# Think 182

Prior to studying discovering mathematics I believed maths to be only about numbers. However following the module, I now understand it is much more than that, it is everything and it is definitely an art form. Mathematics is prominent in many aspects of music such as beats in a bar , notation value ie crotchets and semi quavers etc, tempo and in tablature, the list is infinite.

For those who are unfamiliar with tablature or tab it is a form of musical notation that refers to instrument fingering rather than musical pitches (big guitar , no Date) . For example:

This picture shows musical notation indicating pitch and so requires the guitarist to know where each note is represented on the guitar fret. However the bottom line is the tabs. Tablature tells the individual where to put their fingers so the first note would be open first string as it tells the reader no fingers on fret but to play open first string once, the second note would then be first finger first fret first string played once.

I used to believe those who played tablature to play music were much like artist whos work was made using dot to dot. I didn’t think it required much if any understanding of music. However I recently decided to try playing tab and I think it is possibly the best decision I have made in terms of my musical progression. I already had a conceptual understanding of music or so I believed I recently tried to learn hotel California on guitar however there are quite a lot of sharps and flats in the music and I was struggling to remember it all so I tried using tabs and found it be alot easier. I didn’t need to keep reminding myself where the sharps were as the notation told me where to put my fingers. I learnt to play the song reasonably quick after this I just needed to perfect the pace.  Which brings me to my next point; pacing.

A disadvantage of tablature is that because there is no notes without truly knowing the song or understanding the notation above it indicating the pitch you have no way of determining the speed of the song or the dynamics (how loud a note is played) . However, as I learned to play music before I did tablature I used the notations to help me determine the speed of the song. I also knew that by looking at the start of the music where it tells me how loud to play the song using something I believe could be defined as algebra.

These letters represent more than just their face value they dictate a song massively (Angela ,2018). I believe that the understanding and appreciation of pupils learning about dynamics will increase their appreciation of the beauty of algebra.

In addition to this I used to believe that the f note was first finger first string just because it just was however after comparing the notation to the tablature I noticed the links between the two. I used to wonder why the  g note  was third finger first fret first string and not first finger second fret first string after all G does follow F in the alphabet ?  Tablature has helped me understand by comparing the two it is because there is a semi tone between f and g which Is f sharp and that Is why the second threat first string is f sharp not g.

So this begs the question so what?

I believe that by learning tablature my horizons have been broadened in terms of music as having an understanding of both tablature and notation I have developed a conceptual understanding of music, understanding not only the how but also the why of music. Learning music through the use of numbers have furthered my understanding of a mathematician and a musician and I would love for pupils in my future classes to have the opportunity to experience this and so I will to the best of my abilities ensure I teach maths and music together in hope that both experiences are enlightening and enjoyable for pupils.

# Linears , subtractions and fractions oh Pi!

What is the most terrifying thing you can think?  Common answers may include spiders, heights or horror movies, but what about mathematics?  Many professionals such as Boaler (2009) believes that children find maths fearful. Why is this? Is that just the way it is to be? are people just meant to be afraid of numbers? Or are we doing something wrong?

Maths anxiety comes in many forms and is experienced differently by each person suffering what could be described widespread epidemic. Arem (2010) believes that maths anxiety can cause headaches, sweating, confusion and the inability to concentrate.

So why does this maths anxiety exist?

Many professionals such as Hembree (1990) believes that maths anxiety is defined as a general fear of mathematics.  Why is this allowed to occur? If children experienced this feeling whilst studying any other curricular area it would be pandemonium. So why is little being done to help children overcome this widespread epidemic in schools ? Furthermore, the Scottish government (2009) believe that the most common form of mathematical activity in the classroom is based on pupils using textbooks and working in silence and independently (Scottish Government, 2009). To this day I still remember the feeling of working from textbooks and not for the right reasons, it is the most mind numbing soul-destroying type of work I have ever done. I believe it is our responsibility as teachers to help children develop the notion that maths is fun , enjoyable and most importantly something they are comfortable and good at tackling.

I recently stubbled across a poem that I believe represents what the communication of mathematics is like for some people.

1. One fine day in the middle of the night,
Two dead boys got up to fight,
3. Back to back they faced each other,
4. Drew their swords and shot each other,

5. One was blind and the other couldn’t, see
6. So they chose a dummy for a referee.
7. A blind man went to see fair play,
8. A dumb man went to shout “hooray!”

9. A paralysed donkey passing by,
10. Kicked the blind man in the eye,
11. Knocked him through a nine inch wall,
12. Into a dry ditch and drowned them all,

13. A deaf policeman heard the noise,
14. And came to arrest the two dead boys
15. If you don’t believe this story’s true,
16. Ask the blind man he saw it too!

Whilst this poem is purposely meant to be contradictory. I believe this is how some people see the learning of mathematics. For example, I remember as a child being told that when I multiply by 10 I simply add a zero to my answer. However I was extremely puzzled as to why 1.4 x 10  does not equal  1.40, even though I had been told specifically all I had to do was add a zero. Obviously, I soon learned that when multiplying by ten I move the decimal point one space to the right and when dividing I move it to the left. That’s no big deal to someone who enjoyed learning mathematics. I understand we learn from our mistakes,  but for someone who already has negative associations with maths and finds it confusing it adds additional stress and furthers their negative attitude.

Moreover the understanding of mathematics is crucial Erickson (2008) believes that as conceptual understanding decreases pupils begin to lose interest in maths . So what is conceptual understanding and how do we allow pupils to obtain this ? conceptual understanding is defined by Ben-Hur, (2006)  as a knowledge rich understanding . Therefore it is important that as teachers we ensure that our pupils develop a conceptual understanding of mathematics which will then hopefully enable them to enjoy mathematics and study maths progressively throughout there school career.

To whoever is reading this blog. I usually end my blog posts with a conclusion as to what I will do next to progress, However one of the main points I have taken from the discovering mathematics module is that it is not what I am going to do next but more what we can do next. Everyone is learning and developing as professionals and if everyone works together to ensure a pleasant and enjoyable experience for pupils whilst studying maths and every subject for this matter then pupils will enjoy coming to school and maths anxiety will cease to exist. I believe this to also be an important message to leave for pupils. It is not what I will do as a future teacher to help my pupils overcome maths anxiety but what we all can do to help each other in our learning.

Arem, C. (2010). Conquering math anxiety. Australia: Brooks/Cole Cengage Learning.

Ben-Hur, M. (2006). Concept-rich mathematics instruction. Alexandria, VA: Association for Supervision and Curriculum Development.

Boaler, J. (2009). The elephant in the classroom. London: Souvenir.

Scottish Government (2009) 2008 Scottish Survey of Achievement: Mathematics and Core Skills. Available online at: http://www.scotland.gov.uk/Publications/2009/04/02133043/0 [Accessed 3rd September 2015

Hembree, R. (1990) ‘The nature, effects and relief of mathematics anxiety’, Journal for Research in Mathematics Education, 21, pp.33-46

Metafilter.com. (2018). Two dead boys got up to fight. [online] Available at: https://www.metafilter.com/51472/Two-dead-boys-got-up-to-fight [Accessed 1

# Mathstronomy

Can mathematics be used to allow pupils to explore the world around them and even the entire universe? We often tell our pupils to aim for the stars, but would this metaphor be even more effective if pupils knew exactly how far away these stars actually are?  Many young children I spoke to on placement told me that when they grew up they wanted to be astronauts. However amongst the children that told me this many said they did not like mathematics. Astronomy is a subject that fascinates and engages people of all ages. The idea of an infinite number of galaxies and other planets is extremely interesting.  So can we captivate children’s interest by showing them that within astronomy there is not only an infinite universe to explore but also mathematics? Can we as professionals show children that Space like everything has a story and show them that within this story there is mathematics.

The majority of people are aware that there are many stars in the universe (about ten thousand million million million stars or 10^22).  Numbers are extremely prevalent in space and astronomy and this allows for discussion opportunities with children so that they do not even realise they are doing mathematics. This therefore allows  them to experience maths in a different way that isn’t just from textbooks and is fun engaging and most importantly something they actually can enjoy doing . Boaler (2010) believes that children find mathematics to be boring ,scary and uninteresting. This attitude is vastly spread throughout Scotland It is often referred to as maths anxiety. I argue partially it exists as a result of many children  not being taught that mathematics is more than just numbers.  Mathematics to me is a story told in a variety of different ways. A story communicated in a secret language that is waiting to be decoded with understanding and equations.  Children are constantly trying to decode the world around them why not show them how to do this using mathematics?

In addition to this another example of mathematics in astronomy is a light year; this refers to the distance that light can travel in a year. Light travels at 3.8 x 10^8  metres per second  and there are 365 days in a year,  However  before we do our equation we must first ensure we are using the same units.  we can do this by working out how many seconds there are in a year

365 x 24 (hours in a day) x 60( minutes in an hour)  x 60  (seconds in a minute)

= 31, 536,000

Following this we then use the distance speed and time formula we teach in schools

d = s/t

substitute in

3.8 x 10^8 x 31, 536,000

9,460,000,000,000 kilometres

Therefore light travels 9,460,000,000,000 kilometres in a year.

This concept alone incorporates many of the mathematical concepts covered in the mathematics curriculum for example measurement , distance speed and time,  decimals etc . Hopefully children will be able to form links between what they have learnt in mathematics and see the relevance of it.

To conclude I believe that astronomy offers children with an amazing opportunity to explore mathematics in a completely unique and fun way. I believe it offers a rich and diverse range of  mathematics such as measurement , distance, speed, time and multiplication and many more. Arem(2010) believes that children have the right to enjoy mathematics and question why we do things in maths. He believes that by allowing this we can help children overcome maths anxiety. Therefore I believe it is extremely important to use astronomy to assist children overcome maths anxiety. The universe has so much to offer and will continue to be explored by many for hundreds even thousands of years I know for certain it is something that I look forward to exploring with my class both in relation to the science and  maths behind it.

Arem, C. (2010). Conquering math anxiety. 3rd ed. Australia: Brooks/Cole Cengage Learning.

Boaler, J. (2010) The elephant in the classroom: Helping children learn and love maths. London: Souvenir Press.

# Was the Criticism the 2015 higher maths paper faced justifiable

When I was in fifth year at High School in 2016 I studied higher mathematics and the legacy left behind by the terrifying 2015 Higher Mathematics paper still lingered throughout the school and to a vast extent the entire country.

With all the stigma circulating around the paper I decided to look at it, to see exactly what it was that was responsible for the social media revolution against the SQA. I read the questions and thought to myself if I was in the shoes of the pupils who sat this exam I would have been horrified to be faced with this paper. The exam was “wordy” and didn’t specifically tell you what you had  to do or what formula to use. It instead required you to demonstrate your understanding of the content of the Higher Mathematics course to try and solve the problem and to figure out which equations you needed to use. Looking at this now from a different perspective, I believe that the way most pupils were taught was through relational understanding this, is when the pupils know a process and how to use it but not why it is used. As a result of this they have difficulty in answering questions that do not tell them what process to use or say a question in an unfamiliar way. However, there is also conceptual understanding   and I believe that pupils who scored well in this paper had a conceptual understanding of mathematics meaning that they understood not only the how but also the why of mathematics and as a result of this they knew exactly how to apply it even when it wasn’t completely obvious and required them to problem solve.

As someone who needs to know why we are doing a certain process in maths I recently looked over the paper again and thought to myself yet again. If I was faced with this in an exam situation I would be panicked. However as a result of the understanding I have now developed through studying the why and not just the how of mathematics, I think I would of scored reasonably well. Looking at the paper from a different perspective as a student teacher rather than as a pupil I now believe that the SQA had the right idea. The majority of people on the (MA) Education course  concurred in a lecture that they wished the education system was designed to enhance the understanding  of the subjects  studied and not just about remembering processes in order to pass exams. So as I look at the paper now keeping these desired motifs of education in mind I would now deem the paper as an ideal paper for meeting this criteria. Whilst I agree it was a difficult paper to sit, especially as it was the first year of the new Higher paper, it actually requires pupils to have an indepth understanding of mathematical processes and I believe that is what we want our pupils to develop, the skillset to do.

To conclude I believe that the Higher Maths paper of 2015 was significant in highlighting that as a nation our pupils do not have a conceptual understanding of mathematics and therefore when they are faced with problems where they are required to think about what they are doing, then they struggle. In my opinion this does not equip pupils with the skills they need for work. So how do we move forward? I have used the dreaded exam paper as a rational behind my future way of teaching mathematics, I am not going to rush pupils through work just, so I can tick off the boxes in the CFE experience and outcomes. Instead I am going to spend the time to allow my pupils to develop not only the how’s but also the whys of mathematics. This in turn I believe will change pupils out look of mathematics as it will make more sense to them and will hopefully result in them studying maths further along their education journey and thus further prepare them for the world of work and daily life.

Ali, A. (2018). Exam board admits Higher Maths exam was ‘too hard’. [online] The Independent. Available at: https://www.independent.co.uk/student/news/sqa-exam-results-2015-exam-board-admits-higher-maths-paper-for-scotlands-students-was-too-hard-10436713.html [Accessed 29 Sep. 2018].

Skemp, R. (2009). The Psychology of Learning Mathematics. Hoboken: Taylor and Francis.