Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Category: Reflections

Reflections on learning and teaching and research and life.

  • Past Exam Vision

    Students have just been told their exam results for Semester 1, and some of them are facing replacement exams. So we’ll be trotting out our standard suite of exam advice again, which will be all the more poignant now because these people tried to do it last time and failed!

    One piece of advice we give is not to use past exams as your main study tool. So many students study for their exams by taking a stack of past exams and systematically working their way through each question and making sure they can do all the intimate details. This is a bad idea for several reasons. I’ll list some in dot point form:

    1. The course may have changed over time, so some of the questions will not be relevant to your course content anymore, while still other questions just won’t have appeared in past exams.
    2. The lecturer probably changed, so the style of the exam questions may be quite different to the exam you are about to do.
    3. No one exam can cover every concept in a whole course, and even several exams will miss something between them.
    4. Lecturers are not stupid, and so will generally always put something in that has not been done in an exam for the past several years, in much the same way that they don’t use yesterday’s questions today on a TV quiz show!
    5. You need to save at least a couple exams to do as proper timed exams in exam conditions or you won’t practice the skill of doing exams in exam conditions.

    ​​​But there is one more reason I myself had never really known fully until this last semester. It’s related to point number 3 above, but it’s even more pernicious:

    1. Questions in past exams are often cut-down versions of full problems designed specially to be dealt with in exams, and so will not necessarily help you actually understand the material.

    Let me explain how I fell into the trap of this peculiar kind of “Exam Tunnel Vision”.

    I never studied Differential Equations in a formal course as part of my degree. I managed to avoid all applied maths beyond first year by instead studying statistics, pure maths and Chinese. This means that pretty much everything I know about differential equations has been learned while helping students in the MLC. I have learned a remarkable amount, but there is a problem with my approach: I only see the parts of the course that students ask me about. And since students often study using past exams, the parts of the course I see do not necessarily represent the full picture. Now I do know full well that I should ask questions like “What would happen if it were this way instead?” and “Is there more stuff related to this?” and “Where does this fit in the bigger picture?” and indeed I do ask these things, but sometimes no matter how hard you ask, sometimes you can’t find this information without asking an expert.

    Case in point is the Frobenius method for solving differential equations. What happens is you are supposed to make an indicial equation, which for second-order equations will give you two solutions for r. Then for each value of r, you are supposed to do a process to find a solution. But here’s the catch: this final process is quite long, and so in exams and assignments the lecturer only ever asks students to do one of the solutions. Since my learning about differential equations was based entirely on helping students, I had never seen what you were supposed to do after this point! No-one ever asked, so I didn’t know.

    I had fallen into the very trap I warn students about: I had developed “Past Exam Vision” and couldn’t see beyond the exam to get the full understanding. In future I’ll be more careful, and now I have a good story to tell them to warn them about it. If I can fall into the trap, then anyone can!

  • Numbers don’t change the situation

    The coordinator of first year Chemistry had a chat to me the other day about how to support students in solving word problems. The issue is that students have trouble using the words to help them decide what sorts of calculations need to be done in order to solve the problem. This issue is not new – people have been solving word problems for thousands of years, and the maths education literature is littered with papers discussing the issue. No clear concensus has been reached, of course, because there are any number of factors that affect students’ ability to solve problems.

    One of these many factors I only learned about earlier this year when reading the following paper: A. Af Ekenstam and K. Greger (1983), “Some aspects of children’s ability to solve mathematical problems”, Educational Studies in Mathematics, 14, 369–384. It’s easiest to describe using the following two problems (slightly modified from those presented in the paper):

    Problem 1: A block of cheese weighs 3kg. 1kg costs $28. Find the price of this block of cheese.
    Problem 2: A piece of cheese weighs 0.923 kg. 1kg costs $27.50. Find the price of this piece of cheese.

    The paper reported how students aged 12-13 years were asked these problems, and specifically asked what sort of calculation they would choose to do in order to solve them. What would you choose for each one?

    All of the students in this study chose multiplication for Problem 1. However many of them did not choose multiplication for Problem 2, and some of them did not know at all what to do. To be clear, it wasn’t that the students didn’t know how to actually perform the calculation; it was that they didn’t know what sort of calculation to do. Even when the teacher explicitly pointed out how similar the two problems were, many students still did not know what to do for Problem 2. Upon discussion with the students they discovered that the students were choosing what calculation to perform based on the numbers they saw, rather than on the situation described.

    This was a big surprise to me. Of course, I experience students not knowing what to do and choosing the wrong thing to do all the time, but it had never occurred to me that they were making the choice based on the numbers they saw. To me the situation itself has always told me what to do, regardless of the numbers themselves — if every kilo is worth THIS dollars, then THAT number of kilos ought to be THIS times THAT dollars, regardless of what THIS and THAT actually are. But clearly not everyone thinks this way!

    The authors of the paper have a few theories for why students are confused when the numbers are different.

    One theory is to do with the students’ experience of word problems. For many students, the majority of problems they’ve seen before have involved whole numbers for at least one of the numbers involved, and so seeing decimals in both positions just doesn’t fit with their experience. Moreover, they have succeeded perfectly well on other problems by focussing on the numbers. This says more about the students’ schooling than the student themselves, really.

    Another theory is that their experience of numbers has led them to believe certain things about multiplication and division. With whole numbers, when you multiply the answer can only get bigger, and when you divide the answer can only get smaller. Other research confirms that these ideas are very strong in children and tend to impede them having a fuller picture of what multiplication and division mean for other types of numbers. In this experiment, some students talked about how in the second problem the cheese is less than a whole kilogram and so the answer ought to be smaller than $27.50, which is in fact a perfectly correct and quite sophisticated attack on the problem. But because the answer had to get smaller, they chose to do division, because this is how you make numbers smaller.

    The final theory is that many people view multiplication and division (and most other things in maths) as a procedure, partly because of the focus on procedural fluency in primary school. In this context, the procedure for multiplying decimals by hand actually is different from the procedure for multiplying whole numbers. With decimals there’s all this stuff about shifting decimal places back and forth which makes the procedure much more complicated. And working with fractions is wildly different again! So it’s hardly surprising that students, when faced with a problem involving decimals, will expect that the action to perform should be different.

    Regardless of the reason, one thing is clear: many students are not focussing on the right thing to help them solve the problem! So one way to help those Chemistry students is to help them focus on what the words tell them about the situation, and how the situation tells them what they should be doing, rather than the numbers themselves. Because it’s the situation that tells you what to do, not the numbers, and the numbers don’t change the situation.

  • There is no such thing as “just a quick question”

    We often get students in the MLC saying that they have “just a quick question”: “Finally you’re up to me – it seems like a long time to wait when it’s just a quick question…”; “I know it’s 4:05 and the Centre closed five minutes ago, but it’s just a quick question…”; “I’m sorry to interrupt you when you’re talking to another student, but it’s just a quick question…”. I do understand these students’ need to have their question answered, but the problem is that at the MLC there is no such thing as a quick question. Here’s why…

    The first and most banal reason is that many so-called “quick” questions do not have quick answers. For example, the question of “How do I find where these two lines meet?” is not at all easy to explain quickly, and the question of “Where have I made the error in this working?” can take a good ten to twenty minutes of focussed attention even for the most experienced mathematician (longer if the working is longer).

    However, the real reason is to do with the aims of the MLC. Our aim is to help students learn how to learn and solve problems for themselves, and to fill in missing background knowledge. Even when it is actually possible to answer the question quickly, we wouldn’t be fulfilling our aims if we did!

    Let me give some examples.

    Consider the question, “Have I chosen the right hypothesis test for this assignment question?” Using my experience in this area I could look at the question and say yes or no in a few seconds, but that wouldn’t help the student to know how to make this decision for themselves. Instead, I need to talk through the sorts of information you need to be looking for to decide if it’s the right hypothesis test, and show them how to find that information in their assignment question, and then also discuss how changes in the information might change the hypothesis test. Not to mention the possibility of whether they actually know how to do that hypothesis test. So it’s not a quick question after all!

    Consider the request, “We’ve solved this differential equation. We don’t want you to check if the answer is right, we just want to know if we’ve applied the method in the right way.” I only know a small amount about differential equations, so I couldn’t just tell them if they’ve used the method correctly even if I wanted to! But even if I did know the answer, that wouldn’t help them to know if they’d done it right for themselves. In order to help them learn that, we’d need to go into their textbook and lecture notes to find explanations of the mechanics of this method, and also pick apart some examples to make sure they know how it works. I’d help them make a list of the steps that need to be taken, and then we’d look at their working and check off the list. So it’s not a quick question after all!

    Consider the question, “I’ve done this derivative and set it equal to zero to find the maximum, but it’s not coming out to the answer I expect. What’s going wrong?” As already mentioned, it can take a while to find errors, but even then, me finding the errors for them won’t help the student know how to find errors for themselves. So at the very least I need to talk through the strategies I have for finding errors and fixing them. And it may happen that as we look at the working, I discover that they don’t know how to use the product rule for derivatives, so I would need to explain how that works with various other examples. Or it may happen that looking at their attempt to solve the equation I discover some serious misconceptions about how algebra works which also need attention. So what might have taken 10 minutes even if I just told them the answer, becomes a good half an hour to an hour of serious background knowledge learning. So it’s not a quick question after all!

    Finally, consider the question, “Why do I have to use the right-hand rule to decide the direction of a cross product?” The simple answer would be that it’s just because it’s the definition of the cross product, but that would not be encouraging the student to make connections in order to understand. So at the very least we would need to talk about how the cross product produces a vector perpendicular to both the inputs, and how there are actually two possible directions to choose and we need a consistent way to make that choice, which the right-hand rule supplies. I would probably ask if they knew how to calculate the cross product and under which situations they might use it, in order to strengthen connections to the rest of the topic. I might also talk about how it all boils down to knowing what happens to the standard basis vectors of i, j and k and it seems reasonable for i×j = k. And in order to help them understand how ideas come about in maths I might possibly also discuss Hamilton and how the quest for vector multiplication was actually inspired by the complex numbers. So it’s not a quick question after all.

    So if we are doing our job properly we will always find some way to help students learn more of what they really need, which takes time. (And incidentally I think it’s worth waiting for.) So that’s why in the MLC there is in fact no such thing as a quick question.

    These comments were left on the original blog post:

    apm 12 October 2018:
    Brilliant! You take an approach that is similar to an enrichment exploration activity. “What else do we need to know?” is comparable to “What else can we learn about this?” Funny, but the thought of having a gifted math club going on simultaneously at your center came to mind. I’ve always enjoyed teaching to the extremes but I hadn’t reflected on the exploratory similarities in meeting the needs of struggling learners and gifted math students. Perhaps not the takeaway you intended but well-crafted writing often provides unexpected insights to readers.

    David Butler 18 October 2018:
    Thank you so much for the reply and the insights you brought. I’ve never reflected on it before you said it “aloud”, but yes there are a lot of similarities between helping the struggling and the flying learners. I often say to my strugglers that they are struggling because they are thinking of worry questions that no-one else is bothered about.

  • Two kinds of division

    If you had to explain what the expression “10 ÷ 5” (that is “10 divided by 5”) meant, what would you say? To be clear, I’m not asking for the answer, I’m asking for a story that will give it meaning.

    I’ve been asking people this for the last few days and there are two main stories:

    1. I have 10 things to split into 5 groups; 10 ÷ 5 is how much is in each group.
    2. I have 10 things to split into groups with 5 in each group; 10 ÷ 5 is how many groups there are.

    Most people only say one of these two, which is interesting because only knowing one of them can get you into all sorts of trouble when it comes to solving actual problems.

    If you only know it as “how many groups of 5 fit into 10” then you’re going to have to think quite hard to figure out how many each person gets when you share 10 among 5. And it would be even worse if it wasn’t a whole number of objects shared among a whole number of people but, say, a number of moles of chemical shared across a number of litres of water to make a concentration. Indeed, both perspectives on division are often needed in the same drug calculation problem in nursing and medicine!

    As a teacher you can get into trouble too: consider the meaning of “10 ÷ 1/2”. The first interpretation would give you “I have 10 things and I split them into half a group; 10 ÷ 1/2 is how much in each group.” While this is correct (and quite interesting actually), it makes much less sense than “I have 10 things and I split them into groups with half a thing in each group; 10 ÷ 1/2 is how many groups there are”.

    Mathematicians have the tendency to say that division is simply the inverse of multiplication (so that “10 ÷ 5” means “the solution to 10 = 5x”). But this denies that the understanding and use of maths is deeply connected to how we picture it. When two pictures explain the same maths, we’ve got to be both aware and careful!

    (PS: For those interested in a bit of Maths Education terminology, Meaning 1 listed above is called “Partitive Division” and Meaning 2 is called “Quotative Division“. It took me ages to figure out what they were going on about at my first Maths Education conference! Oh, and there are in fact more ways than these to think about division, corresponding to the many ways there are to think about multiplication!)

  • Elsa’s Freedom

    Disney’s Frozen came out on DVD last week and my family and I watched it on Saturday. It’s a very good movie with an excellent theme about the real nature of true love which is not usually seen in a “princess movie”. There are also two different stories about freedom, which is pretty common in a princess story (consider Rapunzel in Tangled and Jasmine in Aladdin). It’s this I want to talk about today.

    In case you don’t know the story of the film it starts something like this: Princess Elsa is born with the ability to create snow and ice, and while playing with her younger sister Anna there is an accident and Anna is blasted in the eyes with ice. The trolls are able to cure her but warn that Elsa will need to learn to control her power. The King and Queen take this to mean she needs to try not to let her power out at all and so Elsa spends much of her time locked in her bedroom, trying not to feel. Well this all comes to a head at Elsa’s coronation where it all gets too much for her and she lets it out spectacularly and runs off into the mountains. Cue the Oscar-award-winning song Let it go.

    The song is all about how, now that everyone knows about her powers, Elsa is free to let it go and really see what she is capable of. This is where I want to draw my parallel to maths.

    I have been a mathematician from a very young age – my mother says at the age of five I was playing number and letter games in my head. But it wasn’t too long before I realised that a love of maths attracted resentment and even hostility in others. I soon learned that the only way to enjoy maths was in my head, all by myself. I had to work hard to turn off my natural instinct to see the mathematical in things when I was with others for fear of getting the wrong sort of attention.

    In my first year at university, the people I spent time with expressed resentment at being forced to learn useless and boring maths, so even then I felt I had to hide. It wasn’t until my second year at university that I found others who shared a similar joy in maths and was fully free to be myself with them and “let it go”.

    Strangely, I had to go into hiding again as a maths teacher in a high school. Despite the fact that my job was to teach maths, I was looked upon as strange by the other teachers for my love of it. And worse, the many and burdensome responsibilities of day-to-day work in a school meant that I did not have time to play with the maths I was teaching and I struggled having to hold it all in. Coming back to University was like Elsa escaping to the mountains – I felt a great relief that I could play freely.

    Of course, this story is a little sad because there are still throngs of people who don’t share my love of maths and I shouldn’t have to feel trapped when I am with them. Well I’m happy to say I learned a long time ago that it’s not worth worrying about what others think of me, and I am actually free to be myself no matter who I am with. As long as I’m not getting in the way of their joy, I should be able to express mine. (Incidentally this allows me to publicly express my love of other strange things like childrens fiction and singing.)

    The key point I want to make is that it was being part of a community where it was ok to love maths that helped me realise this, and I found that community here at University. There must be scores of students with a similar experience to mine – those who have a love they’ve had to hide and who discover the freedom to express it here at University (and not just maths either). This is much better than Elsa’s so-called freedom, where she still had to be alone to express herself.

    I dare say, however, there are many other students who still don’t find that freedom here because for whatever reason they don’t connect with the right community. Like me in first year, they still feel like they have to hide their love. This is one of the reasons we do public puzzles and sculpture, and talk excitedly in the Drop-In Centre about maths beyond the curriculum. Who knows who might find the freedom to let it go?

  • Life Impact

    The University’s current slogan is “Seek Light”, but the one we used to have before that was “Life Impact”. I have decided that at least for myself I would like to keep the old one, because recent events have shown me its true meaning.

    My mother-in-law Merle died almost two weeks ago, and last week was the funeral and memorial service. She was not just my mother-in-law, but my second mother, and life will never be the same without her. The main reason I feel this way is because my life was already never the same because of her. And not just for me: more than three hundred people came to her memorial service and Dad has been receiving twenty to thirty cards in the mail every day since the news was announced. Clearly, Mum made a huge impact on hundreds of people during her life.

    How did she do this? She wasn’t a well-known public figure famous for her musical skill, or for curing a disease, or  for changing history. She was just an ordinary woman – a wife, a mother and a nurse.

    I think the impact Mum made was at least in part due to the fact that she believed in the power of little things. She believed in doing what she promised to do, in asking people how they were and actually listening, in writing thank-you notes, in telling shop assistants they were doing a good job,  in baking cupcakes just because, in offering to do the dishes, in opening doors for people, in learning her patients names, and in simply giving a smile. I am sure that it was all these little things that made such a huge impact on so many people.

    The University used to post stories about how people’s work at the Uni has had “life impact” because of the big changes it made to the world, but Mum has shown me that the greatest impact you will ever have is on the people around you every day, and that this impact will happen through all the little things you do and say.

    I feel humbled and sometimes overwhelmed to think that I am in a position in the University that puts so many people around me every day who I will impact in this way. I only hope I can continue the legacy she has left, and remember the power of little things. I would feel truly blessed if I had even the tiniest fraction of the positive life impact Mum did.

  • Why I’m not a “lecturer”

    Every so often a student asks me why I am not a “lecturer”. Often it happens after I’ve helped them understand something from their course, or (as it did this week) after I’ve given a revision seminar on some topic from their course.

    Now I do realise that they are giving me a compliment by saying this. They are saying in their roundabout way that I did a good job of my teaching and therefore deserve to be a “real” lecturer. And I’m flattered.

    But even so, I do like to answer the question they actually asked, and it’s high time I did it publicly so everyone knows.

    The first and most banal answer to the question is that, actually, I am a lecturer – “lecturer” is my official title. I don’t need to be a “real” lecturer in the sense of giving lectures to large classes in particular courses because what I do already makes me a real lecturer – it’s a perfectly legitimate activity for an academic (though not everyone in the University agrees with this).

    The second and truest answer is that it’s much more fun teaching in the Maths Learning Centre than it is lecturing! I really love my job and I wouldn’t trade it for anything else!

    As a lecturer in the Maths Learning Centre I get to do things I have always loved doing as legitimate parts of my job. I get to do papier-mache and play-dough and puzzles, and I can be publicly excited about maths on a daily basis. And I can be publicly excited about the learning of maths on a daily basis too and discuss with others who are interested about how people learn maths and how to help them learn maths.

    And even more than these, I get to spend most of my time talking to actual students one-on-one! I get to be there at the moment they learn and see the spark in their eyes as they realise how it all fits together. I get to help them realise they actually can do it themselves, and sometimes see their tears of joy as the walls of their bad experiences with maths begin to fall down. I get to see people progress from terrified at the sight of a mathematical symbol to being able to construct a proof all by themselves to passing on their knowledge to other students.

    And that’s why I’m not a “lecturer”. How could I possibly trade all of that in to “lecture”?

  • Assignments don’t teach people

    It is a well-known truth that assessment drives learning. Students will often not learn a particular topic or concept unless it is assessed by an assignment or exam. Fair enough – often students are not choosing to do a particular course for the sheer love of it, are they?

    However, many lecturers take this truth just a little further and subscribe to the belief that assessment can actually teach. They put quite a bit of faith in what a simple assignment question can do for students: a lot of them believe that a well-chosen assignment question has the ability to teach students amazing truths about maths. They imagine the student doing the assignment question, struggling through it, and coming to an epiphany where suddenly everything makes sense. I have actually had lecturers in the past telling me about the great question they’ve written and how it will teach the students something cool. I think this is just a little unrealistic.

    Through years of observation of students, it seems to me that actually, assignments don’t teach people, people teach people. Let me give you two examples from the last couple of days to illustrate.

    The Maths 1B students are currently studying orthogonal projection and they have a MapleTA (computer-based) assignment due today. In it, there are some questions that give them a basis for a subspace and a vector and ask them to project the vector onto the subspace. The students need to do this and then type their answers into the computer. In the particular question I have in mind, there are two vectors to project onto the subspace, and the second one doesn’t change when you calculate the projection. What this means is that the second vector is already in the subspace, which is why projecting it into the subspace doesn’t do anything. I’m pretty sure that the writer of the question is hoping that students will notice this and wonder why it’s the same and remember that fact about vectors already in the subspace and feel the warm glow of learning.

    But of course they don’t learn. Talking to students yesterday, they didn’t even notice the answer was the same as the input. They just noted their answer was correct and moved on. Luckily for these students, I was there to point it out and ask them why they thought that might happen and help them find the bit of their notes that discussed this concept.

    My second example comes from last week’s written question. It asked the students to prove that each vector in a subspace can be written in terms of the basis in a unique way. This is quite a fundamental idea which is not covered explicitly in the lectures and it’s a pretty safe bet that the writer of that question was hoping that the students, through doing the question, learned this concept. And also, I reckon they also are hoping that the students learned how to prove that something is unique.

    Only they didn’t of course. Almost every student who visited the MLC had dutifully written down the question, but the rest of the page was blank. They had no idea how to even start. Even those who had made a good start by writing down the definition of basis had no clue where to go from there. Since they had no clue how to start, they had no hope of finishing, and absolutely no hope of learning anything! Even those students who only needed a little prompting to solve the problem still had to ask about what was really going on.

    See? The assignment question was certainly the fuel that was needed to learn those things, but it wasn’t the assignment question itself that did the teaching – it was me, or sometimes the students’ friends. It was the discussion with others that helped the students learn. They needed someone there to help them notice what was going on, and to help them turn it into a lesson.

    I’m not saying you can’t choose good assignment questions that make it more likely for students to learn, I’m just saying that without also organising an opportunity to talk to someone as well, students will often not learn anything. Indeed, they often won’t do the question at all. So if you’re a teacher do remember: assignments don’t teach people, people teach peole.

    This comment was left on the original blog post:

    Sophie Karanicolas 21 November 2014:
    David, I have only just come across this amazing space that you are writing and creating in!
    I am really enjoying reading your blogpost on “People teach people”. I couldn’t agree more. Every teaching and learning initiative and every assessment requires people presence. The teacher as the coach and guide. I realised very soon in my career that students need to be well prepared and coached for any kind of assessment. They need to trial assignment and exam type questions and workshops these with teachers and peers to help them develop the critical thinking skills they need to undertake the challenges of assessment with confidence. This also enables them to engage with the content and reach deeper levels of understanding. I came across a student who was so disheartened because she couldn’t seem to understand a topic no ‘matter how many times [she] rewrote the notes’. (There are students out there who still believe that by rewriting notes they will somehow learn. Then they come across a question or scenario where they need to use their knowledge and they have that ‘blank page’ you referred to in your post.)
    Once this student started working with the teacher and her peers to answer these questions she immediately started to ‘get it’! We couldn’t wipe the smile off of her face and couldn’t find enough questions to give her! It opened the flood gates.. she had her ‘aha’ moment. So yes… people teach people!!!!

  • Wrapping up integrals

    I love wrapping presents. I’d like to say it’s because of the warm glow I have inside from giving a gift to someone else – and that feeling is certainly there to an extent – but I’m sorry to say the main reason is because I like the process of wrapping presents itself.

    A rectangular box wrapped in red, gold and green Christmas wrapping

    I like putting the present on the paper and making a judgement of how much paper to cut; I like using the scissors like a knife to cut a clean edge; I like folding the edge of the paper so that it looks nice and clean when you fold it over the present; I particularly like the part where you do the fold-in-the-sides-then-fold-up bit on the sides; and most of all I like the part where it’s all finished and your present is neatly encased in a piece of paper just the right shape with all the bits folded in neatly.

    Yes, I know I’m weird.

    But I reckon I’m not that weird. My daughter at 10 years old, still likes reciting the alphabet, though she learned to do this 6 years ago. My other daughter at 5 years old, will write her name over and over and over and over, seemingly getting pleasure out of the simple act. A musician will sometimes play a song they know well, for the sheer pleasure it, and almost any person will go up to a piano and play chopsticks. Many people I know like the experience of making scrambled eggs, no matter how many times they have done it before.

    It seems that all people derive some pleasure in doing things well that they know how to do well, even though they have done it before. There is something about the repetition that gives you a sense of pleasure. Perhaps your brain likes to have the electrical signals pass down the well-worn paths where it’s not so much effort. Perhaps the experience helps you remember the buzz when you learned it for the first time.

    I think perhaps the second reason is pretty accurate because I see myself doing it all the time in my work as well: guessing eigenvalues, calculating integrals, adding fractions and drawing conics. I love them all. I jump at the chance to do them with students in the MLC because I love doing them, no matter how many times I’ve done them before. And every time I do them, I remember with pleasure the first time I figured out how to do them myself.

    But whatever the reason, I do get pleasure from doing the integral of ex cosh x or (cos x)2 or 1/(x2 – 1) – integrals I have done a hundred times – and it coming out to the answer it ought to. It’s the same pleasure I get from wrapping a present.

    Sometimes you just enjoy doing something you know how to do.

  • Maths is not Science

    Let me say it again a little more emphatically: Maths is NOT Science. The major difference I want to focus on here is the concept of truth. Things are true in Maths, but they are not in Science.

    Let me explain. As far as I can tell, Science is about coming up with models that explain how things work which match closely to what we observe and allow us to make accurate predictions for what might happen next. No model is technically ever “true”, it simply does a super good job at making predictions and matching our observations. There is always the possibility that someone might come up with a better model in the future.

    This is why a very accurate idea accepted by all scientists is only called a “theory”, rather than a “truth”. The word theory doesn’t diminish the decades or even centuries of work that has gone into it or diminish the great job it does of matching the universe we see. It just highlights the fact that the aim of science is to come up with ideas that work, not necessarily ideas that are “true”.

    The “laws” you learn at school are in fact descriptions of observation, which is why they are true – the only things that are true in Science are the actual observations we make ourselves (and even then these are subject to some error).

    (Any scientist reading this, feel free to slap me for simplifying it all too much!)

    On the other hand, Maths is all about truth. A mathematician tries to find out what is true and what is not true, given a set of starting rules (and some of them try to work backwards to find a set of starting rules that will MAKE things true). And once something in Maths is proven true or false, then it always was true or false and it always will be true or false. The sum of the proper divisors of 6 will always be 6, the number pi will always be irrational with no repeating pattern to its digits, an angle of 30 degrees will always be impossible to trisect with only a ruler and compass, and two ovoids in a finite projective space of even order who share all of their tangents will always meet in an odd number of points. No-one is going to come up with a new idea in the future that changes the truth of these things.

    Of course, people may come up with new types of numbers or new types of geometries, or new sets of starting rules. They may decide that there is a better way of writing the maths notation, or that something else is better for solving problems in the “real world”. But that still won’t change the truth of all the maths before it. If you start with the same set of rules you will always get the same results. That’s what maths is about.

    So, don’t try to find a repeating pattern to the digits of pi or prove that it is in fact possible to trisect any angle! Because Maths is NOT Science – things in Maths are true forever.