way of seeing infinity – Benoît Mandelbrot

The natural world has a fondness of the Fibonacci numbers. If you look at flowers most of them have a Fibonacci number of petals.

3 petals = lily & iris

5 petals= buttercups

8 petals = delphinium

13 petals =  marigold & ragwort

21 petals = aster

55/89 petals = daisy

Not all flowers will have these numbers but averagely they do. For instance this is why 4 leaf clovers are so rare as the number 4 is not in the Fibonacci sequence (Bellos, 2010).

Just as pi (π) stands for the ratio of the circumference to the diameter of a circle, Phi (Φ) stands for a special ratio of line segments. When a line is divided in a unique way the ratio Phi happens. “We divide a line into two segments so that the ratio of the whole segment to the longer part is the same as the ratio of the longer part to the shorter part” (Pickover, 2009, p.112).

(a + b)/ b = b/a

The ratio (golden ratio) is 1.61803….

Using this ratio Anna asked us to work in pairs to see how “beautiful” our bodies were. By dividing different measurements by each other we were able to calculate, if our body part = 1.6… then they were “beautiful”. Me and Ellie Kean calculated we both had “beautiful” heights.

The Greeks were amazed by this ‘phi’. They founded the 5-pointed star, it was the admired symbol of the Pythagorean Brotherhood. It was called the ‘extreme and mean ratio’ by Euclid and he was able to make it by a compass and a straightedge method (Bellos, 2010).

Leonardo da Vinci is a prime example of an artist who believed maths and art has a strong bond. This is clearly seen in his most famous drawing of the ‘Vitruvian Man’. The drawing shows mathematically and artistically that the human body is has its perfectly symmetrical measurements and dimensions not by coincidence.

Learn how to see. Realise that everything connects to everything else – Leonardo da Vinci

Overall, there is a true connectedness of mathematics and art. There is proof of this in our worlds nature make-up and has been discovered through history also with the help of Leonardo da Vinci. I can use Leonardo as an inspiration within my future art lessons with students so they can have a more broad understanding of the history of art and how it connects to art we see now in the present day.

References:

Pickover, C. A. (2009) The Math Book From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. London: Sterling.

Bello, Alex (2010) Alex’s Adventures in Numberland London: Bloomsbury

Yan, Tan, Tethera…

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I’m going to be honest I think most of us came out of this particular lecture with a sore head. Today Richard taught us about place value and the binary system. This module is called Discovering Mathematics for a reason as every input I come out as if I’ve dug up gold.

There are lots of number systems that exist in the world, I’m going to talk about the base 10, 12, 20 and 60 systems. Farmers from the North of England commonly use the base 20 system for counting sheep. Very time they count 20 sheep they pick up a stone and put it in his pocket which represents a set of 20, the start again (Bellos, 2010). The way the count isn’t by saying 1, 2, 3, etc. But rather

Yan, Tah, Thethera…

They use words instead of simple numbers. But not all shepherds use the exact same system, in Lincolnshire, Yorkshire, Derbyshire, County Durham and Lancashire they all differ slightly. You may notice for the value of 10 there is a combination of “Dix” and “Dick” which oddly is the same as the French for 10, “Dix”. So it’s not all a load of rubbish, some of the numbers have origins form other languages (Yan, Tan, Tethera, 2017).

Having a base number is essential for any number system to work smoothly with little confusion. If you had a ‘base one’ system, you would only have one number word. So for example, taking yan, tan, tethera, one would be yan, two would be yan yan and three would be yan yan yan. This base system would never work as for 100 you would have to say yan 100 times!! In addition imagine if every number had its own word that no other number had part of. You would have to have a very good memory! (Bellos, 2010).

Some communities that are more isolated haven’t evolved their base systems and still stick to tricky ones. The Arara’s in the Amazon still count using pairs of numbers. The numbers 1-10 as follows:

anane, adak, adak anane, adak adak, adak adak anane, adak adak adak, adak adak adak anane, adak adak adak adak, adak adak adak adak anane, adak adak adak adak adak.

This again isn’t a practical number system to use as Bellos (2010) says it “make haggling at the market rather time-consuming”.

In the UK we use the base 10 system. This is a good system as the base is big enough that we are able to say numbers such as 100 without gasping for breath, but also base 10 isn’t too large that it over complicates our brains (Bellos, 2010). Richard asked us why did we pick this system for counting? We responded with because we have 10 fingers, almost as if both our hands represent the stone in the base 20 system. To say thats the reason, I wasn’t sure until I read chapter 1 in Bellos’s book. He explains that the obvious way to decide on a base number was to look at what markers we could use with our body parts. Looking at our hands we have 5 finger on each hand and again 5 sets of toes on each foot. The sets of 5 also give natural pauses and creates sets of 10; here was invented the base 10 system (Bellos, 2010).

The next base system we looked at was the base 12 system. The base 12 system has numbers 0-9, plus two more numbers that represent ‘ten’ and ‘eleven’. These extra two numbers are called “transdecimals” and represented as an x for ‘ten’ and a backwards, upside down ‘3’ for ‘eleven’. In this system 10 still exists but now represents ‘twelve’. The new digits are given names to avoid confusion with the base 10 system, these are dek for ‘ten’ and el for ‘eleven’, and furthermore ‘twelve’ is given a new name too, which is do, pronounced doh, which is shortened from dozen (Bellos, 2010). Here is a video showing the benefits of the base 12 system by Numberphile:

A very common base system used all around us everyday is the base 60 system. It is the most ancient base system know to man (Bellos, 2010). What could this be used for? It is used in time, angles and geographic coordinates and bearings. Originally back in 1794 watches were produced with numbers only going up to 10. This was quickly changed as an hours with 100 minutes in it was not as fitting as an hour with 60 minutes in it, they figured out that 60 had more divisors than 100 had. It was a little triumph in “dozenal thinking” as 60 divides into 12 and 24, both of which are used to make up a full day of hours (Bellos, 2010).

Binary is another number system I found much more confusing than the rest as it only has place values of 0 and 1… that’s it. It uses 0 and 1 to represent every quantity of numbers. Richard showed us its pattern and I just thought whatttttt???? I eventually became to understand it, my conclusion was that when column 1 was full you move it to the left and set column 1 back to 0. It goes like this…

1 = 0                                    0

2 = 1                                    1

3 = 10                                10

4 = 11                                11

5 = 100                            100

6 = 110                            110

7 = 111                             111

8 = 1000                         1000

      etc.

Still confused here is another video explaining the binary system:

At the end of the lecture we were asked to now make up our own number systems. Me and Beth Arbuckle made up our own and it went a little something like this…

We used shapes and based lines reflecting the numbers around them. a circle was 0-10, a triangle was 11-20, a square was 21-30, etc. This system worked well as by counting the individual lines it made up the exact number it represented. It also work with math equations like addition, subtraction, multiplication and division!

In conclusion, by making up our own number system we were making to check they worked by using Ma’s characteristic of ‘basic ideas’. All base systems connect to basic ideas as numbers are what makes maths, maths in the end. By having a better understanding of base systems I can explore with with my future students and even connect it with foreign languages and their number systems.

 

References:

Yan, Tan, Tethera. (2017). [Website]. Available at: https://en.wikipedia.org/wiki/Yan_Tan_Tethera. (Accessed 04/11/17).

Bello, Alex (2010) Alex’s Adventures in Numberland London: Bloomsbury.

 

“Take A Chance On Me” 🎲

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Is walking out of the casino with a handful of cash based on luck or is there another explanation, perhaps this player has figured out the games winning and losing percentages and which games have the best odds. The subject of gambling was explored in our Discovering Mathematics lecture today. We looked further into the principles of probability and chance and how these effect things happening in our daily lives, and indeed the casino.

But gambling is all about luck i hear you say, well no. Gambling is all rigged from the very start with odds to ensure games never loose money to the players, they are set up against the player before they even have their first ‘go’. In my opinion, after coming out of this lecture, you shouldn’t waist your time gambling as you will come away with an addiction that results in you having less money than you started. But casinos keep you hooked as they set their odds up just right to keep those players coming back for more. The people who make the game give a player a sudden thrill and boost in hope to make them think they can keep winning but in actual fact they will only win every once in a while. So all those loosing games in-between the winning ones equal to a higher lose than win (Bellos, 2010).

Gambling is a dangerous game, people ruin their lives playing the games by spending every last penny on them in hope to get a jackpot miracle in return. In my experience of working in a shop back in high school i would see the same people coming into the shop every Saturday morning to buy lottery tickets and scratch cards, which made me think how much money they are just throwing away every week on these games to get back no where near as much as they spent.

In Richard’s lecture we spoke about the chance and probability of winning the lottery.

Probability is the study of chance (Bellos, 2010).

The particular game called the National Lottery in the UK consists of the player selecting 6 numbers out of all the numbers up to 59. Recently the UK changed the choice of numbers up to 59 from 49. I remember this happening when I still woke in the shop, and i heard a lot of complaints about this change. The change made the odds of winning even harder as people were winning more than the lottery liked for their profits. In order to win the weekly millions you had to have all 6 of your numbers called out. Since having this lecture I now know how to mathematically predict the probability of any 6 numbers being picked out of 59 numbers. If you played the lottery, you would be 1 in 45 million people. This fraction is used to calculating the probability (Poulter, 2015).

Later on in the lecture, Richard got us to test the probability of the innocent game ‘heads or tails’. Surely I thought the chance of head or tails in a equal 50/50 chance, no? As we further researched the matter it turns out that 51% of the time the coin will land on the same side as it starts! We gave some coin flipping a go to test this theory and it was true! Bellos (2010) describes “impossibility” as having a probability of 0 and “certainty” as having a probability of 1.

After the lecture Richard suggested we watch a TV episode call ‘The System’ on chance by Darren Brown. As soon as I got home I watched it straight away as I love watching Derren Brown and find his work very interesting. One part of the episode showed Derren flipping a coin and it landing on the same side 10 times in a row. But what the show didn’t show straight away is that Derren had been flipping the coin all day for hours and hours until eventually it would land on the same side 10 times. This concluded that the more chances you give yourself the more likely you are to succeed.

Randomness is also a topic that links with probability and chance. We as humans like to see patterns, it is a comfort to us, as we like to feel in control. So if an event occurs randomly we as humans feel that we have no control over it and it unsettles us. It is know to be a “deep-rooted survival instinct”. When we come across randomness we often see it as non-random. An example of this was seen in the shuffling of music on an iTunes. People complained that 2 songs by the same artist would come up one after the other but in actual fact randomness is not smooth, it has areas of overlap and areas of empty space. So having 2 songs by the same artists one after the other is complete randomness. True randomness should have absolutely no memory of what came before. (Bellos, 2015).

In the end, probability and chance, I find, is a very interesting subject to discover and research further into. In school I don’t remember doing much on the subject so as a future teacher I would like to teach about probability and chance within my classroom as i think it is am important subject to have knowledge on as the subject is happening all the time in our daily lives. This lecture once again has made me come away with so many questions about the world we live in.

References:

Bellos, A. (2010). Alex’s Adventures in Numberland (Chapter nine). London: Bloomsbury.

Poulter, S. (2015). Banking on a big lotto win? Your odds just got worse: 10 balls will be added to give you only 1 in 45m chance of a jackpot. [Newspaper Article] Available at: http://www.dailymail.co.uk/news/article-3127201/Banking-big-lotto-win-odds-just-got-worse-10-balls-added-1-45m-chance-jackpot.html. (Accessed 03/11/17).

Derren Brown – The System (Full). (2011). [Video] YouTube. https://www.youtube.com/watch?v=wzjaxhEhy4M&t=3. (Accessed 03/11/17).

Maths, Creative? No Way! 🎨

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Up until this point in my education I’ve always thought of maths as a subject full of equations, difficult strategies and complicated rules. Especially in higher maths at school is was a constant routine of go to class, be taught a new subject for half the lesson with vigorous note taking and then the second half of the lesson would be textbook work. To make matters worse if you didn’t finish the textbook work you had to do it as additional homework…so from my point of view there was very little if not no creativity for the most part of my mathematical education in high school. The early years of primary school is probably where creative maths would be found, but I know what I learnt in my lecture today couldn’t be taught to a 6 year old and I think the more advanced maths is better used with creativity as it’s more of an “ah ha” moment.

In my Discovering Maths lecture today I was able to see how creativity is in maths all around us. Artists throughout history have used such maths topics as symmetry, tessellation and proportion. Ancient Greek architects and sculptors used the golden ratio (I’ll speak more about this in another blog!) to make sure that buildings were physically nice to look at. Portrait painters form the renaissance period made sure that proportions of their portraits facial features and head were in proportion to the rest of their body, to do this they followed mathematical ratios. Tessellations and geometric shapes are seen in Islamic art and represents spirituality of the world.

What are tessellations you ask? A tessellation is a pattern of shapes that fit perfectly together, with no gaps, and create a pattern that could repeat for infinity and beyond.

Islamic art avoids the use of human figures or animals in their art. But rather has three principle elements which they include in each piece. These are:

  • Calligraphy
  • Arabesque
  • Geometry

Sacred Geometry gives symbolic meanings to types of geometric proportions/shapes. it is also associated with the belief that “god is the geometer of the world” (Sacred geometry, 2017). Geometry is said to be at the heart of nature, it is seen in all of nature if you look close enough, and therefore makes us all develop so beautifully. And so it is at the heart of Islamic art. “In the world of natural phenomena, it is the underlying patterns of geometric form, proportion and associated wave frequencies that give rise to all perceptions and identifications. Therein lies our fundamental capacity to relate, to interpret and to know.” (Bansal, 2014).  Aspects of nature that geometry can be seen in is honeycombs, spirals in flowers, pine cones, foliage on trees etc. Nature’s design links to the golden ratio, divine proportion, phi and consciousness. The chambered nautilus grows constantly and its shell creates a logarithmic spiral shape and hold the growth without the shell changing its shape, this shape links to the golden ratio which we will learn more about later on (Crystal, no date).

 

 

 

 

 

 

‘The seed of life’ is a concept relating to flower patterns. Within the flowers geometry we start with the seed which includes 6 circles interlinked together which creates with perfect pattern that you may see used in company logos. Once the see germinates it starts to turn into a flower and the geometry continues by adding a further 6 circles, turning the pattern into 12 interlinked circles. This is an fascinating design from natures basic friend, the flower. These patterns are universal in nature and if you were to go to another universe or planet you’d find similar if not the same pattern there too! (McConaghay, 2016).

 

Within Islamic art there are three fundamental shapes used:

  • Equilateral triangle –

This is the simplest shape to draw and fit together and it represents harmony and human consciousness.

  • Square –

Also an easy shape that fits together and it represents the four corners of the earth.

  • Hexagon –

This shape represents heaven.

The star whether it is a 6, 8, 10 or 12 pointed star is also often used in Islamic art.

From this lecture/workshop we got to be fully creative with paints, coloured paper and a load of cutting out (being safe with the scissors at all times!). We made our own versions of tessellations which all turned out very pretty if i may say so myself.

Me, myself am a very creative person in general. I dance, play a musical instrument and took art all the way up to advanced higher at school, therefore my classroom in the future won’t be short of creativity and a splash of colour and art in all aspects of learning!

 

 

 

References:

McConaghay, D. 2016. [Website] Available at: https://www.gaia.com/article/sacred-geometry-nature. (Accessed 02/11/17).

Crystal, E. no date. [Website] Available at: http://www.crystalinks.com/sg.html. (Accessed 02/11/17).

Bansal, A. 2014. [Website] Available at: http://www.archinomy.com/case-studies/1938/geometry-nature-architecture. (Accessed 02/11/17).

Sacred Geometry. (2017). [Website] Available at: https://en.wikipedia.org/wiki/Sacred_geometry. (Accessed 02/11/17).

🐒 Can Animals Count? 🐜

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This question was asked during our Discovering Mathematics lecture and created quite a debate/discussion on the subject of can animals really count, the same way as humans, or are they learning to ‘count’ in a different way? Before the lecture continued past the question I gave my opinion that animals can’t count like a humans can but perhaps an animals owner can teach the animal to recognise the shape of a number or teach it a command which represents a certain number. So the animal doesn’t actually understand the concept of a number but rather a command.

As we went through several animals and their abilities to seemingly count I was still stuck to my initial opinion. The first animal we looked at was a horse called ‘Clever Hans’, who could apparently count using hoof stomps to signal a number. Yet we examined this example closer to conclude that the horse was just understanding that a command from his master that meant to start stomping his hoof. The horse didn’t understand “what’s 2 + 3?” but rather the action the master did with their hand which singled to the horse to start and stop stomping his hoof when the horse got the the correct number.

Clever Hans’s understanding is similar to how a dog learns commands. A dog associates the word “roll over” the the physical action of rolling over. The same with Hans associated a movement from his master with the physical tapping of his hoof.

We looked at lions, ants, bees, robbins, chicks and chimpanzees too. Ants made me question my opinion on if animals can count a little bit. Ants are clever animals as they know exactly how many steps it takes them to get them to their nests. An experiment was done to test their mathematic ability. the first test was to see if the ants would stop short of their nest if a few of their legs were chopped off. The hypothesis was correct, they did indeed stop short. The second test was to see if the ants would go beyond their nest if match stick were to be added/stuck onto their legs. Again the hypothesis was right, the ants carried on past their nests. Scientist put this down to ants “internal pedometers”. The ants could travel back and forth in the dark or blindfolded and they would still make it in exact steps back to their nests (Carey, 2006). Did this change my overall opinion? Not yet.

We finally came onto the most convincing of all examples. The chimpanzees. In particular Ayumu the chimpanzee. We watched a video of the chimpanzee managing to tap the numbers 1-9 in the right order. Now my initial thought was “oh the chimp was simply just learned the look of each number as a symbol and has been taught their order”. But then the video clip continued to show that the chimp can still order the number 1-9 when the numbers only flash up for less than a second, then covered up, and still manages to remember the order they were in. Our class tried this and we definitely didn’t do as well as the chimp did. The video then also showed the chimp doing the same thing but this time there was gaps where numbers were missing…So does the chimp really know how to count or does it simply have a very good memory and has been taught forcefully to remember the order, even with gaps?

Ultimately, I was almost convinced. Trust me I was close to believing it. But I still don’t think that animals see numbers and understand them the same way we do. There’re so many arguments to say they can or can’t. If only they spoke english and we could ask them, then we would know for sure.

References:

Carey, B. (2006) When Ants Go Marching , They Count Their Steps (Accessed: 9/10/17).