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A recent meme circulating on the internet mocked a US government programme (ObamaCare) saying that its introduction cost $360 million when there were only 317 million people in the entire country. It then posed the rhetorical question: "Why not just give everyone a million dollars instead?"

The post Why know any algebra? appeared first on OUPblog.

Try googling 'mathematical gem'. I just got 465,000 results. Quite a lot. Indeed, the metaphor of mathematical ideas as precious little gems is an old one, and it is well known to anyone with a zest for mathematics. A diamond is a little, fully transparent structure all of whose parts can be observed with awe from any angle.

The post Diamonds are forever, and so are mathematical truths? appeared first on OUPblog.

A friend of mine picked an argument with me the other day about how people go on about the beauty of mathematics, but this is not only not obvious to non-mathematicians, it cannot be accessed by those outside the field. Unlike, for example, the modern art, which is also not always obvious, mathematical beauty is elusive to all but the mathematicians. Or so he said.

The post Yes, maths can be for the amateur too appeared first on OUPblog.

What is the purpose of mathematics? Or, as many a pupil would ask the teacher on a daily basis: “When are we going to need this?” There is a considerably ruder version of a question posed by Billy Connolly on the internet, but let’s not go there.

The post Will we ever need maths after school? appeared first on OUPblog.

One of the central tasks when reading a mystery novel (or sitting on a jury) is figuring out which of the characters are trustworthy. Someone guilty will of course say they aren’t guilty, just like the innocent – the real question in these situations is whether we believe them.

The guilty party – let’s call her Annette – can try to convince us of her trustworthiness by only saying things that are true, insofar as such truthfulness doesn’t incriminate her (the old adage of making one’s lies as close to the truth as possible applies here). But this is not the only strategy available. In addition, Annette can attempt to deflect suspicion away from herself by questioning the trustworthiness of others – in short, she can say something like:

“I’m not a liar, Betty is!”

However, accusations of untrustworthiness of this sort are peculiar. The point of Annette’s pronouncement is to affirm her innocence, but such protestations rarely increase our overall level of trust. Either we don’t believe Annette, in which case our trust in Annette is likely to drop (without affecting how much we trust Betty), or we do believe Annette, in which case our trust in Betty is likely to decrease (without necessarily increasing our overall trust in Annette).

Thus, accusations of untrustworthiness tend to decrease the overall level of trust we place in those involved. But is this reflective of an actual increase in the number of lies told? In other words, does the logic of such accusations makes it the case that, the higher the number of accusations, the higher the number of characters that must be lying?

Consider a group of people G, and imagine that, simultaneously, each person in the group accuses one, some, or all of the other people in the group of lying right at this minute. For example, if our group consists of three people:

G = {Annette, Betty, Charlotte}

then Betty can make one of three distinct accusations:

“Annette is lying.”

“Charlotte is lying.”

“Both Annette and Charlotte are lying.”

Likewise, Annette and Charlotte each have three choices regarding their accusations. We can then ask which members of the group could be, or which must be, telling the truth, and which could be, or which must be, lying by examining the logical relations between the accusations made by each member of the group. For example, if Annette accuses both Betty and Charlotte of lying, then either (i) Annette is telling the truth, in which case both Betty and Charlotte’s accusations must be false, or (ii) Annette is lying, in which case either Betty is telling the truth or Charlotte is telling the truth (or both).

This set-up allows for cases that are paradoxical. If:

Annette says “Betty is lying.”

Betty says “Charlotte is lying.”

Charlotte says “Annette is lying.”

then there is no coherent way to assign the labels “liar” and “truth-teller” to the three in such a way as to make sense. Since we are here interested in investigating results regarding how many lies are told (rather than scenarios in which the notion of lying versus telling the truth breaks down), we shall restrict our attention to those groups, and their accusations, that are not paradoxical.

The following are two simple results that constraint the number of liars, and the number of truth-tellers, in any such group (I’ll provide proofs of these results in the comments after a few days).

Result 1: If, for some number m, each person in the group accuses at least m other people in the group of lying (and there is no paradox) then there are at least m liars in the group.

Result 2: If, for any two people in the group p₁ and p₂, either p₁ accuses p₂ of lying, or p₂ accuses p₁ of lying (and there is no paradox), then exactly one person in the group is telling the truth, and everyone else is lying.

These results support an affirmative answer to our question: Given a group of people, the more accusations of untrustworthiness (i.e., of lying) are made, the higher the minimum number of people in the group that must be lying. If there are enough accusations to guarantee that each person accuses at least n people, then there are at least n liars, and if there are enough to guarantee that there is an accusation between each pair of people, then all but one person is lying. (Exercise for the reader: show that there is no situation of this sort where everyone is lying).

Of course, the set-up just examined is extremely simple, and rather artificial. Conversations (or mystery novels, or court cases, etc.) in real life develop over time, involve all sorts of claims other than accusations, and can involve accusations of many different forms not included above, including:

“Everything Annette says is a lie!”

“Betty said something false yesterday!”

“What Charlotte is about to say is a lie!”

Nevertheless, with a bit more work (which I won’t do here) we can show that, the more accusations of untrustworthiness are made in a particular situation, the more of the claims made in that situation must be lies (of course, the details will depend both on the number of accusations and the kind of accusations). Thus, it’s as the title says: accusation breeds guilt!

Note: The inspiration for this blog post, as well as the phrase “Accusation breeds guilt” comes from a brief discussion of this phenomenon – in particular, of ‘Result 2′ above – in ‘Propositional Discourse Logic’, by S. Dyrkolbotn & M. Walicki, Synthese 191: 863 – 899.

The post Accusation breeds guilt appeared first on OUPblog.

        

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Are you worried about catching the flu, or perhaps even Ebola? Just how worried should you be? Well, that depends on how fast a disease will spread over social and transportation networks, so it’s obviously important to obtain good estimates of the speed of disease transmission and to figure out good containment strategies to combat disease spread.

Diseases, rumors, memes, and other information all spread over networks. A lot of research has explored the effects of network structure on such spreading. Unfortunately, most of this research has a major issue: it considers networks that are not realistic enough, and this can lead to incorrect predictions of transmission speeds, which people are most important in a network, and so on. So how does one address this problem?

Traditionally, most studies of propagation on networks assume a very simple network structure that is static and only includes one type of connection between people. By contrast, real networks change in time — one contacts different people during weekdays and on weekends, one (hopefully) stays home when one is sick, new University students arrive from all parts of the world every autumn to settle into new cities. They also include multiple types of social ties (Facebook, Twitter, and – gasp – even face-to-face friendships), multiple modes of transportation, and so on. That is, we consume and communicate information through all sorts of channels. To consider a network with only one type of social tie ignores these facts and can potentially lead to incorrect predictions of which memes go viral and how fast information spreads. It also fails to allow differentiation between people who are important in one medium from people who are important in a different medium (or across multiple media). In fact, most real networks include a far richer “multilayer” structure. Collapsing such structures to obtain and then study a simpler network representation can yield incorrect answers for how fast diseases or ideas spread, the robustness level of infrastructures, how long it takes for interaction oscillators to synchronize, and more.

Recently, an increasingly large number of researchers are studying mathematical objects called “multilayer networks”. These generalize ordinary networks and allow one to incorporate time-dependence, multiple modes of connection, and other complexities. Work on multilayer networks dates back many decades in fields like sociology and engineering, and of course it is well-known that networks don’t exist in isolation but rather are coupled to other networks. The last few years have seen a rapid explosion of new theoretical tools to study multilayer networks.

And what types of things do researchers need to figure out? For one thing, it is known that multilayer structures induce correlations that are invisible if one collapses multilayer networks into simpler representations, so it is essential to figure out when and by how much such correlations increase or decrease the propagation of diseases and information, how they change the ability of oscillators to synchronize, and so on. From the standpoint of theory, it is necessary to develop better methods to measure multilayer structures, as a large majority of the tools that have been used thus far to study multilayer networks are mostly just more complicated versions of existing diagnostic and models. We need to do better. It is also necessary to systematically examine the effects of multilayer structures, such as correlations between different layers (e.g., perhaps a person who is important for the social network that is encapsulated in one layer also tends to be important in other layers?), on different types of dynamical processes. In these efforts, it is crucial to consider not only simplistic (“toy”) models — as in most of the work on multilayer networks thus far — but to move the field towards the examination of ever more realistic and diverse models and to estimate the parameters of these models from empirical data. As our review article illustrates, multilayer networks are both exciting and important to study, but the increasingly large community that is studying them still has a long way to go. We hope that our article will help steer these efforts, which promise to be very fruitful.

The post What do rumors, diseases, and memes have in common? appeared first on OUPblog.

        

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School is back in full swing now, and I’ve returned to my storytelling sessions at J’s school on a Friday afternoon, where I get to read stories and play and craft in what is termed “Golden Time”. It’s a brilliant way to round off the week (definitely much better than the Triple Latin I once had), and it means I’m always on the lookout for picture books which not only lend themselves to creative play, but which also work exceptionally well as class read-alouds.

Big! by Tim Hopgood struck me as one such book the moment I first read it. And given that it is all about growing up, and thinking about being bigger, it was a natural choice for the start of the school year, where all the children have moved up a class and are enjoying being that much “bigger” than they were last year.

What does it mean to be big? And when, exactly, do you become big? Such existential questions are really quite important in young kids’ lives: When will they be big enough to play on your phone? When will they be big enough to have a new bike? When will they be big enough to stay up as late as their older brother or sister? Certainly, J – being the youngest in our home – asks these sorts of questions very often indeed, and finds it very frustrating that she is not yet as big as she would like to be.

And so it was no surprise that she lapped up Hopgood’s observant and giggle-inducing take on being big. Being big partly depends on what you compare it with. Compare yourself to a piece of popcorn and you’re massive! And compare your big sister with a bear, and even she will appear to be tiny

Image: Tim Hopgood. Used with permission.

Hopgood effectively cobines lots of bold blocks of solid colour (there are no white pages anywhere) with visual texture, and draws his questioning boy with such apparent simplicity that it could have been drawn by a child (think Charlie and Lola, and you’ll have the right sort of idea); all this adds further appeal for young readers and listeners. Use of a variety of font sizes lends the book to very expressive reading-aloud – great for groups, but also for young children reading this to themselves.

Full of reassurance about one of life’s BIGGEST questions, Tim Hopgood has created another hit I can warmly recommend.

To go along with reading Big! all the kids in my group at school got to make their own growth chart, using paper measuring tapes stuck onto long lengths of fax paper (used for its convenient width). We talked about tall things which we might draw onto our charts (giraffes, beanstalks, blocks of flats and so on) and then the kids had free rein to decorate their charts how they saw fit. Here are my girls creating their own charts at home:

At school there were two other activities kids could choose to take part in; building the tallest tower they could out of a variety of building blocks, and measuring each other with popcorn (mirroring a suggestion in Hopgood’s book).

I taped a large sheet on the floor of the classroom and kids worked in pairs, whilst one lay down and the other lined up popcorn to see how many pieces of popcorn high they were. This was an incredibly popular activity (especially when I lay down and the kids got to measure me), and was worth every bit of the rather large amount of mess it made!

Whilst making our growth charts at home we listened to:

Other activities which would work well alongside reading Big! include:

Do you have a favourite book about growing up?

If you’d like to make growth charts with your class at school, I do have some spare paper tapes (150cm long, marked in both inches and cm); I’d be happy to post them to you (anywhere in the world), with the proviso that they’re for group use (I don’t want to post fewer than 20 in a go, because they are very difficult to pack!). Let me know, and the first 3 people to contact me will get the tapes!

Disclosure: I received a free review copy of Big! from the author.

By Lara Alcock

Two contrasting experiences stick in mind from my first year at university.

First, I spent a lot of time in lectures that I did not understand. I don’t mean lectures in which I got the general gist but didn’t quite follow the technical details. I mean lectures in which I understood not one thing from the beginning to the end. I still went to all the lectures and wrote everything down – I was a dutiful sort of student – but this was hardly the ideal learning experience.

Second, at the end of the year, I was awarded first class marks. The best thing about this was that later that evening, a friend came up to me in the bar and said, “Hey Lara, I hear you got a first!” and I was rapidly surrounded by other friends offering enthusiastic congratulations. This was a revelation. I had attended the kind of school at which students who did well were derided rather than congratulated. I was delighted to find myself in a place where success was celebrated.

Looking back, I think that the interesting thing about these two experiences is the relationship between the two. How could I have done so well when I understood so little of so many lectures?

I don’t think that there was a problem with me. I didn’t come out at the very top, but obviously I had the ability and dedication to get to grips with the mathematics. Nor do I think that there was a problem with the lecturers. Like the vast majority of the mathematicians I have met since, my lecturers cared about their courses and put considerable effort into giving a logically coherent presentation. Not all were natural entertainers, but there was nothing fundamentally wrong with their teaching.

I now think that the problems were more subtle, and related to two issues in particular.

First, there was a communication gap: the lecturers and I did not understand mathematics in the same way. Mathematicians understand mathematics as a network of axioms, definitions, examples, algorithms, theorems, proofs, and applications.  They present and explain these, hoping that students will appreciate the logic of the ideas and will think about the ways in which they can be combined. I didn’t really know how to learn effectively from lectures on abstract material, and research indicates that I was pretty typical in this respect.

Students arrive at university with a set of expectations about what it means to ‘do mathematics’ – about what kind of information teachers will provide and about what students are supposed to do with it. Some of these expectations work well at school but not at university. Many students need to learn, for instance, to treat definitions as stipulative rather than descriptive, to generate and check their own examples, to interpret logical language in a strict, mathematical way rather than a more flexible, context-influenced way, and to infer logical relationships within and across mathematical proofs. These things are expected, but often they are not explicitly taught.

My second problem was that I didn’t have very good study skills. I wasn’t terrible – I wasn’t lazy, or arrogant, or easily distracted, or unwilling to put in the hours. But I wasn’t very effective in deciding how to spend my study time. In fact, I don’t remember making many conscious decisions about it at all. I would try a question, find it difficult, stare out of the window, become worried, attempt to study some section of my lecture notes instead, fail at that too, and end up discouraged. Again, many students are like this. I have met a few who probably should have postponed university until they were ready to exercise some self-discipline, but most do want to learn.

What they lack is a set of strategies for managing their learning – for deciding how to distribute their time when no-one is checking what they’ve done from one class to the next, and for maintaining momentum when things get difficult. Many could improve their effectiveness by doing simple things like systematically prioritizing study tasks, and developing a routine in which they study particular subjects in particular gaps between lectures.  Again, the responsibility for learning these skills lies primarily with the student.

Personally, I never got to a point where I understood every lecture. But I learned how to make sense of abstract material, I developed strategies for studying effectively, and I maintained my first class marks. What I would now say to current students is this: take charge. Find out what lecturers and tutors are expecting, and take opportunities to learn about good study habits. Students who do that should find, like I did, that undergraduate mathematics is challenging, but a pleasure to learn.

Lara Alcock is a Senior Lecturer in the Mathematics Education Centre at Loughborough University. She has taught both mathematics and mathematics education to undergraduates and postgraduates in the UK and the US. She conducts research on the ways in which undergraduates and mathematicians learn and think about mathematics, and she was recently awarded the Selden Prize for Research in Undergraduate Mathematics Education. She is the author of How to Study for a Mathematics Degree (2012, UK) and How to Study as a Mathematics Major (2013, US).

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Image credit: Screenshot of Oxford English Dictionary definition of mathematics, n., via OED Online. All rights reserved.

The post Memories of undergraduate mathematics appeared first on OUPblog.

Today I’m hosting STEM Friday, over at the STEM Friday blog. I’ve a review of a super, literally deliciously illustrated book for inspiring a love of… fractions! Do head on over to the STEM Friday blog to find out what it is, and if you’ve written a review of a STEM (Science, Technology, Engineering, Maths) book for children, please leave a link to your review over there, so it’s easy for anyone interested in in STEM books to find them all in one place.

As to what my mystery book inspired us to get up to… here are some photos:

Whilst we made our Edible Book version of my mystery book we listened to:

Neither are about fractions or division (both will get your toes tapping though), nor is the next song I had on, but I couldn’t resist:

Finally, a book which ISN’T my mystery book, but which is a fun read alongside the book I’m reviewing on the STEM Friday blog is The Doorbell Rang by Pat Hutchins – the perfect excuse for baking LOTS of biscuits and doing even more maths…

Join STEM Friday!

We invite you to join us!

• Write about STEM each Friday on your blog.
• Copy the STEM Friday button to use in your blog post.

It’s STEM Friday! (STEM is Science, Technology, Engineering, and Mathematics)

• Link your post to the comments of our weekly STEM Friday Round-up. (Please use the link to your STEM Friday post, not the address of your blog. Thanks!)

And do go and see what book gave us such a good excuse to bake cake!