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https://www.onmaths.com/mock_exams/?archiveType=demon

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https://filestore.aqa.org.uk/resources/mathematics/AQA-8365-FS-INS.PDF

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http://www.benjamin-mills.com/maths/Year11/circle-theorems-proof.pdf
NB First 4 especially (5th one I do differently)

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https://corbettmaths.com/more/further-maths/

http://www.mrbartonmaths.com/students/aqa-level-2-certificate-in-further-mathematics/topic-tests.html

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http://www.mrbartonmaths.com/students/aqa-level-2-certificate-in-further-mathematics/past-papers-and-solutions.html#practice1

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https://qualifications.pearson.com/content/dam/pdf/GCSE/mathematics/2015/misc/U219_GCSE_Maths_Poster.pdf

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http://thomaswhitham.pbworks.com/w/file/fetch/63556547/Further%20maths%20GCSE%20revision%20guide.pdf

https://www.thechalkface.net/resources/FML2/FML2_Need_To_Know.pdf

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https://www.desmos.com/calculator

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https://challenges.ukmt.org.uk/index.php?action=anon_join&grp_id=232

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https://www.interactive-maths.com/ukmt-random-question-generator.html

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equation solving tools - wolfram alpha/microsoft

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equation solving tools - wolfram alpha/microsoft

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check non-linear simultaneous and algebraic fraction equation solutions etc

https://www.wolframalpha.com/input?i2d=true&i=y-4x%3D5%5C%2844%29+y%3D4Power%5Bx%2C2%5D%2B3x-16

https://mathsolver.microsoft.com/en/solve-problem/y-4x%20%3D%20%205%2Cy%20%3D%20%204%20%7B%20x%20%20%7D%5E%7B%202%20%20%7D%20%20%2B3x-16

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largest negative integer

-0.083333333333333

0.5

real part of all non-trivial zeros of the Riemann zeta function - if the Riemann hypothesis is true!

0.577215664901532

Euler-Mascheroni constant γ

the limiting difference between the harmonic series and the natural logarithm

0.915965594177219

Catalan's constant G

unknown whether irrational or not

1.059463094359295

proportion between the frequencies of adjacent semitones in the 12 tone equal temperament scale

1.202056903159594

appears in quantum electrodynamics and random spanning trees

1.259921049894873

can't be constructed using ruler a compass, which was one of the three famous geometric problems of antiquity

1.414213562373095

first known irrational number, discovered by Pythagoras or one of his students

ratio of diagonal and the side length in a square

proportion between the sides of DIN paper sizes

1.618033988749894

Golden ratio φ

ratio of the side and one of the diagonals in a pentagon

1.732050807568877

the measure of the fish

length of the space diagonal of a cube with edge length 1

height of an equilateral triangle with side length 2

2.236067977499789

length of the diagonal of a 1 × 2 rectangle

2.502907875095892

second Feigenbaum constant α

ratio between the width of a tine and the width of one of its two subtines

2.718281828459045

Euler's number e

3.141592653589793

circumference of a circle with diameter 1

transcendental number, which means that the famous ancient problem of squaring a circle is impossible

closely approximated by or

4.669201609102990

first Feigenbaum constant δ

limiting ratio of consecutive bifurcation intervals in a one-parameter logistic map

6.283185307179586

circumference of a circle with radius 1

period of the sin, cos and tan functions

9

maximum number of cubes needed to sum to any positive integer

number of muses in Greek mythology

number of significant points in a triangle that lie on a circle: the Nine-Point circle

16

number of pawns in a chess set, and each player starts with 16 pieces

length of credit card numbers

23.140692632779269

24

largest number divisible by all integers less than its square root

number of hours in a day

26

number of sporadic groups

only positive number to be directly between a square and a cube

smallest non-palindromic number whose square is a palindrome:

27

sum of the digits of its cube:

in the Collaz Conjecture you need 112 steps to get from 27 to 1

32

ninth happy number

smallest nontrivial fifth power

sum of the totient functions of the first ten integers

freezing point of water at sea level in Fahrenheit

41

lucky number of Euler, and the largest such prime

number of Mozart's last symphony

45

triangular, hexagonal and 16-gonal number

47

largest number of cubes that cannot tile a cube

atomic number of silver

Carol number

49

multiplying the period of by integers coprime to 49 permutes the digits

type in your calculator and you will be very surprised...

50

represented by the letter L in Roman numerals

57

can be written as 111 in base 7

the 57-cell is a self-dual abstract regular four-dimensional polytope.

57.295779513082320

number of degrees in 1 radian

63

number of partially ordered sets of 5 elements

64

number of tiles on a chess board

maximum number of strokes in any Chinese character

smallest number with 7 divisors

index of Graham's number in the sequence 3, 27, 7625597484987, ...

65

smallest number that becomes a square when its reverse is either added to or subtracted from it

74

number of non-Hamiltonian polyhedra with a minimum number of vertices

78

smallest integer that can be written as the sum of four distinct squares in three ways

81

number of stable chemical elements

only non-trivial number which is the square of the sum of its digits

its inverse repeats all 10 decimal digits in order

89

11th Fibonacci number

91

smallest non-trivial cabtaxi number

sum of the squares of the first six integers.

95

smallest non-interesting number

96

untouchable number, since it cannot be expressed as the sum of some of the proper divisors of another integer

98

the decimal expansion of its reciprocal starts with the powers of two:

100

boiling point of water in Celsius

denoted by the letter C in Roman numerals

113

A113 was the Pixar animation classroom at the California Institute of the Arts

118

smallest number which can be written as the sum of three integers in four different ways, so that the product of every triple is the same: 118 = 14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 + 72

128

smallest number that is the product of 7 prime factors ()

largest number that is not the sum of distinct squares

power of 2, all of whose digits are powers of 2 (unknown if there are any other numbers with this property)

132

sixth Catalan number

145

one of just four numbers to be the sum of the factorials of its digits: 145 = 1! + 4! + 5!

153

sum of the first 5 factorials and the first 17 natural numbers

212

the boiling point of water in Fahrenheit

255

largest number that can be represented using an 8-bit unsigned integer

261

number of hypercube unfoldings

341

smallest base 2 Fermat pseudoprime

365.2425

average number of days in a year, consisting of 365 days, 5 hours 48 minutes and 46.08 seconds

380

the A380 is the world's largest passenger aircraft.

500

denoted by the letter D in Roman numerals

676

smallest palindromic square whose square root 26 is not palindromic

747

the Boeing 747 aircraft is called the "Queen of the Skies"

786

sacred number in Muslim Abjad numerology

836

second weird number, after 70

1 000

1 001

is the product of three consecutive primes 7 × 11 × 13

1 089

1089 × 9 = 9801, the reverse of its digits

1 138

title of George Lucas' first film: THX 1138

1 540

one of just 5 numbers that is both triangular and tetrahedral

1 729

Hardy-Ramanujan number and second taxicab number

2 240

number of pounds in an (English) ton

2 310

2 592

3 003

smallest number to appear 8 times in Pascal's triangle (there is no other number less than 2^23 appearing 8 times)

3 600

number of seconds in one hour

3 999

largest number that can be written in Roman numerals, as MMMCMXCIX

4 900

the only square pyramidal number that is also a square:

5 040

mentioned by Plato as the ideal number of people living in a city, due to its high number of factors

5 913

sum of the first 7 factorials: 5913 = 1! + 2! + 3! + ... + 6! + 7!

6 174

Kaprekar's constant: take any four-digit number, rearrange its digits to make the largest and smallest possible number, subtract these numbers and repeat. Eventually you will arrive at 6174!

8 000

8 128

fourth perfect number

9 694

number of hypercube unfoldings in 5D

12 758

largest number that cannot be represented as the sum of distinct cubes

17 163

largest number that is not the sum of the squares of distinct primes

26 830

number of possible Tic-tac-toe games

40 585

sum of the factorials of its digits: 40,585 = 4! + 0! + 5! + 8! + 5!

40 755

first number, other than 1, that is simultaneously triangular, pentagonal and hexagonal

142 857

smallest base 10 cyclic number: multiplied by 2, 3, 4, 5 or 6, it simply rotates is decimal digits

recurring decimal expansion of

196 560

the number of spheres touching any other sphere in a 24-dimensional Leech lattice

502 110

number of hypercube unfoldings in 6D

666 666

area of the Pythagorean triangle with sides 693, 1924 and 2045

2 598 960

number of distinct, 5-card poker hands

4 729 494

a coefficient in Archimedes' cattle problem

31 557 600

number of seconds in a Julian astronomical year (365.25 days)

33 064 966

number of hypercube unfoldings in 7D

33 550 336

5th perfect number

275 305 224

the number of distinct magic squares of order 5 (excluding rotations and reflections)

299 792 458

speed of light in vacuum

739 391 133

the largest prime number (in base 10), that you can truncate to produce a sequence of smaller primes: 7, 73, 739, ...

2 642 657 228

number of hypercube unfoldings in 8D

3 816 547 290

the only 10-digit number such that, for every k between 1 and 10, its first k digits give a number that is a multiple of k

8 589 869 056

6th perfect number

9 814 072 356

largest perfect power that contains no repeated digits in base ten

137 438 691 328

7th perfect number

158 753 389 900

the probability of receiving a complete suit in the game of Bridge

248 639 631 948

number of hypercube unfoldings in 9D

262 537 412 640 768 743.999999999

a transcendental number which is an almost integer

2 305 843 008 139 952 128

8th perfect number, discovered by Euler in 1772

18 446 744 073 709 551 615

equals

the number of grains of rice required if you place one grain on the first square of a chess board, two on the second, 4 on the third, 8 on the fourth, and so on.

357 686 312 646 216 567 629 137

the largest prime number (in base 10), that you can truncate from its beginning to produce a sequence of smaller primes: 7, 37, 137, 9137, ...

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Almanac of Interesting Numbers - Mathigon

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largest negative integer real part of all non-trivial zeros of the Riemann zeta function - if the Riemann hypothesis is true!

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BLOG CARNIVAL #163....LET'S GO! Fun fact: The number 163 is prime, which we can prove simply by showing that it is not divisible by 2, 3...

The scales below are balanced. Each scale is a different puzzle. For each scale, start by taking off as many blocks as you can from each si...

For the solution, click "Read More" below.

To a topologist, two shapes are "the same" if one can be stretched, twisted, and distorted into the other shape without b...

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A math blog by Sian Zelbo for elementary and middle school. Emphasizes problem solving and mathematical reasoning.

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Fourier transforms are a tool used in a whole bunch of different things. This is an explanation of what a Fourier transform does, and some different ways it can be useful. And how you can make pretty things with it, like this thing:

I'm going to explain how that animation works, and along the way explain Fourier transforms!

By the end you should have a good idea about

What a Fourier transform does

Some practical uses of Fourier transforms

Some pointless but cool uses of Fourier transforms

We're going to leave the mathematics and equations out of it for now. There's a bunch of interesting maths behind it, but it's better to start with what it actually does, and why you'd want to use it first. If you want to know more about the how, there's some further reading suggestions below!

So what is this thing?

Put simply, the Fourier transform is a way of splitting something up into a bunch of sine waves. As usual, the name comes from some person who lived a long time ago called Fourier.

Let's start with some simple examples and work our way up. First up we're going to look at waves - patterns that repeat over time.

Here's an example wave:

This wavy pattern here can be split up into sine waves. That is, when we add up the two sine waves we get back the original wave.

The Fourier transform is a way for us to take the combined wave, and get each of the sine waves back out. In this example, you can almost do it in your head, just by looking at the original wave.

Why? Turns out a lot of things in the real world interact based on these sine waves. We usually call them the wave's frequencies.

The most obvious example is sound - when we hear a sound, we don't hear that squiggly line, but we hear the different frequencies of the sine waves that make up the sound.

Play Full Wave

Play High Frequency

Play Low Frequency

Being able to split them up on a computer can give us an understanding of what a person actually hears. We can understand how high or low a sound is, or figure out what note it is.

We can also use this process on waves that don't look like they're made of sine waves.

Let's take a look at this guy. It's called a square wave.

It might not look like it, but it also can be split up into sine waves.

We need a lot of them this time - technically an infinite amount to perfectly represent it. As we add up more and more sine waves the pattern gets closer and closer to the square wave we started with.

Play Wave
Drag the slider above to play with how many sine waves there are.
Visually, you'll notice that actually the first few sine waves are the ones that make the biggest difference. With the slider halfway, we have the general shape of the wave, but it's all wiggly. We just need the rest of the small ones to make the wigglyness flatten out.

When you listen to the wave, you'll hear the sound get lower, because we're removing the higher frequencies.

This process works like that for any repeating line. Give it a go, try drawing your own!

Play Wave
Move the slider to see how as we add more sine waves, it gets closer and closer to your drawing
Again, aside from the extra wigglyness, the wave looks pretty similar with just half of the sine waves.

We can actually use the fact that the wave is pretty similar to our advantage. By using a Fourier transform, we can get the important parts of a sound, and only store those to end up with something that's pretty close to the original sound.

Normally on a computer we store a wave as a series of points.

What we can do instead is represent it as a bunch of sine waves. Then we can compress the sound by ignoring the smaller frequencies. Our end result won't be the same, but it'll sound pretty similar to a person.

This is essentially what MP3s do, except they're more clever about which frequencies they keep and which ones they throw away.

So in this case, we can use Fourier transforms to get an understanding of the fundamental properties of a wave, and then we can use that for things like compression.

Ok, now let's dig more into the Fourier transform. This next part looks cool, but also gives you a bit more understanding of what the Fourier transform does. But mostly looks cool.

Epicycles

Now at the start, I said it splits things into sine waves. The thing is, the sine waves it creates are not just regular sine waves, but they're 3D. You could call them "complex sinusoids". Or just "spirals".

If we take a look from the side, they look like sine waves. From front on, though, these look like circles.

So far everything we've been doing has only required the regular 2D sine waves. When we do a Fourier transform on 2D waves, the complex parts cancel out so we just end up with sine waves.

But we can use the 3D sine waves to make something fun looking like this:

What's going on here?

Well, we can think of the drawing as a 3D shape because of the way it moves around in time. If you imagine the hand being drawn by a person, the three dimensions represent where the tip of their pencil is at that moment. The x and y dimensions tell us the position, and then the time dimension is the time at that moment.

Now that we have a 3D pattern, we can't use the regular 2D sine waves to represent it. No matter how many of the 2D sine waves we add up, we'll never get something 3D. So we need something else.

What we can use is the 3D spiral sine waves from before. If we add up lots of those, we can get something that looks like our 3D pattern.

Remember, these waves look like circles when we look at them from front on. The name for the pattern of a circle moving around another circle is an epicycle.
Use the slider above to control how many circles there are.
Like before, we get a pretty good approximation of our pattern with just a few circles. Because this is a fairly simple shape, all the last ones do is make the edges a little sharper.

All this applies to any drawing, really! Now it's your chance to play around with it.
Use the slider to control how many circles are used for your drawing
Again, you'll see for most shapes, we can approximate them fairly well with just a small number of circles, instead of saving all the points.

Can we use this for real data? Well, we could! In reality we have another data format called SVG, which probably does a better job for the types of shapes we tend to create. So for the moment, this is really just for making cool little gifs.

There is another type of visual data that does use Fourier transforms, however.

JPEGs

Did you know Fourier transforms can also be used on images? In fact, we use it all the time, because that's how JPEGs work! We're applying the same principles to images - splitting up something into a bunch of sine waves, and then only storing the important ones.

Now we're dealing with images, we need a different type of sine wave. We need to have something that no matter what image we have, we can add up a bunch of these sine waves to get back to our original image.

To do that, each of our sine waves will be images too. Instead of a wave that's a line, we now have images with black and white sections. To represent the size of a wave, each image will have more or less contrast.

We can also use these to represent color in the same way, but let's start with black-and-white images for now. To represent colorless images, we need some horizontal wave images,

Along with some vertical wave images.

By themselves, just horizontal and vertical images aren't enough to represent the types of images we get. We also need some extra ones that you get by multiplying the two together.

For an 8x8 image, here are all the images we need.

If we take the images, adjust their contrast to the right amount, and then add them up we can create any image.

Let's start with this letter 'A'. It's pretty small, but we need it to be small otherwise we'll end up with too many other images.

As we add more and more of these images, we end up with something that becomes closer and closer to the actual image. But I think you'll see the pattern here, as we get a reasonable approximation with just a few of them.

For actual JPEG images there are just a few extra details.

The image gets broken up into 8x8 chunks, and each chunk gets split up separately. We use a set of frequencies to determine how light or dark each pixel is, and then another two sets for the color, one for red-green, and another for blue-yellow. The number of frequencies that we use for each chunk determines the quality of the JPEG.

Here's a real JPEG image, zoomed in so we can see the details. When we play with the quality levels we can see this process happen.

Conclusion

So let's recap:

Fourier transforms are things that let us take something and split it up into its frequencies.

The frequencies tell us about some fundamental properties of the data we have

And can compress data by only storing the important frequencies

And we can also use them to make cool looking animations with a bunch of circles

This is just scratching the surface into some applications. The Fourier transform is an extremely powerful tool, because splitting things up into frequencies is so fundamental. They're used in a lot of fields, including circuit design, mobile phone signals, magnetic resonance imaging (MRI), and quantum physics!

Questions for the curious

I skipped most of the math stuff here, but if you're interested in the underlying principles of how it works, here are some questions you can use to guide your research:

How do you mathematically represent a Fourier transform?

What's the difference between a continuous time Fourier transform and a discrete time Fourier transform?

How do you computationally do a Fourier transform?

How do you do a Fourier transform of a whole song? (Rather than just a single note.)

Further 'reading'

To learn more, some really good resources you can check out are:

An Interactive Guide To The Fourier Transform A great article that digs more into the mathematics of what happens.

But what is the Fourier Transform? A visual introduction. A great Youtube video by 3Blue1Brown, also explaining the maths of Fourier transforms from an audio perspective.

A Tale of Math & Art: Creating the Fourier Series Harmonic Circles Visualization Another article explaining how you can use epicycles to draw a path, explained from a linear algebra perspective.

Fourier transform (Wikipedia) And of course, the Wikipedia article is pretty good too.

The author

I'm Jez! Full time I work at a search company in the Bay Area, and in my spare time I like making games and interactive code things like this!

This webpage is open-source, you can check out the code on GitHub! If you have any feedback or want to ask any questions, feel free to email me at fourier [at] jezzamon [dot] com, or shoot me a tweet on Twitter.

If you want to see more of my work, check out my homepage, and if you want to see what I'm making next, you can follow my Twitter account, @jezzamonn!

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An Interactive Introduction to Fourier Transforms

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Fourier transforms are a tool used in a whole bunch of different things. This is an explanation of what a Fourier transform does, and some different ways it can be useful.

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angle https://www.geogebra.org/m/yc7ayebw

% of circle http://www.amathsteacherwrites.co.uk/percent_estimate_game/

% of square https://www.geogebra.org/m/aZqP2EUX

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integral mathematics resources: log in to the site

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Integral Mathematics Resources: Log in to the site

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Devoted to visualising educational concepts and processes

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Make 'Em Think Maths - Effective Questioning

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Asking the right question at the right time can lead to excellent outcomes and a deeper understanding of the Mathematics they are learning. It encourages articulation and the sharing of ideas. Plan the questions you will ask and then good questioning will become habit. #goodhabits

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YOHAKU JUNIOR BOOKS 1, 2 AND 3 ARE NOW ON SALE!

A yohaku is a new type of number puzzle that will test your number sense and problem solving skills. Each yohaku is either an additive or a multiplicative puzzle (as indicated by the symbol in the bottom right of the grid). Your task is to fill in the empty cells such that they give the sum or product shown in each row and column.

Great for use in classrooms as a lesson starter, or for the morning commute, or for something to accompany an afternoon cup of tea. Some yohakus will take a few minutes to solve, others will have you puzzling for longer (and you're welcome).

Each yohaku is lovingly created by hand!​

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YOHAKU JUNIOR BOOKS 1, 2 AND 3 ARE NOW ON SALE! A yohaku is a new type of number puzzle that will test your number sense and problem solving skills. Each yohaku is either an additive or a multiplicative puzzle (as indicated by the symbol in the bottom right of the grid).

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Interactive fraction wall for use in the classroom. Double click the fractions to rotate through 90 degrees.

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https://mathsbot.com/practicePaper

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https://estimation180.com/days/

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NonExamples.com - Something that is not an example of a rule or definition already shown, used for a clearer explanation.

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Updated for 2021 - Resources Spreadsheet (based on WR)

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In addition to face-to-face events, the Loughborough University Mathematics Education Network offers a growing series of free, online professional development videos, which are available from this page.

Some are presented by colleagues from within Loughborough University's Department of Mathematics Education, and others by external experts within mathematics education. Some videos have been designed mainly for secondary teachers, others mainly for primary/early years teachers, and others for anyone. We hope that there is something here for everyone.

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Professional development | LUMEN | Loughborough University

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In addition to face-to-face events, the Loughborough University Mathematics Education Network offers a growing series of free, online professional development videos, which are available from this page. Some are presented by colleagues from within Loughborough University's Department of Mathematics Education, and others by external experts within mathematics education.

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gradient https://www.geogebra.org/m/bZAkArEt

equation of a line https://www.geogebra.org/m/BhZ6CVZX

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Welcome to my 137 th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

Thank you to Nicola Whiston (@whisto_maths) for sharing a maths guided reading task, using a template created by @MrHand__.

I've not used activities like this in maths before. It looks like it would be great for enrichment, and would also help to develop a very important skill (for everyone, not just for students who go on to take MEI A level maths which involves a comprehension task).

More resources like this were shared in this thread from Katie Pollard (@takepi21). Here's Katie's 'Saving lives with maths' guided reading task:

These teachers have generously shared their tasks here so others can borrow them.

Thank you also to Miss Olive (@missolivemaths) who wrote a series of comprehension exercises for a range of mathematical texts and shared them here back in July.

2. A Level Support
Thank you to @EmporiumMaths for sharing over 120 free videos with practice and exam questions to help students master the key skills that are needed in the transition from GCSE Maths to A Level Maths.

Thank you also to @DavidB52s for pointing me towards MEI's page of Useful resources for students self-studying AS/A level Mathematics and Further Mathematics. If you're setting work remotely for Sixth Form students then this page will be really helpful. It includes links to transition material, A level maths videos, teaching resources, and material for Further Maths. Check out the self-paced online course for students aiming for an A* in A level maths - it's incredible that this is free to access!

I have been thinking about standard form and thought I would share. I Like using the Gattegno chart to represent SF. Also asking pupils to explain why statements are true gets them reasoning and building up understanding. #MakeEmThinkMaths pic.twitter.com/wWCEhh6vUF
- Mr McGuire (@McGuirea499) October 3, 2020

I've shared this before, but it's been updated so it's a good time to share it again. Thank you to @MathigonOrg for the interactive Timeline of Mathematics. Click on each mathematician to read more about them.

Just like everything else on Mathigon, this is excellent.

Thank you to all of these teachers for generously share their resources for free.

CPD
Do check out the free videos on my CPD channel if you get a chance. This is where I share the presentations I have previously done at conferences.

If you enjoy these presentations, then do sign up to Marvellous Maths 2, which is a full online CPD course for maths teachers from me and Craig Barton. For only £30 you get absolutely tonnes of content, and once you've enrolled you can access it at any time, so you don't have to do the whole course at once.

Also check out my newly updated conference listings page, which lists upcoming national conferences for maths teachers.

24 Hour Maths Magic Show
It was an honour to be involved in 24 Hour Maths this weekend. This 24-hour long mathematical magic show featured lots of incredibly talented people sharing mathematical magic tricks and puzzles on YouTube. I really enjoyed taking part (Sophie Maclean did the brilliant Kruskal's Count trick on me! Clever stuff). I also loved watching the rest of the tricks (admittedly I didn't watch the whole 24 hours but will catch up on what I missed!). Well done to the team of organisers, and everyone involved.

If you want to bring some mathematical joy into your life, I highly recommend watching the whole show (the list of guests and timings is here). If you don't have time for the full 24 hours, a good starting point is Chris Smith's (@aap03102) 30-minute guest segment, which had me smiling from start to finish. He shares a trick which is very easy to replicate with students, plus a treat (the Halloween Octagon), plus a brilliant song. Watch here... 👇 Enjoy!

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Resourceaholic - free resources and ideas for teaching secondary school maths

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These beautiful flash cards use spaced repetition to teach multiplication facts. Achieve fluency with just five minutes of practice per day!

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https://www.mathstunners.org/

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https://mathsbot.com/gcse/revisionRecall

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Contains a virtual and easily modifiable triangular prism to help students better conceptually "see" the meanings of the terms "total surface area" a...

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This has been the Centre's main focus over the past few years. The Mathematics Enhancement Programme set out to implement in UK schools the findings of international research. As well as detail of the philosophy and style of teaching advocated, there is also a full set of curriculum resources for Primary and Secondary schools, including pupil texts, lesson plans, classroom resources, assessment materials and on-line interactive resources. Mathematics Enhancement Programme (MEP)

http://www.cimt.org.uk/projects/mep/

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Centre for Innovation in Mathematics Teaching

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This has been the Centre's main focus over the past few years. The Mathematics Enhancement Programme set out to implement in UK schools the findings of international research. As well as detail of the philosophy and style of teaching advocated, there is also a full set of curriculum resources for Primary and Secondary schools, including pupil texts, lesson plans, classroom resources, assessment materials and on-line interactive resources.

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[Official CASIO Scientific & Graphing Calculator website] Choose the best from our Handheld and Software of Scientific/ Graphing Calculators.

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The mission of Gapminder Foundation is to fight devastating ignorance with a fact-based worldview that everyone can understand. We started the Ignorance Project to investigate what the public know and don't know about basic global patterns and macro-trends. We use surveys to ask representative groups of people simple questions about key-aspects of global development.

When we find large knowledge-gaps, we know what teaching materials we should develop. The first results from surveys in UK and Sweden were published in 2013. As the project evolves we will investigate many more countries. The test questions and results will be made freely available under Creative Commons Attribution License.

Background

The Gapminder Foundation started as a spin-off from Professor Hans Rosling's teaching at the Karolinska Institute in Stockholm. He encountered broad ignorance about the rapid health improvement in Asia. He started measuring the ignorance among students and professors and the surprising results from the so called "Chimpanzee Test" were presented in his first TED-talk in 2006.

In the test question Hans combined 5 pairs of countries. Each pair had one Asian country and one European country. He asked the students to pick the country in each pair which had twice the child mortality of the other country. If the country names had been written on 5 pairs of bananas, on average the chimpanzee would score 2.5 correct answers. To Hans's great surprise his Global Health Students performed worse than the chimps, i.e. worse than random. Therefore the wrong answers could not be the results of guessing. They must be due to preconceived ideas that in a systematic way created and maintained ignorance. Only preconceived ideas can make us perform worse than random. When he replicated the test with the professors at the university, he found their results to be equivalent with those of the chimpanzees.

Since then, Gapminder has visualized lots of different data and presented it to lots of different people. During that work we have encountered all kinds of pre-conceived ideas and outdated concepts about our contemporary world. Our priorities have been guided by such ignorance-encounters. With the Ignorance Project we have now decided to start a systematic search for widespread ignorance about the world.

Why is there so much ignorance?

Statistical facts don't come to people naturally. Quite the opposite. Most people understand the world by generalizing personal experiences which are very biased. In the media the "news-worthy" events exaggerate the unusual and put the focus on swift changes. Slow and steady changes in major trends don't get much attention. Unintentionally, people end-up carrying around a sack of outdated facts that you got in school (including knowledge that often was outdated when acquired in school).

What Gapminder does

With the Ignorance Project we identify the specific global statistical trends that have not reached a broad public audience. We ask people questions about major trends and patterns. The questions cover major aspects of global development such as: Environment, Health, Energy, Gender, Economy, Demography and Governance. The most time-consuming part in this work is to write and design the questions and the response alternatives. They need to be precise and clear and yet not complicated. The question need to be about a well-documented fact of great importance, and still be concrete so that there is just one correct response alternative. The questions and the response alternatives need to be understandable to anyone, especially to those with no particular interest in the topic. In Sweden and UK, we have done the initial surveys in close collaboration with a leading survey company, Novus Group International AB.

When we encounter ignorance, we want to find a cure. Sometimes the facts just have to be delivered. But in many cases the facts may be hard to accept, as they don't fit with other misunderstandings. They seem counterintuitive. In these cases we need to invent new simple ways to explain them. Those new explanations are the essence of Gapminder's new free teaching material that make it fun and easy to teach and learn a fact-based worldview.

In addition to spreading the explanations and data, we also give away the facts about the ignorance itself, and the questionnaires and the references to our data sources. In this way we hope to enable others to measure the knowledge about the world among their students, colleagues, employees, applicants or website visitors, without having to look up the data-sources and write the questions from scratch.

A typical question is:
What is the Life expectancy of the world population? A. Around 50 years
B. Around 60 years
C. Around 70 years
The right answer is C. The life expectancy of the world population is 70 years.

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The mission of Gapminder Foundation is to fight devastating ignorance with a fact-based worldview that everyone can understand. We started the Ignorance Project to investigate what the public know and don't know about basic global patterns and macro-trends. We use surveys to ask representative groups of people simple questions about key-aspects of global development.

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3D Pythagoras / Trigonometry visualisation

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3D Pythagoras / Trigonometry visualisation

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This is a draft of a reflections book in geometry

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A growing bank of randomly generated GCSE exam style questions with full worked solutions. Perfect for projecting in the classroom. Choose between single or split screen mode for ‘my turn, your turn’ worked examples.

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https://www.mathsgenie.co.uk/gcse.html

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https://www.mrbartonmaths.com/students/aqa-level-2-certificate-in-further-mathematics/video-guides.html#

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23 people. In a room of just 23 people there's a 50-50 chance of at least two people having the same birthday. In a room of 75 there's a 99.9% chance of at least two people matching.

Put down the calculator and pitchfork, I don't speak heresy. The birthday paradox is strange, counter-intuitive, and completely true. It's only a "paradox" because our brains can't handle the compounding power of exponents. We expect probabilities to be linear and only consider the scenarios we're involved in (both faulty assumptions, by the way).

Let's see why the paradox happens and how it works.

Problem 1: Exponents aren't intuitive

We've taught ourselves mathematics and statistics, but let's not kid ourselves: it's not natural.

Here's an example: What's the chance of getting 10 heads in a row when flipping coins? The untrained brain might think like this:

"Well, getting one head is a 50% chance. Getting two heads is twice as hard, so a 25% chance. Getting ten heads is probably 10 times harder... so about 50%/10 or a 5% chance."

And there we sit, smug as a bug on a rug. No dice bub.

After pounding your head with statistics, you know not to divide, but use exponents. The chance of 10 heads is not .5/10 but $.5^10$, or about .001.

But even after training, we get caught again. At 5% interest we'll double our money in 14 years, rather than the "expected" 20. Did you naturally infer the Rule of 72 when learning about interest rates? Probably not. Understanding compound exponential growth with our linear brains is hard.

Problem 2: Humans are a tad bit selfish

Take a look at the news. Notice how much of the negative news is the result of acting without considering others. I'm an optimist and do have hope for mankind, but that's a separate discussion :).

In a room of 23, do you think of the 22 comparisons where your birthday is being compared against someone else's? Probably.

Do you think of the 231 comparisons where someone who is not you is being checked against someone else who is not you? Do you realize there are so many? Probably not.

The fact that we neglect the 10 times as many comparisons that don't include us helps us see why the "paradox" can happen.

Ok, fine, humans are awful: Show me the math!

The question: What are the chances that two people share a birthday in a group of 23?

Sure, we could list the pairs and count all the ways they could match. But that's hard: there could be 1, 2, 3 or even 23 matches!

It's like asking "What's the chance of getting one or more heads in 23 coin flips?" There are so many possibilities: heads on the first throw, or the 3rd, or the last, or the 1st and 3rd, the 2nd and 21st, and so on.

How do we solve the coin problem? Flip it around (Get it? Get it?). Rather than counting every way to get heads, find the chance of getting all tails, our "problem scenario".

If there's a 1% chance of getting all tails (more like .5^23 but work with me here), there's a 99% chance of having at least one head. I don't know if it's 1 head, or 2, or 15 or 23: we got heads, and that's what matters. If we subtract the chance of a problem scenario from 1 we are left with the probability of a good scenario.

The same principle applies for birthdays. Instead of finding all the ways we match, find the chance that everyone is different, the "problem scenario". We then take the opposite probability and get the chance of a match. It may be 1 match, or 2, or 20, but somebody matched, which is what we need to find.

Explanation: Counting Pairs (Approximate Formula)

With 23 people we have 253 pairs:

(Brush up on combinations and permutations if you like).

The chance of 2 people having different birthdays is:

Makes sense, right? When comparing one person's birthday to another, in 364 out of 365 scenarios they won't match. Fine.

But making 253 comparisons and having them all be different is like getting heads 253 times in a row -- you had to dodge "tails" each time. Let's get an approximate solution by pretending birthday comparisons are like coin flips. (See Appendix A for the exact calculation.)

We use exponents to find the probability:

Our chance of getting a single miss is pretty high (99.7260%), but when you take that chance hundreds of times, the odds of keeping up that streak drop. Fast.

The chance we find a match is: 1 - 49.95% = 50.05%, or just over half! If you want to find the probability of a match for any number of people n the formula is:

Interactive Example

I didn't believe we needed only 23 people. The math works out, but is it real?

You bet. Try the example below: Pick a number of items (365), a number of people (23) and run a few trials. You'll see the theoretical match and your actual match as you run your trials. Go ahead, click the button (or see the full page).

As you run more and more trials (keep clicking!) the actual probability should approach the theoretical one.

Examples and Takeaways

Here are a few lessons from the birthday paradox:

$\sqrt{n}$ is roughly the number you need to have a 50% chance of a match with n items. $\sqrt{365}$ is about 20. This comes into play in cryptography for the birthday attack.

Even though there are 2¹²⁸ (1e38) GUIDs, we only have 2⁶⁴ (1e19) to use up before a 50% chance of collision. And 50% is really, really high.

You only need 13 people picking letters of the alphabet to have 95% chance of a match. Try it above (people = 13, items = 26).

Exponential growth rapidly decreases the chance of picking unique items (aka it increases the chances of a match). Remember: exponents are non-intuitive and humans are selfish!

After thinking about it a lot, the birthday paradox finally clicks with me. But I still check out the interactive example just to make sure.

Appendix A: Repeated Multiplication Explanation (Exact Formula)

Remember how we assumed birthdays are independent? Well, they aren't.

If Person A and Person B match, and Person B and C match, we know that A and C must match also. The outcome of matching A and C depends on their results with B, so the probabilities aren't independent. (If truly independent, A and C would have a 1/365 chance of matching, but we know it's a 100% guaranteed match.)

When counting pairs, we treated birthday matches like coin flips, multiplying the same probability over and over. This assumption isn't strictly true but it's good enough for a small number of people (23) compared to the sample size (365). It's unlikely to have multiple people match and screw up the independence, so it's a good approximation.

It's unlikely, but it can happen. Let's figure out the real chances of each person picking a different number:

The first person has a 100% chance of a unique number (of course)

The second has a (1 - 1/365) chance (all but 1 number from the 365)

The third has a (1 - 2/365) chance (all but 2 numbers)

The 23rd has a (1 - 22/365) (all but 22 numbers)

The multiplication looks pretty ugly:

But there's a shortcut we can take. When x is close to 0, a coarse first-order Taylor approximation for $e^x$ is:

so

Using our handy shortcut we can rewrite the big equation to:

But we remember that adding the numbers 1 to n = n(n + 1)/2. Don't confuse this with n(n-1)/2, which is C(n,2) or the number of pairs of n items. They look almost the same!

Adding 1 to 22 is (22 * 23)/2 so we get:

Phew. This approximation is very close, plug in your own numbers below:

Good enough for government work, as they say. If you simplify the formula a bit and swap in n for 23 you get:

and

With the exact formula, 366 people has a guaranteed collision: we multiply by $1 - 365/365 = 0$, which eliminates $p(\text{different})$ and makes $p(\text{match}) = 1$. With the approximation formula, 366 has a near-guarantee, but is not exactly 1: $1 - e^{-365^2 / (2 \cdot 365)} \approx 1$ .

Appendix B: The General Birthday Formula

Let's generalize the formula to picking n people from T total items (instead of 365):

If we choose a probability (like 50% chance of a match) and solve for n:

Voila! If you take $\sqrt{T}$ items (17% more if you want to be picky) then you have about a 50-50 chance of getting a match. If you plug in other numbers you can solve for other probabilities:

Remember that m is the desired chance of a match (it's easy to get confused, I did it myself). If you want a 90% chance of matching birthdays, plug m=90% and T=365 into the equation and see that you need 41 people.

Wikipedia has even more details to satisfy your inner nerd. Go forth and enjoy.

Other Posts In This Series

1. A Brief Introduction to Probability & Statistics

2. An Intuitive (and Short) Explanation of Bayes' Theorem

3. Understanding Bayes Theorem With Ratios

4. Understanding the Monty Hall Problem

5. How To Analyze Data Using the Average

6. Understanding the Birthday Paradox

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23 people. In a room of just 23 people there's a 50-50 chance of at least two people having the same birthday. In a room of 75 there's a 99.9% chance of at least two people matching. Put down the calculator and pitchfork, I don't speak heresy.

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6. Product Engineer

Want to make something round in a factory? From tin cans to ball bearings, there are plenty of products which are round. A product engineer works in manufacturing to make production possible and they need to use Pi to get their calculations correct.

7. Artist or sculptor

Some artists and sculptors like to focus on using geometrical forms in their work. Sculptors may use Pi when they are making works of art based on the circle, as this allows them to estimate the volume of material needed. Having a deep understanding of geometry can be a big advantage for an artist. Would you use Pi if you wanted to create a sculpture like this?

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Want to make something round in a factory? From tin cans to ball bearings, there are plenty of products which are round. A product engineer works in manufacturing to make production possible and they need to use Pi to get their calculations correct. Some artists and sculptors like to focus on using geometrical forms in their work.

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Here is a list of books that we would recommended for young people who are interested in mathematics. They are grouped into three different categories:

History of Mathematics

Recreational

Thinking Mathematically

We have given each book a lower age band based on the content and language used in the book and have presented them in ascending order of age. This is merely a guideline and you are welcome to make your own judgements on whether or not a book is suitable.

If you'd like to see some book recommendations for younger learners, take a look at the list of books compiled by our Primary team.

Students considering studying Maths or a related subject at university might be interested in the Cambridge Mathematical Reading List, which contains a wide range of suggested books including historical, recreational and technical maths books.

There are lots of maths podcasts you can listen to. Why not try: BBC's More or Less podcast Numberphile's videos and postcasts.

History of Mathematics

The Boy Who Loved Math: The Improbable Life of Paul ErdÃ¶s

Age 7+ Most people think of mathematicians as solitary, working away in isolation. And, it's true, many of them do. But Paul ErdÃ¶s never followed the usual path. At the age of four, he could ask you when you were born and then calculate the number of seconds you had been alive in his head. But he didn't learn to butter his own bread until he turned twenty. Instead, he traveled around the world, from one mathematician to the next, collaborating on an astonishing number of publications. With a simple, lyrical text and richly layered illustrations, this is a beautiful introduction to the world of math and a fascinating look at the unique character traits that made "Uncle Paul" a great man.

Blockhead: The Life of Fibonacci

Age 7+ As a young boy in medieval Italy, Leonardo Fibonacci thought about numbers day and night. He was such a daydreamer that people called him a blockhead. When Leonardo grew up and traveled the world, he was inspired by the numbers used in different countries. Then he realized that many things in nature, from the number of petals on a flower to the spiral of a nautilus shell, seem to follow a certain pattern. The boy who was once teased for being a blockhead had discovered what came to be known as the Fibonacci Sequence!

Infinity and Me

Uma can't help feeling small when she peers up at the night sky. She begins to wonder about infinity. Is infinity a number that grows forever? Is it an endless racetrack? Could infinity be in an ice cream cone? Uma soon finds that the ways to think about this big idea may just be . . . infinite.

On a Beam of Light: A Story of Albert Einstein
Age 7+
Travel along with Einstein on a journey full of curiosity, laughter, and scientific discovery. Parents and children alike will appreciate this moving story of the powerful difference imagination can make in any life.

50 Mathematical Ideas You Really Need to Know by Tony Crilly

In this book, Professor Tony Crilly explains in 50 clear and concise essays the mathematical concepts - ancient and modern, theoretical and practical, everyday and esoteric - that allow us to understand and shape the world around us. Packed with diagrams, examples and anecdotes, this book is the perfect overview of this often daunting but always essential subject. For once, mathematics couldn't be simpler.

The Math Book by Clifford A Pickover

Age 12+

Mathematic's infinite mysteries and beauty unfold in this book. Beginning millions of years ago with ancient ant odometers and moving through time to our modern-day quest for new dimensions, prolific polymath Clifford Pickover covers 250 milestones in mathematical history. Among the numerous concepts readers will encounter as they dip into this inviting anthology: cicada-generated prime numbers, magic squares, the discovery of pi and calculus, and the butterfly effect. Each topic is presented in a lavishly illustrated spread, including formulas, fascinating facts about scientists' lives and real-world applications of the theorems.

Alex's Adventures in Numberland by Alex Bellos

Age 13+

The world of maths can seem mind-boggling, irrelevant and, sometimes, boring. This groundbreaking book reclaims maths from the geeks. Mathematical ideas underpin just about everything in our lives: from the surprising geometry of the 50p piece to how probability can help you win in any casino. In search of weird and wonderful mathematical phenomena, Alex Bellos travels across the globe and meets the world's fastest mental calculators in Germany and a startlingly numerate chimpanzee in Japan. Packed with fascinating, eye-opening anecdotes, Alex's Adventures in Numberland is an exhilarating cocktail of history, reportage and mathematical proofs that will leave you awestruck.

The Monty Hall Problem: Beyond Closed Doors by Rob Deaves

Age 14+

This short book explores the Monty Hall dilemma, a well known mathematical puzzle. The original problem, the controversy surrounding it and its solution are discussed. Further, the boundaries of the problem are expanded to consider prior knowledge and host intention. This book should be of interest to those who enjoy problem solving.

The Language of Mathematics by Keith Devlin

Age 14+

In The Language of Mathematics, award-winning author Keith Devlin reveals the vital role mathematics plays in our eternal quest to understand who we are and the world we live in. More than just the study of numbers, mathematics provides us with the eyes to recognize and describe the hidden patterns of life. Devlin shows us what keeps a jumbo jet in the air, explains how we can see and hear a football game on TV, allows us to predict the weather, the behavior of the stock market, and the outcome of elections. Far from a dry and esoteric subject, mathematics is a rich and living part of our culture. An exploration of an often woefully misunderstood subject, this book celebrates the simplicity, the precision, the purity, and the elegance of mathematics.

The Music of the Primes by Marcus Du Sautoy

Age 14+

How can one predict when the next prime number will occur? Is there a formula which could generate primes? These apparently simple questions have confounded mathematicians ever since the Ancient Greeks. In 1859, the brilliant German mathematician Bernhard Riemann put forward a hypothesis which finally seemed to reveal a magical harmony at work in the numerical landscape. The promise that these eternal, unchanging numbers would finally reveal their secret thrilled mathematicians around the world. Yet Riemann never publicly provided a proof for his hypothesis and his housekeeper burned most of his personal papers on his death. Whoever cracks Riemann's hypothesis will go down in history, for it has implications far beyond mathematics. In business, it plays a central role in security and e-commerce. In science, it brings together vastly different areas, with critical ramifications in Quantum Mechanics, Chaos Theory and the future of computing. Pioneers in each of these fields are racing to crack the code and a prize of $1 million has been offered to the winner. As yet, it remains unsolved.

Journey Through Genius: The Great Theorems of Mathematics by William Dunham

Age 14+

In this book Dunham treats mathematical theorems as creative works of art. He places each theorem within its historical context and explores the very human and often turbulent life of the creator. He studies such great mathematicians as Archimedes, Gerolamo Cardano and Georg Cantor. He also provides step-by-step proofs for the theorems, each easily accessible to readers with no more than a knowledge of high school mathematics.

The Mathematical Universe: Alphabetical Journey Through the Great Proofs, Problems & Personalities by William Dunham

Age 14+

In this book, Dunham takes us through a tantalizing selection of the great proofs, notorious disputes, and intriguing unsolved mysteries of the mathematical universe. Subjects range from the golden age of Greek geometry to the furthest frontier of infinite series. Dunham explores more than five thousand years of mathematical history, digging into the earliest records in Egypt, Babylon, India, and China, and turning up surprising tales and tidbits from modern times. All along the way, Dunham portrays the great masters of mathematics at their work. In colorful anecdotes, the brilliant - often eccentric - luminaries chart the course of mathematical progress. This book is accessible to any reader with a basic knowledge of algebra and geometry. You will come away from this exhilarating book with a keen sense of the power and splendor of the magical mathematical world.

Chaos by James Gleick

Age 14+

Chaos is what happens when the behaviour of a system gets too complicated to predict; the most familiar example is the weather, which apparently cannot be forecast accurately more than five days ahead. This book tells the story so far in the study of this new field of Physics.

Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace by Leonard Mlodinow

Age 14+

Anyone who thought geometry was boring or dry should prepare to be amazed. Despite its worthy cover this book is exactly what its title says - a story - and the plot of this story involves life, death and revolutions of understanding and belief. It stars the some of the most famous names in history, from Euclid who laid the logical foundations, to Albert Einstein, who united space and time in a single non-Euclidean geometry. It offers an alternative history of mathematics, revealing how simple questions anyone might ask about space - in the living room or in some other galaxy - have been the hidden engines of the highest achievements in science and technology.

Closing the Gap: The Quest to Understand Prime Numbers by Vicky Neale

Age 14+

Prime numbers have intrigued, inspired and infuriated mathematicians for millennia. Every school student studies prime numbers and can appreciate their beauty, and yet mathematicians' difficulty with answering some seemingly simple questions about them reveals the depth and subtlety of prime numbers. In this book, Vicky Neale charts the recent progress towards proving the famous Twin Primes Conjecture, and the very different ways in which the breakthroughs have been made: a solo mathematician working in isolation and obscurity, and a large collaboration that is more public than any previous collaborative effort in mathematics. Interleaved with this story are highlights from a significantly older tale, going back two thousand years and more, of mathematicians' efforts to comprehend the beauty and unlock the mysteries of the prime numbers.

Fermat's Last Theorem by Simon Singh

Age 14+

The story of the solving of a puzzle that has confounded mathematicians since the 17th century. In 1963, a schoolboy browsing in his local library stumbled across the world's greatest mathematical problem: Fermat's Last Theorem, a puzzle that every child can understand but which has baffled mathematicians for over 300 years. Aged just ten, Andrew Wiles dreamed that he would crack it. Wiles's lifelong obsession with a seemingly simple challenge set by a long-dead Frenchman is an emotional tale of sacrifice and extraordinary determination. In the end, Wiles was forced to work in secrecy and isolation for seven years, harnessing all the power of modern maths to achieve his childhood dream. Many before him had tried and failed, including a 18-century philanderer who was killed in a duel. An 18-century Frenchwoman made a major breakthrough in solving the riddle, but she had to attend maths lectures at the Ecole Polytechnique disguised as a man since women were forbidden entry to the school.

The Code Book by Simon Singh

Age 14+

The Code Book is a history of man's urge to uncover the secrets of codes, from Egyptian puzzles to modern day computer encryptions. As in Fermat's Last Theorem, Simon Singh brings life to an astonishing story of puzzles, codes, languages and riddles that reveals man's continual pursuit to disguise and uncover, and to work out the secret languages of others. Codes have influenced events throughout history, both in the stories of those who make them and those who break them. The betrayal of Mary Queen of Scots and the cracking of the enigma code that helped the Allies in World War II are major episodes in a continuing history of cryptography. In addition to stories of intrigue and warfare, Simon Singh also investigates other codes, the unravelling of genes and the rediscovery of ancient languages and most tantalisingly, the Beale ciphers, an unbroken code that could hold the key to a 20 million dollar treasure.

The Penguin Dictionary of Curious and Interesting Numbers by David Wells

Age 14+
Infinite Powers by Steven Strogatz
Look up 1729 to see why it is 'among the most famous of all numbers'. Look up 0.7404 ($= \frac{\pi}{18}$) to discover that this is the density of closely packed identical spheres in what is believed by many mathematicians (though it was at that time an unproven hypothesis) and is known by all physicists and greengrocers to be the optimal packing. Look up Graham's number (the last one in the book), which is inconceivably big: even written as a tower of powers ($9 ^{9^ {9 ^\cdots}}$) it would take up far more ink than could be made from all the atoms in the universe. It is an upper bound for a quantity in Ramsey theory whose actual value is believed to be about 6. A book to be dipped into at leisure.

Calculus has a fierce reputation as complicated, abstract and difficult, but Steven Strogatz tells the true story of its beauty and simplicity. Starting with the Ancient Greeks, calculus has tantilised and fascinated humanity for millenia. Its applications grew from teaching us how to determine the area of a circle with only sand and a stick to safely launching rockets into space. Without it, we wouldnâ€™t have mobile phones, TV, GPS, or ultrasound. This book explains the history of our relationship with calculus, and its relationship with infinity.

GÃ¶del, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter

Age 16+

Douglas Hofstadter's book is concerned directly with the nature of maps or links between formal systems. However, according to Hofstadter, the formal system that underlies all mental activity transcends the system that supports it. If life can grow out of the formal chemical substrate of the cell, if consciousness can emerge out of a formal system of firing neurons, then so too will computers attain human intelligence. GÃ¶del, Escher, Bach is a wonderful exploration of fascinating ideas at the heart of cognitive science: meaning, reduction, recursion, and much more.

The Great Mathematical Problems by Ian Stewart

Age 17+

There are some mathematical problems whose significance goes beyond the ordinary - like Fermat's Last Theorem or Goldbach's Conjecture - they are the enigmas which define mathematics. This book explains why these problems exist, why they matter, what drives mathematicians to incredible lengths to solve them and where they stand in the context of mathematics and science as a whole. It contains solved problems - like the PoincarÃ© Conjecture, cracked by the eccentric genius Grigori Perelman, who refused academic honours and a million-dollar prize for his work, and problems which, like the Riemann Hypothesis, remain baffling after centuries. Stewart is the guide to this mysterious and exciting world, showing how modern mathematicians constantly rise to the challenges set by their predecessors, as the great mathematical problems of the past succumb to the new techniques and ideas of the present.

Recreational

Can you Solve my Problems? by Alex Bellos

Age 11+

This book contains 125 of the world's best brainteasers from the last two millennia, taking us from ancient China to medieval Europe, Victorian England to modern-day Japan, with stories of espionage, mathematical breakthroughs and puzzling rivalries along the way. Some solutions rely on a touch of cunning, others call for creativity, others need mercilessly logical thought. All are guaranteed to sharpen your mind!

Snowflake Seashell Star by Alex Bellos and Edmund Harriss

Age 11+

This book is the first in a series of mathematical colouring books by Alex Bellos and mathematical artist, Edmund Hariss. It is full of intricate and divinely beautiful illustrations, using patterns such as loops, waves and spirals. Stretch your artistic talents and embrace mathematics' aesthetic and conceptual exquisiteness. In 80 glorious images, Alex will teach you how to convey the wonder of mathematics through colour and design. Snowflake, Seashell, Star is an incredible celebration of the overlaps between mathematics and art, learning and play, concentration and calm.

A Mathematical Pandora's Box by Brian Bolt

Age 11+

Through his own experience, Brian Bolt has discovered a worldwide interest in mathematical puzzles. Bolt not only uses them to stimulate creative thinking, but also to open up new areas of mathematics to the reader. This book contains 142 activities: in addition to puzzles, there are games, tricks, models and explanation of various phenomena. They range from number manipulation, through happy and amicable numbers, coin puzzles, picnicking bears and pentominoes, to building shapes with cubes. There is a detailed commentary at the end of the book, giving solutions and explanations, together with the occasional follow-up problem.

The Number Devil by Hans Magnus Enzensberger

Age 11+

The quirky and unusual story of a young boy who hates maths at school, but who discovers a new side to the subject when he meets an unusual mathematician in a dream. This book takes you on an adventure through creative mathematical thinking, with great illustrations along the way.

The Indisputable Existence of Santa Claus by Hannah Fry and Thomas OlÃ©ron Evans

Age 11+

How do you apply game theory to select who should be on your Christmas shopping list? Can you predict Her Majesty's Christmas Message? Will calculations show Santa is getting steadily thinner - shimmying up and down chimneys for a whole night - or fatter - as he tucks into a mince pie and a glass of sherry in billions of houses across the world? Full of diagrams, sketches and graphs, beautiful equations, Markov chains and matrices, this book brightens up the bleak midwinter with stockingfuls of mathematical marvels. Mathematics has never been merrier.

Aha! Insight & aha! Gotcha by Martin Gardner

Age 11+

Previously published separately, the two books aha! Gotcha and aha! Insight have been combined as a single volume. The aha! books, as they are referred to by fans of Martin Gardner, contain 144 wonderful puzzles from the reigning king of recreational mathematics. In this combined volume, you will find puzzles ranging over geometry, logic, probability, statistics, number, time, combinatorics, and word play. Gardner calls these puzzles aha! problems, that 'seem difficult, and indeed are difficult if you go about trying to solve them in traditional ways. But if you can free your mind from standard problem solving techniques, you may be receptive to an aha! reaction that leads immediately to a solution. Don't be discouraged if, at first, you have difficulty with these problems. After a while you will begin to catch the spirit of offbeat, nonlinear thinking, and you may be surprised to find your aha! ability improving.'

Entertaining Mathematical Puzzles by Martin Gardner

Age 11+

Only an elementary knowledge of math is needed to enjoy this entertaining compilation of brain-teasers. It includes a mixture of old and new riddles covering a variety of mathematical topics: money, speed, plane and solid geometry, probability, topology, tricky puzzles and more. Carefully explained solutions follow each problem.

My Best Mathematical and Logic Puzzles by Martin Gardner

Age 11+

Over a period of 25 years as author of the Mathematical Games column for Scientific American, Martin Gardner devoted a column every six months or so to short math problems or puzzles. This volume contains a rich selection of 70 of the best of these brain teasers, in some cases including references to new developments related to the puzzle. Now enthusiasts can challenge their solving skills and rattle their egos with such stimulating mind-benders as The Returning Explorer, The Mutilated Chessboard, Scrambled Box Tops, The Fork in the Road, Bronx vs. Brooklyn, Touching Cigarettes, and 64 other problems involving logic and basic math. Solutions are included.

The Thrilling Adventures of Lovelace and Babbage by Sydney Padua

Age 11+

In this book Sydney Padua transforms one of the most compelling scientific collaborations into a hilarious set of adventures, starring Ada Lovelace and Charles Babbage. This book presents a delightful alternate reality in which Lovelace and Babbage build the Difference Engine and use it to create runaway economic models, battle the scourge of spelling errors, explore the wider realms of mathematics and, of course, fight crime - for the sake of both London and science. Extremely funny and utterly unusual, this book comes complete with historical curiosities, extensive footnotes and never-before-seen diagrams of Babbage's mechanical, steam-powered computer. And ray guns.

The 'Uncle Albert' Series by Russell Stannard

Age 11+

A best selling science/adventure series, beginning with 'The Time and Space of Uncle Albert'. Uncle Albert and his intrepid niece, Gedanken, enter the dangerous and unknown world of a thought bubble. Their mission: to unlock the deep mysteries of Time and Space... Discover why you can't break the ultimate speed barrier, how to become older than your mother, how to put on weight without getting fat, and how to live forever without even knowing it. Other books in the series include: 'Black Holes and Uncle Albert' and 'Uncle Albert and the Quantum Quest'.

Professor Stewart's Cabinet of Mathematical Curiosities by Ian Stewart

Age 11+

A selection of mathematical puzzles, stories, tricks and short articles - great to read all in one go, or to dip into. The content varies between simple logic puzzles to introductions to more advanced topics such as the Four Colour Theorem, which tells us that we can colour in any map using only four colours, so that no bordering countries have the same colour.

Book of Curious & Interesting Mathematics by David Wells

Age 11+

A collection of strange mathematical facts and stories. This anthology covers a whole range of ages, maths and mathematicians, and includes probability paradoxes, jumbled Shakespearean sonnets, record-breaking monkeys and typewriters, and theories of big game hunting. Also featured are stories of people who looked for logical loopholes in the American Constitution or calmed their nerves with algebra.

Book of Curious & Interesting Puzzles by David Wells

Age 11+

This collection by best-selling author David Wells, a Cambridge math scholar and teacher, includes more than 560 puzzles, from the "mind sharpeners" of a medieval monk to the eitheenth-century Ladies' Diary, the Hindu Bhakshali manuscript, and riddles and popular rhymes. None require any mathematics beyond the most elementary algebra and geometry - and few require even that. Complete answers appear at the end.

Mathematics, Magic and Mystery by Martin Gardner

Age 12+

Famed puzzle expert Martin Gardner explains the mathematics behind a multitude of mystifying tricks: card tricks, stage "mind reading," coin and match tricks, counting out games, geometric dissections, etc. Each of these are actually demonstrations of probability, sets, number theory, topology and other braches of mathematics. No skill at sleight of hand is needed to perform the more than 500 tricks described in this book because mathematics guarantees their success.

How Many Socks Make a Pair? by Rob Eastaway

Age 13+

Can maths be creative? This book sets out to prove that it can, through a selection of short articles on surprising maths in everyday life. Through lots of intriguing problems, involving card tricks, polar bears and, of course, socks, Rob Eastaway shows shows how maths can demonstrate its secret beauties in even the most mundane of everyday objects.

Why do Buses Come in Threes? by Rob Eastaway and Jeremy Wyndham

Age 13+

With a foreword by Tim Rice, this book will change the way you see the world. Why is it better to buy a lottery ticket on a Friday? Why are showers always too hot or too cold? And what's the connection between a rugby player taking a conversion and a tourist trying to get the best photograph of Nelson's Column? These and many other fascinating questions are answered in this entertaining and highly informative book, which is ideal for anyone wanting to remind themselves - or discover for the first time - that maths is relevant to almost everything we do. Dating, cooking, travelling by car, gambling and even life-saving techniques have links with intriguing mathematical problems, as you will find explained here. Whether you have a PhD in astrophysics or haven't touched a maths problem since your school days, this book will give you a fresh understanding of the world around you.

Flatterland by Ian Stewart

Age 13+

In 1884, Edwin A. Abbott published â€œFlatlandâ€Â; a brilliant novel about mathematics and philosophy that charmed and fascinated all of England. Now, Ian Stewart has written a fascinating, modern sequel to Abbott's book. Through larger-than-life characters and an inspired story line, "Flatterland" explores our present understanding of the shape and origins of the universe, the nature of space, time, and matter, as well as modern geometries and their applications.

The Liar Paradox and the Towers of Hanoi: 10 Greatest Math Puzzles of All Time by Marcel Danesi

Age 14+

Ever since the Sphinx asked his legendary riddle of Oedipus, riddles, conundrums, and puzzles of all sizes have kept humankind perplexed and amused. The Liar Paradox and the Towers of Hanoi takes die-hard puzzle experts on a tour of the world's most enduringly intriguing braintwisters, from KÃƒÂ¶nigsberg's Bridges and the Hanoi Towers to Fibonacci's Rabbits, the Four Color Problem, and the Magic Square. Each chapter introduces the basic puzzle, discusses the mathematics behind it, and includes exercises and answers plus additional puzzles similar to the one under discussion. Here is a veritable kaleidoscope of puzzling labyrinths, maps, bridges, and optical illusions that will keep aficionados entertained for hours.

The Number Mysteries by Marcus du Sautoy

Age 14+

An exploration of surprising ways maths occurs in our everyday lives, centred around five famous unsolved problems in mathematics. Topics include how to detect an art forgery, winning strategies in Monopoly, and how to crack a code. Sprinkled with games and links to interactive online content so you can try out some of the ideas for yourself!

Things to Make and Do in the Fourth Dimension by Matt Parker

Age 14+

This is the complete guide to exploring the fascinating world of maths you were never told about at school. Stand-up comedian and mathematician Matt Parker uses bizarre Klein Bottles, unimaginably small pizza slices, knots no one can untie and computers built from dominoes to reveal some of the most exotic and fascinating ideas in mathematics. Starting with simple numbers and algebra, this book goes on to deal with inconceivably big numbers in more dimensions than you ever knew existed. And always with something for you to make or do along the way.

How to Cut a Cake: and Other Mathematical Conundrums by Ian Stewart

Age 14+

In this book are twenty more curious puzzles and fantastical mathematical stories from one of the world's most popular and accessible writers on mathematics. This is a strange world of never-ending chess games, empires on the moon, furious fireflies, and, of course, disputes over how best to cut a cake. Each chapter - with titles such as, "How to Play Poker By Post" and "Repealing the Law of Averages" - presents a fascinating mathematical puzzle that is challenging, fun, and introduces the reader to a significant mathematical problem in an engaging and witty way. Illustrated with clever and quirky cartoons, each tale will delight those who love puzzles and mathematical conundrums.

Games and Mathematics by David Wells

Age 14+

The appeal of games and puzzles is timeless and universal. In this book, David Wells explores the fascinating connections between games and mathematics, proving that mathematics is not just about tedious calculation but imagination, insight and intuition. The first part of the book introduces games, puzzles and mathematical recreations, including knight tours on a chessboard. The second part explains how thinking about playing games can mirror the thinking of a mathematician, using scientific investigation, tactics and strategy, and sharp observation. Finally the author considers game-like features found in a wide range of human behaviours, illuminating the role of mathematics and helping to explain why it exists at all. This thought-provoking book is perfect for anyone with a thirst for mathematics and its hidden beauty; a good high school grounding in mathematics is all the background that is required, and the puzzles and games will suit pupils from 14 years.

Mathematical Puzzles: A Connoisseur's Collection by Peter Winkler

Age 16+

Collected over several years by Peter Winkler, dozens of elegant, intriguing challenges are presented in this book. The answers are easy to explain, but without this book, devilishly hard to find. Creative reasoning is the key to these puzzles. No involved computation or higher mathematics is necessary, but your ability to construct a mathematical proof will be severly tested - even if you are a professional mathematician. For the truly adventurous, there is even a chapter on unsolved puzzles.

Thinking Mathematically

Elastic Numbers by Daniel Griller

Age 11+

Beautifully crafted and immensely enjoyable, the problems in this book require minimal technical knowledge, being accessible to young secondary school pupils. However, there is an astonishing range in difficulty; while some of the problems are fairly straightforward, others are significantly tougher, with a great deal of ingenuity and clarity of thought needed to make progress. Whether you are a student preparing for a maths competition, an educational establishment seeking to supplement your problem solving resources, or an individual looking for a different sort of challenge, Elastic Numbers is a unique collection, and will push you to the very edge of your abilities.

Thinking Mathematically by Mason, Burton & Stacey

Age 11+

Thinking Mathematically is perfect for anyone who wants to develop their powers to think mathematically, whether at school, at university or just out of interest. This book is invaluable for anyone who wishes to promote mathematical thinking in others or for anyone who has always wondered what lies at the core of mathematics. Thinking Mathematically reveals the processes at the heart of mathematics and demonstrates how to encourage and develop them. Extremely practical, it involves the reader in questions so that subsequent discussions speak to immediate experience.

Finding Moonshine: A Mathematician's Journey Through Symmetry by Marcus Du Sautoy

Age 13+

This book tells the story of one of the biggest adventures in mathematics: the search for symmetry. This is the story of how humankind has come to its understanding of the bizarre world of symmetry - a subject of fundamental significance to the way we interpret the world around us. Our eyes and minds are drawn to symmetrical objects, from the sphere to the swastika, from the pyramid to the pentagon. 'Symmetry' is all-pervasive: in chemistry the concept of symmetry explains the structure of crystals; in evolutionary biology, the natural world exploits symmetry in the fight for survival; symmetry and the breaking of symmetry are central to ideas in art, architecture and music; the mathematics of symmetry is even exploited in industry, for example to find efficient ways to store more music on a CD or to keep your mobile phone conversation from cracking up through interference.

Mathematics for the Curious by Peter Higgins

Age 13+

When do the hands of a clock coincide? How likely is it that two children in the same class will share a birthday? How do we calculate the volume of a doughnut? Mathematics for the Curious provides anyone interested in mathematics with a simple and entertaining account of what it can do. Author Peter Higgins gives clear explanations of the more mysterious features of childhood mathematics as well as novelties and connections that prove that mathematics can be enjoyable and full of surprises. Topics include: the truth about fractions, ten questions and their answers, and the golden ratio. Higgins poses entertaining puzzles and questions tempting the reader to ponder math problems with imagination instead of dread. Mathematics for the Curious is an accessible introduction to basic mathematics for beginning students and a lively refresher for adults.

Mathematics for the Imagination by Peter Higgins

Age 13+

Mathematics for the Imagination provides an accessible and entertaining investigation into mathematical problems in the world around us. From world navigation, family trees, and calendars to patterns, tessellations, and number tricks, this informative and fun book helps you to understand the maths behind real-life questions and rediscover your arithmetical mind. This is a highly involving book which encourages the reader to enter into the spirit of mathematical exploration.

Mathematics and the Physical World by Morris Kline

Age 13+

A stimulating account of development of basic mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations and non-Euclidean geometries. Also describes how maths is used in optics, astronomy, motion under the law of gravitation, acoustics, electromagnetism, and other aspects of physics.

Mathematics: A Very Short Introduction by Timothy Gowers

Age 14+

Tim Gowers is a Fields Medalist (the Fields medal is the mathematical equivalent of the Nobel prize), so it is not at all surprising that what he writes is worth reading. What is surprising is the ease and charm of his writing. He touches lightly many areas of mathematics, some that will be familiar (Pythagoras) and some that may not be (manifolds) and has something illuminating to say about all of them.

Excursions in Geometry by C. Stanley Ogilvy

Age 14+

In this book, Professor Ogilvy demonstrates the mathematical challenege and satisfaction to be had from geometry, the only requirement being two simple implements (straight-edge and compass) and a little thought. Topics including harmonic division and Apollonian circles, inversive geometry, the hexlet, conic sections, projective geometry, the Golden Section and angle trisection are addressed in a way that brings out the true intellectual excitement inherent in each. Also included are some unsolved problems of modern geometry.

Excursions in Mathematics by C. Stanley Ogilvy

Age 14+

This book offers a fascinating glimpse into the world of mathematics and mathematicians. It is designed for the reader who has no advanced mathematical background of special aptitude, but who wants to acquaint him or herself with the intellectually stimulating and aesthetically satisfying aspects of the subject. After illuminating the role of the mathematician and dispelling several popular misconceptions about the nature of mathematics, Professor Ogilvy takes you on a lively tour of the four basic branches of the subject: number theory, algebra, geometry and analysis. Focusing on the interesting, and even amusing, aspects of mathematics, he points out the interconnections between the branches and presents mathematics as a vital subject whose frontiers are continually expanding. Many illustrations and examples illuminate classic aspects of the subject as well as recent advances.

Excursions in Number Theory by C. Stanley Ogilvy and John Anderson

Age 14+

This delightful volume, by two well-known mathematicians, invites readers to join a challenging expedition into the mystery and magic of number theory. No special training is needed - just high school mathematics, a fondness for figures and an inquisitive mind. Beginning with familiar notions, the authors skillfully transport the reader to higher realms of mathematics, developing the necessary concepts along the way. Included are thorough discussions of prime numbers, number patterns, irrationals and iterations and calculating prodigies, among other topics.

Q.E.D. - Beauty in Mathematical Proof by Burkard Polster

Age 14+

Which famous proof did Archimedes inscribe on his tombstone? How and why do knots make perfect pentagons? Have you ever seen a proof so completely that it is just obvious? In this delicious little book, top down-under mathemagician Dr. Polster presents many of the most visually intuitive and exciting proofs from the dusty annuals of mathematical history. You can test your ability to follow the logic, leap into mathemagnosis and experience eureka-moment after eureka-moment.

How to Solve It by George Polya

Age 14+

A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out - from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft - indeed, brilliant - instructions on stripping away irrelevancies and going straight to the heart of the problem.

The Joy of X: A Guided Tour of Mathematics from One to Infinity by Steven Strogatz

Age 14+

Maths is everywhere, often where we don't even realise. Award-winning professor Steven Strogatz acts as our guide as he takes us on a tour of numbers that - unbeknownst to the unitiated - connect pop culture, literature, art, philosophy, current affairs, business and even every day life. In The Joy of X, Strogatz explains the great ideas of maths - from negative numbers to calculus, fat tails to infinity - with clarity, wit and insight. He is the maths teacher you never had and this book is perfect for the smart and curious, the expert and the beginner.

Problem-Solving Strategies In Mathematics: From Common Approaches To Exemplary Strategies by Alfred S Posamentier and Stephen Krulik

Age 15+

This book introduces ten problem-solving strategies by first presenting the strategy and then applying it to problems in elementary mathematics. In so doing, the common less efficient approach is first mentioned and then the effective strategy is shown. Elementary mathematics is used so that the reader can focus on the strategy and not be distracted by some more sophisticated mathematics.

1089 and All That: A Journey into Mathematics by David Acheson

Age 15+

David Acheson's extraordinary little book makes mathematics accessible to everyone. From very simple beginnings he takes us on a thrilling journey to some deep mathematical ideas. On the way, via Kepler and Newton, he explains what calculus really means, gives a brief history of pi, and even takes us to chaos theory and imaginary numbers. Every short chapter is carefully crafted to ensure that no one will get lost on the journey. Packed with puzzles and illustrated by world famous cartoonists, this is one of the most readable and imaginative books on mathematics ever written.

An Introduction to Mathematical Reasoning by Peter Eccles

Age 15+
Mathematics for Human Flourishing by Francis Su
The purpose of this book is to introduce the basic ideas of mathematical proof to students. The emphasis is on helping the reader in understanding and constructing proofs and writing clear mathematics. This is achieved by exploring set theory, combinatorics and number theory, topics which include many fundamental ideas which are part of the tool kit of any mathematician. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the classic proofs.

Can the study of mathematics enrich the heart as well as the mind? Francis Su explains how humans crave logic, beauty, truth and play, and that searching for these in mathematics is not only rewarding, but builds virtues in us that help us be better individuals and members of society. The book includes puzzles to draw everyone in to the enjoyment of mathematics, as part of his vision that mathematics should be shared by everyone.

What is Mathematics? by Richard Courant, Herbert Robbins and Ian Stewart

Age 16+

Written for beginners and scholars, for students and teachers, for philosophers and engineers, this book is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.

Towards Higher Mathematics: A Companion by Richard Earl

Age 16+

Containing a large and varied set of problems, this rich resource will allow students to stretch their mathematical abilities beyond the school syllabus, and bridge the gap to university-level mathematics. Many proofs are provided to better equip students for the transition to university. The author covers substantial extension material using the language of sixth-form mathematics, thus enabling students to understand the more complex material. There are over 1500 carefully graded exercises, with hints included in the text, and solutions available online. Historical and contextual asides highlight each area of mathematics and show how it has developed over time.

The Art of the Infinite by Robert and Ellen Kaplan

Age 16+

This book unlocks the secrets of maths - revealing it to be our lost, native language, as much a part of us as the words we use every day. Number and form are the essence of our world: from the patterns of the stars to the pulses of the market, from the beats of our hearts to catching a ball or tying our shoelaces. Drawing on science, literature, history and philosophy, this book makes the rich patterns of maths brilliantly clear.

Algorithmic Puzzles by Anany & Maria Levitin

Age 16+

In this book, Anany and Maria Levitin use many classic brainteasers as well as newer examples from job interviews with major corporations to show readers how to apply analytical thinking to solve puzzles requiring well-defined procedures. The book's unique collection of puzzles is supplemented with carefully developed tutorials on algorithm design strategies and analysis techniques intended to walk the reader step-by-step through the various approaches to algorithmic problem solving. Mastery of these strategies - exhaustive search, backtracking, and divide-and-conquer, among others - will aid the reader in solving not only the puzzles contained in this book, but also others encountered in interviews, puzzle collections, and throughout everyday life. Each of the 150 puzzles contains hints and solutions, along with commentary on the puzzle's origins and solution methods. Readers with only middle school mathematics will develop their algorithmic problem-solving skills through puzzles at the elementary level, while seasoned puzzle solvers will enjoy the challenge of thinking through more difficult puzzles.

The Mathematics of Games and Gambling by Edward Packel

Age 16+

This book introduces and develops some of the important and beautiful elementary mathematics needed for rational analysis of various gambling and game activities. Most of the standard casino games (roulette, craps, blackjack, keno), some social games (backgammon, poker, bridge) and various other activities (state lotteries, horse racing) are treated in ways that bring out their mathematical aspects. The mathematics developed ranges from the predictable concepts of probability, expectation, and binomial coefficients to some less well-known ideas of elementary game theory. Game-related exercises are included and solutions to some appear at the end of the book.

How to Study for a Mathematics Degree by Lara Alcock

Age 17+

Every year, thousands of students go to university to study mathematics. Many of these students are extremely intelligent and hardworking, but even the best will, at some point, struggle with the demands of making the transition to advanced mathematics. The mathematics shifts in focus from calculation to proof, so students are expected to interact with it in different ways. These changes need not be mysterious - mathematics education research has revealed many insights into the adjustments that are necessary - but they are not obvious and they do need explaining. This book translates these research-based insights into practical advice for a student audience. It covers every aspect of studying for a mathematics degree, from the most abstract intellectual challenges to the everyday business of interacting with lecturers and making good use of study time.

How to Think Like a Mathematician by Kevin Houston

Age 17+

Looking for a head start in your undergraduate degree in mathematics? This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.

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Here is a list of books recommended for young people who are interested in mathematics.

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Tim Harford explains the numbers and statistics used in everyday life