# Reflecting on the Discovering Mathematics Module

Throughout this module we have been looking topics and how mathematics is used in each of them. We have looked at topics from medicine to art and more mathematical based topics such as time and probability. In the lectures we have been looking at the origins of the topics, how maths is used in them and activities that help us to see the how maths influences the topic. In my blog posts I have been looking at some of the topics in greater detail focusing on how a basic knowledge of mathematical concepts is fundamental to understanding these topics and how they can be applied to our own lives.

Liping Ma talks about how the key elements of [inter]connectness, multiple perspectives, basic principles and longitudinal concepts are essential for a good maths teacher. Ma also talks about how teachers need to teach their pupils these key elements for them too to have a sound knowledge of mathematical concepts and be able to apply them to their own lives.

As a future teacher I think it is important that I teach my pupils about how Liping Ma’s key elements will help them with their maths skills and how they can apply mathematics to different elements of their lives. It is important for pupils to understand how the basic mathematical skills of number, measurement, arithmetic, position and quantity etc. influence all the other maths they will learn as they make their way up the school. It is also important for pupils to know how mathematics influences lots of other topics in school such as art and science and can also be applied to their lives outside school such as in sport.

Through this module I have learnt how important it is to fully understand the basic mathematical principles in order to continue further with harder mathematical concepts. It has also opened my eyes to how much mathematics influences other aspects of our lives in ways I had never thought off such as puzzles and games. The module has affected my teaching practice as I now understand just how important it is for pupils to understand the fundamental principles of mathematics in order to fully understand maths.

# Counter-Intuitive Maths

We recently had a lecture on counter-intuitive maths. Walking into this lecture I had no idea what the term counter-intuitive maths meant. Counter-intuitive maths is something that goes against what you thought it would be or your intuition. The aim of the lecture was for us to be less confused about why somethings happen in maths that we don’t expect and how a knowledge in the fundamental concepts in mathematics can assist us in our understanding of counter-intuitive maths.

We started off the lecture by looking at the concept of coin flipping. When you flip a coin you expect that there is a 50/50 chance that you will get the outcome that you want. However if you flipped a coin 100 times would the outcome be 50 heads and 50 tails. We tested out this ourselves by flipping a coin 20 times.

I found that the coin landed on heads 8 times whilst it landed on tails 12 times. Which is a ratio of 2:3. Most people think of randomness within the context of a coin flip being the same amount of heads as tail however it is much more random than that. There are different factors that influence the outcome a coin toss such as if the coin is tossed, spun or allowed to land on the floor. All of these things influence the outcome and change the probability. For example if a coin is tossed and caught, it has about a 51% chance of landing on the same face it was launched. The video below explains how coin flips aren’t always 50/50

We went on to look at the Monty Hall problem. Monty Hall was the host of the game show Let’s Make a Deal. On the show Monty would show a contestant 3 doors. Behind two doors are goats and behind one door is a car. The contestant would pick a door. Monty would then reveal one of the doors which has a goat behind it leaving two doors left over. Monty would then ask the contestant if they would like to stick with their original decision or switch to the other door. Most people believe that because one door is taken away their 1/3 chance of picking the correct door would change to a ½ chance. This however is not true. At the beginning of the game each door has a 1/3 of being the door with a car behind it. Meaning that there is a 1/3 chance that the door you picked initially has a car behind it. When a door is taken away you still have a 1/3 chance of a car being behind the door you picked but the chance of the door you didn’t pick now has a chance of 2/3. This concept is easier to understand when it is on a larger scale. Imagine there were 100 doors and 1 of those 100 doors had a car behind it and the other 99 had goats. Say you picked door number 1 and Monty revealed 98 of the doors that had goats behind them and asked if you wanted to change your answer would you? Of course you should because the door you initially picked has a 1/100 chance of being the right door whilst the remaining door now has a 99/100 chance of being the door with a car behind it as it has absorbed all the chances of the other doors. In short if one door is taken away and you are asked if you want to switch you should switch every single time. The video below explains this way better than I ever could.

But how does counter-intuitive maths relate back to the fundamental basics of mathematics? A knowledge of basic mathematic principles is needed to fully understand counter-intuitive maths. Chance and probability play a big role in counter-intuitive maths and so without a basic knowledge of it you would never be able to understand why a coin flip is completely random or why you double your chances when you change your answer.

I started playing the piano when I was 7 and I picked up the oboe when I was 8. Through the oboe I’ve been playing in school orchestras, in church and my local school board orchestra as well as singing in school choir. I would consider myself quite a musical person and I really enjoy both listening to and playing music. Maths has been a massive help to my musical journey. The basic mathematical skill of counting is so important in music especially to me as my oboe teacher would often exclaim “Megan, do I need to send you back to P1 to learn how to count properly!” Or when I was starting out playing the piano and counting up the notes to figure out what notes they were in the bass clef. Without a basic understanding of mathematical concepts I wouldn’t have gotten very far.

In our lecture we looked at some aspects of music which maths are essential for. Some examples we came up with were; note values, rhythms, beats in a bar, tuning, pitch, chords, counting songs, fingering on music, time signature, scales, musical intervals and modulating the key. We went on to look in detail about the maths used in different types of scales. This reminded me of a key musical concept that I used to help me with my grade 5 theory: the circle of fifths.

The circle of fifths is used to help composers find the relative major or minor keys of the key they’re using as well as giving the sharps and flats in the key signature. The circle of fifths is show in order of fifths (hence the name). For example between C and G there is an interval of 5 notes. I used the circle of fifths for my grade 5 theory to help to remember the order of sharps and flats in key signatures. Without maths a musical concept such as this would not be possible as you need a basic understanding of counting, ordering and sequencing to be able to understand the circle of fifths.

Another thing we looked at in our lecture was the musical concept of tuning. This is something that I’ve always found hard to wrap my head around. The oboe is quite a hard instrument to tune. Most oboists use an electronic tuner to make sure if they are playing at the right pitch especially when they are tuning an orchestra. However sometimes using an electronic tuner isn’t possible and so oboists have to train themselves to knowing when they are in tune and how to fix it if they aren’t. Another instrument which is hard to tune is the piano. Pianos can never be completely in tune. We watched the video below on how it is impossible to tune a piano perfectly to the right pitch. Tuners use mathematics to tune pianos and without a knowledge in the basic mathematical concepts this task would be impossible.

Music and mathematics go hand in hand. Without mathematical concepts and ideas music would be a lot harder to understand especially to those who don’t play musical instruments. Maths helps musicians to understand what they are playing and how they compose music. Maths has certainly helped me in my own ability to play and understand music.

# Puzzles, Games and Fun

In our most recent lecture we looked at how mathematics can be used in various puzzles and games. Most of the time we don’t take into consideration the mathematical logic that can be used to complete puzzles or win games. We started off by looking at one of my favourite puzzle games; Sudoku.

My grandmother taught me how to play Sudoku when I was younger and so I’m very familiar with the rules and concepts. However I had never stopped to think of how basic mathematic principles can be applied to the game. Most people are familiar with the standard Sudoku which consists of 9X9 square and has 81 cells. The 81 cells are filled with in with numbers ranging from 1-9. The goal is to fill in the whole grid using the nine numbers so that each row, column and block contains each number exactly once. The basic principles of mathematics that you need to solve a Sudoku puzzle are number, sequencing and problem solving. After doing research online I found that there are many different mathematical ways that people have created to complete Sudoku puzzles. I also learned that there is 6670903752021072936960 different grids that can be created with Sudoku.

We went on to look at games that we had researched ourselves. I had researched into one of my favourite cards games called Spit. Spit is a 2 player game and requires one pack of cards. The card are split evenly between the two players. Each player creates 5 piles of cards. The first pile has 1 card, the second has 2, the third has 3, the fourth has 4 and the fifth has 5. The top card of each pile is turned around so it is facing upwards. The remaining cards are set above the piles to the left of the player. Both players turn over one card from their remaining pile and the game begins. The object of the game is to get rid of all your cards. The first player to get rid of all of their piles of cards then must say spit on the pile of remaining cards with the smallest amount of cards. They then add these cards to their pile of remaining cards and the game starts again. The game ends once a player has got absolutely no cards left.

There are various basic mathematical principles that can be applied to Spit. For example knowledge of sequencing and order are needed when attempting to get rid of your cards properly. Players must understand that they can work up or down the numbers and understand which numbers come next in the sequence. Problem solving is needed as players have to decide which pile is the smallest one to say Spit on. Players also need to be fast as even if they have not got rid of all their cards if their opponent has they can spit on the smallest pile before their opponent does and get that pile. There is also an element of chance involved as it all depends on which cards you are dealt and when they come up in your piles or when they can be dealt.

Even though arithmetic may not be used in many games and puzzles Knowledge of basic mathematical concepts is needed to successfully understand how games such as Sudoku and card  games like Spit work and how you can win them.

# The next Alan Sugar?

Wednesday nights in my house currently consist of settling down on the sofa at 9pm with a mug of hot chocolate to watch the newest episode of The Apprentice. So finding out we were doing an apprentice-style business simulation I was very excited. Everyone gets caught up in thinking that if they were in The Apprentice they’d know exactly what to do so I was curious to see how well we would actually do in a business style exercise whilst applying our knowledge of the fundamental principles of mathematics.

The rules of the exercise were that in teams of three we each had €5000 to spend each quarter on a various number of items which were: xmas selection boxes, champagne, soft drink, beer, whole frozen turkeys, ice cream wafers, bunch of bananas, celebration luxury hampers, crisps, sherbet dip dabs, bread, milk, tins of beans, luxury biscuit selection and premium durian. We could pick up to 5 items per quarter and any items not sold, if not perishable, could be carried on to the next quarter.

We started off the activity by dividing the items into different categories such as Christmas and summer items so we could decide which items would be best to buy and when. This proved to be a good idea as some items such as xmas selection boxes didn’t sell very well between April- June but 100% of the selection boxes sold between October- December. We stuck with this method throughout the task and it proved to work very well. However it was pointed out to the class at the end that we should have been paying close attention to the difference between the purchase price per unit and the seasonal selling price. For example tins of beans were bought for 25p each but they were sold at a seasonal selling price €2.50 which 10 times more than they were sold for and in January- March 100% of the tins of beans sold. If we had put all our money on the beans we would have come out on top.

There were various mathematical strategies we used during this business simulation. Basic mathematical concepts such as addition, subtraction, division and multiplication were applied to different parts of the simulation. Addition and multiplication was used when deciding how many items we should buy and how much these items would cost. Addition also helped us to keep track of what items we had left and what could be carried over as well as adding on €5000 onto the profit each quarter. Dividing and subtraction were used when figuring how many items were sold and how much profit was made on each item.

Overall we did pretty well in this task we started off with €5000 and ended with €78,771. This task helped me to understand the maths which is used in business situations like this and how without a basic understanding of the fundamental concepts of mathematics this task would be unsolvable. This task could be transferred into the classroom to help pupils see the real life relevance of mathematics as well as sparking their business interests. I really enjoyed the task and the thrill of making money through mathematical concepts but I might not be ready for The Apprentice just yet.

# Smack-ball

During our lecture on maths in sport we were instructed to either improve the rules of an existing sport or to create our own sport. We decided that since none of us knew enough about one particular sport we would create our own.

We created a game called smack ball. The aim of the game is to smack he ball against the opposing team’s wall. The players have to keep the ball in the air by hitting it upwards with your hands. You can defend by smacking the ball out of the opposing team’s hands. Once the ball is dropped the first person to pick it up again gets the ball.

There are three players on each team and the teams play on a court which is 10m by 5m with the shorter walls being the goals. We wanted the ball to be hand-span sized, weigh 196g and be kind of like a smaller volleyball. Each game lasts around 10 minutes and there is no extra time or substitutions.

We discovered that there are many ways maths could be used in smack ball. For example Pythagoras could be used to determine what the perfect pass looks like. Maths could also be used to figure out the force needed to hit the ball off the wall from a distance or how to create the perfect ball spin. Maths could even be used for figuring out when is the perfect time to hit the ball out of someone’s hand.

After presenting our game to the class we discovered that a similar game was played by the ancient Mayans who used a rubber ball which they hit off their elbows to make passes.

# Maths is not a Spectator Sport

My experience of sports has always been about avoiding falling over, getting hit in the face with a ball or accidentally scoring a goal in my own nets so I’ve never really stopped to think about the maths that comes with sports. Both of my younger brothers are really into playing rugby and I often spend most of my time during their matches not watching them but instead trying to wrap my head how many points a try is worth. Sports and I have never really got along well so when I walked into a lecture all about maths in sports I intended to completely get lost in all the sporting rules but I surprisingly found that I actually learnt a lot and are more comfortable with the concept of maths in sport.

Prior to the lecture we had to research a sport and some of the maths that surrounds it. I decided to pick tennis because I understand the rules and the scoring system of the game and was interested to learn more. Whilst doing research I found that mathematics can be used in many ways that I would never have thought of e.g. to rank players, plan tournaments, evaluate strategies, track and predict tennis ball trajectories (Pragear, 2016). One aspect of maths in tennis I was very keen to learn more about was how to hit the perfect serve. A mathematician from University College London Nick Ovenden said that “the perfect tennis serve relies on keeping your opponent guessing about how the ball may bounce out of the service box- this can be achieved by putting spin on the ball”. Ovenden looked at the speed the ball flies through the air at, the axis of the angle the ball spins at and the force that the racket hits the ball to create the perfect serve (Thomas, 2012).

During the lecture we looked at the first football league table from 1888. The only experience I’ve had with football league tables is when my brother tries to explain to me how he’s going to win his fantasy football league. So at first I was very confused. We were instructed to re-organise the information from the matches into a modern league table so after some googling we worked out the best way to organise the teams. From the 1888 table we found that there was 132 games played by 12 different football teams. We had to organise the information from the 1888 table into 10 columns; position, team, matches played, matches won, matches drawn, matches lost, goals for, goals against, goal difference and overall points. The most difficult one to figure out was the goal difference so we came up with an equation to work it out: Points = (No. of wins x 2) + no. of draws.

After this lecture I have a new found appreciation of how maths is used in sports. Maths is pretty much used in every sport to work out the size of courts or tracks, the size of equipment and the timings used for different sports. But it is also used to work out predictions of the outcomes of games and comparison for different athletes. We watched an interesting video on the history of the 100 metre Olympic sprint and how the runners have gotten faster over time (Wimp, 2012) . This made me think of how we still haven’t reached a peak in sporting performance and how much maths can help improve athletic performance.

References

Wimp. (2012). Usain Bolt vs. 116 years of Olympic sprinters. [Online Video]. 5 August 2012. Available at: https://www.wimp.com/usain-bolt-vs-116-years-of-olympic-sprinters/?dm_i=LQE,25SE0,3LDIRH,7T51A,1/ (Accessed: 14 November 2017).

Thomas, R (2012) Spinning the Perfect Serve. Available at: https://plus.maths.org/content/spinning-perfect-serve (Accessed: 14 November 2017).

Pragear, C (2016). Maths point—the mathematics of tennis. Available at: https://www.science.org.au/curious/everything-else/tennis-maths (Accessed: 14 November 2017).

# Can you be-leave it?

This week in discovering mathematics we looked at number systems and place value. When I was in primary school I found it very difficult wrapping my head around place value I just kind of accepted that when you move up the number system you just add a zero instead of actually thinking about how you’re actually exchanging it for another number. Going into this week’s lectures I was nervous that I wouldn’t fully understand how number systems or place value works but I was surprised to find that it wasn’t as difficult as I thought.

We started off by looking at what are numbers and numerals (which I didn’t actually realise were two different things). Numbers are a mathematical object used to count, measure and label (Number, 2017) whilst numerals are symbols which are used to represent a number. After briefing looking at the question of can animals count (I was quite surprised to hear that there is a chimpanzee out there that can remember number patterns better than I ever could) we looked at several different number systems.

Today we use a 10 base number system which also corresponds with our metric system. However some tribes use different systems. For example the ancient Mayans use a base 20 number system which is represented by dots and dashes (Number systems, no date). Or the ancient Greeks who created a system where the 27 symbols used to represent their alphabet also represented different numbers (Number systems, no date). After looking at the binary system (that one really confused me) we created our own number system.

We decided that our number system would be a base 10 system as it was the one that we understood that best. We decided that the symbols that represented our numbers would be leaves (hence the bad pun used as my post title) and the number of points on the leaf represented what the number was. For example the leaf that represented the number 4 had four points. After trying out some simple questions using our base 10 number system we decided that it worked quite well.

However there are some people that believe that our base 10 number system isn’t the most efficient number system. Some people believe that a base 12 system has more advantages over base 10 (most of which only confused me further). For example “The dozen, and the dozen dozen, or gross, have shown their usefulness in packing and packaging over many, many years” (dozenal society, 2016). However changing our number system now would create great confusion for people wh
o are so used to the base 10 system.

Going forward in being a student teacher I think these lessons on number systems and place value would be very useful to pupils learning about number and place value. As a pupil who was very confused about these things when I was younger looking at the reasons at why we use our system and the history of it and other systems would be very beneficial.

References

Dozenal Society (2016) available at: http://www.dozenalsociety.org.uk/ (accessed: 7th October 2017)

Holme, R (2017) ‘Origin of Number Systems 2017 MyDundee’ [PowerPoint presentation] ED21006: Discovering Mathematics (year 2) (17/18). Available at: https://my.dundee.ac.uk/webapps/portal/execute/tabs/tabAction?tab_tab_group_id=_2093_1  (accessed: 7th October 2017)

Holme, R (2017) ‘Place value MyDundee’ [PowerPoint presentation] ED21006: Discovering Mathematics (year 2) (17/18). Available at: https://my.dundee.ac.uk/webapps/portal/execute/tabs/tabAction?tab_tab_group_id=_2093_1  (accessed: 7th October 2017)

‘Numbers’ (2017) Wikipedia. Available at: https://en.wikipedia.org/wiki/Number (accessed: 7th October 2017)

Number systems (no date) available at: http://www.math.wichita.edu/history/topics/num-sys.html#mayan (accessed: 7th October 2017)

# Reflecting on Semester 1

Semester one of university brought about a lot of key moments in my journey to becoming a teacher. From the values module, which encouraged me to think about my personal and professional values, to having to choose an elective subject where I got to learn further about a topic that I was interested in. However the standout moment for me from semester one had to be the working together module and what it taught me about reflection.

At the start of the module I was apprehensive. I was so eager to get stuck straight into teaching and so was a little confused about why we were doing this module. On reflection I can see the importance of working together. The module involved working together in a group of ten people. I am quite a naturally shy person and so being thrown into a group of big personalities was very daunting to me. At the start I was very content to sit in the background occasionally agreeing with someones opinion and not really fully investing. However as the weeks progressed our group got closer and started to have more banter with each other. I began to get more comfortable contributing to the group and by the presentation I was confident enough to present alongside my group.

Through the working together module we learnt a lot about reflection and how important it is to your professional development. Reflection is about assessing what you did right, what you did wrong and what you would do different next time to improve. It is important to not just reflect after the incident but during the incident in order to assess your next move. This will be key for me as I begin my journey as a teacher and start professional practice later in the semester.

When reflecting on the working together module I can see the positives and negatives in my work. Even though I am naturally shy I should’ve let go of any reservations I had and put 100% effort into my group discussion. However I am proud of how when the weeks progressed I went out of my comfort zone in order to contribute to the group. I am now eagerly looking forward to learning more about reflection in the upcoming semester and my placement.

# Managing my learning

I will update the tables below as I continue to learn what helps my learning.

 Recognition/ Reflection Action What helps my learning? How can I utilise this? Discussing the topic with peers Set up a study group or group chat of peers Organisation Use to do lists in order to manage my time effectively Reading Setting out time to do the readings Removing distractions Turn off my phone and place it in another room whilst I do my room Self-confidence By not being afraid to ask questions if I don’t understand  topic

 Recognition/Reflection Action What hinders my learning? How can I address this factor? Easily distracted By taking away the distractions such as my phone Not preparing for lectures Look over the slides before the lectures Not doing the readings on time Do the readings before the lectures