Category Archives: 2 Prof. Knowledge & Understanding

Over and Out.

What do I Think?

I chose the Discovering Maths module as an elective in my second year, because I was afraid of maths and always avoided even attempting any mathematical problems. Despite coming through primary and secondary school in the ‘top maths group’ and being very competent and confident in the subject, it all went downhill when i decided to drop the subject after gaining my National 5 A grade. From then on, I began to experience maths anxiety and my confidence in my own mathematical skills, which were previously very good, began to wither. For that reason, I thought it would be a good idea to face my mathematical fears head on and attempt to regain my confidence through this module.

This module has been great for me. I have thoroughly enjoyed looking at different perspectives of maths and even just becoming aware that a huge proportion of adults and children also experience maths anxiety made me feel at ease again.

Through exploring maths I can now see the importance of it for every day life and in wider society, which I had not recognised before. I was not previously aware of how often I actually use maths in real life and this is something I want to demonstrate to my class when I am a teacher.

Including a variety of contexts in the lectures has been an effective way to make mathematics relative to us as students and I feel I have engaged more with it due to this.

My competences in maths have greatly increased from engaging with this module and I feel more prepared to teach this in schools. I have finally regained the maths skills I once had and can apply them to real life rather than just equations in a textbook.

My Thoughts on the Necessary Blogs

Blogging was a big part of this module and i feel it has been a great benefit to me. Blogging about the lectures and mathematical concepts has encouraged me to research more in depth about the topics and in return, has helped my subject knowledge as i know stuff I otherwise wouldn’t have known without researching and questioning it. It has also allowed me to take on more responsibility for my own learning and create blogs which are personal to me.

Through engaging with writing blogs, it has increased my confidence in putting my work out there in public for anyone to see. I am now more relaxed about who views my blogs and i feel i could continue to keep my blog up to date after this module has been completed.

My Future in Maths

In my future profession as a teacher, I want to show children that maths is fun and is all around us every day. I will specifically use Liping Ma’s theory of Profound Understanding of Fundamental Mathematics to aid me in the way I teach maths to pupils.

Overall, this maths module has been very influential for me and i hope to pass on the teaching to another generation of children.

Would you Take a Chance?

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Probability & Randomness

As we are all aware, when we can’t decide what to do, we flip a coin. It is well known that by flipping a coin you have a 50/50 chance of either getting heads or tails.

https://www.flickr.com/photos/nicubunuphotos/5262645427

However, that’s not true.

It is a common misconception that the toss of a coin offers a 1 in 2 chance of landing on either side but we must also take into consideration that the coin will actually land on its edge. Despite the chances of this happening being so slim, it still nonetheless alters the probabilities.

However, on top of that, if we were to remove the chance of it landing on its side, there is still a 51% chance of the coin landing on the same face as it started off on. This is due to the fact that sometimes the coin doesn’t actually fully rotate therefore ends up just falling back down still facing the same way. Seems complicated for just a simple toss of a coin.

Tossing a coin gives a random result, whether it be heads, heads, tails, heads or tails, tails, tails, tails. The chance of the next flip is random yet as humans, we would expect the next turn in the second example to produce a heads as it has not appeared for a while. But if it were to be a tails people would struggle to believe there was no ‘magic’ going on. This demonstrates humans inabilities to truly grasp the concept of randomness.

References

Eastaway, R. (2010) How Many Socks Make a Pair? –Surprisingly Interesting Everyday Maths. London: JR Books

Do I Need to be a Mathematician to Play Games?

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Everyone loves a little bit of Sudoku!

As much as I hate to admit it, I have always been a lover of Sudoku. Despite never thinking highly of my math skills, I always managed to complete puzzle after puzzle – I have even been gifted a 365 puzzle book, one Sudoku puzzle for every day of the year! (Needless to say I still have the majority of them to work through). When I was introduced to the idea that maths is involved in these puzzles I struggled to identify the principles behind it, except the obvious of being able to count from one to nine. Nonetheless, after a bit of conferring and questioning, I realised that there were math skills involved in this particular puzzle. Without being aware of it, I had been problem solving, using the process of elimination, differentiating between the numbers, using sequences and strategies while attempting these puzzles. Newspapers say there is no maths involved in Sudoku, however what they really mean is there is no arithmetic because evidently there is many mathematical concepts involved. To clarify, there is no arithmetic involved in this particular puzzle because the numbers in the puzzle can be removed and replaced by symbols, ultimately removing the arithmetic.

According to Maki Kaji, over 100 million people now regularly indulge in Sudoku puzzles, meaning these people all use these mathematical skills, usually without even realizing it.

“Mathematical riddles, rhymes and games are now collectively known as recreational maths. It s a wide-ranging and vibrant field, an essential feature of which is that the topics are accessible to the dedicated layperson” (Bellos, A. 2010). There are underlying mathematical concepts in nearly every game and puzzle available. For example, the game ‘Battleships’ which I’m sure most are familiar with, involves strategic reasoning and estimations. To play this game you must be confident with locating coordinates on a grid and working out the next point to hit. Also, ‘Connect 4’ is a childhood classic that supports mathematical concepts including geometric thinking, planning and pattern recognition without even realising your doing these things.

Recreational mathematics remains in very good shape today. It is an exciting and diverse field that continues to give pleasure to people of all ages around the world, as well as inspiring serious research. Next time you find yourself playing a game or attempting a puzzle, try work out the underlying mathematical principles involved in playing the game and being successful.

In my future career as a teacher, I know for certain I will be reaching for a board game or puzzle to reinforce mathematical skills in my class. Games are so much fun for everyone and you don’t even realise you are practicing maths!

References

Bellos, A. (2010) Alex’s Adventures in Wonderland. London: Bloomsbury Publishing Plc.

The Sound of Maths

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Maths and music go hand in hand. “Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry.” Marcus du Sautoy (2011).

There are many links between maths and music such as the value of notes, the rhythm, the amount of beats in a bar, the tempo of a piece of music, the pitch, musical intervals even scales and arpeggios are mathematical patterns. Maths is very much a big part of music and to learn a musical instrument you must have a basic understanding of mathematical concepts.

As the great 17th century German mathematician Gottfried Leibniz stated, “music is the sensation of counting without being aware you were counting”.

The “grammar” of music – rhythm and pitch – has mathematical foundations. When we hear two notes an octave apart, we feel we’re hearing the same note which is why they’ve been given the same name. The frequencies of the two notes are in an exact 1:2 ratio.

Just as music is not about reaching the final chord, mathematics is about more than just the end result. It is the journey that excites the mathematician. This is much the same way as we listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don’t realise about mathematics is that it involves a lot of choice: not about what is true or false, but from deciding what piece of mathematics is worth“listening to”.

Musical Symmetry

Symmetry is not restricted to only the visual arts. Music is usually very symmetrical and predictable. Symmetry is the repetition of sounds in music. Symmetry has been used as a formal constraint by many composers, such as Steve Reich and James Tenney. In classical music, Bach also used concepts of symmetry.

Fibonacci Sequence in Music

The Fibonacci sequence beginning 1 1 2 3 5 8 13 can be seen in music. The following video uses visuals to show how the numbers appear in a musical context.

References

Du Sautoy, M. (2011). ‘Listen by numbers: music and maths’ Guardian. Available: http://theclassicalsuite.com/2011/06/listen-by-numbers-music-and-maths-via-guardian/ (Accessed: 04 November 2013)

The Guardian. 2017. Listen by numbers: music and maths | Music | The Guardian. [ONLINE] Available at:https://www.theguardian.com/music/2011/jun/27/music-mathematics-fibonacci. [Accessed 15 November 2017].

Doctors or Mathematicians?

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Statistics and Data in Medicine

Statistics can be defined as a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data. It can be a very complicated area of mathematics but essential in the medical world.

Medicine has become increasingly reliant on mathematics in recent years. Statistics are used every single day in the hospital to ensure the correct dosage of medicine is administered to patients. While most adults are given the same dosage, the dose for children must be carefully calculated because it is much easier to poison a child. Prescriptions need to be calculated very specifically based on the weight of the patient, how many times the medication must be taken each day, how long the prescription will need to last etc. Doctors must also consider how long the medication will stay in the patient’s body as this will determine how often the patient needs to take their medication in order to keep a sufficient amount of it in their body. Doctors must be able to do these calculations mentally with speed and accuracy. Therefore, the understanding of mathematical concepts are a necessity when practising medicine.

IV drips also require careful calculating to ensure it wont do more harm than good. The maths involved in fluid prescribing include calculating how much fluid has ben lost, how much is needed for the patient and not too much that the electrolytes become diluted.

Mathematics is at the centre of interpreting research and probabilities in medicine. Medical professionals use graphs to record patients’ deterioration/improvement which allow others to read the information and analyse it to know the next steps in their treatment. Graphs can be used to look closely at infectious diseases and make realistic predictions as they use patterns of distribution, how fast it manifests and the areas located to work out the future impact of the disease.

A new approach to improve the health care system, which is based around mathematics, aims to predict influxes in patient admittance to hospitals. Doctors and hospitals use twitter to search for certain symptoms of an illness/outbreak in tweets to predict when they will seek medical advice as symptoms continue or become worse. This can help prepare doctors and nurses to deal with this.

Mathematics is absolutely crucial in medicine as one small mistake could cost a life.

References

Medicine and Math – Math Central. 2017. Medicine and Math – Math Central. [ONLINE] Available at:http://mathcentral.uregina.ca/beyond/articles/medicine/med1.html. [Accessed 15 November 2017].

The Nature of Mathematics

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It is a common belief that nature can be understood using mathematics. Many scientists have discovered mathematical concepts and patterns in nature for example from sunflowers to snowflakes to hurricanes and galaxies.

Symmetry

A great example of mathematical concepts in nature is symmetry which is found in abundance in the natural world. A snowflake exhibits a six-fold radial symmetry with unique and identical patterns on each arm. Each snowflake is different as when they fall from the sky, each experiences its own unique atmospheric conditions such as humidity and wind, which ultimately affect how the crystals on the snowflake forms. Due to each arm of the snowflake experiencing the same environmental conditions, it crystallises in the exact same way, producing a symmetrical snowflake.

https://www.flickr.com/photos/chaoticmind75/10152925944

Honeycombs are an example of wallpaper symmetry. It is believed by mathematicians that bees build their honeycombs in a hexagonal pattern as it is the best shape for storing the greatest amount of honey while using the least amount of wax to create the structure.

Orb spiders build their webs using radial symmetry. They create near-perfect circular webs that have near-equal-distanced radial supports coming out of the middle and a spiral that is woven to catch prey. Scientists believe that these spiders build their webs in this way for strength and to evenly distribute the force of impact when a fly or other prey becomes entangled in the web.

Symmetry in nature appears to have multiple benefits.

The Fibonacci Sequence

There is a simple sequence of numbers that appears in many places in nature called the Fibonacci sequence. Named after Leonardo of Pisa, also known as Fibonacci. This sequence of numbers involves starting at 0 then 1 and adding the previous two number together to get the next digit in the sequence, ultimately equating to 0 1 1 2 5 8 13 21 34 and so on.

The Fibonacci Spiral

Image result for fibonaccinumbers

The pattern in sunflower seeds and many other flowers are arranged in a Fibonacci spiral which keeps the seeds uniformly distributed regardless of the size of the seeds. “A Fibonacci spiral is a series of connected quarter-circles drawn inside an array of squares with Fibonacci numbers for dimensions. The squares fit perfectly together because of the nature of the sequence, where the next number is equal to the sum of the two before it.” (Live Science, 2017) – see diagram opposite. The spirals pack florets as tight as can be, maximising their ability to gather sunlight for the plant.

Image result for fibonaccinumbers The Fibonacci spiral can be found in many different places in nature for example the spiral is very evident in the chambers of a nautilus shell. The spiral occurs as the shell grows outwards and tries to maintain its proportional shape. The benefit of the nautilus’s growth pattern allows it to maintain its shape throughout its entire life.

In some plants, the leaves appear to be arranged in a Fibonacci spiral. The reason behind the staggered leaves in a spiral shape is to maximise the space each leaf has and to ensure the greatest absorption of sunlight for the plant’s growth.

Therefore, maths is everywhere in nature and has a specific purpose, either to ensure a snowflake is strong enough, a plant receives optimum conditions or even for structural purposes to save waste. Nature is spectacular in the way it uses mathematical concepts to create its own benefits.

References

Fibonacci in Nature. 2017. Fibonacci in Nature. [ONLINE] Available at:http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm. [Accessed 31 October 2017].

Live Science. 2017. What is the Fibonacci Sequence?. [ONLINE] Available at: https://www.livescience.com/37470-fibonacci-sequence.html. [Accessed 31 October 2017].

Planet Dolan | Obscure Facts About Life. 2017. 15 Beautiful Examples of Mathematics in Nature – Page 2 – Planet Dolan | Obscure Facts About Life. [ONLINE] Available at: http://www.planetdolan.com/15-beautiful-examples-of-mathematics-in-nature/2/. [Accessed 31 October 2017].

Science | AAAS. 2017. ScienceShot: Sunflowers Do the Math | Science | AAAS. [ONLINE] Available at:http://www.sciencemag.org/news/2013/06/scienceshot-sunflowers-do-math. [Accessed 31 October 2017].

Conquering Maths Anxiety

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Many of us will be familiar with the shear panic, the sweaty palms, the headaches and the confusion that comes alongside attempting mathematical problems but not many will be aware that this is a diagnosable condition and is extremely prevalent among the population. Maths anxiety has been defined as “a feeling of tension, apprehension, or fear that interferes with math performance” (Ashcraft, M.H 2002, p. 1) and is believed to affect around a quarter of the population including thousands of school teachers.

Why do we have maths anxiety?

For me, the anxiety I have for maths came as a surprise as i was always very competent and able in the subject all through primary and up to secondary school. This anxiety first began when i stopped studying it at high school and saw others around me become more competent in the subject. This created a fear that i wasn’t as good as them and couldn’t understand the concepts they were learning. I felt i was being left behind in my learning and was not as smart as them. Therefore, I opted to avoid any mathematical problems that i perceived as too difficult or simply used my calculator to prevent making mistakes and looking bad.

For others, maths anxiety can often be caused by influential adults such as parents who have negative attitudes towards mathematics because they project their fears onto their children. This leads to the children adopting the same negative feelings, such as “maths is a waste of time” and taking them into school. It can also come from the teachers themselves. If a teacher is negative or afraid of maths then their teaching will influence how the children in the class feel towards it. Maths anxiety can often come from the fear of embarrassment or failure which puts the children off from even attempting the subject. If a child has been embarrassed or made to feel wrong in front of the class, it can destroy their confidence and discourage them to explore mathematical concepts in the future.

Maths is a difficult subject and needs strong support from teachers, not more pressure of succeeding in tests or appearing to be always right.

How does maths anxiety impact us?

Learners with maths anxiety will repeatedly go above and beyond to avoid doing maths in class and outside class, which in turn results in even poorer competences in the subject and applications of maths to real life. Children are unfortunately led to believe maths is about having right answers and are therefore too afraid of making mistakes to even try. This means as children grow up avoiding maths and not trying they are likely to have the same poor abilities in their adulthood.

As adults, a lifetime of maths anxiety can have devastating consequences. Some avoid applying for jobs which may involve some mathematical reasoning, therefore limiting their career opportunities. They may have a lack of confidence dealing with their own personal finances, their mortgage, buying/running a car etc. From these examples we can see just how important maths skills are in the real world outside of the classroom. This is why, as future teachers, we need to promote the positivity and relevance of maths to children.

Conquering Maths Anxiety!

As teachers, it is vital that we help children understand maths and not be afraid of it. Allowing children to experiment and discover maths in their own time is so important for their futures and the quality of life they will receive from doing so. We need to show them the relevance of maths and ensure that mathematical problems are set in meaningful contexts to the children. This will make the questions worth solving!

Making maths fun in the classroom will encourage children to enjoy discovering concepts in maths and be curious in their discoveries.

Make children feel comfortable to ask questions and explore as children should not be afraid of getting it wrong – focus more on the discovery and on the excitement!

This Ted Talk by Robert Ahdoot is a great place to start on learning how to conquer maths anxiety in the classroom!

References

Ashcraft, M.H. (2002), Math anxiety: Personal, educational, and cognitive consequence, Current Directions in Psychological Science, 11: 181–185

Atkinson, E (1992) Mathematics with Reason, Oxon: Bookpoint Ltd.

Brian, K (2012) Maths Anxiety: the numbers are mounting The Guardian [Online] Available at: http://www.theguardian.com/education/2012/apr/30/maths-anxiety-school-support [Accessed: 7 October 2017]

“Maths is hard”

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Why is Maths Hard?

What is it that makes maths appear to be a hard subject? Is it all the rules you must follow or all the formulas you need to remember? Is it the focus on accuracy or all the strange unfamiliar symbols?

Most people you ask will tell you they found maths difficult at school and they were never very good at it. In the UK, it has now become culturally acceptable to be ‘bad’ at mathematics and we more than often overhear the words “I can’t do maths” (Kowsun,2008). Negativity towards the subject is, unfortunately, something that people take pride in and boast about. This may be due to the stereotype that maths has been given, thus leading people to believe that only the rare and gifted people are capable of doing maths. It is seen as male dominated and ‘nerdy’, all of which are untrue stereotypes.

A research study by BAE Systems found that approximately one in six adults admitted to being embarrassed by how difficult they find mathematics and one in five adults required the use of a calculator to work out simple sums. The proportion of adults struggling with mathematics has greatly increased as today, 49% of adults have the maths skills expected of an 11 year old child still at primary school (Garner, R. 2012). Low adult numeracy skills lead to many disadvantages in life such as difficulty getting a job, struggling to do their job as it may entail some form of mathematics or even just simply difficulty dealing with home finances and everyday shopping. Low skills in mathematics costs the nation around £20.2 billion (National Numeracy).

Importance of Maths

Little do these people know how important mathematics is to them and their everyday lives and perhaps, even how much they really use maths on a day- to-day basis. For example, a routine task of crossing the road requires many mathematical processes including estimating the speed of oncoming vehicles, the speed you can walk across, the time it will take you to cross and the distance to the other side. These are all common calculations our brain carries out each day however not many would include this as being ‘maths’ .The preconception is that mathematics is all about finding a ‘right answer’ whereas in reality, “mathematical discovery relies on the same guesswork that informs our everyday maths” ( Pound, L 2008). This meaning that maths is all around us and is vitally important to our lives.

Maths is for Everyone!

Despite all these factors insinuating that maths is hard, we as human beings are born mathematical which makes learning maths significantly easier than it would be otherwise.

Everyone can do maths and we all carry out calculations in our head every single day without our realisation. The only boundary we have stopping us succeeding in maths is our own attitude towards it. “Doing mathematics does not require any special ability not possessed by every one of us” (Devlin, 2000 p253).

References

Devlin, K. (2000) The Maths Gene London: Weidenfeld & Nicolson.

Garner, R. Independent. 2012. Almost 50 per cent of adults can’t do basic maths. [ONLINE] Available at: http://www.independent.co.uk/news/education/education-news/almost-50-per-cent-of-adults-cant-do-basic-maths-that-means-half-7469119.html. [Accessed 6 October 2017].

Hall, J (2013). Adults Struggling with Basic Maths. [ONLINE] Available at:http://www.independent.co.uk/news/uk/home-news/adults-struggling-with-basic-maths-with-one-in-five-requiring-a-calculator-for-even-the-most-simple-8532488.html. [Accessed 6 October 2017].

Kowsun, J. (2008). This innumerate isle – Article – TES [Online] Available at: http://www.tes.co.uk/article.aspx?storycode=2033102 (Accessed 6 October 2017)

National Numeracy. Attitudes Towards Maths. [ONLINE] Available at:https://www.nationalnumeracy.org.uk/sites/default/files/attitudes_towards_maths_-_updated_branding.pdf. [Accessed 6 October 2017].

Pound, L (2008) Thinking and Learning about Mathematics in the Early Years. New York: Routledge