Talking To Teens: Expert Tips for Parenting Teenagers

Andy:
You have this book that I've just finished reading called Thinking Better, The Art of the Shortcut in Math and Life. What got you interested in shortcuts? And why did you think that writing a whole book on this topic would be a good idea?

Marcus:
Well, I'm a mathematician, and actually I decided I wanted to be a mathematician when I became a teenager actually. It was about the age of 13 that my maths teacher at my school, or math teacher as you say in America, weird, why do we have more maths than you do? I don't know.

Andy:
No, there's only one.

Marcus:
But yeah, my maths teacher just told us this wonderful story about a young German mathematician, illustrating the power of mathematical thinking. And I mean, you can do the little challenge yourself. The challenge was, how do you add up the numbers from one to a hundred. Now, there's a really boring, slow, laborious way to do that, which is you start at the beginning, one plus two-

Andy:
One plus two, right.

Marcus:
Yeah. Plus three is six. And you're likely to make mistakes. It's going to take you ages to do. But my teacher told me this story of this young German mathematician who was challenged with this at school, beginning of the 19th century, he was at school. And the teacher thought, I'll set this problem, take the kids ages to solve.

Andy:
This will keep them busy.

Marcus:
Yeah, exactly. I can get a good bit of rest at the front of the class. And before he even finished posing the problem, this kid, Carl Friedrich Gauss was his name had written down a number on his little slate pad and put it down in front of the teacher. And the teacher thought, gosh, what a strange kid that's just writing random numbers down. But then he saw the right answer there.

Marcus:
So he asked this young Gauss, "How did you do that so quickly?" And he said, "Well, there's a shortcut because you don't have to start at the beginning and just add the numbers up in turn. You can actually take them in pairs. So you can begin at the beginning and take the end at the same time. So one plus a hundred adds up to 101. But two plus 99 also adds up to 101."

Marcus:
So he saw this clever pattern that if you look at the numbers in pairs from the start and the end of the calculation, you've actually just got 50 pairs of numbers adding up to 101. And so the answer is 50 times 101, which is 5,050.

Marcus:
And so my teacher told us this story. And I was like, wow, what a brilliant way of thinking. Teenagers, look, we're all lazy as teenagers, we're all looking to avoid doing hard work. And it's universal across ... there aren't any teenager who doesn't want to avoid doing hard work.

Marcus:
So this just really appealed to me. I said, okay, wow. What an amazing shortcut. You don't have to do any hard work. You just think cleverly and you get the answer. And my teacher at the time said, "Look, this is what mathematics is all about. It isn't about hard work. This is a subject." And he called it the art of the shortcut. And he said, "Look, at school, I'm going to be teaching you all of these really clever ways of thinking about complex problems, which is just going to do away with all the laborious hard slog. And just you'll be able to do these questions really quickly once you've got these techniques. And then you can go off and play football or soccer."

Marcus:
So that really, I was like, oh my gosh, I'm going to dedicate myself to this amazing subject, mathematics, the art of the shortcut.

Marcus:
So in a way, this book that I've written is my thank you in a way to my teacher, first of all, for really showing me this amazing subject. And also just I hope a guide for all of those people out there who don't really realize that mathematics is this amazing tool to look at problems in this fast, efficient, cunning way. And to get rid of all the hard work and to think better. You can get to the solution much more easily by using these techniques.

Marcus:
So the book's the collection of clever ways that we've come up with over 5,000 years of coming up with mathematics. It's your shortcut to the art of the shortcut. The book hopefully gives you all of these clever ways of thinking.

Andy:
But I mean, isn't laziness that, don't we want our kids to be hardworking and not just searching for the easy way out of everything?

Marcus:
I think that laziness gets a bad press actually. And that actually we should be celebrating laziness to a certain extent. It's down there as one of the seven deadly sins. But I think our laziness is often the secret to us coming up with clever ways of thinking. These clever ways to avoid the hard slog.

Andy:
It breeds efficiency.

Marcus:
Yeah, well, Babe Ruth, the great baseball player, he actually credited his laziness for his great ability at baseball. Because he said, "Look, I really hate running. I don't want to run around the bases. I'm just going to knock the ball out of the stadium, home run. Then I just wander around slowly, get to clear the bases."

Marcus:
And there are a lot of executives as well that talk about, sometimes hard work can get in the way of new, innovative ideas. So if you go to any startup, you'll find that it's full of games and pool tables and table tennis because they realize that just encouraging a playful attitude to solving problems and allowing people to have downtime and just time to sit and look out the window. I mean, that's really can be a very powerful way of coming up with new ideas. So, I think sometimes, perhaps the easy option is actually to just start doing a lot of hard work.

Andy:
Right. To just jump in and say one plus two plus three plus four-

Marcus:
Exactly. That's the easy way. But in the end it turns out to be the long way. So, I think it's much better to go, okay ... And it does cause a bit of anxiety because you've got quite a lot of period of thinking, oh, I don't quite know what I'm doing, but I'm getting my space to think about it. And so yeah, I think laziness gets a bad press. I think we tend to want to control society. And so we say hard work is the way to do it. And I think actually, you take a risk by allowing people space to be creative and think. But actually, I think we should do that more.

Andy:
And so one of the ways that we often can look for shortcuts is by recognizing patterns and seeing where patterns exist. Which is what Gauss saw, I guess, that this pattern that you could add the last number with the first number and they always kept adding up to the same thing. And you point out that though this is looking for patterns and being able to find patterns is helpful in lots of areas of life. And I thought it was interesting, you were talking about how even a lot of the way that we learn things like music is about practicing arpeggios and practicing scales, training ourselves to see these patterns. So that then when we are playing a piece of music, we can just jump right into it and we start to see, oh, I see this is A minor blues, yep, no problem. I know what to do here. I hadn't really thought about that before as pattern recognition, but yeah, it's a cool of thinking about it.

Marcus:
It's funny because actually as a teenager, I fell in love with maths about the age of 13. But it was also the same age that I really started enjoying playing music. I started learning the trumpet around 12/13. And I think that those two subjects have always run parallel for me. And a lot of people have talked about connections between mathematics and music. And I think you're absolutely right, the idea of a pattern is common to both of them. I actually sometimes call mathematics the science of patterns, and music, the art of patterns.

Marcus:
And I do think you're absolutely right. The point is that, why do we learn our scales and arpeggios, as a teenager learning a musical instrument, that feels like the really boring part.

Andy:
This sucks. Yeah.

Marcus:
Yeah, it does, it does. What to do. But actually if you were to explain, no, this will ultimately give you an amazing shortcut when you come to read music for the first time. And as you say, you'll just see a pattern on the page and go, oh, that's an arpeggio and I know that. And so my fingers run it off without any effort at all.

Marcus:
I talked to a cellist during the book about shortcuts in music and things. And she compared it very nicely to reading. You don't want to read each individual letter as you are reading a book, that's just going to take you forever. So, actually you start to read collections of letters and see them as words. And so in music as well, these patterns of notes become words that then facilitate you playing your music really quickly.

Marcus:
In music, you've got to do your scales and arpeggios, but you know what you're heading towards because you can hear a wonderful piece of music, a pop song you love, or a piece of jazz or a great concerto. And I think our problem in mathematics, and something I've been trying to do in a way with this book, is, a school, I think kids are learning their mathematical scales and arpeggios, multiplication tables, algebra and things, but they never realize the big picture, the big stories, the big music. And we often fail I think in our mathematical education to spark that excitement about mathematics by playing them some big ideas.

Marcus:
So I think that's what my teacher did for me as a teenager. He opened up this magical world and showed us that there are exciting things outside of what was in school. Exciting things about prime numbers, about high dimensional geometries, about topology.

Marcus:
And I think that's what's missing in our mathematical education. What I hope, any parents out there who are hoping to excite their kids about the importance of mathematics, I think a lot of the, hopefully my book, but other people's books and the videos that are online, can just help to play the big ideas that are out there. Which will hopefully motivate them to then say, okay, well, I'm prepared to do the hard work at school of learning these mathematical scales and arpeggios. Because yeah, I'd quite like to make amazing technology using these mathematical ideas.

Andy:
So how do we get better at spotting those patterns? Or put ourself in the right mindset to start seeing things as patterns? Instead of just feeling lost, adding all the numbers together or trying to read every letter of the sentence.

Marcus:
Yeah. I think with all of these things, it's really an element of practice and just getting an experience of the patterns to look for and building up a range of techniques to try and test a series of numbers. Oh, what's the difference between these numbers? Is there a pattern there?

Marcus:
So I think a lot of people get frightened that maybe you have to be born with a brain or eyes or a way of thinking. And that's really not true. The more time you spend experiencing patterns, being shown patterns in numbers or patterns in geometry and shapes or whatever, even in music, you begin to get an experience that you can tap into later on.

Marcus:
So I think that's really important. It's about practice in a way and that your brain starts to really get better and better at spotting these patterns. So, don't expect it all to happen at once. Just spending your time in this world gradually gives you the experience that will help you to spot a pattern in a new set of data that you will perhaps want to look at.

Andy:
You talk about how perfectionism is considered essential when it comes to mathematics. But that also, a lot of times in life perfectionism can be a killer for success. I'm interested in that dichotomy and how you think those two sides of the coin come together?

Marcus:
Yeah. I think it is one of the difficulties with mathematics. It's very unforgiving. So if you make a mistake, it just can be really deadly. But that said, I think, again, one of the faults of our education system is often that we want our students to get everything right first time. But actually we perhaps learn more from the moments when we fail. And then we look back and we understand, well, why did I get something wrong? And I think that often is a far richer learning experience than just actually getting everything right first go.

Marcus:
So I wish in education, we actually celebrated a bit more those moments when people get things wrong. As I think those are really true learning moments.

Marcus:
And it's very interesting, my previous book that I wrote before this one was about artificial intelligence and creativity. And there's new code that's emerging, machine learning, where code changes and mutates from its learning process. And the fascinating thing is, the code only changes and becomes better if it gets something wrong. That's the moment when it actually changes. So I think we should take this into our education.

Marcus:
So yeah, ultimately we were looking for people getting the right answers, but on the way maybe we should be more forgiving and more celebrating the moments when people get things wrong because now you can learn from what you just did.

Marcus:
And that said, even some of the shortcuts in the book are quite interesting because sometimes it's important to know what information you can throw away. You might say perfectionism, well, the perfect thing is to know everything about the setting you're in. But often there's a lot of things which aren't important, and that's a really important shortcut. What can I throw away that isn't important here? And what do I need to retain because that's key to understanding?

Marcus:
So a nice example of that is a map that we use every day in the city, which is a map of the subway or the underground. Because, I mean, we have this wonderful thing, the London underground map, and any big city probably has their own map. But it isn't a real map because the distances between the stations are not-

Andy:
Just dot, dot, dot, dot.

Marcus:
It's just dots and how they're connected. So this is a new sort of map we came up with in mathematics and it's called a topological map, not a geometric map. Because what's important is, we don't really care how long it takes to get from one station to the other.

Andy:
It doesn't matter. Right.

Marcus:
We just want to know that there is a connection between the two. So actually this is the wonderful ... There was this guy, Beck, in London, in the thirties who came up with this new map of the London underground. Because people were finding it really difficult to use a geometric map with all the distances, it was too messy. They couldn't see what was going on. But with this new ... They pushed and pulled the map around, and made this network map. And then suddenly you can navigate this.

Marcus:
So, that's a case of, I need to throw away the unimportant stuff, which is the distances and time between the stations. I just want to know how they're connected. So that's very interesting. I think that's a very important skill to learn. To look at a problem and say, you know what? I don't need to know any of this stuff, this is the key. So finding the essence of the problem is often a really important shortcut.

Andy:
That's cool. Yeah. And we can only consider so much information at one time before it just starts to get overwhelming. And a lot of things in math especially are so complicated, you can't think about the entire problem all at once. You really have to figure out what's important and where can I focus my attention. And that's such a life skill.

Andy:
You have a whole book on shortcuts, but are there ever times when it makes sense to take a long cut?

Marcus:
Yeah, that's interesting because I talked to people from other professions because I was interested whether there were shortcuts that they had that were similar to my ones. Or sometimes whether a shortcut is not really what you want.

Marcus:
And I really enjoy hiking, and you think about it, a shortcut's a bit pointless if you are going out for a day's hike up to the top of a mountain. So I talked to a mountaineer, and he said, "Well, I could take a helicopter to the top of the mountain, but that defeats the object."

Marcus:
So actually the Greek philosopher, Aristotle, had this nice division of different sorts of work. Poiesis which is work to get to a goal. So all you are interested in is actually getting to the finishing point. So any shortcut there is, you don't mind, because it's just hard work getting there. You just want to get to the goal.

Andy:
Totally.

Marcus:
But then he said there's a different sort of work which is called praxis, which is work for its own sake, the work that you enjoy doing. So I think it's important to know, yeah, absolutely, maybe I don't want a shortcut because I'm really enjoying the time spent doing this work. So, my feeling is, these shortcuts I'm proposing are trying to help you to avoid the work you don't want to do. So you can get to the point where you can do the work you want to do.

Marcus:
So it's a bit like, if going on a hike, I don't want to do the boring walk to the beginning of the hike because that's probably, cross lots of roads and stuff like that. So I'm quite happy to get to the beginning of the walk using my car or using the local bus. But then I get out and I want to use my legs and I want to take time doing that.

Marcus:
So, I think you're right. Sometimes the long way is what you enjoy doing. I never watch trailers for movies because that's basically [inaudible 00:19:23]. And I often find, I don't need to watch the movie anymore.

Andy:
Well, now I know what's going to happen. Yeah. All the funniest parts.

Marcus:
Exactly. They show me all the best bits. All the best gags. Exactly. So actually there, you don't want to shortcut the movie because you want to spend the time in it.

Marcus:
But on the other hand, you don't want to see every single detail of the main character's life. So there is already a shortcut, the movie's taken the best bits and put it together. So, that's the kind of balance.

Marcus:
I tell you, at the end of the book, I actually talk about problems that we would like a shortcut for. But that we actually believe mathematically that there isn't a clever way, that you can't avoid doing the hard work.

Marcus:
So there's one very famous one called the traveling salesmen problem. So this is a salesman who wants to go around a map of different cities say in America. And he's got, or she's got distances between all of the cities. And has got to find the best way, the shortest way to visit all-

Andy:
How do I optimize this route?

Marcus:
Yeah. And a lot of companies are absolutely faced with that problem, if they're shipping goods, they want to find the shortest way to use the least amount of fuel. So we would love a shortcut which would say, okay, this is the way you find the shortest way around this map.

Marcus:
But we think this problem, actually there isn't a shortcut beyond actually trying all of the different possibilities, listing them, and then seeing which one is the shortest one.

Andy:
Interesting.

Marcus:
Yeah. With 10 cities, that's already a huge number of possibilities. So, it's interesting that mathematics, it is the art of the shortcut, but it also might be possible to show when a problem, actually there isn't the shortcut beyond doing all the hard work. And that's as important as ... sometimes you need to know, well, I shouldn't waste my time looking out the window trying to think of a clever idea because this one, I just have to do the hard work.

Marcus:
So yeah, it's interesting that we do have these problems. And in fact, there's a million dollar prize, if you want to incentivize your teenagers to do maths, then a good monetary prize might be a good one. But there's a million dollar prize to prove actually that there isn't a shortcut to that traveling salesman problem. Because we don't know, we think that ... There could well be, there might be some cunning way that we haven't [crosstalk 00:21:42].

Andy:
That;'s possible. Right.

Marcus:
So either you have to come up with a cunning way to solve that problem. Or prove there is no solution. And either way-

Andy:
Definitively prove it can't be done.

Marcus:
Yeah. If you can do either of those, you win a million dollars.

Andy:
That doesn't sound bad. Yeah. Set your kids loose.