Mathematical Wonders as Seen by an Autistic Mind

Step into the world of numbers as seen through the eyes of someone who doesn't just understand mathematic but sometimes gets excited about it. Here, I take you on an adventure, drawing you into an understanding of mathematics intertwined with the nuanced perspectives of an autistic mind. Imagine carving up a pizza not just to savour its flavour but to digest fractions, or handing out chocolates to friends, each leftover piece a tangible reminder of division with remainders. It's a session where trigonometry escapes the confines of textbooks, scaling the heights of real-world edifices, and where special interests like the solar system and Lego transform into living, breathing mathematical playgrounds.

Now, let's translate that mathematical magic into your everyday life. We venture beyond theory to practice as we uncover the algebra in sports scores and sums in the shopping aisles. Ever consider how integrating 3x² can lead to sharper strategies, or how to come up with and solve equations when watching the next golf swing or snooker shot? Or how mathematical reasoning could make you a savvier shopper, one unit price comparison at a time? From decoding league tables to enhancing decision-making, this episode isn't just about numbers—it's a celebration of how they shape our understanding of the world. Join me as I aim to piece together some of the mathematical puzzles that not only intrigue but also empower us in our daily lives.

Thanks for listening. You can find me on my website stephensevolution.com, or on twitter here. You can sign up to receive news of new episodes when they're released here.


Artwork produced by Elena Designe
Music composed by Nela Ruiz

Transcript

Mathematical Wonders as Seen by an Autistic Mind

Intro:

[Background music fades in]

Hi there everyone, and welcome back to another episode of the Stephen's Evolution podcast, where I share my personal experiences with autism, aiming to inspire hope and understanding in others. I'm Stephen McHugh, your host, and in this episode I'm going to be diving into a fascinating topic unravelling the language of numbers and patterns. As someone on the autism spectrum, I've always had a unique relationship with mathematics. 

In this episode, I aim to explore the intricate world of mathematical thinking, from visual representations to special interests that have helped to enhance my own mathematical abilities. Join me on this journey as I aim to unravel the language of numbers and patterns. 

I aim to unravel the language of numbers and patterns, delving into things like special interests and the intersection of autism and mathematical proficiency. So, without further ado, let's get stuck into this episode and discover the fascinating ways in which mathematics can come to life for individuals like us. 

Part 1: Incorporate Practical Demonstrations  1:49 - 7:30

And now, here in segment one, we're going to explore together how visual thinking processes can enhance mathematical proficiency. Visual thinking processes can enhance mathematical proficiency, especially for individuals like myself on the autism spectrum. 

Let's get stuck in with some examples. Imagine you have a scenario where you need to divide a number of objects among a group and you're left with the remainders. Imagine a group of five friends who have to share out 23 chocolates amongst them. 

Once they've shared out 20, they'll all have four each leaving three. And if the remaining three went to three of them, two of them would only have four chocolates and the other three will have five. Now, as you can imagine, the two friends who only have four will think and say hey, wait a minute, we've only got four and you three have got five. Hey, wait a minute, we've only got four and you three have got five. How can that be fair? This is an example of a scenario that can represent division with remainders, a very common concept in mathematics. 

Now let's visualise fractions. Imagine having a pizza in front of you. That is like representing one as in one whole. You then slice it up into, let's say, six pieces, so that each piece represents a sixth of the pizza. One thing I can see here is if you add like one sixth each time, six times, you get six over six, which means one. 

Visualising fractions here can help us to understand how they can compare to one another. You could, like get two slices and add them together to get two sixths, which cancels down to a third. And if you add three slices together, you get three over six or three sixths, which cancels down to a half, so you can see that a half is bigger than a third. 

And now let's explore trigonometry, where sides of triangles are labelled as opposite, adjacent and hypotenuse in right-angled triangles, adjacent and hypotenuse in right angled triangles. The hypotenuse is always the longest side of a right angled triangle. To find the opposite side, just draw a narrow from the angle straight until it meets with the side of the triangle it's facing, which is the opposite. 

With the hypotenuse and opposite sides now labelled, that only leaves one remaining side, the adjacent. This is one way I found easiest to identify the sides of a triangle, a right angled one that is. What I found here was this visual labelling can be crucial when trying to understand trigonometric functions like sine, cosine and tangent. 

What I found easy to identify here was you'd use sine when the opposite and hypotenuse sides were involved. In the case of cosine, you'd use this trigonometric function if the adjacent and hypotenuse sides were involved. And onto the tangent, you'd use this trigonometric function if the opposite and adjacent sides were being worked with. 

When I think about it now, one interesting thing about trigonometry is you can use it to work out how high tall buildings are and other tall objects, and I've always had a fascination with measurements, including how tall some tall-looking objects are. 

So imagine standing at the base of a tall building or tree. The side directly facing you, which is the object going up, is the opposite side. Trigonometry uses these relationships to help solve problems like these and other complex geometric problems. Through these examples that I've talked about, I hope you now have enough of an idea of how visual representations can aid in mathematical thinking and problem solving.
 

Part 2: Expand on Special Interests  7:35 - 13:36

Now let's delve deeper into how special interests can intersect with mathematics, into how my special interests have intertwined with mathematics, by using examples that helped to enrich my understanding and abilities in unique ways. One example I can give was a time in the past when I did my own personal project on the solar system and one thing I wanted to do was calculate the speed of planets orbiting the Sun, since I was fascinated with speed.

Eventually, after some research, I came across a formula to help me. It was by multiplying the diameter of a planet's orbit around the Sun by pi, a mathematical symbol which represents 3.142. For example, to calculate the Earth's speed around the Sun, knowing that the average distance of the Earth is 150 million kilometres from the Sun, we can work out that the diameter of the Earth's orbit is 300 million kilometres. So we multiply 300 million by 3.142 to give us a value just over 900 million, and we divide that value by the number of hours in a year, which is 8,760. You get the value of the speed in kilometres per hour. 

Another interest I've always had was in Lego. That to an extent helped to enhance my grasp of multiplication. There are many different Lego pieces. There is one Lego piece which has two rows of four studs. We can see that four multiplied by two here is eight, four being the number of columns going across and the two rows of four being the two which we use to multiply the four to get the eight. So here a Lego piece has a specific number of studs arranged in rows and columns, a tangible representation of multiplication concepts. 

And now, moving on to the world of music, let's try and calculate the number of keys on the standard piano For each octave. On a piano there are seven white keys and five black keys, and on the piano there are seven full octaves. Therefore, we can use this to try and calculate the number of total keys on the piano.

In addition, there is a partial octave towards the end of the piano, on the right hand side. So here, to start with, we can multiply seven octaves by seven white keys to give 49 white keys. Following that, we multiply five black keys by seven octaves, giving us 35.

 And finally, for the partial octave on the right-hand side at the end of the piano, there are an extra three white keys and one extra black key. So doing 49, add 35, will give us 84, and then add the extra four keys gives us a total of 88 keys on a standard piano. 

As well as that, understanding musical notation can involve mathematical concepts to a degree. An example here can be to do with the number of beats per bar and note values. By calculating the number of beats per bar based on written manuscripts, we can apply mathematical thinking to interpret musical compositions. 

There are many different notes we can use, all with different beat values. Music composers can choose to have a certain number of beats per bar. To do this, they can use a certain number of notes or even a combination of notes to add up to the intended number of beats in the bar so that it matches the number of beats they want in the bar itself. 

By exploring these intersections between special interests and mathematics, I have found we can gain insights into how diverse passions can enhance mathematical proficiency and foster a deeper appreciation for the interconnectedness of knowledge connectedness of knowledge.

Part 3 - Offer Concrete Learning Strategies:  13:41 - 17:01

And now that we've explored these fascinating examples, let's now go on to discuss learning strategies and tips for individuals on the autism spectrum to enhance their mathematical skills. And now, here in segment three, I'll talk about examples of learning strategies that proved beneficial for me here. 

One example was differentiation. When I studied A-level maths, what I found easy about this was when differentiating a term involved a variable raised to a power, for example ax number by the power n, and then you subtract 1 from the power to get the new power. So, for example, 3x², you multiply 3 by 2 to get 6x, after taking 1 away from the power of two. 

With differentiation, I was fascinated by some of its applications, certainly in terms of being able to work out gradients. So, going back to 3x², a curve with that equation, getting 6x from  3x² squared, you multiply 6 times, let's say, where x equals 2 on the curve, you multiply 6 by 2 to get 12, which would equal the gradient at where x equals 2. 

Another interesting application of differentiation was using maximum and minimum values to calculate volumes of shapes outlined in particular mathematics questions. 

Another interesting topic was integration. This involved raising the power of a variable. For example, for 3x squared, you'd add 1 to the power and then divide the coefficient by the new power, by the new power. So for 3x squared, you'd add 1 to the power to get 3x³ and then divide that by the new power, which is 3. So you'd get 3x³ cubed divided by 3, which would cancel down to x³, and then add c, which is the constant of integration.

What I found interesting about integration was that you could use it to calculate areas under graphs between certain points. For me, these strategies demonstrate key principles of differentiation and integration. By gaining an understanding and then applying these strategies, we can efficiently calculate derivatives and integrals of various functions for many different applications. 

Part 4: Relate Mathematics to Everyday Scenarios  17:06 - 25:23

And now on to segment four the use of mathematics in everyday scenarios. Let's explore how mathematical thinking extends beyond academic settings into everyday life, making complex concepts practical and applicable. Whenever I've watched golf on TV, I like to analyse the scores of players to calculate the path for a particular course on which they're playing.

This can involve using mathematical calculations to determine the standard score expected for a golf course. An example of applying mathematics to sports analysis Golf tournaments on TV I've seen have involved players playing four rounds on the same course. Imagine one player has played three rounds and their scores are, let's say, 68, 70 and 72. Now let's suppose their overall score after three rounds is minus three. 

What we can do here is put a bit of algebra to some use. We could use x to equal any of the three previous scores. So, for example, let x equal 70. For the score of 68, it is x minus 2. And for the round of 72, it is x plus 2. Therefore we can come up with the equation of x minus 2, add x, add the other x, add 2, to equal the overall score of minus 3. 

At the end of it we come up with 3x equals minus 3, which means x is equal to minus one, and as 70 is x, the minus one means 70 is one less than the path of the course, which we can now work out to be 71. 

Similarly, whenever I'm watching snooker, I like to use players' scores within a particular frame to assess their position and predict potential outcomes. This strategic analysis can involve mathematical calculations to understand the dynamics of the game. Again, as before, we can use algebra here. Suppose a player has 54 points in the frame and their opponent has 39. That means they're 15 points ahead. 

Now let's assume there are only 25 points remaining on the table. Here the information we have is one player in the frame is 15 points ahead. We can use x to represent any points scored. Therefore we can come up with the equation for the player who is 15 points ahead 15 add x, and whenever they score any more points that can be taken away from the 25 remaining. 

To work out the value of x. Here we take 10 away from both sides to leave x = 10 minus x. The next stage is to add x to both sides so that we have 2x is equal to 10, and 10 divided by 2 is 5, giving the value of x. So the player who's 15 points ahead has to score more than 5 points to put themselves in a strong position to win the frame, with their opponent likely to need at least one or more snookers. 

Sometimes during shopping trips, I like to apply mathematical reasoning to make cost-effective decisions. For example, if I see a similar fish in two different shops, what I do here is, in each case I calculate the weight of each fish per unit of currency, which is pound sterling in the UK here. By doing this, I can determine which of the two shops offers the particular fish the best value for money. 

And back to the realm of sports. Mathematics can play a crucial role in league tables. Whether it's calculating the number of points needed for staying in a league or determining when a team has secured a title. 

Mathematics can provide insights into the dynamics of sports competitions. Provide insights into the dynamics of sports competitions. For example, if a team in first place is eight points ahead of the team in second place and there are four games remaining, that means there are 12 points to play for. That's assuming there's three points for a win and one for a draw. Here we can again use algebra. 

We can use x to represent points gained. For the team in first place who's, let's say, eight points ahead of the team in second place, we can come up with 8. Add x. And for the 12 points still to play for in the four games remaining, we can come up with 12 minus x. That's assuming the team in first place at least matches the results of the team in second place. Therefore, we can come up with 12 minus x. Here we now have 8. Add x equals 12 minus x. 

The next stage here is to take away 8 from both sides to come up with x equals 4 minus x. Then we add x to both sides to come up with 2x equals 4, and then 4 divided by 2 to give the value of 2, which is x. Therefore, the team in first place only needs two more points to claim the title. That's assuming they match the results of the team in second place. So, for instance, they could draw their next two games and, providing the team in second place also draws their next two or loses at least one of them. 

By integrating mathematical thinking into everyday scenarios, we can gain a deeper appreciation for the practical applications of mathematics and its role in decision-making and problem-solving. Before I wrap up this episode, let's take time to reflect on the diverse ways in which mathematical thinking can enrich our lives and empower us to navigate the world with more confidence and clarity. 

Conclusion:  25:28 - End 

Thank you for joining me on this journey through the world of mathematics. We've explored together fascinating examples and insights into the intersection of mathematics and personal growth, particularly focusing on my experiences with the autism spectrum and my passion for solving mathematical puzzles. In this episode, I've dove into things like visual thinking and special interests, discovering how these elements can contribute to good mathematical abilities. 

Among the things we've explored include learning strategies, including differentiation and integration, which have been instrumental in my own mathematical journey instrumental in my own mathematical journey. As I conclude, I now invite you to reflect on how these concepts and other concepts may resonate with you. Have you found similarities in your own experiences? Do you share a passion for mathematics and mathematical puzzles, share a passion for mathematics and mathematical puzzles, or have unique strategies that enhance your own learning? I would love to hear from you. 

You can share your experiences, insights or questions with me. You can find a link to me on Twitter via a link at the footer of my website, stephensevolution.com, or you can contact me via the contact page on the website itself. You can connect with me on social media. You can connect with me via these channels to engage further and continue the conversation.

Furthermore, if you enjoyed today's episode and want to stay updated on future episodes, be sure to subscribe to my podcast. You can find a link towards the footer of the homepage of my website, the home page of my website, stephensevolution.com here. By clicking on this link, you'll be taken to a form where you can sign up to receive notifications and exclusive content, ensuring that you never miss a new release. 

To finish, what I'd like to do in this particular episode is give you a challenge that will put your problem solving skills to the test. So get ready to sharpen your minds and embrace the world of logic puzzles. Whether you're a seasoned puzzle enthusiast or new to the world of logic games, this puzzle promises to be both fun and intellectually stimulating.

 I've prepared a separate file available on my website, stephensevolution.com. It will be available on stephensevolution.com/activities, and it will be under the title of Mathematics Activities. So by the end of this activity, it is hoped that you will have exercised your logical thinking and might even discover new strategies for tackling complex problems. 

Thank you once again for tuning in. Together, let's continue unravelling the language of numbers and patterns, celebrating the beauty of mathematics and its profound impact on personal growth. Mathematics and its profound impact on personal growth. Until next time, keep exploring, keep learning and keep discovering the endless possibilities within mathematics. Thank you.