Oxford handbook of numerical cognition

Promoting Maths to the General Public Oxford Handbooks Online Promoting Maths to the General Public Chris J Budd The Oxford Handbook of Numerical Cognition (Forthcoming) Edited by Roi Cohen Kadosh and Ann Dowker Online Publication Date: Feb 2015 Subject: Psychology, Cognitive Psychology, Educational Psychology DOI: 10.1093/oxfordhb/9780199642342.013.47 Abstract and Keywords In this chapter I will address the issue that whilst mathematics is vital to all of our lives, playing a vital role in modern technology and even helping us to understand the brain, it is often perceived to be a dry, boring, and useless subject The chapter will explore various ways that mathematics can be presented to the general public in a way that makes it seem to be exciting and relevant, and captures its essence without dumbing it down In particular it will look at strategies that have been shown to work well with the public including the use of careful real life examples and relating maths to people, hands on maths at science fairs, and maths in the media and on the Internet The chapter includes some case studies of what does and does not work in the field of maths communication Keywords: Mathematics, popularization, formulae, science fairs, masterclasses What’s it all about? Mathematics is all around us, it plays a vital role in much of modern technology from Google to the Internet and from space travel to the mobile phone It is central to every school student’s education, and anyone needing to get a mortgage, buy a car, sort out their household bills, or just understand the vast amount of information now thrown at them, needs to know some maths Maths is even used to help us understand, and image, the complex networks and patterns in the brain and many of the processes of perception However, like the air around us, the importance of mathematics is often invisible and poorly understood, and as a result many people are left unaware of the vital role that it could, and does, play in their lives In an increasingly technology and information driven world this is potentially a major problem However, we have to be honest, mathematics and its relevance, is a difficult subject to communicate to the general public It certainly doesn’t have the instant appeal of sex and violence that we find in other areas (although it does have applications to these) and there is a proud cultural tradition in the UK that it is good to be bad at maths For example when I appeared once on the One Show, both presenters were very keen to tell me that they were rubbish at maths and that it didn’t seem to have done them any harm! (I do wonder whether they would have said the same to a famous author, artist, or actor.) Maths is also perceived as a dry subject without any applications (this is also very untrue and I will discuss this later) and this perception does put a lot of school students (and indeed their teachers) off Finally, and (perhaps this is what makes it especially hard to communicate), maths is a linear subject, and a lot of background knowledge, and indeed investment of time, is required of any audience to whom you might want to communicate its beauty and effectiveness For example, one of the most important way that maths affects all of our lives is through the application of the methods of calculus But very few people have heard of calculus, and those that have are generally scared by the very name It also takes time and energy to communicate maths well and (to be honest), most mathematicians are not born communicators (in fact rather the opposite) However, it Page of 12 Promoting Maths to the General Public is a pleasure to say that there are some gifted maths communicators out there who are making a very positive impact, as well as university courses teaching maths communication skills Indeed, the popularization of mathematics has become an increasingly respectable and widespread activity, and I will describe some of this work in this chapter So why do we bother communicating maths in the first place, and what we hope to achieve when we attempt to communicate maths to any audience, whether it is a primary school class, bouncing off the walls with enthusiasm, or a bored class of teenagers on the last lesson of the afternoon? Well, the reason is that maths is insanely important to everyone’s lives whether they realize it directly (for example through trying to understand what a mortgage percentage on an APR actually means) or indirectly through the vital role that maths plays in the Internet, Google, and mobile phones to name only three technologies that rely on maths Modern technology is an increasingly mathematical technology and unless we inspire the next generation then we will rapidly fall behind our competitors.1 However, when communicating maths we always have to tread a narrow line between boring our audience with technicalities at one end, and watering maths down to the extent of dumbing down the message at the other Ideally, in communicating maths we want to get the message across that maths is important, fun, beautiful, powerful, challenging, all around us and central to civilization, to entertain and inspire our audience and to leave the audience wanting to learn more maths (and more about maths) in the future, and not to be put off it for life Rather than dumbing down maths, public engagement should be about making mathematics come alive to people This is certainly a tall order, but is it possible? While the answer is certainly YES, there are a number of pitfalls to trap the unwary along the way In this chapter I will explore some of the reasons that maths has a bad image and/or is difficult to communicate to the general public I will then discuss some general techniques which have worked for myself, and others, in the context of communicating maths to a general audience I will then go on to describe some initiatives which are currently under way to do this Finally I will give some case studies of what works and what does not What’s the problem with maths? Let’s be honest, we do have a problem in conveying the joy and beauty of mathematics to a lay audience, and maths has a terrible popular image A lot of important maths is built on concepts well beyond what a general audience has studied Also mathematical notation can be completely baffling, even for other mathematicians working in a different field Here for example is a short quote from a paper, authored by myself, about the equations describing the (on the face of it very interesting) mathematics related to how things combust and then explode: Let − Δϕ = f (ϕ) A weak solution of this PDF satisfies the identity ∫ ∇ϕ∇ψdx = ∫ f (ϕ) ψdx ∀ψ ∈ H01 (Ω) Assume that f (ϕ) grows sub − critically it is clear from Sobolev embedding that ∃ϕ ∈ H01 This quote is meaningless to any other than a highly specialist audience Trying to talk about (say in this example) the detailed theory and processes involved in solving differential equations with an audience which (in general) doesn’t know any calculus, is a waste of everyone’s time and energy As a result it is extremely easy to kill off even a quite knowledgeable audience when giving a maths presentation or even talking about maths in general The same problem extends to all levels of society Maths is perceived by the greater majority of the country as a boring, uncreative, irrelevant subject, only for (white, male) geeks All mathematicians know this to be untrue Maths is an extraordinarily creative subject, with mathematical ideas taking us well beyond our imagination It is also a subject with limitless applications without which the modern world would simply not function Not being able to do maths (or at least being numerate) costs the UK an estimated £2.4B every year according to a recent Confederation of British Industry report (CBI Report, 2010) Uniquely amongst all (abstract) subjects, mathematicians and mathematics teachers are asked to justify why their subject is useful Not only is this unfair (why is maths asked to justify itself in this way, and not music or history), it is also ridiculous given that without maths the world would starve, we would have no mobile phones and the Internet would not function I have thought very hard about why the popular image and perception of maths is so different from reality and why it is culturally fine to say that you are bad at maths There are many possible reasons for this Page of 12 Promoting Maths to the General Public Firstly, the obvious Maths is really hard, and not everyone can do it Fair enough However, so is learning a foreign language or taking a free kick, or playing a musical instrument, and none of these carry the same stigma that maths does Secondly, maths is often taught in a very abstract way at school with little emphasis on its extraordinary range of applications This can easily turn an average student off ‘what’s the use of this Miss’ is an often heard question to teachers Don’t get me wrong, I’m all in favor of maths being taught as an abstract subject in its own right It is the abstractness of maths that underlies its real power, and even quite young students can be captivated by the puzzles and patterns in maths However, I am also strongly in favor of all teaching of maths being infused with examples and applications Mathematicians often go much too far in glorifying in the ‘uselessness’ of their subject (witness the often quoted remarks by Hardy in ‘A Mathematicians Apology’ (Hardy 1940) see for example his concluding remark in that book,2 which was certainly not true, given Hardy’s huge impact on many fields of science) However this is sheer nonsense Nothing in maths is ever useless I think that it is the duty of all mathematicians to understand, and convey, the importance and applications of the subject to as broad an audience as possible, and to teachers in particular Thirdly, we have structural problems in the way that we teach maths in English schools (less so in Scotland) Most UK students give up maths at the age of fifteen or sixteen and never see it again These students include future leaders in government and in the media What makes this worse is that the huge majority of primary school teachers also fall into this category The result is that primary level maths is taught by teachers who are often not very confident in it themselves, and who certainly cannot challenge the brightest pupils They certainly cannot appreciate its creative and useful aspects (Indeed when I was at primary school in the 1960s maths lessons were actually banned by the headmistress as ‘not being creative’.) Students at school are thus being put off maths far too early, and are given no incentive to take it on past GCSE Even scientists (such as psychologists!) who need mathematics (and especially statistics) are giving up maths far too early Perhaps most seriously of all, those in government or positions of power, may themselves have had no exposure to maths after the age of 15, and indeed there is a woeful lack of MPs with any form of scientific training How are these policy makers then able to cope with the complex mathematical issues which arise (for example) in the problems associated with climate change (see the example at the end of this chapter) We urgently need to rectify this situation, and the solution is for every student to study some form of maths up to the age of 18, with different pathways for students with different abilities and motivation (See the Report on Mathematical Pathways post 16 (ACME Report 2011 and also Vorderman 2011) Finally, and I know that this is a soft and obvious target, but I really do blame the media With notable (and glorious) exceptions, maths hardly ever makes it onto TV, the radio, or the papers When it does it is often either extremely wrong (such as the report in the Daily Express about the chance of getting six double-yoked eggs in one box) or it is treated as a complete joke (the local TV reports of the huge International Conference in Industrial and Applied Mathematics at Vancouver in 2011 are a good example of this, see ) Sadly this type of report is the rule rather than the exception, or is given such little airtime that if you blink then you miss it Contrast this with the acres of time given to the arts or even to natural history, and the reverence that is given to a famous author when they appear on the media Part of this can be explained by the ignorance of the reporters (again a feature of the stopping of mathematics at the age of 15), but nothing I feel can excuse the antagonistic way in which reporters treat both mathematicians and mathematics I have often been faced by an interviewer who has said that they couldn’t do maths when they were at school, or they never use maths in real life, and that they have done really well To which my answers are that they are not at school anymore and that if they can understand their mortgage or inflation or APR without maths then they are doing well Worst of all are those journalists that ask you tough mental arithmetic questions live on air to make you look a fool (believe me your mind turns to jelly in this situation) It is clearly vital to work with the media (see later), but the media also needs to put its own house in order to undo the damage that it has done to the public’s perception of mathematics How can maths be given a better image? As with all things there is no one solution to the problem of how to communicate to the broader public that maths Page of 12 Promoting Maths to the General Public isn’t the irrelevant and scary monster that they (and the media) often make it out to be Many different maths presenters have adopted different (and equally successful) styles However some techniques that I have found to have worked with many audiences (both young and old) include the following • Starting with an application of maths relevant to the lives of the audience, for example Google, iPods, crime fighting, music, code breaking, dancing (yes, dancing) Hook them with this and then show, and develop, the maths involved (such as in the examples above, network theory, matrix theory, and group theory) Science presenters can often be accused of ‘dumbing down’ their subject, and it is certainly true that it is impossible to present higher level maths to a general audience for the reasons discussed above However, a good application can often lead to many fascinating mathematical investigations • Being proud not defensive of the subject Maths really DOES make a difference to the world If mathematicians can’t be proud and passionate of it then who will be? Be very positive when asked by any interviewer ‘what’s the point of that’ • Showing the audience the surprise and wonder of mathematics It is the counter-intuitive side of maths, often found in puzzles or ‘tricks’, that often grabs attention, and can be used to reveal some of the beauty of maths The public loves puzzles, witness the success of Sudoku, and many of these (such as Griddler, Killer Sudoku, and problems in code breaking) have a strong mathematical basis (Those that say that Sudoku has nothing to do with maths simply don’t understand what maths really is all about!) There are also many links between maths and magic (as we shall see later); many good magic tricks are based on theorems (such as fixed point theorems in card shuffling and number theorems in mind-reading tricks) Indeed a good mathematical theorem itself has many of the aspects of a magic trick about it, in that it is amazing, surprising, remarkable, and when the proof is revealed, you become part of the magic too • Linking maths to real people Many of our potential audiences think that maths either comes out of a book, or was carved in stone somewhere Nothing could be further from the truth One of the problems with the image of maths in the eyes of the general public is that it does not seem to connect to people Indeed a recent letter in Oxford Today () the Oxford alumni magazine (which really should have known better!) said that the humanities were about people and that science was about things (and that as a consequence the humanities were more important) What rubbish! All maths at some point was created by a real person, often with a lot of emotional struggle involved or with argument and passion No one who has seen Andrew Wiles overcome with emotion at the start of the BBC film Fermat’s Last Theorem produced by Simon Singh and described in his wonderful book (Singh 1997), can fail to be moved when he describes the moment that he completed his proof Also stories such as the life and violent death of Galois, the recent solution of the Poincare Conjecture by a brilliant, but very secretive Russian mathematician, or even the famous punch up surrounding the solution of the cubic equation or the factorization of matrices on a computer, cannot fail to move even the most stony-faced of audiences • Not being afraid to show your audience a real equation Stephen Hawking famously claimed that the value of a maths book diminishes with every formula This is partly true as my earlier example showed There are, however, many exceptions to this Even an audience that lacks mathematical training can appreciate the elegance of a formula that can convey big ideas so concisely Some formulae indeed have an eternal quality that very few other aspects of human endeavor can ever achieve Mind you, it may be a good idea to warn your audience in advance that a formula is coming so that they can brace themselves So here goes: π 1 1 1 =1− + − + − + −… 11 13 Isn’t that sheer magic You can easily spend an entire lecture, or popular article, talking about that formula alone If I am ever asked to ‘define mathematics’ then that is my answer Anyone who does not appreciate that formula simply has no soul! You can find out more in my article (Budd 2013) Whole (and bestselling) books (Nahin 2006) have been written on arguably the most important and beautiful formula of all time eiπ = −1 which was discovered by Euler and lies behind the technology of the mobile phone and also the electricity supply industry For more fabulous formulae see the book 17 Equations that Changed the World, by Ian Stewart (2012) Page of 12 Promoting Maths to the General Public • Above all, be extremely enthusiastic If you enjoy yourself then there is a good chance that your audience will too So, what’s going on? As I said earlier, we have seen a rapid increase in the amount of work being done to popularize maths Partly this is a direct result of the realization that we do need to justify the amount of money being spent on maths, and to increase the number of students both studying maths and also using it in their working lives I also like to think that more people are popularizing maths because it is an exciting thing to do which brings its own rewards, in much the same way that playing an instrument or acting in a play does Maths communication activities range from high profile work with the media, to writing books and articles, running web-based activities, public lectures, engaging with schools, busking, stand up events, outreach by undergraduates, and science fairs In all these activities we are trying to reach three groups; young people, the general public, and those who control the purse strings The Media As I described above, the media is a very hard nut to crack, with a lot of resistance to putting good maths in the spotlight However, having said that we are very fortunate to have a number of high profile mathematicians currently working with the media in general and TV/radio in particular Of these I mention in particular Ian Stewart, Simon Singh, Matt Parker, Marcus du Sautoy, and Sir David Spiegelhalter, but there are many others The recent BBC4 series by Marcus du Sautoy on the history of maths was a triumph and hopefully the DVD version of this will end up in many schools) and we mustn’t also forget the pioneering work of Sir Christopher Zeeman and Robin Wilson Marcus du Sautoy, Matt, and Steve Humble (aka Dr Maths) also show us all how it can be done, by writing regular columns for the newspapers It is hard to underestimate the impact of this media work, with its ability to reach millions, although it is a long way to go before maths is as popular in the media as cooking, gardening and even archaeology Popular Books Ian Stewart, Robin Wilson, Simon Singh, and Marcus du Sautoy are also well-known for their popular maths books and are in excellent company with John Barrow, David Acheson, and Rob Eastaway, but I think the most ‘popular’ maths author by quite a wide margin is Kjartan Poskitt If you haven’t read any of his Murderous Maths series then do so They are obstensively aimed at relatively young people and are full of cartoons, but every time I read them I learn something new Certainly my son has learnt (and become very enthusiastic about maths) from devouring many of these books The Internet Mathematics, as a highly visual subject, is very well-suited to being presented on the Internet and this gives us a very powerful tool for not only bringing maths into peoples homes but also being able to have a dialogue between them and experienced mathematicians via blog sites and social media The (Cambridge-based) Mathematics Millennium Project (the MMP) has produced a truly wonderful set of Internet resources through the NRICH and PLUS websites and the STIMULUS interactive project Do have a look at these if you have time I have personally found the PLUS website to be a really fantastic way of publishing popular articles which reach a very large audience The Combined mathematical Societies (CMS) have also set up the Maths Careers website, , showcasing the careers available to mathematicians I mustn’t also forget the very popular Cipher Challenge website run by the University of Southampton Direct engagement with the public There is no substitute for going into schools or engaging directly with the public A number of mechanisms exist to link professional mathematicians to schools, of which the most prominent are the Royal Institution Mathematics Masterclasses I am biased here, as I am the chair of maths at the Royal Institution, but the masterclasses have an enormous impact Every week many schools in over 50 regions around the country will send young people to take Page of 12 Promoting Maths to the General Public part in Saturday morning masterclasses on topics as various as the maths of deep sea diving to the Fibbonacci sequence These masterclasses are often run (and are based in) the university local to the region and are a really good way for university staff to engage with young people Of course it is impossible to get to every school in the country and it is much more efficient to bring lots of schools to really good events One way to do this is through the LMS Popular Lectures, the Training Partnership Lectures, and the Maths Inspiration series () The latter (of which I’m proud to be a part) are run by Rob Eastaway and deliver maths lectures in a theatre setting, often with a very interactive question and answer session A recent development has been the growth of ‘Maths Busking’ () This is really busking where maths itself is the gimmick and reaches out to a new audience who would otherwise not engage with mathematics or mathematicians Closely related are various stand up shows linked to maths such as the Festival of the Spoken Nerd or Your Days are Numbered These link maths to comedy and reach out to a very non-traditional maths audience, appearing, for example, at the Edinburgh Fringe Science fairs are a popular way of communicating science to the public Examples range from the huge, such as the British Science Association annual festival, the Big Bang Fair, and the Cheltenham festival of science, to smaller local activities such as Bath Taps Into Science and Maths in the Malls (Newcastle) I visit and take part in a lot of science fairs and it is fair to say that in general maths has traditionally been very much under-represented Amongst the vast number of talks/shows on biology, astronomy, archaeology, and psychology you may be lucky to find one talk on maths The problems we referred to earlier of a resistance to communicate maths in the media often seem to extend to science communicators as well Fortunately things are improving, and the maths section of the British Science Association has in recent years been very active in ensuring that the annual festival of the BSA has a strong maths presence Similarly, the maths contribution towards the Big Bang has grown significantly, with the IMA running large events since 2011, attended by approximately 50 000 participants Hopefully mathematics will have a similar high profile presence at future such events Indeed 2014 marks the launch of the very first Festival of Mathematics in the UK A related topic is the presence of mathematics exhibits in science museums It is sad to say that the maths gallery in the Science Museum, London, is very old and is far from satisfactory as an exhibition of modern mathematics Fortunately it is now in a process of redesign Similarly the greater majority of exhibits in science museums around the UK have no maths in them at all There seems to be a surprising reluctance from museum organizers to include maths in their exhibits However, our experience of putting maths into science fairs shows that maths can be presented in an exciting and hands on way, well-suited to a museum exhibition It is certainly much cheaper to display maths than most other examples of STEM (Science, Technology, and Mathematics) disciplines The situation is rather better in Germany where they have the ‘Mathemtikum’ () which contains many hands-on maths exhibits as well as organizing popular maths lectures, and in New York with the Museum of Maths Plans are underway to create ‘MathsWorld UK’ which will be a UK-based museum of maths Maths Communicators Finally, my favorite form of outreach are ambassador schemes in which undergraduates go into the community to talk about mathematics They can do this for degree credit (as in the Undergraduate Ambassador Scheme () or the Bath ‘Maths Communicators’ scheme), for payment as in the Student Associate Scheme, or they can act as volunteers such as in the Cambridge STIMULUS programme which encourages undergraduates to work with school students through the Internet The undergraduates can be mainly based in schools, or can have a broader spectrum of activities Whatever the mechanism Student Ambassador Schemes have been identified as one of the most effective activities in terms of Widening Participation and Outreach They combine the enthusiasm and creative brilliance of the pool of maths undergraduates that we have in the UK, with the very need no only to communicate maths but to teach these undergraduates communication skills which will be invaluable for their subsequent careers Everybody wins in this arrangement The students often describe these courses as the best thing that they do in the degree, and they create a lasting legacy of resources and a lasting impression amongst the young people and general public who they work with The recent IMA report on Maths Student ambassador Case studies () gives details on a number of these schemes What doesn’t work Page of 12 Promoting Maths to the General Public I repeat the fact that maths can remain hard to communicate, and it is very easy to fall into a number of traps For the sake of a balanced chapter (and to warn the unwary) here are a few examples of these Too much or too little We have already seen an example of where too much maths in a talk can blow your audience away It is incredibly easy to be too technical in a talk, to assume too much knowledge and to fail to define your notation We’ve all been there, either on the giving or the receiving end The key to what level of mathematics to include is to find out about your audience in advance In the case of school audiences this is relatively easy—knowing the year group and whether you are talking to top or bottom sets should give you a good idea of how much maths they are likely to know Yet too often I have seen speakers standing in front of a mixed GCSE group talking about topics like dot products and differentiation and assuming that these concepts will be familiar It is equally dangerous to put in too little maths and to water down the mathematical content so that it becomes completely invisible, or (as is often the case) to talk only about arithmetic and to miss out maths all together With a few notable exceptions, most producers and presenters in the media, think that any maths is too much maths and that their audience cannot be expected to cope with it at all But this only highlights the real challenge of presenting maths in the media where time and production constraints make it very hard indeed to present a mathematical argument In his Royal Institution Christmas Lectures in 1978, Prof Christopher Zeeman spent 12 minutes proving that the square root of two was irrational It is hard to think of any mainstream prime time broadcast today where a mathematical idea could be investigated in such depth A couple of minutes would probably be the limit, far too short a time to build a proof Perhaps at some point in the future this will change, but for the time being, maths communicators have to accept that television is a very limited medium for dealing with many accessible mathematical ideas The curse of the ‘formula’ As I have said, one of the ways of engaging audiences in maths is by relating it to everyday life and done correctly this can be very effective This can, however, be taken too far Taking a topic that is of general interest—romance, for example—and attempting to ‘mathematize’ it in the hope that the interest of the topic will rub off on the maths, can backfire badly Much of the maths that gets reported in the press is like this Although we love the use of formulae when they are relevant, the use of irrelevant formulae in a talk or an article can make maths appear trivial For example, I was once rung up by the press just before Christmas and asked for the ‘formula for the best way to stack a fridge for the Christmas Dinner’ The correct answer to this question is that there is no such formula, and an even better answer is that if anyone was able to come up with one they would (by the process of solving the NP-hard Knapsack problem) pocket $1 000 000 from the Clay Foundation However the journalist concerned seemed disappointed with the answer No such reluctance however got in the way of the person that came up with K= F (T + C) − L S Which is apparently the formula for the perfect kiss All I can say is: whatever you do, don’t drop your brackets For the mathematician collaborating with the press this might seem like a great opportunity to get maths into the public eye To the journalist and the reading public, however, more often it is simply a chance to demonstrate the irrelevance of the work done by ‘boffins’ Such things are best avoided And what does work I will conclude this chapter with some examples of topics that contain higher level of maths in them than might be anticipated and communicate maths in a very effective way More examples of case studies can be found in my article (Budd and Eastaway 2010), or on my website , or on the Plus maths website Example 1 Asperger’s Syndrome In the book The Curious Incident of the Dog in the Night-time by Mark Haddon (2004) the reader was invited to find an example of a right-angled triangle in which its sides could not be written in the form n2 + 1, n2 - 1 and 2n Page of 12 Promoting Maths to the General Public (where n > 1) On the face of it this was quite a high level of mathematics for a popular book (which has now been turned into a play) The Curious Incident is a book about Asperger’s Syndrome, written from a personal perspective Millions of people have read this book, and many of these (who are not in any sense mathematicians) have read this part of it and have actually enjoyed, and learned something, from this The reason this worked was twofold First, the maths was put into the context of a human story, which made it easier for the reader to empathize with it The second was that the author used a clever device whereby he allowed the lead character to speak for maths, while his friend spoke for the baffled unmathematical reader As a result, Haddon (a keen mathematician) managed to sneak a lot of maths into the book without coming across as a geek himself Example 2 Maths Magic Everyone (well nearly everyone) likes the mystery and surprise that is associated with magic To a mathematician, mathematics has the same qualities, but they are less well appreciated by the general public One way to bring them together is to devise magic tricks based on maths I have already alluded to some of these The general idea is to translate some amazing mathematical theorem into a situation which everyone can appreciate and enjoy These may involve cards, or ropes, or even mind-reading As an example, it is a well-known theorem that if any number is multiplied by nine, then the sum of the digits of the answer is itself a multiple of nine Similarly, if you take any number and subtract from it the sum of its digits then you get a multiple of nine Put like this these results sound rather boring, but in the context of a magic trick they are wonderful ambassadors for mathematics The first leads to a lovely mind-reading trick Ask your audience to think of a whole number between one and nine and then multiply it by nine They should then sum the digits and subtract five from their answer If they have a one they should think A, two think B, three think C, etc Now take the letter they have and think of a country beginning with that letter Take the last letter of that country and think of an animal beginning with that letter Now take the last letter of the animal and think of a color beginning with that letter Got that Well hopefully you are now all thinking of an Orange Kangeroo from Denmark The reason that this trick works, is that from the first of the above theorems, the sum of the digits of the number that they get must be nine Subtract five to give four, and the rest is forced This trick works nearly every time and I was delighted to once use it for a group of blind students, who loved anything to do with mental arithmetic For a second trick, take a pack of cards and put the Joker in as card number nine Ask a volunteer for a number between 10 and 19 and deal put that number of cards from the top Pick this new pack up and ask for the sum of the digits of the volunteer’s number Deal that number of cards from the top Then turn over the next card It will always be the Joker This is because if you take any number between 10 and 19 and subtract the sum of the digits then you always get nine With a collection of magic tricks you can introduce many mathematical concepts, from primary age maths to advanced level university maths The best was to do this, is to first show the trick, then explain the maths behind it, then get the audience to practice the trick, and then (and best of all) get them to devise new tricks using the maths that they have just learned You never knew that maths could be so much fun! Example 3 How Maths Won the Battle of Britain It may be unlikely to think of mathematicians as heroes, but without the work of teams of mathematicians the Allies would probably have lost the Second World War Part of this story is well-known The extraordinary work of the mathematical code breakers, especially Alan Turing and Bill Tutte, at Bletchley Park has been the subject of many documentaries and books (and this is one area where the media has got it right) This has been described very well in the Code Book (also) by Simon Singh (1999) However, mathematics played an equally vital role in the Battle of Britain and beyond One of the main problems faced by the RAF during the Battle of Britain was that of detecting the incoming bombers and in guiding the defending fighters to meet them The procedure set up by Air Vice Marshall Dowding to do this, was to collect as much data as possible about the likely location of the aircraft from a number of sources, such as radar stations and the Royal Observer Corps, and to then pass this to the ‘Filter Room’ where it was combined to find the actual aircraft position The Filter Room was staffed by mathematicians who’s job was to determine the location of the aircraft by using a combination of (three-dimensional) trigonometry to predict their height, number, and location from their previous known locations, combined with a statistical assessment of their most likely position given the less than reliable data coming from the radar stations and other sources Once Page of 12 Promoting Maths to the General Public the location of the aircraft was known further trigonometry was required to guide the fighters on the correct interception path (using a flight direction often called the ‘Tizzy’ angle after the scientific civil servant Tizard) An excellent account of this and related applications of maths is given in Korner (1996) In a classroom setting this makes for a fascinating and interactive workshop in which the conditions in the filter room are recreated and the students have to do the same calculations under extreme time pressure One of the real secrets to popularizing maths is to get the audience really involved in a hands-on manner! (It is worth saying that the same ideas of comparing predictions with unreliable data to determine what is actually going on are used today both in Air Traffic Control, meteorology and robotics.) Whilst it might be thought that this is a rather ‘male oriented’ view of applied mathematics, it is well worth saying that the majority of the mathematicians employed in the filter rooms were relatively young women in the WAAF, often recruited directly from school for their mathematical abilities In a remarkable book, Eileen Younghusband (2011) recounts how she had to do complex three-dimensional trigonometric under extreme pressure, both in time and also knowing how many lives depended on her getting the calculations right After the Battle of Britain she ‘graduated’ to the even harder problem of tracking the V2 rockets being fired at Brussels When I tell this story to teenagers, they get incredibly involved and there is not a dry eye in the house No one can ever accuse trigonometry of not being useful or interesting! Example 4 Weather and Climate One of the most important challenges facing the human race is that of climate change It is described all the time in the media and young people especially are very involved with issues related to it The debates about climate change are very heated From the perspective of promoting mathematics, climate change gives a perfect example of how powerful mathematics can be brought to bear on a vitally important problem, and in particular gives presenters a chance to talk about the way that equations can not only model the world, but are used to make predictions about it Much of the mathematical modeling process can be described and explained through the example of predicting the climate and the audience led through the basic steps of: (1) Making lots of observations of pressure, temperature, wind speed, moisture, etc (2) Writing down the (partial differential) equations, which tell you how these variables are related (3) Solving the equations on a computer (4) Constantly updating and checking the computer simulations with new data (5) Assessing the reliability of the prediction (6) Informing policy bodies about the results of the simulations There are plenty of mathematics and human elements to this story, starting from Euler’s derivation of the first laws of fluid motion, the work of the mathematicians Navier and Stokes on fluids or Kelvin in thermodynamics (the latter was a real character), the pioneering work of Richardson (another great character) in numerical weather forecasting, and the modern day achievements and work of climate change scientists and meteorologists However, the real climax of talking about the climate should be the maths itself which comes across well as being an impartial factor in the debate, far removed from the hot air of the politicians As a simple example, if ‘T’ is the temperature of the Earth, ‘e’ is its emmisivity (which decreases as the carbon dioxide levels in the atmosphere increase), ‘a’ is its albedo (which decreases as the ice melts), and ‘S’ is the energy from the sun (which is about ⅓kW per m2 on average) then: eσT = (1 − a) S This formula can be solved using techniques taught in A level mathematics, and allows you to calculate the average temperature of the Earth The nice thing about this formula is that unlike the formula for the perfect kiss, this one can be easily checked against actual data From the perspective of climate science its true importance is that it clearly shows the effects on the Earth’s temperature (and therefore on the rest of the climate) of reducing the emmisivity ‘e’ (by increasing the amount of Carbon Dioxide in the atmosphere) or of reducing the albedo ‘a’ (by reducing the size of the ice sheets This leads to a frightening prediction The hotter it is the less ice we have as the ice sheets melt As a consequence the albedo, ‘a’, decreases, so the Earth reflects less of the Sun’s radiation Our formula then predicts that the Earth will get hotter, and so more ice melts and the cycle continues Thus we can see the possible effects of a positive feedback loop leading to the climate spiraling out of control This is Page of 12 Promoting Maths to the General Public something that any audience can connect with, and leads to fierce debates! It may come as a surprise, but I have always found that audiences generally like the ‘unveiling’ of this equation, and seeing how it can be used to make predictions A talk about mathematics can be exactly that, i.e ‘about’ mathematics If the audience gains the impression that maths is important, and that the world really can be described in terms of mathematical equations and that a lot of mathematics has to be (and still is being done) to make sense of these equations, then the talk to a certain extent has achieved its purpose Talks on climate change often lead to a furious email (and other) correspondence, which goes against the implicit assumption in the media that no one is really interested in a mathematical problem At another level, climate change is exactly the sort of area where mathematicians and policy makers need to communicate with each other as clearly as possible, with each side understanding the language (and modus operandi) of the other Example 5 Maths and Art Click to view larger Figure 1 (a) A Circular Celtic Knot (b) The Chased Chicken Sona pattern One of the aspects of mathematics which tends to put people off is that it is perceived as a dry subject, far removed from ‘creative’ subjects such as art and music Of course this is nonsense, as maths is as creative a subject as it is possible to get (I spend my life creating new mathematics), but it is worth making very explicit the wonderful links between mathematics and art (When faced with the question: is maths an art or a science? The correct answer is simply ‘Yes’.) Some of these links run very deep, for example the musical scale is the product of many centuries of mathematical thought (started by Pythagoras) The subject of origami was for many years treated simply as an art form However, working out the folding pattern to create a three-dimensional object (such as a beetle) from a single sheet of paper is fundamentally a mathematical problem This was realized recently by Robert Lang amongst others, and the fusion of mathematics with Origami leads to sublime artistic creations Another area where art meets maths in a multicultural setting is in Celtic Knots and the related Sona drawings from Africa Examples of both of these are illustrated in Figure 1a, b, with Figure 1a showing a circular Celtic Knot created by a school student, and Figure 1b a Sona design called the ‘Chased Chicken’ Celtic Knots are drawn on a grid according to certain rules These rules can be translated into algebraic structures and manipulated using mathematics By doing this, students can explore various combinations of the rules, and then turn them into patterns of art This is an incredibly powerful experience for them as they see the direct relation between quite deep symmetry patterns in mathematics and beautiful art work Usually when I do Celtic Art workshops I have two sessions, one where I describe the maths and then I wait for a month whilst the students work with an art department By doing this they learn both maths and art at the same time As I said, a very powerful experience all round A nice spin-off is the related question of investigating African Sona patterns Mathematically these are very similar to Celtic Knots, and in fact the ideas behind them predate those of Celtic Knots An excellent account of these patterns along with many other examples of the fusion of African mathematics and art, is given in Gerdes (1999) Doing a workshop on Celtic Knots and Sona patterns, demonstrates the fact that maths is not a creation of the Western World, but is a truly international and multi-cultural activity And finally I hope that I have demonstrated in this chapter that although maths is hard and has a terrible public image, it is a subject that can be presented in a very engaging and hands on way to the general public Indeed it can be used to bring many ideas together from art to engineering and from music to multi-culturalism By doing so, everyone can both enjoy, and see the relevance, of maths There is still a long way to go before maths has the same popularity (and image) on the media as (say) cooking or gardening (or even astronomy or archaeology), but significant Page 10 of 12 How Informal Learning Activities Can Promote Children’s Numerical Knowledge Oxford Handbooks Online How Informal Learning Activities Can Promote Children’s Numerical Knowledge Geetha B Ramani and Robert S Siegler The Oxford Handbook of Numerical Cognition (Forthcoming) Edited by Roi Cohen Kadosh and Ann Dowker Online Publication Date: Mar 2014 Subject: Psychology, Cognitive Psychology, Educational Psychology, Developmental Psychology DOI: 10.1093/oxfordhb/9780199642342.013.012 Abstract and Keywords Before children begin school, there is a wide range of individual differences in children’s early numerical knowledge Theoretical and empirical work from the sociocultural perspective suggests that children’s experiences in the early home environment and with informal number activities can contribute to these differences This article draws from this work to hypothesize that differences in the home explain, in part, why the numerical knowledge of children from low-income backgrounds trails behind that of peers from middle-class backgrounds By integrating sociocultural perspectives with a theoretical analysis of children’s mental number line, the authors created an informal learning activity to serve as an intervention to promote young children’s numerical knowledge Our studies have shown that playing a simple number board game can promote the numerical knowledge of young children from low-income backgrounds The authors discuss how informal learning activities can play a critical role in the development of children’s early maths skills Keywords: maths, preschoolers, informal learning activities, interventions, board games Acknowledgments We would like to thank the Institute of Educational Sciences, which supported this research, Grants R305A080013 and R305H050035, and the Teresa Heinz Chair at Carnegie Mellon University Providing children with a strong foundation of mathematical knowledge is critical for success in school and beyond Children’s early mathematical knowledge predicts their rate of growth in mathematics (Aunola, Leskinen, Lerkkanen, & Nurmi, 2007; Jordan, Kaplan, Locuniak, & Ramineni, 2007), as well as mathematics achievement test scores in later elementary school and even into the high school years (Duncan et al., 2007; Jordan, Kaplan, Ramineni, & Locuniak, 2009; Locuniak & Jordan, 2008; Mazzocco & Thompson, 2005) Furthermore, mathematical achievement can impact children’s performance in college and choice of careers (National Mathematics Advisory Panel, 2008) Given the importance of maths education, the development of maths skills has been extensively researched over the past 20 years (Dehaene & Brannon, 2011; Geary, 2006) One specific area of interest is the wide range of individual variations in children’s mathematical knowledge Of particular concern to educators, policymakers, and parents is that many of these differences are evident prior to children beginning school Individual differences in the numerical knowledge of preschool and kindergarten children have been demonstrated on a variety of foundational maths skills, such as counting, identifying written numerals, and simple arithmetic (see National Research Council of the National Academies, 2009, for a review) These early differences tend to increase the further children move through school (Alexander & Entwisle, 1988; Geary, 1994, 2006) Therefore, understanding Page of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge the kinds of experiences that promote early numerical knowledge is of critical importance because it can impact children’s long-term maths achievement in school and possibly their long-term career opportunities In this article, we review recent research on how children’s early experiences in the home and, with informal learning activities, can influence their mathematical understanding and performance First, we present a growing body of literature that demonstrates from a sociocultural perspective how contextual factors and early experiences, mainly with parents in the early home environment, can shape children’s numerical development and can explain individual differences in numerical knowledge Second, we review theories and empirical work regarding a specific aspect of numerical knowledge, the mental number line, which likely underlies children’s number sense Finally, we argue that by integrating this theoretical analysis of the mental number line with a sociocultural perspective can help inform the design of activities to promote children’s numerical knowledge To support our argument, we review research that we have conducted that shows that playing a linear number board game can promote the numerical knowledge of preschool-age children from low-income backgrounds Sociocultural Perspectives on the Development of Numerical Knowledge The sociocultural perspective provides an impetus for examining how children’s early home environment and interactions with adults can influence children’s mathematical development A central tenet of sociocultural theory is that social interactions with adults play a critical role in children’s cognitive development (Gauvain, 2001; Rogoff, 1990; Vygotsky, 1976, 1978) Play and other informal activities are considered particularly important contexts in which adults provide children with new information, support their skill development, and extend their conceptual understanding Every-day, informal activities can provide children with extensive numerical information in the home environment (Saxe, 2004) Furthermore, much of the development of mathematical understanding in early childhood is social in nature, occurring during activities with parents, such as meals, chores, and shopping For example, a young child could learn about one-to-one correspondence, while setting the table, fractions while cooking, and arithmetic while at the grocery store The sociocultural perspective has also motivated research on how children from different cultural and socioeconomic status SES backgrounds can vary in numerical knowledge and how these differences are influenced by their early home experiences Differences in mathematical knowledge between children from China and the United States have been found in preschoolers as young as age 3 years (Miller, Kelly, & Zhou, 2005) The advanced number skills of young Chinese children have been shown on familiar tasks, such as arithmetic problems, and also on novel tasks, such as a number line estimation task (Siegler & Mu, 2008) These differences are related to early home experiences – Chinese parents practice skills such as arithmetic at home much more than US parents do (Zhou, Huang, Wang, Wang, Zhao, Yang, L., et al., 2006) – and also to general societal differences, such as variations in number names (Miller et al., 2005) Even within the United States, young children’s maths achievement and their mathematical experience vary widely Specifically, the numerical knowledge of young children from low-income backgrounds trails far behind that of their peers from middle-income backgrounds These differences do not extend to all numerical tasks On non-verbal numerical tasks, the performance of young children from low-income backgrounds is equivalent to that of age peers from wealthier backgrounds (Jordan, Huttenlocher, & Levine, 1992; Jordan, Levine, & Huttenlocher, 1994) However, the same studies find large differences on tasks with verbally stated numbers, story problems, written numerals, and higher-level maths problems Specifically, differences have been found in areas such as knowing the cardinality principle when counting, identifying written numbers, and solving arithmetic problems (Lee & Burkam, 2002; Klibanoff, Levine, Huttenlocher, Vasilyeva, & Hedges, 2006; Saxe, Guberman, & Gearhart, 1987; Starkey, Klein, & Wakeley, 2004) These symbolic tasks are foundational for later mathematical concepts and, therefore, are the focus of the research presented in this article Numerical Activities in the Home One source of individual differences in numerical knowledge is the early home environment In this section, we focus on the kinds of informal learning activities children engage in at home, variations in exposure to these activities, and consequences of these variations for children’s developing numerical knowledge Page of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge Number-related Activities and Parent–child Interactions The number-related activities and support for learning that parents provide to children has been examined through self-reports, naturalistic observations in the home, and structured observations in the home and the laboratory Questionnaires and interviews suggest that parents engage their young children in both formal and informal numerically relevant activities These activities include formal teaching through number-related activity books and worksheets, explanations of number concepts, and practicing number skills, such as identifying written numbers (Huntsinger, Jose, Larson, Balsink Krieg, & Shaligram, 2000; Skwarchuk, 2009) Parents also engage their children in informal activities involving numbers, such as playing board games and card games, singing songs and nursery rhymes, and measuring ingredients while cooking (Blevins-Knabe & Musun-Miller, 1996) Naturalistic observations indicate that parents begin exposing their children to numbers at a young age In a longitudinal study with primarily middle-income families (Durkin, Shire, Riem, Crowther, & Rutter, 1986), parents were observed using number words with children as young as 9 months Over the next 2 years, parents more frequently used number words and engaged their children in number-related activities, such as counting and reading books involving numbers A more recent longitudinal study of parental input (Levine, Suriyakham, Rowe, Huttenlocher, & Gunderson, 2010) found that variations in parent number talk is related to children’s developing numerical knowledge Families from diverse backgrounds were observed in their homes for five visits between the ages of 14 and 30 months At 46 months, children were administered a measure of understanding of cardinality, in which children were shown two cards and asked to point to the one with a specific number (e.g point to 6) The amount of number talk that the parents engaged in from 14 to 30 months predicted their children’s understanding of cardinality at 46 months, even after controlling for SES A follow-up study revealed that particular types of number talk seemed better at promoting children’s cardinality understanding Specifically, parents’ number talk involving counting or labelling the cardinal value of visible objects and their talk about large sets of objects were the most predictive of children’s later number knowledge (Gunderson & Levine, 2011) Studies using structured observations have indicated that the kinds of activities, materials, and games in which parents engage their children influences parental talk and support about numbers and numerical concepts For example, Vandermaas-Peeler, Nelson, Bumpass, and Sassine (2009) asked parents to read a book with their children about a shopping trip, and then provided them with materials related to the story, such as pretend money and food, as well as cooking materials They found that providing parents with these everyday materials elicited talk about the materials’ general properties, such as appropriate uses for money, but elicited little talk about their numerical properties, such as the value of the money In contrast, other studies using such methods have found that many parents engage their preschoolers in counting and other numerical activities, while reading books, playing with blocks, and working on a mathematical workbook Materials that had fewer applications to daily life elicited greater talk about numbers (Anderson, 1997; Anderson, Anderson, & Shapiro, 2004) Others have also found that different materials and contexts elicit different kinds of talk and support from parents For example, Bjorklund, Hubertz, and Reubens (2004) observed parents and their preschoolers while they played a modified version of Chutes and Ladders together, and while they solved maths problems together Parents were more likely to give cognitive directives, such as modelling the correct answer and providing instruction on strategies, when solving the maths problems, whereas they were more likely to use simple prompts when playing the board game Together, these findings suggest that parents engage in a variety of activities and discussions with their children that could support early numerical knowledge Furthermore, the type of material influences parents’ mathematically relevant interactions with their children Relations between Familial Numerical Activities and Early Maths Knowledge Parents vary considerably in the frequency with which they engage in mathematically relevant activities with their children, and these variations influence children’s developing numerical knowledge Observational studies of the early home environment suggest that the absolute frequency with which parents engage their children in mathematical activities in the home tends to be quite low For example, Tudge and Doucet (2004) conducted an observational study of preschool children from both White and Black families from diverse SES backgrounds Children were observed during their daily routines for 18 hours, distributed over many days, in places such as Page of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge home, childcare centers, and parks On average, children were observed engaging in a mathematical lesson or play activity in less than 1 out of 180 observations One reason for this low absolute frequency is that parents tend to place a greater emphasis on literacy development than on mathematics development for their young children (Barbarin, Early, Clifford, Bryant, Frome, Burchinal, et al., 2008; Cannon & Ginsburg, 2008).Observations of homes and preschools, as well as reports of teachers and parents, suggest that the home and preschool environments provide children with fewer mathematical than literacy-orientated experiences (LeFevre, Skwarchuk, Smith-Chant, Fast, Kamawar, & Bisanz, 2009; Plewis, Mooney, & Creeser, 1990; Tizard & Hughes, 1984; Tudge & Doucet, 2004) Although the average amount of mathematical activity is low, there is considerable variation in children’s experience with number-related activities For example, Levine et al (2010), the quantity of number words that parents used varied from 4 to 257 words over the 7.5 hours of observation As would be expected, children whose parents present them with more mathematical activities generally have greater mathematical knowledge and maths fluency This relation is present for both amount of direct instruction and amount of informal learning activities involving numbers (Blevins-Knabe & Musun-Miller, 1996; Huntsinger et al., 2000; LeFevre et al., 2009) Frequency of engaging in non-mathematical informal learning activities at home also is predictive of subsequent maths achievement Parents’ reports of 3- and 4-year-olds’ engagement in informal learning activities, such as rhyming and singing songs, as well as providing direct instruction about letters and numbers, predicts children’s mathematical achievement at age 10 (Melhuish, Sylva, Sammons, Siraj-Blatchford, Taggart, Phan, M et al., 2008) Differences in the mathematical knowledge of children from lower- and higher-income backgrounds also reflect differences in environmental support for maths learning Lower SES families have numerous stressors, such as financial constraints and lower education, which can limit their ability to support their children’s academic development (Rouse, Brooks-Gunn, & McLanahan, 2005) These factors likely influence SES-related differences in the types of resources and number-related activities parents report engaging in with their children at home Interviews with parents of 2- and 4-year-olds revealed that middle-income parents reported engaging their children in numerical activities that were rated as higher in complexity than working class parents (Saxe et al., 1987) An example of a complex activity involved arithmetic operations and a simple activity involved songs involving numbers Others also have found that parents of middle-class children report engaging their children in a wider range of number-related activities, and engaging in such activities more frequently, than families from lowerincome backgrounds (Starkey et al., 2004; Starkey & Klein, 2008) However, even within lower-income families, the early home environment and the type of support they offer their children for preparing them for school vary greatly and in ways that influence their mathematics proficiency (Burchinal, McCartney, Steinberg, Crosnoe, Friedman, McLoyd, V et al., 2011; Holloway, Rambaud, Fuller, & Eggers-Pierola, 1995) For example, within Head Start populations, children’s numerical knowledge is positively related to the frequency of parents’ engagement with them in both formal maths activities, such as practicing simple arithmetic, and informal numerical activities, such as board games and card games (Ramani, Rowe, Eason, & Leech, 2013) Thus, informal learning activities and the early home environment appear to play a critical role in the development of children’s number skills Having more opportunities for practicing number skills, through both informal learning activities and direct instruction, is positively related to children’s maths skills Integrating Sociocultural Theory with a Theoretical Analysis of Numerical Sense In this section, we discuss how we integrated the sociocultural orientation and research studies that followed from this orientation with a theoretical analysis of the mental number line, which is hypothesized to underlie children’s number sense (e.g Dehaene, 2011) Our goal was to understand individual differences in children’s numerical magnitude knowledge and to identify ways of improving this understanding We were interested in three specific questions: How does numerical magnitude knowledge vary between children, especially children from different SES backgrounds What are the sources of these differences How can we promote young children’s numerical magnitude knowledge? Page of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge Before discussing answers to these questions, however, we review research on the development of numerical magnitude knowledge and the importance of this knowledge for maths learning Representations of Numerical Magnitudes Accurate estimation of numerical magnitudes is critical for mathematical achievement and is central to the concept of number sense (Berch, 2005; Siegler & Booth, 2005; Sowder, 1992) Estimation can involve approximating the answer to arithmetic problems (e.g 124 + 272), the distance between two objects or places (e.g about how many miles is it to school), or the number of objects in a set (e.g about how many cookies are in the jar?) Children with more advanced numerical estimation skills in first grade show faster growth in maths skills over the elementary school years, even after controlling for other predictive factors, such as intelligence and working memory (Geary 2011) Having a strong understanding of numerical magnitudes can lay the foundation for learning later, more complex, mathematics The cognitive structure thought to underlie numerical magnitude knowledge is the mental number line, which is based on the hypothesis that numbers are represented spatially on a continuum In cultures that use left-to-right orthographies, numerical magnitudes increase from left to right on the continuum (Dehaene, 2011) Both behavioural and neural research indicates the importance of the mental number line (Ansari, 2008; Hubbard, Piazza, Pinel, & Dehaene, 2005) One body of evidence comes from research on the numerical magnitude task Specifically, people more quickly answer ‘Which is bigger, 8 or 3’ when correct responses require pressing a key on the right rather than on the left However, people more quickly answer ‘Which is smaller, 8 or 3’ by pressing a key on the left rather than on the right (Dehaene, Bossini, & Giraux, 1993) This spatial-numerical association of response codes (SNARC) effect provides evidence of how representations of quantities are ordered horizontally in a left to right array The form of mental number line representations can be measured using a number line estimation task The number line estimation task involves presenting people lines with a number at each end (e.g 0 and 100) and a third number, printed above the line, in that range The task is to estimate the location of the third number on the line (e g., ‘Mark where 36 would go on the line’) There are three major benefits of the task The task can be used with any range of numbers, because any two numbers can be used at the ends of the lines Any type of number – whole, fraction, decimal, percentage, positive, negative – can be located on the line The number line task parallels the ratio characteristics of the number system That is, just as 60 is twice as large as 30, the distance of the estimated position of 60 from 0 should be twice as great as the distance of the estimated position of 30 from 0 Performance on the number line estimation task correlates strongly with mathematics achievement test scores at all grade levels from kindergarten through eighth grade (Booth & Siegler, 2006; Geary, Hoard, Nugent, & Byrd Craven, 2008; Holloway & Ansari, 2009; Schneider, Grabner, & Paetsch, 2009; Siegler & Booth, 2004; Siegler, Thompson, & Schneider, 2011) Causal connections between number line estimation and maths learning also are present Improving the numerical magnitude knowledge of children improves their learning of arithmetic and other mathematical skills (Booth & Siegler, 2008) Numerical magnitude knowledge develops over several years and involves knowledge of a range of numerical skills and concepts Knowledge of the counting sequence likely contributes to the development of numerical magnitude knowledge, but is not sufficient Children count correctly from 1 at least a year before they show much knowledge of numerical magnitudes in the same range or even know the rank order of those numbers (Le Corre, Brannon, Van de Walle, & Carey, 2006; Le Corre & Carey, 2007; Lipton & Spelke, 2005) For example, not until the age of 5 years do children’s number line estimates become accurate and linear for the numbers 0–10, even though they can count to 10 at least a year earlier (Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, 2010) Similarly, not until second grade do number line estimates become accurate for the numbers between 0 and 100, despite children being able to count to 100 a year or two earlier (Ebersbach, Luwel, Frick, Onghena, & Verschaffel, 2008; Geary et al., 2008; Siegler & Booth, 2004) Page of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge Variations in Numerical Magnitude Knowledge To answer our first question of whether numerical magnitude knowledge varies between children from different SES backgrounds, we compared the performance of middle- and lower income preschoolers on a 0–10 number line estimation task (Siegler & Ramani, 2008) Children from lower-income backgrounds tended to have poorer numerical magnitude knowledge than children from middle-income backgrounds Analyses of individual children’s estimates showed the best fitting linear function accounted for a mean of 60% of the variance in the estimates of the individual children from middle-income backgrounds, but only 15% of the variance among children from lowincome backgrounds Children from lower-income background also had a poorer understanding of the order of numbers Specifically, we compared each child’s estimate of the magnitude of each number with the child’s estimate for each of the other numbers, and calculated the percentage of estimates that were correctly ordered Children from higher-income backgrounds correctly ordered 81% of the estimates compared with 61% correct from the children from lowerincome backgrounds Sources of SES-related Variations in Numerical Magnitude Knowledge Our second question concerned potential sources of these SES-related differences in numerical magnitude knowledge Based on the above review of the literature on the role of home activities in preschoolers’ maths learning, informal activities seem to be critical for promoting early numerical knowledge One common activity that seems ideally designed for producing linear representations of the mental number line is playing linear, numerical, board games – that is, board games with linearly-arranged, consecutively numbered, equal-sized spaces (e.g Chutes and Ladders.) As noted by Siegler and Booth (2004), linear board games provide multiple cues to both the order of numbers and the numbers’ magnitudes Specifically, the greater the number in a square, the greater: The distance the token from the origin to its present location The number of discrete moves the child has made The number of number names the child has spoken The number of number names the child has heard The amount of time since the game began Thus, children playing the game have the opportunity to relate the number in each square to the time, distance, and number of manual and vocal actions required to reach that number In other words, the linear relations between numerical magnitudes and these visuospatial, kinaesthetic, auditory, and temporal cues provide a broadly-based, multi-modal foundation for a linear representation of numerical magnitudes To test our hypothesis that such board games could account for SES-related differences in numerical knowledge, Ramani and Siegler (2008) asked young children from lower- and higher-income backgrounds directly about their experiences with board games at home – whether they play board games, card games, and video games at their own home and those of other friends and relatives, and which games they play As predicted, more preschoolers from middle-income backgrounds reported playing board games and card games than children from low-income backgrounds Specifically, 80% of the children from middle-income backgrounds reported playing board games at home or at other people’s home, whereas only 47% of the children from low-income backgrounds did A similar pattern was found for card games, but not for video games; 66% of the preschool children from lower-income backgrounds reported playing video games at home, whereas only 30% of the middle-income children did Children from low-income backgrounds who had more experience playing board games at their own and other people’s homes exhibited greater numerical knowledge (Ramani & Siegler, 2008) The relationship was even present when only experience playing the single number board game Chutes and Ladders was considered Reported experience playing card games and video games was not closely related to numerical knowledge, thus indicating that the correlations with board game experience were not due to numerically advanced children having better memory for their game playing experience or being more willing to report it Playing a Linear Number Board Game to Improve Numerical Magnitude Knowledge Page of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge Click to view larger Figure 1 The (a) number and (b) colour linear board games (Ramani & Siegler, 2008; Siegler & Ramani 2008) To address our third question of how to improve the numerical knowledge of children from low-income backgrounds, Siegler and Ramani (2008) randomly assigned preschoolers from Head Start classrooms to either play a linear numerical board game with squares numbered from 1 to 10 or a colour board game that was identical, except for the squares varying in colours, rather than numbers (Figure 1) At the beginning of each session, children were told that, on each turn, they would spin a spinner with a ‘1’ or ‘2’ on it and move their token that number of spaces; the first person to reach the end would win Children in the colour board condition were given the identical instructions, except their spinner varied in colour The children were told to say the number (colour) on each space as they moved For example, children who played the number board game who were on 3 and spun a 2 would say, ‘4, 5’ as they moved their token Similarly, children who played the colour board game who were on a red space and spun a purple, would say, ‘green, purple’, as they moved Children played one of the two games one-on-one with an experimenter for four 15–20-minute sessions distributed over a 2-week period Each game lasted approximately 2–4 minutes, so that children played their game roughly 20 times over the four sessions Children estimated the positions of the numbers 1–10 on a number line prior to Session 1 as a pretest and at the end of Session 4 as a post-test Children who played the number board game considerably improved their numerical magnitude knowledge On the pretest, the best-fitting linear function accounted for an average of 15% of the variance in individual children’s estimates; on the post-test, it accounted for 61% In contrast, for children in the colour board game condition, the best fitting linear function accounted for 18% of the variance on both pretest and posttest Children who had played the numerical board game also ordered correctly the magnitudes of more numbers on the posttest than on the pretest Peers who played an identical game, except for the squares varying in colour, rather than number, did not show comparable improvements on the task Improving Foundational Number Skills over Time We then tested the generality of the benefits of playing the number board game across various number tasks and over time (Ramani & Siegler, 2008) Playing the linear board game provides children with practice at counting and at numeral identification, because players are required to name the squares through which they move (e.g saying ‘6, 7’ after starting on the 5 and spinning a 2) Thus, playing such games would be expected to improve counting and numeral identification skills, as well as performance on tasks that require understanding of numerical magnitudes We also wanted to examine whether children’s learning remained apparent many weeks after the last game playing session Page of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge Ramani and Siegler (2008) presented 124 Head Start children several measures of numerical knowledge of the numbers 1–10 Children were give the number line estimation task, a magnitude comparison task (‘Which is bigger: N or M?’), a numeral identification task (‘Can you tell me the number on this card?’), and rote counting (‘Count from one to 10’) These tasks were presented on a pretest before the game playing began in Session 1, on a post-test immediately after the final game was played in Session 4, and in a follow-up session 9 weeks after the post-test Click to view larger Figure 2 Performance of preschoolers from low-income backgrounds on four numerical tasks: (A) number line estimation; (B) magnitude comparison; (C) numerical identification; and (D) counting, before playing the number or colour board game (pretest), immediately after the fourth and final session of playing the game (post-test), and 9 weeks after the final game playing session (follow-up) (Ramani & Siegler, 2008) After playing the number board game, children showed improvements on all four measures of numerical knowledge These improvements were stable over time After 9 weeks of not having played the board game, improvements on all four tasks remained significant, and they were at least as large on three of the four tasks as on the immediate post-test (Figure 2) As in the previous study, children who played an identical game, except for the squares varying in color, did not show comparable improvements Identifying Important Features of the Board Game’s Design Click to view larger Figure 3 The circular number board games (Siegler & Ramani, 2009; Ramani & Siegler, 2011) Testing the effects of specific features of the board game on children’s learning is critical for understanding why the game works and for creating future informal learning activities One feature that seemed likely to be important was how the linearity of the game board influenced children’s learning about numerical magnitudes To test this feature, Siegler and Ramani (2009) assigned children to play either the linear board game or a circular board game (Figure 3) The linear game was expected to be more effective than the circular one on the magnitude tasks (number line and magnitude comparison), because the linear board is easier to translate into a linear mental number line There was no obvious reason to predict that the linear board would promote greater improvement than the circular board in numeral identification because that skill does not depend in any obvious way on a linear representation Instead, the linear board and the circular board were expected to be equally effective in promoting numeral identification skills As predicted, playing the linear board game for roughly an hour increased low-income preschoolers’ proficiency on numerical magnitude comparison and number line estimation Playing the game with the circular boards did not improve children’s performance on these measures Also as predicted, playing the linear and circular board games Page of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge improved children’s numeral identification skills by an equal amount Counting was at ceiling in this sample, so no comparisons were possible Another major finding of Siegler and Ramani (2009) was that preschoolers who earlier had played the linear board game learned more from subsequent training on arithmetic problems than peers who had played the linear board game We predicted that learning answers to arithmetic problems in part reflects appropriate numerical magnitude representations, and that playing the linear board game produces such representations During the pretest and post-test, children were administered four simple arithmetic problems, 2 + 1, 2 + 2, 4 + 2, and 2 + 3 After playing the board game for the four sessions, in a fifth session children were given a brief training with feedback on the two easiest arithmetic problems that they answered incorrectly on the pretest Then they were asked to answer the two problems without feedback Among children who had previously played the linear numerical board game, the percentage of correct addition answers was higher than in the circular board game condition or in a control condition Especially relevant to the idea that the gains came about through improving children’s numerical magnitude representations, children’s errors in the linear board condition, but not in the circular board condition, tended to become closer to the correct answer from pretest to post-test A second feature of the board game that we tested was whether the context of the game was important for promoting children’s numerical knowledge To determine whether playing numerical board games had effects above and beyond those of common number activities, Siegler and Ramani (2009) provided a third group of children practice with counting and numeral naming tasks, but not in a game context We found that having children engage in a continuing cycle of the tasks – number string counting, numeral identification, and object counting – for the same length of time as children played the board game did not influence children’s numerical knowledge These children also did not improve their ability to learn the answers to novel arithmetic problems Scaling Up the Board Game Intervention Determining whether an intervention can be used in everyday settings requires evidence beyond that the intervention is effective under controlled, laboratory conditions Three types of evidence relevant to the present context are whether the board game is effective with different populations, when played with small groups of children, and when implemented by a paraprofessional from the children’s classroom One variable that is important for scaling up the intervention is whether children from some groups benefit more than others Preschoolers from low-income backgrounds at times show greater gains from mathematical interventions than preschoolers from middle-income backgrounds (Starkey et al., 2004) Because children from middle-income backgrounds have greater prior board game experience than children from low-income backgrounds, this greater experience might make playing the present board game redundant with the middleincome children’s prior experience, and thus less effective for improving their numerical knowledge We tested whether playing the linear board game would improve the numerical knowledge of preschoolers from middleincome backgrounds, or whether the benefits were unique to children from low-income backgrounds Specifically, we compared the relative benefits of playing a linear number board game for two groups of children who before the experience had equal numerical knowledge: 3- and 4-year-olds from middle-income backgrounds and 4- and 5year-olds from low-income backgrounds Children from low-income backgrounds learned at least as much, and on several measures more, than preschoolers from middle-income backgrounds with comparable initial knowledge Within each group, children who initially knew less tended to learn more Overall, the findings suggest that playing the linear board game is effective at promoting the numerical knowledge of children from both lower- and middle-SES backgrounds Interventions that are effective in lab settings are often ineffective in classrooms (Newcombe, Ambady, Eccles, Gomez, Klahr, Linn, M et al., 2009; White, Frishkoff, & Bullock, 2008) Two likely reasons are that the 1:1 experimenter–child interactions that are typical of laboratory studies are often impossible in classrooms, and that the people executing the intervention in classrooms usually have fewer opportunities for training in executing the interventions than research personnel To better understand these challenges, we have begun to examine whether the number board game could be translated into a practical classroom activity that improves Head Start children’s numerical knowledge Participants Page of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge within a condition were randomly divided into groups of three children and were presented six 20–25-min sessions within a 3- or 4-week period Children played a similar linear number board game or colour board game to those used in previous studies The procedures and game playing were also similar except the children played the game with each other, and the experimenter facilitated the game, instead of playing Playing the number board game as a small group learning activity promoted low-income children’s number line estimation, magnitude comparison, numeral identification, and counting Children who played the colour board game only improved in counting skills Improvements were also found when paraprofessionals from the children’s classrooms played the game with small groups of children The paraprofessionals were given roughly an hour of training prior to the start of the study During this time, they were given the board game materials and an opportunity to practice with them They were also given a short booklet that included the rules for the number and colour games, scripts for how to explain the games to the children, and standard prompts for correcting errors The paraprofessionals also watched a demonstration video of a group of children playing the board games, and were told to correct errors or omissions by prompting children to say the required number or colour If the errors continued, the paraprofessionals were to model the correct move and ask the child to repeat the move We found that playing the number board game as a small group activity supervised by a paraprofessional from the classroom improved children’s numerical knowledge on four measures Observations of the game playing sessions revealed that paraprofessionals adapted the feedback they provided to reflect individual children’s improving numerical knowledge over the game playing sessions and that children remained engaged in the board game play even after multiple sessions (Ramani, Siegler, & Hitti, 2012) Thus, the linear number board game can be used to promote the numerical knowledge of children from a range of SES backgrounds and can be used effectively in preschool classrooms Preschool Mathematics Interventions and Curricula Other targeted interventions also have been shown to be effective in improving young children’s numerical understanding An adaptive software program called ‘The Number Race,’ aimed at improving young children’s number sense, has improved the skills of children with mathematical difficulties (Wilson, Revkin, Cohen, Cohen, & Dehaene, 2006; Wilson, Dehaene, Dubois, & Fayol, 2009) Other experimental interventions that have incorporated number board games have also produced improvements in young children’s numerical knowledge (e.g Malofeeva, Day, Saco, Young, & Cianco, 2004) More comprehensive curricula for improving low-income preschoolers’ and kindergartners’ mathematical knowledge have also shown large positive effects These programmes integrate informal learning activities with direct classroom instruction One such curriculum is Number Worlds, which includes a wide range of numerical activities – songs about numbers, counting games, games involving money, and board games The goal of the curriculum is to provide children with a strong foundation with numbers before teaching them more advanced concepts Researchers have found that following participation in the Number Worlds curriculum, low-income kindergarteners had significantly better basic numerical skills than peers who did not receive the curriculum (Griffin, 2004) Another early childhood curriculum, Pre-K Mathematics, combines school- and home-based activities to promote children’s numerical knowledge Children participate in small-group activities in the classrooms and parents are also provided with activities to do at home that link to the small-group activities in the school Participation in the programme led to kindergartners from low-income backgrounds having mathematical knowledge at the end of the programme equivalent to that of middle-income peers who did not participate in it (Starkey et al., 2004) Similarly, the Building Blocks curriculum (Clements & Sarama, 2007) includes classroom activities, with small group activities and computer games Randomized control trials of preschoolers from low-income backgrounds indicated that children given the Building Blocks curriculum made much greater progress than a control group in number, geometry, measurement, and recognition of patterns Overall, these curricula have found success by combining direct instruction, home involvement, and informal learning activities as ways to promote the numerical knowledge of young children Page 10 of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge Conclusions The early home environment and children’s experiences with informal learning activities play a critical role in mathematical development Understanding that these experiences contribute to individual differences in numerical knowledge helped to inform the design of an intervention that can be used to promote the numerical knowledge of young children from lower-income backgrounds A large body of evidence, including our research on board games, provides good reason to advocate that parents and teachers more frequently engage preschoolers in mathematical activities This evidence includes the finding that early numerical knowledge lays the groundwork for later mathematical achievement, and that young children’s number skills can be improved through a variety of informal and formal numerical activities Many of these activities, including the linear number board game, are inexpensive or can be created at home, and require minimal time to play Thus, it seems critical to play such games, engage in other informal maths activities, and make available mathematics curricula of proven effectiveness to a wider range of preschoolers, especially preschoolers from low-income backgrounds References Alexander, K.L., & Entwisle, D.R (1988) Achievement in the first 2 years of school: patterns and processes Monographs of the Society for Research in Child Development, 53(2, Serial No 157) Anderson, A (1997) Families and mathematics: a study of parent–child interactions Journal for Research in Mathematics Education, 28(4), 484–511 Anderson, A., Anderson, A., & Shapiro, J (2004) Mathematical discourse in shared storybook reading Journal for Research in Mathematics Education, 35(1), 5–33 Ansari, D (2008) Effect of development and enculturation on number representation in the brain Nature Reviews Neuroscience, 9(4), 278–291 Aunola, K., Leskinen, E., Lerkkanen, M.K., & Nurmi, J.E (2007) Developmental dynamics of math performance from preschool to grade 2 Journal of Educational Psychology, 96, 699–713 Barbarin, O.A., Early, D., Clifford, R., Bryant, D., Frome, P, Burchinal, M et al (2008) Parental conceptions of school readiness: relation to ethnicity, socioeconomic status, and children’s skills Early Education and Development, 19(5), 671–701 Berch, D B (2005) Making sense of number sense: implications for children with mathematical disabilities Journal of Learning Disabilities, 38, 333–339 Berteletti, I., Lucangeli, D., Piazza, M., Dehaene, S., & Zorzi, M (2010) Numerical estimation in preschoolers Developmental Psychology, 46, 545–551 Bjorklund, D.F., Hubertz, M.J., & Reubens, A.C (2004) Young children’s arithmetic strategies in social context: how parents contribute to children’s strategy development while playing games International Journal of Behavioral Development, 28(4), 347–357 Blevins-Knabe, B., & Musun-Miller, L (1996) Number use at home by children and their parents and its relationship to early mathematical performance Early Development and Parenting, 5, 35–45 Booth, J.L., & Siegler, R.S (2006) Developmental and individual differences in pure numerical estimation Developmental Psychology, 42, 189–201 Booth, J.L., & Siegler, R.S (2008) Numerical magnitude representations influence arithmetic learning Child Development, 79, 1016–1031 Burchinal, M., McCartney, K., Steinberg, L., Crosnoe, R., Friedman, S L., McLoyd, V et al (2011) Examining the Black–White achievement gap among low-income children using the NICHD study of early child care and youth development Child Development, 82(5), 1404–1420 Page 11 of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge Cannon, J., & Ginsburg, H.P (2008) ‘Doing the math’: maternal beliefs about early mathematics versus language learning Early Education and Development, 19(2), 238–260 Clements, D.H., & Sarama, J (2007) Early childhood mathematics learning In F K Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp 461–555) New York: Information Age Publishing Dehaene, S (2011) The Number Sense: How the Mind Creates Mathematics (revised edn) New York: Oxford University Dehaene, S., Bossini, S., & Giraux, P (1993) The mental representation of parity and numerical magnitude Journal of Experimental Psychology: General, 122, 371–396 Dehaene, S., & Brannon, E (Eds) (2011) Space, Time, and Number in the Brain: Searching for the Foundations of Mathematical Thought London, Academic Durkin, K., Shire, B., Riem, R., Crowther, R D., & Rutter, D R (1986) The social and linguistic context of early number word use British Journal of Developmental Psychology, 4, 269–288 Duncan, G J., Dowsett, C J., Claessens, A., Magnuson, K., Huston, A C., Klebanov, P, et al (2007) School readiness and later achievement Developmental Psychology, 43, 1428–1446 Ebersbach, M., Luwel, K., Frick, A., Onghena, P., & Verschaffel, L (2008) The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year-old children: evidence for a segmented linear model Journal of Experimental Child Psychology, 99, 1–17 Gauvain, M (2001) The Social Context of Cognitive Development New York, NY: Guilford Geary, D.C (1994) Children’s Mathematics Development: Research and Practical Applications Washington, DC: American Psychological Association Geary, D.C (2006) Development of mathematical understanding In W Damon & R M Lerner (Series Eds), and D Kuhn & R S Siegler (Volume Eds), Handbook of Child Psychology: Volume 2: Cognition, Perception, and Language (6th edn, pp 777–810) Hoboken, NJ: Wiley Geary, D.C (2011) Cognitive predictors of individual differences in achievement growth in mathematics: a five year longitudinal study Developmental Psychology 47(6), 1539–1552 Geary, D.C., Hoard, M.K., Nugent, L., & Byrd Craven, J (2008) Development of number line representations in children with mathematical learning disability Developmental Neuropsychology, 33, 277–299 Griffin, S (2004) Building number sense with Number Worlds: a mathematics program for young children Early Childhood Research Quarterly, 19(1), 173–180 Gunderson, E.A., & Levine, S.C (2011) Some types of parent number talk count more than others: relations between parents’ input and children’s cardinal-number knowledge Developmental Science, 14, 1021–1032 Holloway, I.D & Ansari, D (2009) Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children’s math achievement Journal of Experimental Child Psychology, 103, 17–29 Holloway, S.D., Rambaud, M.F., Fuller, B., & Eggers-Pierola, C (1995) What is ‘appropriate practice’ at home and in child care? Low-income mothers’ views on preparing their children for school Early Childhood Research Quarterly, 10(4), 451–473 Hubbard, E.M., Piazza, M., Pinel, P., & Dehaene, S (2005) Interactions between numbers and space in parietal cortex Nature Reviews Neuroscience, 6(6), 435–448 Huntsinger, C.S., Jose, P.E., Larson, S.L., Balsink Krieg, D., & Shaligram, C (2000) Mathematics, vocabulary, and reading development in Chinese American and European American children over the primary school years Journal of Educational Psychology, 92, 745−760 Page 12 of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge Jordan, N.C., Huttenlocher, J., & Levine, S.C (1992) Differential calculation abilities in young children from middleand low-income families Developmental Psychology, 28, 644–653 Jordan, N.C., Kaplan, D., Locuniak, M.N., & Ramineni, C (2007) Predicting first-grade math achievement from developmental number sense trajectories Learning Disabilities Research and Practice, 22, 36–46 Jordan, N.C., Kaplan, D., Ramineni, C., & Locuniak, M.N (2009) Early math matters: kindergarten number competence and later mathematics outcomes Developmental Psychology, 45(3), 850–867 Jordan, N.C., Levine, S.C., & Huttenlocher, J (1994) Development of calculation abilities in middle- and low-income children after formal instruction in school Journal of Applied Developmental Psychology, 15, 223–240 Klibanoff, R.S., Levine, S.C., Huttenlocher, J., Vasilyeva, M., & Hedges, L.V (2006) Preschool children’s mathematical knowledge: the effect of teacher ‘Math Talk’ Developmental Psychology, 42(1), 59–69 Le Corre, M.L., Brannon, E.M, Van de Walle, G., & Carey, S (2006) Re-visiting the competence/performance debate in the acquisition of the counting principles Cognitive Psychology, 52, 130–169 Le Corre, M.L & Carey, S (2007) One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal counting principles Cognition, 105, 395–438 Lee, V.E., & Burkam, D (2002) Inequality at the Starting Gate: Social Background Differences in Achievement as Children Begin School Washington, DC: Economic Policy Institute LeFevre, J., Skwarchuk, S., Smith-Chant, B., Fast, L., Kamawar, D., & Bisanz, J (2009) Home numeracy experiences and children’s math performance in the early school years Canadian Journal of Behavioural Science, 41, 55–66 Levine, S.C., Suriyakham, L.W., Rowe, M.L., Huttenlocher, J., & Gunderson, E.A (2010) What counts in the development of young children’s number knowledge? Developmental Psychology, 46, 1309–1319 Lipton, J.S., & Spelke, E.S (2005) Preschool children’s mapping of number words to nonsymbolic numerosities Child Development, 76, 978–988 Locuniak, M.N & Jordan, N.C (2008) Using kindergarten number sense to predict calculation fluency in second grade Journal of Learning Disabilities, 41(5), 451–459 Malofeeva, E., Day, J., Saco, X., Young, L., & Ciancio, D (2004) Construction and evaluation of a number sense test with Head Start children Journal of Educational Psychology, 96(4), 648–659 Mazzocco, M.M.M., & Thompson, R.E (2005) Kindergarten predictors of math learning disability Learning Disabilities Research and Practice, 20(3), 142–155 Melhuish, E., Sylva, K., Sammons, P., Siraj-Blatchford, I., Taggart, B., Phan, M et al (2008) Preschool influences on mathematics achievement Science, 321(5893), 1161–1162 Miller, K.F., Kelly, M.K., & Zhou, X (2005) Learning mathematics in China and the United States: cross-cultural insights into the nature and course of mathematical development In J I D Campbell (Ed.), Handbook of Mathematical Cognition (pp 163–173) New York: Psychology National Mathematics Advisory Panel (2008) Foundations for Success: The Final Report of the National Mathematics Advisory Panel Washington, DC: US Department of Education National Research Council of the National Academies (2009) Mathematics learning in early childhood: paths toward excellence and equity In C T Cross, T A Woods, & H Schweingruber (Eds), Committee on Early Childhood Mathematics Washington, DC: National Academies Newcombe, N.S., Ambady, N., Eccles, J., Gomez, L., Klahr, D., Linn, M et al (2009) Psychology’s role in mathematics and science education American Psychologist, 64(6), 538–550 Plewis, I., Mooney, A., & Creeser, R (1990) Time on educational activities at home and education progress in Page 13 of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge infant school British Journal of Educational Psychology, 60, 330−337 Ramani, G.B., Rowe, M.L., Eason, S.H., & Leech, K.A (2013, April) Parent talk about math during informal learning activities in Head Start families In G Ramani (Chair), The role of input and interaction in early numeracy development Symposium conducted at the biennial meeting of the Society for Research in Child Development, Seattle, WA Ramani, G.B., & Siegler, R.S (2008) Promoting broad and stable improvements in low-income children’s numerical knowledge through playing number board games Child Development, 79, 375–394 Ramani, G.B., & Siegler, R.S (2011) Reducing the gap in numerical knowledge between low- and middle-income preschoolers Journal of Applied Developmental Psychology, 32, 146–159 Ramani, G.B., Siegler, R.S & Hitti, A (2012) Taking it to the classroom: Number board games as a small group learning activity Journal of Educational Psychology, 104, 661–672 Rogoff, B (1990) Apprenticeship in Thinking: Cognitive Development in Social Context New York: Oxford University Rouse, C.E., Brooks-Gunn, J., & McLanahan, S (2005) Introducing the issue Future of Children, 15(1), 5–14 Saxe, G.B (2004) Practices of quantification from a sociocultural perspective In K A Demetriou & A Raftopoulos (Eds), Developmental Change: Theories, Models, and Measurement (pp 241–263) New York, NY: Cambridge University Saxe, G.B., Guberman, S.R., & Gearhart, M (1987) Social processes in early number development Monographs of the Society for Research in Child Development, 52 (Serial No 216) Schneider, M., Grabner, R.H., & Paetsch, J (2009) Mental number line, number line estimation, and mathematical school achievement: their interrelations in Grades 5 and 6 Journal of Educational Psychology, 101, 359–372 Siegler, R.S., & Booth, J (2004) Development of numerical estimation in young children Child Development, 75, 428–444 Siegler, R.S., & Booth, J (2005) Development of numerical estimation: a review In J I D Campbell (Ed.), Handbook of Mathematical Cognition (pp 197–212) New York, NY: Psychology Siegler, R.S., & Mu, Y (2008) Chinese children excel on novel mathematics problems even before elementary school Psychological Science, 19, 759–763 Siegler, R.S., & Ramani, G.B (2008) Playing board games promotes low-income children’s numerical development Developmental Science, Special Issue on Mathematical Cognition, 11, 655–661 Siegler, R.S & Ramani, G.B (2009) Playing linear number board games – but not circular ones – improves lowincome preschoolers’ numerical understanding Journal of Educational Psychology, 101, 545–560 Siegler, R.S., Thompson, C.A., & Schneider, M (2011) An integrated theory of whole number and fractions development Cognitive Psychology, 62, 273–296 Skwarchuk, S-L (2009) How do parents support preschoolers’ numeracy learning experiences at home? Early Childhood Education Journal, 37, 189–197 Sowder, J.T (1992) Estimation and number sense In D A Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp 371–389) New York, NY: Macmillan Starkey, P., & Klein, A (2008) Sociocultural Influences on Young Children’s Mathematical Knowledge: Contemporary Perspectives on Mathematics in Early Childhood Education Charlotte, NC: Information Age Starkey, P., Klein, A., & Wakeley, P (2004) Enhancing young children’s mathematical knowledge through a prekindergarten mathematics intervention Early Childhood Research Quarterly, 19, 99–120 Page 14 of 15 How Informal Learning Activities Can Promote Children’s Numerical Knowledge Tizard, B., & Hughes, M (1984) Children Learning at Home and in School London: Fontana Tudge, J.R.H., & Doucet, F (2004) Early mathematical experiences: observing young black and white children’s everyday activities Early Childhood Research Quarterly, 19, 21–39 Vandermaas-Peeler, M., Nelson, J., Bumpass, C., & Sassine, B (2009) Numeracy-related exchanges in joint storybook reading and play International Journal of Early Years Education, 17(1), 67–84 Vygotsky, L S (1976) Play and its role in the mental development of the child In J S Bruner, A Jolly, & K Sylva (Eds), Play (pp 537–559) New York, NY: Harper and Row Vygotsky, L.S (1978) Mind and Society: The Development of Higher Psychological Processes Cambridge, MA: Harvard University White, G., Frishkoff, G., & Bullock, M (2008) Bridging the gap between psychological science and educational policy and practice In S Thurman, C A Fiorello, S Thurman, C A Fiorello (Eds), Applied Cognitive Research in K– 3 Classrooms (pp 227–263) New York, NY: Routledge/Taylor and Francis Group Wilson, A.J., Dehaene, S., Dubois, O., & Fayol, M (2009) Effects of an adaptive game intervention on accessing number sense in low-socioeconomic-status kindergarten children Mind, Brain, and Education, 3, 224–234 Wilson, A.J., Revkin, S.K., Cohen, D., Cohen, L., & Dehaene, S (2006) An open trial assessment of ‘The Number Race’, an adaptive computer game for remediation of dyscalculia Behavioral and Brain Functions, 2, 20 Zhou, X., Huang, J., Wang, Z., Wang, B., Zhao, Z., Yang, L., et al (2006) Parent-child interaction and children’s number learning Early Child Development and Care, 176, 763–775 Geetha B Ramani Geetha B Ramani, Department of Human Development, University of Maryland Robert S Siegler Robert S Siegler, Department of Psychology, Carnegie Mellon University, Pittsburgh, PA Page 15 of 15 ... Budd, FIMA, Cmath, University of Bath Page 11 of 12 Philosophy of Number Oxford Handbooks Online Philosophy of Number Marcus Giaquinto The Oxford Handbook of Numerical Cognition (Forthcoming) Edited by Roi Cohen Kadosh and Ann Dowker... Cohen Kadosh & A Dowker (Eds.), The Oxford Handbook of Numerical Cognition Oxford: Oxford University Press Campbell, J.I.D (1994) Architectures for numerical cognition Cognition, 53(1), 1–44 Campbell, J.I.D... detain us, but an essential element is that things are regarded as falling into exclusive layers or ‘types’ – ordinary individual items are of type 0, sets of individuals are of type 1, sets of sets of individuals are of type 2, and in general sets of things of type n are of type n + 1 Then each number k splits into many, the set of all k-membered