Tag: problem-solving

Inspiration, not instructions

We have a big problem-solving poster on the MLC wall that gives students advice for solving problems. One of those pieces of advice is that to decide what to do for your current problem, you could look at other problems for inspiration. Yesterday I saw the dangerous results of what happens if you look at other problems for instructions rather than inspiration.

Across the day I talked to several students about Statics, which is an engineering physics course concerning situations where objects are not moving (ie “static”). At least three of them had put their answers into the computer-marking system and found that they were wrong, and so they wanted help to figure out why. As always, I asked them to tell me more about the problem and how they solved it.

The students showed me the problem they were trying to solve, and then pulled out their lecture notes and showed me how they found a problem that was similar and followed the same procedure. Normally, this would make me extremely happy – students actually looking at their actual lecture notes independently? Bliss! But unfortunately, what they had done was notice how the example had the same letters in it as their problem and put the values of those letters from their problem in the formulas they saw in the example.

The big problem was that the forces in the book’s examples were perfectly horizontal, but the ones in their assignment were at an angle, so they couldn’t just put things in the formula. Oh dear.

These students had interpreted the example as a list of instructions for precisely what to do if they saw a similar picture with similar letters in it. They had interpreted the process of solving Statics problems as finding the right formula and putting things in the right places. Instead, Statics problems are more about making your own formula from the structure of the problem situation itself. When you read a Statics example, you are looking for inspiration for how to think about the problem, rather than for instructions for how to do it.

I am very glad now that when we chose the words for our poster, we used the word “inspiration”. In the future I will try to highlight that aspect of it a bit more, so that students can be looking for the right things when they perform the good practice of looking for examples in the notes.

This comment was left on the original blog post:

Terry Bennett 10 April 2015:

Great post/observation David,

I’ve come across the same thing in class – “.. are you trying to reuse the equations on pg44 of my notes? Did you draw a free body diagram of this particular problem?”.

Next step: the reflection “what can I do better next year to avoid/reduce this misconception?” (answers on a postcard please …). Thanks for the help in identifying this “recipe trap” I’ve inadvertently introduced.

Terry

10 April, 2015
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.

12 June, 2014
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.

3 June, 2014
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!!!!

21 January, 2014
You will never see this problem again

“Now you understand that you’ll never see this problem again, don’t you?” I said, after a particularly productive problem-solving session at the MLC whiteboard with a group of students.

And then the world ended.

At least, I would have gotten the same reaction from the students if it had. They were all staring at me with wild expressions of undisguised horror. It was such a sudden and strong reaction that I almost turned around just to check if the four horsemen of the apocalypse were riding through Hub Central at us.

They weren’t, of course. It was just that I had effectively pinpointed something fundamentally wrong with their way of studying maths.

You see, when we were doing the problem, I noticed that they were all furioulsy writing down everything I had done, and asking me about the exact wordings. This worried me a little, but I continued working with them until the problem was done. And then they were all staring at the whiteboard with looks of intense concentration on their faces, but they were not the looks of people trying to understand what I had done. No, they were the looks of people trying to remember what I had done.

It occurred to me that they were trying to remember how to do a problem like this if they ever saw one again. I suddently had a vision of them poring over the exam solutions from past semesters trying to remember the solution to each one, only to find that many of the questions in their real exam were not at all like the ones in past exams. I couldn’t knowingly set them up for that sort of failure, so I said the statement that ended their worlds.

Then I proceeded to explain how at least a quarter of all the things we lecturers ask them to do are things that they’ve never seen before, but we are confident they’ll be able to figure it out from the information we’ve given them. Then I asked them what this problem could teach them about the concepts in their course, and what it could teach them about how to solve problems they’ve never seen before.

I think perhaps we could say this sort of thing to students a little more often when we show them the solutions to problems. Then perhaps they won’t react so strongly when they realise that maths isn’t just a list of different types of problems and how to solve each one!

11 September, 2013
Does it matter that roosters don’t lay eggs?

There is a particularly annoying puzzle that goes something like this:

“A rooster sits on the apex of a barn roof. The roof pitches at an angle of 43 degrees above the horizontal and is made of wood painted red. On the northern side of the roof, there is a large tree which casts a shadow over most of the roof. On the southern side, there is a duck pond. There is a very light rain shower falling, and the wind speed is 20 km/h from the Southwest. It is 10:47am on the 19th of August and the current temperature is 14 degrees celsius. The rooster lays an egg. Which way does it roll?”

The correct answer to this puzzle is usually given as: “Roosters don’t lay eggs.”

I take offense at this for several reasons:

Firstly, whenever I have seen anyone ask this puzzle, they have seemed to delight in watching the other person squirm, and they have had a superior “You’re so stupid that you haven’t figured it out” expression on their face. In my book, you are the worst sort of asshole if you choose to do things simply to make yourself feel superior to someone else.

Secondly, the puzzle didn’t ask to evaluate whether it was possible for a rooster to lay an egg in this situation! No, it asked to find which way the egg rolls if indeed the rooster did lay an egg. Sure you could discuss whether it was possible, but that doesn’t change what you’ve been asked to do.

My goodness, I know so many puzzles where the situation described is impossible or extremely unlikely, and it never stops people from figuring out the answer anyway. Just how likely IS it that a man has to take a cabbage, a goat and a wolf across a river in a small boat? Yet people have been posing and solving this problem for hundreds of years!

On a more serious note, almost none of the problems we give students are 100% realistic. In Physics courses, the students get problems about objects being dropped down holes drilled to the centre of the Earth. In Economics courses, they do problems that concern people who only consume nuts and bananas. In Architecture, they get volume and area problems with only whole-number answers. None of these situations is actually possible in real life, but that doesn’t stop us from expecting the students to do them anyway, because we know it will help them learn if they do. Are we really going to allow people to declare the imposibility of the situation and on that basis refuse to do the problem?

And finally, as a pure mathematician, I am always solving problems in situations that are so-called impossible – imaginary numbers, four dimensional space, projective space where there are no parallel lines, finite fields where 1+1 = -1 …

So in the end, I don’t think it matters that roosters don’t lay eggs!

19 August, 2013
Who tells you if you’re correct?

At our uni, the first year maths students do the majority of their assignments online using MapleTA, and this week MapleTA was having problems. As always happens with technology glitches, it was an absolute schemozzle. It was bad enough for students that it was intermitently not working at all, but what made it worse was that even when it was working, the “preview” and “how did I do” functions were both failing. This meant that students could not use the computer to check if they were right, and a lot of them were extremely distressed by this.

And this has confirmed something that had been bothering me for a while: many students are not learning the skill of telling if they’re right for themselves. I had suspected this to be true, considering how many students in our survey on “cheat” sheets mentioned how useful it was to have a way of checking they were correct, and also considering how many students in the MLC ask us the question “can you check if this is right?”. It seems that a lot of students need an authority outside themselves to tell them they are right, whether it be the computer, the cheat sheet, or the MLC staff.

It’s strange, but I remember as a student spending hours trying my assignment questions in multiple ways, or reproving a result to make sure it was really true and I really could use it in my assignment, or simply subbing my answers back into the original equation to see if they worked. And in our puzzle gathering, One Hundred Factorial, we often ask ourselves how we can tell if we’re right and sort out ways to be sure. It’s clear that at some point I learned to be sure I was right on my own.

When will our current students learn these to do this? Because right now they seem to be relying on others to tell them they’re right. Yet at some point there won’t be a higher authority, and they’re just going to have to know if they’re right for themselves – most imminently, during the exam!

I’m not suggesting going back to the frustrating days of not being able to check if your syntax is correct, or not being able to submit the assignment multiple times. These things encourage students to at least try and retry the questions, rather than see it as all too hard, and this has very positive learning benefits. But what this means is that it falls to us, their teachers, to encourage the confidence and skills to know they are right for themselves. We can always show them how to check their work, rather than check it for them when they ask us. We can always help them check their work on paper even before they put it into the computer. And we can keep building their confidence so they will be more independent, and won’t feel as strong a desire for someone outside to tell them they’re right.

This comment was left on the original blog post:

Carol Matthews 18 January 2013:
That’s a very good point David. I remember in my high school doing a mock maths exam at which we were permitted calculators which were new and, as it eventuated, gave wrong answers to a value of 0.10. What stunned the teachers was that the vast majority of us accepted the calculator’s answer, without question either not noticing the error or assuming that we, not the calculator, must be wrong.

17 January, 2013
Statistics and Insomnia

Some years ago, I saw a snippet on the ABC science show Catalyst about insomnia – in particular, the flavour of insomnia where a person has trouble falling asleep at all. They reported on a trial study investigating the effectiveness of a tortuous new treatment for chronic insomnia. (You can find the published research here: Click here to go to insomnia article .)

The usual way to cure insomnia is to retrain your brain and your body to associate the bed with sleep rather than wakefulness. What they recommend is to only go to sleep when you’re really really tired, and if you don’t fall asleep in quarter of an hour, to get up and go to some other room until you feel tired enough to go to sleep again. Eventually, you’ll fall asleep in bed. Then you try again tomorrow night, and the next night, and the next night… Usually it takes a month.

The big problem with it is that people just don’t have the stamina to put themselves through all this for four weeks. Here’s where the radical treatment comes in: you compress the month of practice into 24 hours. The poor participant is put in a windowless room and practises going to sleep, and when they finally do fall asleep, they only get four minutes to sleep before they are woken up to try and fall asleep again. In this way you fit a month’s worth of falling-to-sleep practice in one day. Imagine how desperate you would have to be to sign up for this sort of thing!

Recently, it occurred to me that there are a lot of other skills that take a lot of practice to learn and this practice is usually drawn out over such a long period that people just don’t get through it all. One of these is statistics – in particular, the process of deciding which statistical procedures should be used to analyse your data.

In your standard stats course, the approach to teaching students to make decisions is to get them to do a project. This gives them practice at making decisions a grand total of once. And so students need a whole degree’s worth of projects, and probably years of working as a statistician, to learn how to make decisions. Hence, very few people ever get very good at making them. It’s just like the poor insomniac trying to cure their own insomnia once a night.

But what if you could, like the new insomnia treatment, compress all that practice into a short amount of time? What if you could pick out just the part where you make the decision and get students to make a lot of decisions all at once? Then they might get the necessary experience rather more quickly than the standard approach.

I tried it out last year with the med students. I gave them a quick lecture about how you make the decision of which hypothesis test to use. Then, I gave them 30 research questions and got them to make a decision for each one. They seemed to get the idea of how it worked. So much so that they actually had intelligent questions to ask afterwards!

I’m trying again this year, only this time the Medical School is letting me help design the whole stats teaching program, not just one lecture. Here’s hoping that a little bit of torture for a short time can alleviate months of pain later…

Theis comment was left on the original blog post:

Richard Knowling 27 January 2012:
This is an awesome idea David! I only wish Mike Roberts had still been alive to hear about it!

27 January, 2012
Discounting your problem-solving

As I was leaving the other day, a student said that she would come to see us the next day to ask some questions about her assignment. She said she had tried to do as much of it herself as she could, and had only done 70% of it.

The “only” made me start – she had done most of it herself but that wasn’t good enough because she still had to ask for help. And somehow in the hurry of the moment, this came out of my mouth: I said, “And how much did you do on your own last time?”

It was her turn to start – “Oh!” she said, “A lot less I suppose.” And then I had to keep walking or risk missing my train. But as I walked the incident ran around my mind: it’s amazing how many people can discount the evidence of their own problem-solving ability, simply because they still need help.

I’ve seen it before, but I’ve never seen a way to fight against it. I’ve always tried to tell them that they can do it, and point out the bits they did do, but it always seemed to wash over them without leaving an impact. I’ve been focussing on a single moment of problem-solving.

What I’ve learned from my thirty second conversation is that perhaps I should help people focus on more than just today. Instead, maybe I can help them look at their journey so far and focus on the improvement.

But my student has given me even more: she’s given us something concrete to focus on: the amount they have done on their own. This is so much more tangible than “problem-solving ability” because it’s plain numerical data. The student can compare this time to last time and feel success as long as they’ve done that little bit more on their own.

12 April, 2011
Books in the 22nd Century

I’ve just read a book called “Written for Children” by John Rowe Townsend. It was published in 1974 and gives the history of writing for children (in English) up to that time. It was very interesting reading. What I’d like to comment on here is the final chapter, where the author talks about the future of books (p333 onwards):

The question that arises next is whether changes in the book world might be overtaken by technological developments which would make the book itself, or at any rate reading for pleasure, obsolete. … Myself, I have an instinctive faith in the ability of the book to keep going. It is a tough old bird, after all. People thought that the cinema and radio and television would kill it, but they have not done so yet. Perhaps it is not too wildly optimistic to hope that in the twenty-first century, when all the modern miracles and some we have not yet dreamed of have come to pass, a child will still be found here and there, lying face down on the hearthrug or whatever may be then have replaced the hearthrug, light years away from his surroundings, lost in the pages of a book.

It makes me happy to know that Mr Townsend’s vision did in fact come to pass and that children can still be found lost in a book even here in the twenty-first century. And it gives me hope that in the same way that the book was not killed by cinema or radio or television, that it will also survive the internet and the ipad.

And finally it makes me think of a parallel situation in mathematics. I have heard people say that the computer is forever changing the way mathematics is done. This is definitely true, but I don’t believe that the “old ways” will die. I believe that there is a certain joy that comes from doing something yourself, from scratching out a problem yourself on paper, from playing with symbols and pictures, from visualising things in your own mind, from dreaming about new ideas – a joy that is absent when the computer does things for you. So I hope that even in the 22nd century you’ll still see people sitting down with a pencil and paper scribbling as they try joyfully to solve a problem all on their own.

10 January, 2011