Category Archives: Discovering maths

Play to win!

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In workshop, we were looking at the mathematics behind puzzles and board games. One of the games we looked at was Sudoku. People argue that there is no arithmetic in Sudoku as there’s no adding or calculations necessary. However, surely being able to look at patterns and sequences across the Sudoku square is using basic mathematical principles since maths is about seeing patterns in nature (Sawyer, 1955, cited in Boaler (2010, p.17). Furthermore, it does involve problem solving. Alternatively, it could be viewed that it Is just a process of elimination since you could use shapes, animals or letters instead of numbers but this would still involve analysing patterns.

The board game that I chose to look at for the workshop was Monopoly as there have been many summers when my friends and I have spent hours trying to win at Monopoly and now I’ve discovered that maths can help me win!

When I played monopoly, the only I thought I gave to the mathematics behind it was to buy Park lane, Mayfair and the train stations as once I purchased these, I could build houses. Then when someone landed on my property, I would get a considerate amount of money from people owing me rent (although I wasn’t fully aware that this was going back to basic mathematical principles (Ma, 2010, p. 104). The thrill of this was one of the most exciting parts of the game! I was looking at buying properties that were worth the most instead of looking into the chance and getting the properties which are the most landed on (standupmaths, 2016)

The basic mathematical concepts that link with Monopoly are chance and probability. This is because when you throw the dice it is up to chance what the outcome is. If you want to learn more about chance and probability visit a previous blog post of mine (Noble, 2017). Other basic concepts used in the game are adding and subtracting as the cashier when calculating rent as well as counting number of squares as you move your object (in my case that was always the dog).

A woman girl from Northern Ireland was crowned the UK champion for Monopoly and she gave some tips as to how you can win at Monopoly which relate to fundamental mathematical concepts (Telegraph, 2015). The basic mathematics of probability explains that Park Lane is actually the least landed on property therefore, I shouldn’t purchase it!

However, I was on the right path to success by buying up all the Railway stations because “…they deliver a more constant level of avenue over the course of the game.” (ShortList, no date). In addition to the stations, it is also best to buy the red properties and the orange ones as these are the top two most landed on properties on the board. They also do not cost much but bring in more money once you put hotels on them (ShortList, no date). However, is it beneficial to always place a hotel on your properties?

In the past, always would have bought four houses and then moved to buying hotels for properties. However, the UK champion of Monopoly says that “Once you’ve reached 3 houses, the amount that the rent increases with each house maxes out.” (Telegraph, 2015). Therefore, fundamental mathematics was used to help figure out the best way to win!

Furthermore, jail is the most commonly landed on place on the board, this is probably because there are several ways in which you can go to jail: landing on go to jail; community chest or chance (standupmaths, 2016). Thus, when you are in jail you should try and stay there for the longest time possible later on in the game (ShortList, no date). This is because other people will have bought up properties, put up houses or hotels. Therefore, the longer you stay in the jail the zero probability or chance you will have of landing on their properties and losing money since you are not moving.

Another interesting point to note is that according to probability, the most rolled number in Monopoly is 7. So, if you are going to place houses on your property you should wait until they are 7 spaces away from your property so that you can win more money (Robitzski, 2017).

(Business Insider, 2015)https://

However, you might think using maths to win ruins the fun but these are still interesting observations. I know in the future when I play monopoly, I am going to apply my mathematical knowledge of chance and probability and what I have learnt from this blog to try and win. For example, I am going to try to purchase the red and orange properties and stay clear of buying park lane!

References:

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

Robitzski, D. (2017) How to Crush Your Friends at Monopoly Using Math. Available at: https://www.inverse.com/article/32781-win-monopoly-using-math (Accessed: 1 December 2017).

Business Insider (2015) How to use math to win at Monopoly. Available at: https://www.youtube.com/watch?time_continue=108&v=7_SXFtdf65s (Accessed: 1 December 2017).

Ma, L. (2010) Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and United States. (Anniversary Ed.) New York: Routledge.

Noble, R. (2017) ‘Win with maths?’, My teacher journey, 17 October. Available at: https://blogs.glowscotland.org.uk/glowblogs/rsneportfolio/2017/10/17/win-with-maths/ (Accessed 13 November 2017).

ShortList (no date) 17 ways to win at Monopoly every single time. Available at: https://www.shortlist.com/news/17-tips-to-become-a-monopoly-master/50673 (Accessed: 1 December 2017).

standupmaths (2016) The Mathematics of Winning Monopoly. Available at: https://www.youtube.com/watch?v=ubQXz5RBBtU (Accessed: 1 December 2017).

Telegraph Reporters (2015) ‘How to always win at Monopoly: seven essential tips’, The Telegraph, 21 July. Available at: http://www.telegraph.co.uk/men/the-filter/how-to-always-win-at-monopoly-seven-essential-tips/ (Accessed: 1 December 2017).

 

 

 

Reflection post on science and maths!

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What links does maths have with science?

In class, we drew graphs to plot y=x, y=x^2, y=1/x and y=1/x^2. In order to draw these graphs, we first needed a fundamental understanding of basic concepts such as numbers, the sequence of numbers and being able to understand patterns in order to compare results and see relationships. Furthermore, we needed knowledge of the measurement of the squares on the page first, we needed to know measurements as one square is 1mm therefore, 10 of them make a centimeter so we can use 1mm to represent one of somethingg out of 10. Therefore, we demonstrated knowledge of fundamental understanding of mathematics as we used basic concepts and connecting them together in order to make relationships and associations between the graphs (Ma, 2010, p.104).

Science is defined by Oxford Living Dictionaries (2017) as “The intellectual and practical activity encompassing the systematic study of the structure and behaviour of the physical and natural world through observation and experiment”. So, science involves observation and experiment and these could not be carried out without maths because in order to observe for example the magnetic forces between magnets it involves recording observations on a graph which as I have stated above involves many basic mathematical concepts. In science, you are applying your use of mathematics to study the world. However, if science is the application of maths how is it that even Einstein (1943, quoted in Letters from and to children, 2004 ) one of the greatest scientists in history said that even he had great difficulty with maths?

Through our use of maths and science together to find the relationship between distance and  forces with magnets. We found that distance and force has an inverse relationship which also links to electricity and gravity (Taylor, N. (2017). Therefore, once we understand these links we can explore what we can do more with our knowledge. For example, with electricity once this discovery was made it was used to make phones, dishwashers, washing machines, laptops, televisions and apple watches which are all everyday things that we may use in society but we don’t think about how the science and maths behind them all started out from basic concepts and relationships.

In order to develop children’s fundamental knowledge of mathematics in the future, I would need to start off teaching fundamental mathematical concepts such as measurement, how to draw a graph, the skills of observation used in science and how to plot a graph in order to show relationships between science and maths which links to connections between topics which Ma (2010, p.104) refers to as an element in developing your understanding of fundamental mathematics. Cross-curricular links could also be made to technology!

List of references:

  • Taylor, N. (2017) Science and maths. [Lecture to Discovering Mathematics Year 2], ED21006: Discovering Mathematics (year 2) (17/18). University of Dundee. 20th November.

Application of maths for a business simulation.

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In a workshop, I took part in a business situation which demonstrated an application of Profound understanding of fundamental mathematics in the real world!

An overview of what was involved in this business simulation:  

  • We had 5,000 Euros to spend for each quarter of the year
  • We had to choose a maximum of 5 items to sell for each quarter
  • We had to choose what quantity of items we wanted (quantity being a basic concept which is needed to have a profound understanding of fundamental mathematics according to Ma (2010, p.104))
  • We had to be careful and look at how much we were buying the item for to ensure we were not paying more for an item than it was being sold for such as the Christmas selection boxes in January.
  • We had to keep track of how many items were sold and whether they could be carried over into the next quota or whether it was a ‘write off’
  • We calculated how much we had spent by adding the prices together and subtracting what we had spent from the 5000 Euros or money we had to see what was left over. Therefore, through this process we were building upon each stage and adding to it over a period time which links to longitudinal coherence (Ma, 2010, p.104). Furthermore, we connected basic concepts together such as adding, subtracting, multiplying, percentages and decimals

  • We had to look at multiple perspectives to see what we were going to buy and sell however, we didn’t really look at multiple perspectives from a mathematical point as we continued to use the same methods of calculations throughout the session.

However, although we tried to be tackle by what we bought by guessing what items were more likely to be bought around a certain time of year, we could have looked ahead and seen that for January there was a sale price mark-up which mattered more. For example,  beans only cost 25p to buy but they were being sold for 2.50 euro and 100% were sold. Therefore, there was a ten times mark up. Consequence, if I was to do this again i would pay more attention to the price per unit that we were buying it for and the seasonal price in order to make a bigger profit.

In the future, I would like to use this activity or at least one similar to further pupils understanding and application of fundamental mathematics in the real world or in the wider society. Therefore, they could connect and link various topics together for a meaningful purpose (Ma, 2010, p.104). It would be a fun and creative task or game for the children to do and could potentially due to the relevance of the task, it could give them an interest in business as a future career. However, in order to improve this activity I would like to further gain the children’s interest as well as emphasis the relevance by having the packaging of the actual items out on desks to further emphasise the relevance.

 

References:

  • Ma, L. (2010) Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and United States. (Anniversary Ed.) New York: Routledge.

Statistics can save lives!

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From the lecture presentation I developed my understanding of the fundamental mathematics behind statistics as statistics couldn’t be understood without starting with fundamental basic mathematics. Also, statistics are an example of longitudinal coherence as they give a full picture and can be broken down to basic concepts that were built upon each other which Ma (2010, p.104) states is a profound understanding of fundamental mathematics. In primary school, you are taught how to do tally and other sorts of graphs. The knowledge gained of recording and creating these graphs is built upon in order to create charts for medical reasons. Thus, a profound understanding of fundamental mathematics has major implications, having this knowledge can keep people alive or it can be fatal as child can be poisoned if too high of dose is given to a child (Hothersall, 2016). These drug doses are given per kg.  Therefore, children are weighed so the drug doses can be worked out. Here is a basic concept in maths that is required for use in medicine, weight which involves measurement and this is applied or links to medicine to save lives (Ma, 2010, p. 104).

Furthermore, junior doctors are expected have a profound understanding of fundamental mathematics as they are to know ratios and statistics to do with for example, the risk and probabilities of what one single cigarette can do to you (Hothersall, 2016). Therefore, this part of medicine involves several basic concepts of mathematics such as probabilities, ratio, change and recording (Ma, 2010, p.104). Even breaking your leg requires knowledge of the basic concepts as it’s due to forces and the angle, the speed that something might hit your leg at to break it (Porta, D.J).

However, maths should be taught in a fun and relevant way as it’s important to show the relevance of maths and helping them see that maths in science. So, when you are doing a fun experiment in class, help them see its fun. Maths is fun so inspire their love and interest in maths. I also want to help develop pupil’s knowledge of profound understanding by demonstrating this example of longitudinal coherence to them. The importance and relevance of statistics or why we learn the basics in maths like making graphs. I know that when I went to school I was learning how to draw tan graphs but I saw no point in learning it because I was not told the relevance.

An interesting point is that statistics links to social media. For public health reasons, there is an “Ailment Topic Aspect Model” that has prior knowledge of ailments. This model the analyses tweets to search and track sickness or illness over a period of time by “…measuring behaviour risk factors, locating illness by geographical region, and analysing symptoms and medication usage.” (Paul and Dredze, no date, p.1). However, a weakness of using twitter for statistics is that surely many people don’t tweet seriously, they may exaggerate their problems, they could be lying for a laugh or attention seeking. Furthermore, many account are private so how are all the tweets from these accounts accounted for in their data? What if the majority of information about a recent illness is on these accounts? Additionally, the symptoms that people might state on twitter could potentially be too vague.

A drawback should be noted about statistics. Although they can be useful and save lives, they are not always correct. There is such a thing as bias statistics especially in advertisement. For example, Colgate toothpaste claimed that 80% of dentists recommended Colgate but this was misleading as dentists were given a list of options to choose which toothpaste in comparison to the other competitors (Derbyshire, 2007). Therefore, fundamental mathematics can be used in a negative way in wider society. Therefore, In the future I would like to teach children that they need to critically evaluate statics and use multiple perspectives to look at the different ways statistics could be looked at as different perspectives can tell you different things (Ma, 2010, p. 104).

The video below gives some examples of negative uses of statistics (TED-Ed, 2016).

 

In conclusion, statistics are an example of how fundamental mathematics such as longitudinal coherence can be applied to wider societal issues.  A drawback to statistics is that they can be misleading however, they can also help save lives and this is why learning mathematics is important!

List of references:

Derbyshire, D. (2007) ‘Colgate gets the brush off for ‘misleading’ ads’, The Telegraph, 17 January. Available at: http://www.telegraph.co.uk/news/uknews/1539715/Colgate-gets-the-brush-off-for-misleading-ads.html (Accessed: 15 November 2017).

Hothersall, E. (2016) ‘Numeracy: Every contact counts (or something)’ [PowerPoint presentation]. ED21006: Discovering Mathematics (Year 2) (17/18) Available at: https://my.dundee.ac.uk/webapps/blackboard/content/listContent.jsp?course_id=_56905_1&content_id=_4941433_1&mode=reset (Accessed 9 November 2017).

Ma, L. (2010) Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and United States. (Anniversary Ed.) New York: Routledge.

Paul, M. J., and Dredze, M. (no date) You Are What You Tweet: Analyzing Twitter for Public Health. Available at: http://www.cs.jhu.edu/~mpaul/files/2011.icwsm.twitter_health.pdf (Accessed: 9 November 2017).

Porta, D.J. (no date) Biomechanics of Impact Injury Available at: http://eknygos.lsmuni.lt/springer/659/279-310.pdf (Accessed: 9 November 2017).

TED-Ed (2016) How statistics can be misleading – Mark Liddell. Available: https://www.youtube.com/watch?v=sxYrzzy3cq8 (Accessed: 15 November 2017).

 

 

Maths causes major happiness!

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There’s a link between music, maths and our emotions. How can this be and why do major chords in songs make us happy? This is what I am going to explore in this blog post!

The links between maths and music:

  • beats (the value of each note)
  • timings
  • rhythm
  • string numbers
  • if you are changing key, you go up or down
  • scales: scales follow the same pattern, no matter what note
  • tempo: speed of the music (metronome)

Music makes us happy. I don’t know about you but I love listening to music and it makes me feel better no matter what mood I’m in. Kim (2015) states that the Feel Good Index (which is an equation itself) is the “… sum of all positive references in the lyrics, the song’s tempo in beats per minute and its key.” Having a fast tempo in a song makes us want to dance, we want to move and it makes us happy. Apparently this is because “Beats automatically activate motor areas of the brain.” Fernández-Sotos, Fernández-Caballero  and Latorre (2016) also agree that the tempo impacts whether the music makes us happy or sad.

However, sad music can make us happy. Sometimes I’ll be in the mood to listen to sad and slow songs but they make me happy however, this contradicts what is said above? The lyrics are sad, not pleasant. This is because whilst the emotion of sadness is seen as negative, in artistic form, sadness can be felt, sensed or understood differently. Therefore, songs that are in a minor key which are identified as being sad songs do not always cause a negative emotion (Kawakami et al. (2013, pp. 1-2). Personally I think the reason for this is more than just the links to the maths behind the music such as a slower tempo or a minor key. It’s also because we connect with the feeling, the emotion through the song or lyrics and this connection is pleasant to us (Nield, 2016). The emotions are caused as the music brings back previous experiences that can make us happy or sad. (Konečni, 2008, cited in Hunter, Schellenberg, and Schimmack, 2010, p.54).

Additionally, why do certain pop songs in the charts make us feel happy because their tempos are fast, around 116 beats per minutes and have a major third musical key (Kim, 2015)? You might be thinking, what does music have to do with maths? Well, music is made up of “…pleasurable patterns of rhythm, beat, harmony and melody” (Gupta, 2009). If you are still asking what do tempo and beats have to with music well, according to Fernández-Sotos, Fernández-Caballero and Latorre (2016),

Tempo is “…the speed of a composition’s rhythm, and it is measured according to beats per minute.”  

“Beat is the regular pulse of music which may be dictated by the rise or fall of the hand or baton of the conductor, by a metronome, or by the accents in music.”

So, what makes a pop song catchy? Why do we enjoy hearing the next new hit song on the radio and want to sing and dance along? The University of Bristol asked the same questions. They formed a mathematical equation to work out what makes a popular song popular. They created a diagram to compare pop songs patterns. The coloured parts of it represented beats and the connections seen in the diagram were sections of music that join. What they found was that many pop songs had a very similar pattern Seeker (2014).

A drawback to the equation that The University of Bristol developed (which they recognise themselves), is that it will need adapted as what becomes popular changes since, over time the songs that have become popular are ones that are getting louder (University of Bristol, 2017). Why is it that what is popular changes over time?

Why do these pop songs make us feel good? In one of my previous posts (Noble, 2017), I talked about how according to Burkeman (2011) gambling is satisfying as we are addicted to the potential of getting a reward and the satisfaction is due to the chemical dopamine being released (How the Brain Gets Addicted to Gambling, 2017). This same chemical is released when we listen to these pop songs therefore, making us feel satisfied, please, happy and feel good. However, why do we not get bored of these songs? In the video linked below it demonstrates how a lot of the famous pop songs are repetitive, they use the same four chords (random804, 2009).

So, we seem to be  satisfied by the same types of songs. These songs that are popular, are popular because they fit “[I]mplicitly learned patterns…” or their patterns only differ by a slight bit (Wheatley, no date, cited in Hughes, 2013). This slight difference must be what keeps us satisfied as we would be bored if it was the exact same every time.

Critically, maths can be non-existent in music according to Sangster (2017). For example, every pitch has a different frequency but a piano note can’t be tuned to the exact frequency that it mathematically should be or it doesn’t sound right musically. This is where maths cannot be applied to music. This video below demonstrates why it’s impossible to tune the piano notes to the exact mathematical frequency although it demonstrates why this is so using maths (Minutephysics, 2015).

In the future, I would like to apply my knowledge and understanding of how maths underpins music to teach pupils about maths. Pupils could make their own short songs, using their maths skills such as counting, timings, rhythms and beats! This would be a fun activity showing the relevance that maths has in the wider environment and in everyday society.

In conclusion, maths is behind music. Music is composed by using basic concepts in maths such as the speed of the music, timings and counting the beats per minute. In order to count beats, people need to know how to count in a sequence, and how long a minute is. All these basic concepts are put together and built upon to make music over a long period of time. All of which relate to what Ma (2010, p. 104) says are the principles of fundamental mathematics. Furthermore, the type of music that is produced from using and building upon all these basic mathematical concepts can have an affect on peoples emotions, making them either happy or sad and formulas can even be created to determine or predict what songs will be big hits!

List of references:

 

 

 

 

 

 

 

Reflection on maths in sports!

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Previously, it hadn’t occurred to me about maths in sports. If I was playing a sport, my main aim and line of thought would be to try and score, or win, along with using some tactics. Therefore, today’s workshop about maths in sports really opened up my eyes to see how the fundamentals of mathematics can be used to help people in everyday life activities such as playing sports.

Prior to today workshop, I researched about the maths in the sport Cricket. Interestingly, I found that even the weight of the bat can affect your performance in the game, the angle at which the ball hits the bat, the vertical bounce of the ball, the speed the ball is bowled at, the mass of the ball and even the ball spin (Physics of Cricket, 2005). Angles, weight and speed are all basic concepts used in mathematics which links to what Ma (20 10, p.104) refers to. All of these basic concepts link together, can be tested over a period of time and thus, the game performance can benefit over a period of time, by building upon the results and doing research. (Ma, 2010, p.104).

In addition to maths and cricket, today in workshop we redesigned the football league table from 1888-1889 by including the goals against, goals for, goal difference, points, games played, games won, games drawn and games lost. Below is an example of the original football league table.

 

 

 

 

 

 

 

We decided to order the football clubs from top to bottom according to the most points a club had won. We used basic mathematical principles (Ma. 2010, p. 104) of counting to find out how many games each team had won, drawn or lost. To work out the goal difference we used the basic mathematical concept of subtraction, subtracting the goals against from goals for.

After creating our new football league table, we discovered that we could have used simple  algebra to work out the points if we multiple the wins by 2 and add the draws.

Winn x2 + draws = the points (simple algebra). 

We could have designed this table differently by taking at the average of the goals scored (goal average)  and sorted them that way.

Next, we designed our own sports game and looked at how maths was applied to the sport. Our sport was called Smack-ball. It was based on the memory of a game many people have played. This being, when you have a balloon you try and keep the balloon up in the air by hitting the balloon as it comes back down and you can’t let it touch the ground. So, essentially, smack-ball involves using a ball that is the size of your hand-span and it weighs 196 grams. The court will be rectangular, 10m by 5m, the shorter walls are the goals, there are three players on each team and the game lasts 10 minutes. The aim is to try and smack the ball (involving passes to your other team members) towards your teams wall without letting the opposite team intercept the ball.

Applying maths to our sport:

  • using Pythagoras to try and perfect the perfect pass (Tohi, 2016)
  • force and distance for hitting the wall
  • when is the best time to intercept the ball out of someones hand
  • ball spin and speed

Reflection point:

In the future, I would like demonstrate to pupils how basic mathematical principles can be applied to sport! Pupils could choose one of their favourite sports therefore, creating interest and they could play the sport whilst others record results to illustrate how speed, distance and points are all relevant to sports. Demonstrating how maths can link to their interests, making maths enjoyable and relevant (The Scottish Government, 2008, p. 30)!

In summary, maths can be used to help benefit peoples performance in sport and can actually help people win! Many fundamental basic concepts were involved to create a league table and to look into how to improve in sport such as examining size, weight, position, location, counting, adding, subtraction, multiplication, simple algebra, force, distance, speed and time (Ma, 2010, p.104). Therefore, the fundamentals of mathematics can be used to help people in everyday life activities such as playing sports.

 

Mathematics is in everything!

List of references:

  • Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.) New York: Routledge.
  • Physics of Cricket (2005) Available at: http://www.physics.usyd.edu.au/~cross/cricket.html (Accessed: 5 November 2017).
  • The Scottish Government (2008) curriculum for excellence building the curriculum 3 a framework for learning and teaching. Available at:  http://www.gov.scot/resource/doc/226155/0061245.pdf (Accessed: 6 November 2017).
  • Tohi, K. (2016) Maths in Field Hockey. Available at: https://prezi.com/3mk2gu_u74ph/maths-in-field-hockey/ (Accessed: 6 November 2017).

When in doubt, change your answer?

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Have you ever taken a test and put one answer down but the more you think about it, the more you think you should change your answer? Should I stick with what I originally thought or should I change it? What if I change it and my first answer is the right one, that would be so frustrating but what if I do change it and I get the correct answer? What should I do? This is what I am going to explore in this blog, what should you do?

Below is a video that will give you an overview of what I am going to discuss.

Supposedly, if you change your answer from what your initial first instinct was, then you are more likely to be correct or successful in your answer (Kruger, Wirtz, and Miller, 2005, p. 725). I think that this argument can be argued against if a person takes a longer time to think of their answer first, before putting their initial answer down. However, I guess that if you are thinking for a long enough time, your answer may well have changed from the original, first instinct to a counter answer therefore, proving this argument to be true.

Although, Brownstein and Green (2000, cited in Kruger, Wirtz, and Miller, 2005, p. 725) give a counter argument to Kruger, Wirtz, and Miller. They state that if you change your original answer you are more likely to be wrong.

The potential reason why people prefer sticking with their instinct is due to memory bias. Times when people have changed their answer, they have ended up not doing well or it has turned out unsuccessful and this negativity surrounding changing their answer is more memorable than times when they have changed their answer and the result being successful. For example, if you’re on the motorway, when you change lanes because the other lane seems to be going faster and as soon as you do, the lane you used to be in moves faster, which is even more irritating. It is due to this irritation that we remember the event more therefore, in the future, we decide to stick with our instinct to not change and to just stick in the lane we are in (Kruger, Wirtz, and Miller, 2005, p. 726).

In the first study that (Kruger, Wirtz, and Miller, 2005, p. 726) mention, pupils were informed prior to the test about what would happen. That they were doing a study on changing answers or sticking to the original. I think that this could have influenced pupils as to whether they changed their answers or kept them the same. Maybe these pupils were ones who never changed answers or vice versa and now thought about changing them which could have impacted this study. The result concluded that there were more right to wrong answers by changing answers therefore, disagreeing with the idea that when in doubt, you should change your answer (Kruger, Wirtz, and Miller, 2005, p. 726) but there were various limitations in this study that could have caused this outcome such as, the examiners didn’t see the original answer as it was rubbed out, their rubbed out answers were not their first answer or maybe the eraser mark is from accidently selecting the wrong answer and therefore, it wasn’t their first instinct (Kruger, Wirtz, and Miller, 2005, p. 727).  Although, study three agreed that changing your answer is more beneficial (Kruger, Wirtz, and Miller, 2005, p. 729).

In workshop, we tested this theory. I was in a group of three. One student was going to always choose to always stick with her first answer, I was going to change my answer each time and the other student moved the cups. Our aim was to try and guess which cup the toy car was under. To us, it seemed that this would be a 50/50 chance. However, the result was that the student who stuck with her answer, never got it right and I got it right three out of four times by changing my original answer. The response in our group was, “This is so weird”, “This is so odd” and “unreal”. It seemed that this theory was correct and we were blown away. However, afterwards I thought to myself, there are limitations to our experiment. We were being told to change our answers and maybe if I took more time to consider my first answer (rather than not taking any time at all because I knew I was to change my answer) the result could have been different.

Bauer, Kopp, and Fischer (2007) believe that those who have been told the benefits of changing their answers, when in doubt, had scored higher upon changing their answers than those who were not told about the benefits. However, if this is only for when a person is in doubt, how beneficial is it to change your answer when you are not in doubt but know it’s the other answer or you are in semi-doubt.

Therefore, something to consider for my future development is should teachers inform pupils of the potential benefits of changing their answers from their initial ones. This article seems to say so but, the amount by which the test results increased was not that significant. As long as you only change your answer once, this is the case (Bauer, Kopp, and Fischer, 2007). Although, if teachers tell pupils of these benefits, some pupils may not use their actually intelligence and knowledge to answer correctly in tests but instead, they may choose another answer just because they might score higher.

According to Merritt (2009), you are three times more successful if you change your initial answer. Contrastingly, this article says that you should only change your initial answer if you have logical reason to or you can back it up. You should not change your answer just because you are in doubt or you worry your answer might be wrong which, conflicts with Bauer, Kopp, and Fischer’s (2007) research.

Nevertheless, two other experiments were done to test the theory of whether to change or stay and the results where that in both college exams, sticking with their original answer and changing their answer resulted in more correct answers (Couchman, 2015). This result seems to have come about because the students were instructed to put their confidence level beside each answer. Pupils appear to have chosen carefully if they were choosing to stick or change. They had reason to because of how confident they were on whether they were sure of their answer being correct. Hence, teachers need to be careful what they inform students of such as, telling students to stick with their first answer or to always change it (Couchman, 2015).

In essence, statistics show that you are more likely to have correct answer if you change your answer from the initial one you out down. This links to the basic mathematical concepts that connect to one another such as counting, quantifying links to the basic ideas of chance and probability of getting the answer correct, as this idea looks at which choice is most likely to lead to the correct answer or success.

I would like to further pupils understanding of fundamental mathematics in the future by carrying out this demonstration in the classroom and children could use tally charts to see which method (changing or sticking) is more successful than the other. Furthermore, pupils could look at the different perspectives that this issue can be looked at. For example, do the toy car test that we did in workshop or role play a grocery store. This would involve standing in a busy line for the till, and deciding whether to change to a seemingly faster queue or to stick in the one you’re in (Ma, 2010, p. 104).

So, what should you do? Should you change your answer or keep the initial answer you put down? Well, statistically, you should change your answer if you are unsure of the first answer because you are three times more likely to get the answer right if you do change (Merritt, 2009). Although, you should take time to think about why you might be changing, think of it logically, do not change just out of self-doubt but only if you can back up the reason why you are going to change your answer (Merritt, 2009). After all, it was in the experiment that pupils were not pressured to either change or keep their first answer that they ended up have more correct answers than in the experiments where they were instructed to do one or the other (Couchman, 2015).

List of references:

Win with maths?

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Chance and probability link to the basic ideas or concepts of mathematics (Ma, 2010, p. 104). The basic understanding of probability is that it is the “[L]ikelihood or chance that something will happen.” (Tuner, no date, p.5). The “Probability of an event happening = Number of ways it can happen over the total number of outcomes.” (Probability, 2014). For example, when obtaining a two when rolling one dice, there are 6 outcomes possible. I can only get the two on the dice by landing on the two therefore, the probability is 1 out of 6.  In this blog post, I am going to look at how probability can affect business and I am going to explore the links between gambling, social media likes, addiction and mathematics.

In our workshop, we looked at the different combinations possible when ordering meals (Holme, 2017). The fundamental maths is that there are multiple perspectives (various ways) to figure out the number of choices that could be made (Ma, 2010, p. 104).  McDonald’s became a big business because they had a small variety of options on their menu. These options could all be cooked in a similar way and could be cooked fast. However, at one point, it was said that the McDonald’s menu was getting too big, which caused a decline in business. With more options available, there were complaints that the fast food restaurant, was not fast enough! Therefore, the less choice there is the more customers and sales. (Lutz, 2014). Restaurants could look into the probability of more choice vs the number of customers.

Gambling

Gambling in the UK is now at two billion pounds. However, gambling is addictive and produces a moral and an ethical issue. Addiction to gambling has led some people to steal due to the amount of debt they are in. The number of gambling addicts have doubled in six years and MPs aim to reduce this. It could have a knock-on impact on economy especially if many people are entering debt and Gambling is increasingly prevalent (as it can even be played at home now) (Gallagher, 2013).

Skinner did research on training people with rewards, this being the variable ratio schedule. Skinner suggests that we should not give rewards every time but to give them at random, to change up when we give a reward. He states that to train someone, we should keep rewarding them at the start which gets them hooked, then vary when the reward is given once the behaviour has been established. Casinos have researched this about rewards and in turn have used this information to get people to play and to keep them hooked (Weinschenk, 2013).

Rewards link to social media notifications such as emails or Twitter. We do not continually access these because we have got a notification or because we have a message but because we might have one, it’s the unpredictability. There is a potential reward. This is why we get addicted (Burkeman, 2011). I know I have opened my social media apps, knowing that sometimes a notification doesn’t pop up on my home screen but if I go into the app there is a possibility or a chance that I might get one if I open it. Therefore, this theory about rewards by Skinner can be applied to slot machines because if they rewarded someone every time they played, people wouldn’t get addicted. So, in order to get people addicted or hooked they vary when the rewards are given.

 

Teacher note: Could we use this idea or concept of varied rewards to have more positive behaviour in the classroom? This is something I would like to look into to develop my understanding of fundamental mathematics in the future.

 

Click here to watch a video on addiction to instagram likes. (Yates, 2017)

A reward such as a like on your Instagram causes a release of a chemical called dopamine. This gives us a ‘high’ which encourages our addiction to the likes further. We want more likes and we watch them roll in, creating an addiction because we like that feeling of satisfaction (How the Brain Gets Addicted to Gambling, 2017).

This theory of random rewards causing addiction, connects to the concept of randomness which is explored in the basic mathematical concept of probability and chance. Randomness according to De Finetti (1990, cited in Turner, no date, p. 3) is defined as an individual will not know what a result will be. However, can we determine what the outcome will be using the mathematical concept of probability? Therefore, if the outcome could be more predictable could this mean a reduced likelihood of addiction since it’s the unpredictability of rewards that gets us addicted (Burkeman, 2011).

Gambling relies on randomness and probability (Turner, no date, p. 3). However, it seems that we humans have a different view to randomness since we expect a random outcome to not have the same result in a row, when tossing a coin for example (Holme, 2017).

I thought that a random result wouldn’t have so many tails or heads in a row. When predicting, I tried to mix it up the result, I varied it to make the outcome what I thought to be random. Therefore, supporting the “Naïve Concept of Random Events” as I didn’t think the outcome would be in clumps (Turner, no date, p. 4). Thinking along the lines of, if this result hasn’t occurred much or at all, it’s going to next time. I did not expect a pattern (Turner, no date, p. 5). Even Apple had to make the random shuffle option for selecting songs, less random in order for it to seem more random to humans as the songs ended up being clustered (Bellos, 2010).

Therefore, it’s interesting that Howard (2013, cited in Smart Luck, 2017) contradicts that random outcomes can be in patterns even though the coins toss has just proven that there can be patterns in randomness. One tip that he recommends people use to win the lottery is to follow the human thinking of randomness. Meaning that you should not pick numbers that have a pattern or consecutive numbers. He also suggests that numbers shouldn’t be picked from the same group as its unlikely you’ll win. Although, this would contradict Apples random shuffle option, which often picked songs from the same album (same group) (Bellos, 2010). Furthermore, even though through analysing patterns, he has concluded that, “That which is most possible happens most often. That which is least possible happens least often.” However, the challenge with using probability to help you win in the lottery, is that it isn’t certain that it will occur (Probability, 2014).

This concept of randomness needs to be addressed in school to remove the misconception about randomness. Doing experiments such as the coin toss would be a great example for the class to see as it demonstrates a real-life example of randomness in probability such as the one shown below which we did in a workshop.

Regarding the point I made earlier about addiction being removed or reduced when gambling if the outcome is more predictable, this could be done by utilizing fundamental maths and probability. One example of this is by Darren Brown, who tried to estimate the speed of moving objects, looked at velocity and rate, using mathematics and engineering to guess where the ball might land on the roulette table (Bob, 2009). This being an example of different areas of maths connecting to others to solve a problem (Ma, 2010, p.104). However, even though he used all his knowledge and connected it together, he still was one out from being correct and winning. Similar to probability, it is only a “guide”, it isn’t certain that what is probable will occur (Probability, 2014).

In summary, probability could be used to help restaurants such as fast food places to work out the likelihood of how many customers they will have if they have a certain amount of options on their menu. Additionally, gambling is addictive because people love the satisfaction of the unpredictable rewards, the possibility of getting rewards which is similar to social media likes (Burkeman, 2011). Therefore, gambling machines would need to give rewards at random to keep people hooked which links to the basic concept of probability and chance in maths (Weinschenk, 2013). However, surprisingly there are patterns seen in randomness that humans do not expect. Furthermore, I explored how there is potential to reduce the addiction if the outcomes could be predicted using probability, chance and other areas of maths that could connect (Ma, 2010, p.104). Although, it seems that because probability cannot give an exact answer of the outcomes, its unreliable to use in this case (Probability, 2014).

 

List of references:

Bellos, A. (2010) ‘And now for something completely random, by Alex Bellos’, The Daily Mail, 7 December. Available at: http://www.dailymail.co.uk/home/moslive/article-1334712/Humans-concept-randomness-hard-understand.html (Accessed: 12 October 2017).

Bob (2009) Derren Brown: How to Beat a Casino. Available at: https://learningonscreen.ac.uk/ondemand/index.php/prog/011B9829?bcast=34497126 (Accessed: 12 October 2017).

Burkeman, O. (2011) ‘Can ‘intermittent variable rewards’ help you become addicted to more positive behaviours?’, The Guardian, 23 April. Available at: https://www.theguardian.com/lifeandstyle/2011/apr/23/this-column-change-life-random-rewards (Accessed: 12 October 2017).

Gallagher, P. (2013) ‘Addiction soars as online gambling hits £2bn mark’, Independent, 27 January. Available at: http://www.independent.co.uk/news/uk/home-news/addiction-soars-as-online-gambling-hits-2bn-mark-8468376.html (Accessed: 12 October 2017).

Holme, R. (2017) ‘Chance & probability’ [PowerPoint presentation]. ED21006: Discovering Mathematics (Year 2) (17/18) Available at: https://my.dundee.ac.uk/webapps/blackboard/execute/displayLearningUnit?course_id=_56905_1&content_id=_4941456_1 (Accessed: 12 October 2017).

How the Brain Gets Addicted to Gambling (2017). Available at: https://www.scientificamerican.com/article/how-the-brain-gets-addicted-to-gambling/ (Accessed: 12 October 2017).

Lutz, A. (2014) McDonald’s Menu Is Completely Out Of Control. Available at: http://uk.businessinsider.com/mcdonalds-menu-is-too-large-2014-12?r=US&IR=T (Accessed: 12 October 2017).

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

Probability (2014) Available at: http://www.mathsisfun.com/data/probability.html (Accessed: 12 October 2017).

Smart Luck (2017) United Kingdom National Lottery Suggestion. Available at: http://www.smartluck.com/free-lottery-tips/england-nationallotto-659.htm (Accessed: 12 October 2017).

Tuner, N. (no date) Probability, Random Events, and the Mathematics of

Gambling. Available at: https://www.problemgambling.ca/EN/Documents/HPG%20Probabilty%20Final.pdf (Accessed: 12 October 2017).

Weinschenk, S. (2013) Use Unpredictable Rewards To Keep Behavior Going. Available at: https://www.psychologytoday.com/blog/brain-wise/201311/use-unpredictable-rewards-keep-behavior-going (Accessed: 12 October 2017).

Yates, E. (2017) What happens to your brain when you get a like on Instagram. Available at: http://uk.businessinsider.com/what-happens-to-your-brain-like-instagram-dopamine-2017-3 (Accessed: 12 October 2017).

 

Can animals count?

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Can animals count? This question I posed to my friend. She responded with, “Yes! No! Wait… maybe?” I had the same reaction to this question. In this blog post I will share my thoughts and findings with you and maybe you too might wonder, can animals actually count?

Horses?

In the 1900’s, there was supposedly a horse that was able to learn basic maths and could count. The owner would ask the horse maths questions that involved adding and dividing. The horse would tap its hoof with the answer. For example, if the answer was two, the horse would tap its hoof twice. However, although it seemed that the horse was counting, I think that it wasn’t, It must have been trained to react that way. As I continued to watch the video about this horse, I came to the discovery that indeed, it was found through experiments, that the horse responded the way it did, due to cues (Bourassa, 2012).

Lions?

There was an experiment carried out to see if lions could count. It seemed that the lions in this experiment were able to count if the number of lions coming (who were intruders) were greater than those in their own group. They were able to do so by listening to the lions roars. Males from the intruders would quantify the amount of females that were in the group before attacking. They could distinguish if there were one or three lionesses by the sound of their roars (BBC News, 2003). Maybe these lions are not aware that they are counting  or actively counting, as they are not thinking, one, two, three like us.  However, maybe they are counting since they could work out the exact amount of lions in prides accurately (BBC News, 2003). If lions can count, why is it that their counting is limited to five or six? (Silver, 2015).

Baby chicks?

An experiment was carried out with baby chicks to see if they could count (these chicks had not been trained). Three balls were hid behind a screen and two behind another. Chicks went to the screen that had the largest amount of balls. Then it was changed up, when some of the balls had been moved, the chicks could “count” that another screen had more balls there. It was believed that they could add and subtract (Gill, 2009).

Chimps?

It is argued that chimps can count as they could understand originality since the chimps could remember the pattern of numbers. In this experiment, I think it was purely a “memory test”, the chimps had learned symbols or, the chimps did so because they wanted their peanuts (Muldertn, 2008).

Dogs?

It seems that dogs cannot discern numbers over one. They know the difference between something and nothing. This may be because they don’t need to “count”. There is no reason to, they do not need to “count” for survival whereas wolves do and they can “count” (Silver, 2015).

Parrot?

Alex, a parrot could allegedly understand colour, shape and could count but, it had been through years of training. Therefore, parrots do not have an innate ability to count like chicks believably have (Silver, 2015). Therefore, maybe animals can count if they are trained.

However, I think that these animals are able to recognise that there are more or less of something. They can recognise if a member of their pack is missing. For the chicks, maybe they just used their memory to know that there were more behind one screen than another and again, they could recognise the difference in quantity, they know what more than one looks like. Animals who have been trained can maybe count. Could we humans count if we were not  taught or trained to? Surely I would be able to recognise the concept of more or less like the chicks? However, isn’t the topic of quantity, one of the basic concepts or principles in maths? (Ma, 2010, pp.24-25).

Although, Gill argues that animals can count, “It is already known that many non-human primates and monkeys can count, and even in domestic dogs…” (Gill, 2009). 

Burns (no date, cited in Tennesen, 2009) accepts a similar idea as I do, stating that animals have an “[I]nnate ability to discern between small numbers”.  In contrast, Burns (no date, cited in Tennesen, 2009) also believes that animals can train themselves to recognise not just small numbers but numbers up to 12!

Interestingly, counting connects to longitudinal coherence, ” Counting is a combination of several skills, each building on the other” (Silver, 2015). Therefore, if animals could count, they would understand the basic elements of the key fundamental principles of mathematics! Additionally, being able to count requires an understanding of ordinality meaning, having an understanding of a series one, two, three. For example, two comes after three and three after two (Oxford Living Dictionaries, 2017).

It seems that animals can count and some are better than others (Silver, 2015). On the other hand, the idea that animals can count is contradicted by Whorf (no date, cited in Bredow, 2006). Whorf’s theory states that if you don’t have any words for numbers, you cannot understand maths and numbers (Silver, 2015).  Therefore, if animals don’t have words for numbers can they really understand numbers, can they count? The Pirahã people are a tribe that did not have words for numbers but, they could understand quantity of more or less and had words for these. When Daniel Everett tried to teach them to count to ten in Portuguese, he found that they could not count and it is not because they are any less intelligent than others. This theory is backed up by an example of The Warlpiri group in Australia who had words, “one-two-many”  for counting and when they were taught numbers past two in English, they were able to (Bredow, 2006).

So, from developing my understanding of fundamental maths and applying it to my research, do I think animals can count? Well, from the evidence above it seems that animals have the ability to recognise a difference in quantity. It is necessary for example, for survival and other animals can count if they are trained. Although, if animals can recognise and understand quantity that demonstrates a basic understanding of a basic principle or idea in mathematics, quantity (Ma, 2010, p. 104).  However, after looking at Whorf’s theory it suggests that animals cannot count because there need to be words for numbers first in order to understand numbers and to be able to count.

Please comment any thoughts you might have on whether animals can count!

References:

  • BBC News (2003) ‘ Counting lions roar for help’, 19 September. Available at: http://news.bbc.co.uk/1/hi/sci/tech/3119456.stm (Accessed: 7 October 2017).
  • Bourassa, P. (2012) Clever Hans. Available at: https://www.youtube.com/watch?v=C0LKN2lFWI4 (Accessed: 7 October 2017).
  • Bredow, R. (2006) Living without Numbers or Time. Available at: http://www.spiegel.de/international/spiegel/brazil-s-piraha-tribe-living-without-numbers-or-time-a-414291.html (Accessed: 7 October 2017).
  • Gill, V. (2009) ‘Baby chicks do basic arithmetic’, BBC News, 1 April. Available at: http://news.bbc.co.uk/1/hi/7975260.stm (Accessed: 7 October 2017).
  • Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.)New York: Routledge.
  • Oxford Living Dictionaries (2017) Definition of ordinal number in English. Available at: https://en.oxforddictionaries.com/definition/ordinal_number (Accessed: 7 October 2017).
  • Tennesen, M. (2009) More Animals Seem to Have Some Ability to Count. Available at: https://www.scientificamerican.com/article/how-animals-have-the-ability-to-count/ (Accessed: 7 October 2017).
  • Muldertn (2008) Chimps counting. Available at: https://www.youtube.com/watch?v=0cKg9D4QKCM (Accessed: 7 October 2017).
  • Silver, K. The animals that have evolved the ability to count. Available at: http://www.bbc.co.uk/earth/story/20150826-the-animals-that-can-count (Accessed: 7 October 2017).

Maths in art? Art in maths?

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Previously, I’ve heard people say, “I cannot do art” or “Art is something that I just wasn’t gifted in”. This is why one particular maths workshop came as surprise to me as it demonstrated that you can do art using maths.

In class we created tessellations. Little did I know that this type of art links to basic ideas and concepts in maths such as properties of 2D shapes, angles, triangles, quadrilaterals, symmetry, proportion. It also links to longitudinal coherence since we can use these basic concepts of maths and build upon them to teach area and transformation (Ma, 2010,p. 104).

To create our tessellation, we had to look at the size of the shapes, how the regular shapes of an equilateral triangle, squares and triangles could fit together and why they did. Throughout the workshop we used a whole range of mathematical languages such as edges, sides, angles, pattern and symmetry.

We looked at the tiles in front of us and thought of what pattern we wanted to create. Once we had chosen our pattern, we placed the tiles down on the paper to see if the 2D, regular shapes would fit together. We found that there were going to be gaps however, we solved this challenge as we found that if we rotated some of the triangles, they would fit, leaving no gaps. Once we completed our tessellation, we painted it, using colours to emphasise the pattern created.

This workshop was useful as in the future if there are some pupils who for example say, “I can’t do maths but they say they are good at art, I can show them how both connect to one another, making maths or alternatively art more interesting, fun, engaging and relevant to the pupils as art can be created using mathematical concepts.

 

In the future, this activity would useful for teaching maths and for developing pupils understanding of fundamental mathematics. To start out, pupils could look at shape, angles and explore how shapes fit together. Pupils could be challenged by looking at why these shapes fit together. Pupils could explore this by using a protractor to see how all the angles add up to 360 degrees where the angles meet. Children could be further challenged by making their own shapes. They then could measure the angles to discover how and why they fit together. Additionally, I would like to make shapes relevant to pupils by showing them shapes in real life contexts because pupils might not think of shapes when looking at buildings for example, they could look at photos of the Pantheon in Rome, as the front of it is the shape of a pentagon.

References:

  • Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.)New York: Routledge.
  • Valentine, E (2017) Maths, creative? – No way! [PowerPoint Presentation], ED21006: Discovering Mathematics (year 2) (17/18). University of Dundee. 26 September.