… For most people anyway. Because I missed my video post yesterday, here’s a very short TED talk from Arthur Benjamin on why we should teach statistics before calculus.

Of course, calculus has it’s place and is extremely important, but statistics, probability, and data analysis are so much more useful in everyday life. It would be a tremendous leap forward in promoting scientific literacy in the US. After all, if more people knew what a p-value was or what a confidence interval actually is, society as a whole would be a lot better equipped to understand the numbers that are constantly thrown at us. Reporters might realize that if one study finds a statistically significant result which is unable to be replicated, most likely they made a type I error and the null was probably rejected by random chance error.

I hope that in my lifetime, all students will…

This week Year 10 (UK age 14-15) have been exploring different graph types and also transformations and graphs.

For homework I asked them to draw just a small number of graphs by hand but wanted them to check their work and explore further graphs using the Desmos graphing calculator. Early in the week I made sure they could all use Desmos including the use of tables so in an IT room they used the slideshow here and created several graphs of their own.

Once all the students were confident to use Desmos to create various lines and curves I asked them to explore a series of graphs so that this coming week we can discuss transformations and graphs. Using Desmos allowed them to explore many graphs in a short space of time and several students chose to take screenshots and make notes for themselves.

Having used sliders they were able…

http://www.math.niu.edu/%7Ebeachy/abstract_algebra/study_guide/contents.html

The Study Notes on 600 problems and solutions:

http://www.math.niu.edu/~beachy/abstract_algebra/guide/contents.html

Here’s a link. Check it out! It says that Chinese are better in math than americans.

Okay, I just got myself a new goal: find out why Chinese are so much better than everyone else in math!

This is what a Chinese math class looks like. Keep in mind this is only grade 8 level. They don’t have the same methods of learning, at ALL!

It’s likely that you’ve seen this book reviewed elsewhere. There has been a lot of buzz about it in the kid-lit world. And for good reason. The Boy Who Loved Math: The Improbable LIfe of Paul Erdos is a wonderful biography of a fascinating man. In case you’re ignorant like me, Paul Erdos was a Hungarian mathematician known for his work in number theory and for his eccentric personality.

Deborah Heligman strikes a perfect balance  in this book between the story of Erdos’ life and an explanation of the mathematical problems that so intrigued and consumed him.The main focus of the text is on the life of Erdos: from a childhood where he was kicked out of school for not following the rules; to his ability at the age of four to quickly tell a person how old they were in seconds once he knew their birthdate and time; to…

https://www.udacity.com/courses#!/All

Read the review of MOOC challenges:

http://readwrite.com/2014/01/28/open-online-education-and-the-trend-towards-legitimacy#awesm=~ouoDz7h5ZAFTbR

The best MOOC (Massive Open Online Course) model will be free of exams / tests, only assignments / projects, forum discussions. 

The 2 current MOOC  – Coursera and Udacity –  are changing the ‘bottle’ (campus-based to online virtual campus),  but not the ‘wine’ (same old teaching methodology through quizzes, tests, exams, which are hated by students who are mostly working adults). 

A better example will be the Khan Academy by a MIT graduate Salman Khan teaching school kids in Math and Sciences. Bill Gates and Google founder sponsored him > $6 million. His successful model is “free +no exams“.

The Chinese sage Laozi 老子 said 3,000 years ago “Wu Wei” (无为: Do something with no specific purpose) is actually “You Wei”(有为: With potential great achievement). Attending MOOC with no paper-chasing (and money making)…

The Garden Project:

Over the past couple of weeks, my 3rd graders have been working with our new set of iPads on a Garden Project. Since our school has put in learning gardens this year, I thought it would be an applicable project for them.

The premise of the problem: The school wanted to build a garden with the most space to plant our vegetables. Each group was given 18 feet of fencing (18 toothpicks) to use as their perimeter. They were to design each garden, record the dimensions, and take a picture to save in their photos. After they designed all possible rectangular gardens, they had to create a presentation in Numbers to show me which garden they wanted to build.

The instruction page looked like this (Since this was our first project, I put the app pic next to each direction to help them along the way):

I…

As you’re probably aware, I’m a big believer in using stories to bring math to life. Especially when you’re teaching tricky concepts, using a story can be the “magic switch” that flicks on the light of understanding. Armed with story-based understanding, students can recall how to perform difficult math processes. And since people naturally like stories and tend to recall them, skills based on story-based understanding really stick in the mind. I’ve seen this over and over in my tutoring.

The kind of story I’m talking about uses an extended-metaphor, and this way of teaching  is particularly helpful when you’re teaching algebra. Ask yourself: what would you rather have? Students scratching their heads (or tearing out their hair) to grasp a process taught as a collection of abstract steps? Or students grasping  a story and quickly seeing how it guides them in doing the math? I think the answer is…

This will be a quick post because I have a student-posed math problem that I need some time to reason through!

Today, students found the area and perimeter of squares that increase in side length by one each time. Students used a variety of models when building their squares from Minecraft carpets, to Geoboards to graph paper. Here is the completed activity sheet from their work: I then gave them a few minutes to talk to their tablemates about things they notice in their work. Here are the answers they shared as a class and I recorded on the board:

“An even dimension by even dimension = an even area”

“An odd dimension by odd dimension = an odd area”

“The perimeter goes up by 4 every time the square gets bigger”

“The areas are square numbers.”

“The areas go up by odd skip counting: +3, +5, +7…”
I was…

This 23-year-old Chinese ‘Rain Man’ Chou Wei has IQ 46, dropped out from Primary school at age 12 because his teachers and classmates discriminated him as an idiot.

This video shows he can mentally do very complex math calculations:

The full video (in Chinese):

Today I realized something about being a tutor.

A big part of it — maybe as much as half it — involves nothing more than  …   being nice.

By that I mean being kind.

By which I mean that if someone looks at you, as a young man did today, shaking his head and saying, “It’s crazy … I don’t know what 3 x 6 is,” I don’t laugh or chuckle or say anything remotely mean or mocking. Instead I just say, “It’s o.k. Look, I tutor people every day who don’t know what 3 x 6 is. Who cares, really? Let’s just try to figure it out … or use a calculator, as long as your teacher doesn’t mind.”

Really. That is a lot of what being a math tutor is about. Being nice. Really nice. Really understanding. And being there to be accepting of people no matter…

Have you ever been befuddled by the rules for logs?

More specifically, have you ever looked at this rule:

log (v w) = log v + log w

and thought: Now why in the world is that true?! What exactly is this saying? I know that I, myself, have had that thought. And for me the desire to understand this rule never went away. Till I got it some time ago.

[By the way, keep in mind that the v and the w in the parentheses are multiplying each other, so that v w actually means: v times w]

And the good news is: I think I can explain this rule in a way so that pretty much everyone who knows basic algebra can grasp it.

O.K., first, I knew that this log rule was related to another rule, the  exponent rule that says:

(a^b) x (a^c) = a^(b…

This term I shall be giving Cambridge’s course Analysis I, a standard first course in analysis, covering convergence, infinite sums, continuity, differentiation and integration. This post is aimed at people attending that course. I plan to write a few posts as I go along, in which I will attempt to provide further explanations of the new concepts that will be covered, as well as giving advice about how to solve routine problems in the area. (This advice will be heavily influenced by my experience in attempting to teach a computer, about which I have reported elsewhere on this blog.)

I cannot promise to follow the amazing example of Vicky Neale, my predecessor on this course, who posted after every single lecture. However, her posts are still available online, so in some ways you are better off than the people who took Analysis I last year, since you will have…

I’m encountering a sporadic bug over the past few months with the way WordPress renders or displays its LaTeX images on this blog (and occasionally on other WordPress blogs).  On most computers, it seems to work fine, but on some computers, the sizes of images are occasionally way off, leading to extremely distorted and fairly unreadable versions of the images appearing in blog posts and comments.  A sample screenshot (with accompanying HTML source), supplied to me by a reader, can be found here (in which an image whose dimensions should be 321 x 59 are instead being displayed as 552 x 20).  Is anyone else encountering this issue?  The problem sometimes can be resolved by refreshing the page, but not always, so it is a bit unclear where the problem is coming from and how one might mitigate it.  (If nothing else, I can add it to the bug collection…

The Leap Forward

We are very proud to present you the new Mathist. It is a real sneak peak into the future, that allows you to easily solve and compute anything you write with a single click!

Here are some examples;

• Simplify any expression to speed up problem solving,
• Plot graphs,
• Evaluate expressions and get approximate values,
• Solve equations and inequalities (with visual representation),
• Solve definite and indefinite integrals (with visual representation),
• Crunch difficult logarithms,
• Find derivatives and much more…

You can do all of this with just one button: 

Why is this important?

Have you ever worked on a problem, that is easy to understand, but just requires too much number crunching?– So did we!

We understand that learning a million new things every day as a technology or a science student doesn’t leave too much time for crunching math. And yet in most cases you cannot solve…

What is The Mathist?

TheMathist is a very unique social math note taking application. It works on every device, and is free for everyone. You can finally focus on the math, not the software.

Do I have to know TeX or similar syntax?

There is no need to learn TeX or save the notes as pictures, they will always be available to you as regular online documents that you can edit anywhere and anytime. We have built a revolutionary simple math editor, a lot of hours went research on this. Go ahead and give it a try!

Why should I use The Mathist?

Because it is simple and straightforward to use, a true breeze… Because you already take all your notes in digital form and save them in the cloud. Because it is fast and works on all devices. It does not require any software knowledge to start, and…

After many days of discovering my HUGE learning curve with Minecraft, I am finally starting to feel relatively comfortable in Creative mode…I can build a house without flooding it, planted a few trees and I no longer have random blocks floating in the sky around my world!  My class has been staying with me during recess to teach me how to play and I am amazed at how fast and detail-oriented they are in their designs, such as putting lava rocks under the water blocks to form a hot tub and putting glass windows in their new greenhouses. I just kept thinking that I would love for them to use this same precision and perseverance in math class.

I must have Minecraft on the brain, because I as I was planning this weekend for the upcoming week (multiplying fractions w/arrays), all of the scenarios were about planting on an acre…

This week I am in San Diego for the annual joint mathematics meeting of the American Mathematical Society and the Mathematical Association of America. I am giving two talks here. One is a lecture (for the AMS “Current Events” Bulletin) on recent developments (by Martel-Merle, Merle-Raphael, and others) on stability of solitons; I will post on that lecture at some point in the near future, once the survey paper associated to that lecture is finalised.
The other, which I am presenting here, is an address on “structure and randomness in the prime numbers“. Of course, I’ve talked about this general topic many times before, (e.g. at my Simons lecture at MIT, my Milliman lecture at U. Washington, and my Science Research Colloquium at UCLA), and I have given similar talks to the one here – which focuses on my original 2004 paper with Ben…

Math, art, English, science, history, sports, music, plumbing, dancing in the rain all have one thing in common: to do it well you need to practice, practice, practice.

Most people think that ‘cramming’ and ‘nap osmosis’ are the best ways to study. Just cause you sit at Starbucks or the library or go to tutoring for x hours a day does not equal success. Here are some tips:

Here is a website from MIT’s Center for Excellence on how best to study: http://web.mit.edu/uaap/learning/study/breaks.html. MIT, they know what they are talking about. If you don’t know, MIT is one of the top universities on Earth- perhaps even in the universe…

As you may already know, Danica McKellar, the actress and UCLA mathematics alumnus, has recently launched her book “Math Doesn’t Suck“, which is aimed at pre-teenage girls and is a friendly introduction to middle-school mathematics, such as the arithmetic of fractions. The book has received quite a bit of publicity, most of it rather favourable, and is selling quite well; at one point, it even made the Amazon top 20 bestseller list, which is a remarkable achievement for a mathematics book. (The current Amazon rank can be viewed in the product details of the Amazon page for this book.)

I’m very happy that the book is successful for a number of reasons. Firstly, I got to know Danica for a few months (she took my Introduction to Topology class way back in 1997, and in fact was the second-best student there; the class web…

Firstly, apologies for the long gap.  Very far from being Theorem of the Week, I know.  Here’s another theorem for now, and I’ll do what I can to revert to a weekly post.

So, to this week’s theorem.  I have previously promised to write about Fermat‘s Little Theorem, and I think it’s time to keep that promise.  In that post (Theorem 10, about Lagrange’s theorem in group theory), I introduced the theorem, so I’m going to state it straightaway.  If you haven’t seen the statement before, I suggest you look back at that post to see an example.

Theorem (Fermat’s Little Theorem) Let $latex p$ be a prime, and let $latex a$ be an integer not divisible by $latex p$.  Then $latex a^{p-1} \equiv 1\mod{p}$.

If you aren’t comfortable with the notation of modular arithmetic, you might like to phrase the conclusion of the theorem as saying that $latex…

Excellent Notes on Group Theory.

http://lishen.files.wordpress.com/2012/07/groups6.pdf

Senior Wrangler is the First position in the Math Tripos in Cambridge. Singapore Prime Minister Lee Hsien Loong was the Senior Wrangler in 1973, the first Singaporean student with such great honors, among other senior wranglers like Arthur Cayley (Group Theory), J.J. Sylvester (Inventor of Matrix, private tuitor of the “inventor of Nursing” Florence Nightingale), J.E. Littlewood (partnered in a twin research team with G.H. Hardy), Frank Ramsey (Ramsey’s Theorem), Stokes, Pell, etc.

Some great mathematicians like Bertrand Russell (Logician, Nobel Litterature Prize) , G.H. Hardy (20th century greatest Pure Mathematician, mentored 2 geniuses: Indian Ramanujian and Chinese Hua Luogeng 华罗庚*) were not Senior Wrangler. Prof Hardy hated Math Tripos syllabus (revealed in his autobiography: “A Mathematician’s Apology“).

1914 Brian Charles Molony
1923 Frank Ramsey
1928 Donald Coxeter
1930 Jacob Bronowski
1939 James Wilkinson
1940 Hermann Bondi
1952 John Polkinghorne
1953 Crispin Nash-Williams
1959 Jayant Narlikar
1970 Derek…

By Simon Singh

Synopsis: ‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.’

It was with these words, written in the 1630s, that Pierre de Fermat intrigued and infuriated the mathematics community. For over 350 years, proving Fermat’s Last Theorem was the most notorious unsolved mathematical problem, a puzzle whose basics most children could grasp but whose solution eluded the greatest minds in the world. In 1993, after years of secret toil, Englishman Andrew Wiles announced to an astounded audience that he had cracked Fermat’s Last Theorem. He had no idea of the nightmare that lay ahead.

In ‘Fermat’s Last Theorem’ Simon Singh has crafted a remarkable tale of intellectual endeavour spanning three centuries, and a moving testament to the obsession, sacrifice and extraordinary determination of Andrew Wiles: one man against all the odds.

First Published: 1997, Reissued: 2002| ISBN-13: 978-1841157917

Author’s…

► See the top performers in reading, mathematics and science  (on this page).

Chart A2·1 [ page 42] ranks countries, in descending order, according to the percentage of adults who have completed an upper secondary education (the most recent data in the 2013 report is from 2011).

PISA

• Singapore Math emphasizes the development of strong number sense, excellent mental-math skills, and a deep understanding of place value.
• The curriculum is based on a progression from concrete experience—using manipulatives—to a pictorial stage and finally to the abstract level or algorithm. This sequence gives students a solid understanding of basic mathematical concepts and relationships before they start working at the abstract level.
• Singapore Math includes a strong emphasis on model drawing, a visual approach to solving word problems that helps students organize information and solve problems in a step-by-step manner.
• Concepts are taught to mastery, then later revisited but not re-taught. It is said the U.S. curriculum is a mile wide and an inch deep, whereas Singapore’s math curriculum is said to be just the opposite.
• The Singapore approach focuses on developing students who are problem solvers.

I suppose every reader of this ‘ere blog will know Heron’s formula for the area $latex K$ of a triangle with sides $latex a,b,c$:

$latex K = \sqrt{s(s-a)(s-b)(s-c)}$

where $latex s$ is the “semi-perimeter”:

$latex \displaystyle{s=\frac{a+b+c}{2}.}$

The formula is not at all hard to prove: see the Wikipedia page for two elementary proofs.

However, I have only recently become aware of Brahmagupta’s formula for the area of a cyclic quadrilateral. A cyclic quadrilateral, if you didn’t know, is a (convex) quadrilateral all of whose points lie on a circle:

And if the edges have lengths $latex a,b,c,d$ as shown, then the formula states that the area is given by

$latex K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$

where as above $latex s$ is the semi-perimeter:

$latex \displaystyle{s=\frac{a+b+c+d}{2}.}$

This can be seen to be a generalization of Heron’s formula. Although the formula is named for Brahmagupta (598 – 670), who does indeed seem to…

Recently, in The Conversation, the Vice Chancellor of Monash University, wrote an article discussing MOOCs. He made some criticisms about the nature of assessment and grading that MOOCs offer. However, my attention was grabbed by two sentences:

The other major problem the MOOCs haven’t solved is assessment. They work very well for subjects like maths, which have objectively right and wrong answers, and can therefore be pretty easily marked by computers.

Now, here we have the Vice Chancellor of one of Australia’s leading universities – and indeed, one of the world’s leading universities (and incidently the University where I did both my Masters and my PhD) demonstrating an extraordinary lack of understanding about the fundamental nature of mathematics. He seems to think that mathematics is all about teaching students (in the fine words of John Power from Leeds University) about “finding ‘x'”. I suppose he thinks this…

WordPress.com supports LaTeX, a document markup language for the TeX typesetting system, which is used widely in academia as a way to format mathematical formulas and equations. LaTeX makes it easier for math and computer science bloggers and other academics in our community to publish their work and write about topics they care about.

If you’re a math genius — many of you are! — and you’ve blogged about equations you’ve worked on, you’ve probably used LaTeX before. If you’re just starting out (or simply curious to see how it all works), we’ve gathered a few examples of great math and computing blogs on WordPress.com that will inspire you.

In general, to display formulas and equations, you place LaTeX code in between $latex and $, like this:

$latex YOUR LATEX CODE HERE$

So for example, inserting this when you’re creating a post . . .

$latex i\hbar\frac{\partial}{\partial…

Prove

$latex x^{2} \equiv 3411 \mod 3457 $
has no solution?

Legendre Symbol:

$latex \displaystyle
x^{2} \equiv a \mod p
\iff
\boxed{
\left( \frac {a}{p} \right)
= \begin{cases}
-1, & \text{if 0 solution} \\
0 , & \text{if 1 solution} \\
1, & \text{if 2 solutions} \\
\end{cases}
}
$

Hint: prove $latex \left( \frac{3411}{3457} \right) = -1$

Using the Law of Quadratic Reciprocity, without computations, we can prove there is no solution for this equation.

Solution:

1.
3411 = 3 x 3 x 379 = 9 x 379

$Latex \displaystyle
\boxed{
\left(\frac{a}{p}\right)
\left(\frac{b}{p} \right)=
\left(\frac{ab}{p}\right)
}
$

$latex \displaystyle
\left(\frac{3411}{3457} \right)=
\left(\frac{9}{3457} \right).\left(\frac{379}{3457} \right)=
\left(\frac{379}{3457} \right)
$
since
$latex \displaystyle\left(\frac{9}{3457} \right)=1 $
because 9 is a perfect square, 3457 is prime.

2. By Quadratic Reciprocity,
$latex \displaystyle
\boxed{
\text{If p or q or both are } \equiv 1 \mod 4 \implies
\left(\frac{p}{q} \right)=
\left(\frac{q}{p} \right)}
$

Since
$latex…

The famous Singapore Math for children in primary schools is based on  visual models.

The Singapore Ministry of Education has published a new 2013 Math syllabus for primary and secondary schools, which will roll out in examinations within 4 to 6 years. Todate only Primary 1 and Secondary 1 Math syllabuses are published here:

http://www.moe.gov.sg/education/syllabuses/sciences

Who wins?

This comic video illustrates Singapore Math’s Arithmetics Polya-style problem solving process vs Algebra’s mechanical method.

The problem is as follow:
R is 3 times older than S two years ago. From now 2 years later, their total age is 32. How old is R now ?

See my previous blog (search “Monkey”) the Nobel Physicist Paul Dirac’s problem “The Monkeys and Coconuts“, 3 methods are used: 2 adanced modern math (by Sequence, eigenvector & eigenvalue), and the easiest & intuitive method (by Singapore Modelling Math). High-school Algebra method is impossible, if not cumbersome, to solve the Monkey problem !

I find Khan Linear Algebra video excellent. The founder / teacher Sal Khan has the genius to explain this not-so-easy topic in modular videos steps by steps, from 2-dimensional vectors to 3-dimensional, working with you by hand to compute eigenvalues and eigenvectors, and show you what they mean in graphic views.

If you are taking Linear Algebra course in university, or revising it, just go through all the Khan’s short (5-20 mins) videos on Linear Algebra here:

In 138 lessons sequence:

http://theopenacademy.com/content/linear-algebra-khan-academy

or random revision:

Relationship-Mapping-Inverse (RMI)
(invented by Prof Xu Lizhi 徐利治 中国数学家 http://baike.baidu.com/view/6383.htm)

Find Z = a*b

By RMI Technique:
Let f Homomorphism: f(a*b) = f(a)+f(b)

Let f = log
log: R+ –> R
=> log (a*b) = log a + log b

1. Calculate log a (=X), log b (=Y)
2. X+Y = log (a*b)
3. Find Inverse log (a*b)
4. ANSWER: Z = a*b

Prove:

$latex \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= 2$

1. Take f = log for Mapping:
$latex \log\sqrt{2}^{\sqrt{2}^{\sqrt{2}}} $
$latex = \sqrt{2}\log\sqrt{2}^{\sqrt{2}}$
$latex = \sqrt{2}\sqrt{2}\log\sqrt{2} $
$latex = 2\log\sqrt{2} $
$latex = \log (\sqrt{2})^2 $
$latex = \log 2$

2. Inverse of log (bijective):
$latex \log \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= \log 2$
$latex \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= 2$