Saturday, October 3, 2015

Starting Up our Algebra Game

Software is eating the world. Yet, it is a marvel of how readily the younger generation eats up software like Angry Bird, Storybird, Youtube, digital games etc.

We have learned an enormous insight to the Terence Tao's algebra game and want to make this fun game accessible to more people especially the younger generation.  

In the next few months, we will be launching various software revolving around the Algebra Game that, I hope, become interesting for the younger generation and also for us to learn some mathematics and computer science. 
We have started the Facebook App playable on a website at

The goal of our work is twofold. We plan to establish a rigorous study of learning via mobile and cloud computing that supports Advance Personalized Learning at scale. And we plan to walk the ground to a few primary schools in Hong Kong (and some parts of Asia) soon to introduce our software and to get feedback.

We will post pictures of our learning endeavours on our Algebra Game Facebook Page and release a few more software  — so stay tuned!

Saturday, July 11, 2015

Terence Tao's Algebra Game

Terence Tao, a mathematics professor at the UCLA, blogged about various issues on Gamifying High-School Mathematics in 2012. He also introduced an interesting mathematical puzzle to gamify elementary algebra, i.e., solving a single linear algebra equation with one variable, on his blog.

The idea of Tao's algebra game is to reduce a given linear algebra equation to "x = numerical_solution" through a selection of a finite number of given clues. This algebra game was created using the MIT Scratch software tool and can be played here (requires Java). Let's give an example using a screenshot of the game (taken from Tao's blog) as shown below. Initially, the puzzle is the algebra equation "5x + 3 = x + 11" and the given clues are the three possibilities "Subtract 1", "Divide by 2" and "Subtract x". A player chooses one of the possibilities by clicking on the avatar icon. Say, suppose the player chooses "Subtract 1", the algebra equation changes to "5x + 2 = x + 10" (since both sides of the original equation "5x + 3 = x + 11" get subtracted by one).

Of course, one possible "solution" to the puzzle below is the sequence of "Subtract 1" then "Subtract x" then "Divide by 2" then "Subtract 1"and then finally "Divide by 2" to yield "x = 2". A total of five moves to reach the desired state. What matters is not the final value of x (which can be eye-balled easily), rather, it is the inquisitive problem-solving process while playing.

Tao's algebra game is absolutely fun (and some of the higher levels are fairly challenging, even for an adult like me). The benefit is obvious: children learn a real subject (e.g., elementary algebra) while playing. Particularly pleasant is the mathematical depth behind the game. The twelve levels (from easy to hard) in Tao's Scratch mock-up version were cleverly hand-crafted. Yet, there are a few intriguing questions: first, how to engineer the difficulty level of the game automatically? Second, how does a computer (not human player) solve a given puzzle efficiently, i.e., with the fewest number of moves? And, third, how to engage the human players in an entertaining manner so that they keep on playing it and, unknowingly, develop a better number sense or mathematical intuition and that such an improvement can be measured?

Another practical question is the means by which these kind of educational games can be readily deployed on existing computing platforms to reach the target audience players (in this case, the target audience is presumably 8 to 12 years old primary school students, and why not adults? I like playing it too). The MIT Scratch software tool is relatively easy (even for 6 years-old) to create games and animations on desktop PC's and laptops, and the end-products are largely accessible through a web browser interface. If this were a Facebook game, can it reach a fairly broad audience of age 8 to 12 (since Facebook requires users to be older than 13)? If this were a mobile app game, can it reach the target audience who may not have personal mobile devices or audience who only have desktop PC's and laptops?

Friday, June 26, 2015

Claude Shannon: Centennial and Machines

Claude Shannon and his electromechanical mouse Theseus. Retrieved from

Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was a distinguished American electrical engineer and mathematician who founded the field of Information Theory when he published his landmark paper "A Mathematical Theory of Communication" in 1948. He worked on anti-aircraft and cryptography systems during World War II, and later on had a career at the AT&T Bell Labs and as a Professor at MIT. The MIT Technology Review Magazine ran an article on this Reluntant Father of the Digital Age shortly after his death.

In fact, while Shannon was still a student at MIT, he wrote what has been touted as the Most Important Master's Thesis of the Twentieth Century (putting forward the idea that Boolean algebra can be used for computing). Many technological fields that we know of now (such as the digital revolution, secrecy and cryptography, artificial intelligence, wearable computing) can all be traced back to Shannon's pioneering ideas and work.

Despite all these achievements, Shannon is surprisingly little known. Next year 2016 is Shannon's centenary. To get the public know more about Shannon, the Information Theory Society is on a drive to make a movie about this scientific genius of the twentieth century. Incidentally, Shannon did appear in a movie once - back in 1961 - in a MIT Centennial Film called The Thinking Machine. At 50:09 mins of the film, he shared his vision of a future machine capable of learning by experience. The Information Theory Society also aims to organize events worldwide to celebrate the Shannon's centenary. 

I wonder: What little bits of things can we prepare or do to celebrate Shannon's centenary?

Perhaps, particularly endearing is Shannon's fun-loving and unconventional side: To amuse himself, Shannon created very clever mechanical and electrical toys -- a juggling robot, a maze-solving mouse, a chess-playing machine, a mind-reading machine, a manipulator to solve the Rubik's cube game etc. The Ultimate Machine is another such curiously uncanny toy -- we have one built from a small motor, off-the-shelf electronics and recyclable. See video below.

In a 1987 interview, the Omni Magazine asked Shannon: Do you find it depressing that chess computers are getting so strong? 

Shannon replied: I am not depressed by it. I am rooting for the machines! I have always been on the machines' side. Ha-ha! 

Tuesday, June 16, 2015

Googling Arithmetic Limerick

While using Google to scour the Internet for poems on arithmetic and algebra, I came across the following three limericks. The first is similar in spirit to the Carroll's poem and the next two are similar to Diophantus puzzle in my earlier posts. 

Leigh Mercer, a noted British wordplay, wrote the following (excerpt from Word Ways, 1980, pp. 36):

A dozen, a gross, and a score 
Plus three times the square root of four 
Divided by seven 
Plus five times eleven 
Is nine squared and not a bit more.

Interestingly, Google's omnipotent search engine can "solve" this limerick automatically by entering the following slightly modified verse into the Google search bar:

(a dozen, and a gross, and a score plus three times the square root of four) divided by seven plus five times eleven

The following two are by David Pleacher, a retired mathematician:

Said a certain young lady named Gwen 
of her tally of smitten young men, 
"One less and three more 
Divided by four 
Together give one more than ten." 

How many boyfriends had she? Answer here.

Some freshmen from Trinity Hall 
Played hockey with a wonderful ball; 
They found that two times its weight, 
Plus weight squared, minus eight, 
Gave "nothing" in ounces at all. 

What was the weight of the ball? Answer here.

Friday, June 12, 2015

Learning and Testing

In many Asian countries, standardized testing is the hallmark of primary and secondary school education. Outside of regular schools, numerous after-school coaching activities, preparation schools and commercial tuition centers thrive in training young students to excel in standardized testing. This is big business - involving enormous education spending in the hope of getting a good enough grade for these pressure-cooker standardized tests.

In Hong Kong and Singapore, the sense of anxiety and dread lingers among primary and secondary school children in the lead-up to sitting en masse for UK-style standardized exams (primary school leaving exams, O-level, A-level). And in the hot summer month of June, in mainland China, massive number of students travel across the country to sit for the national higher education entrance examination - a high-stakes game - that requires gadgets like multi-rotor drones as invigilators.

I recently came across an Edweek article "Rethinking the Emphasis on Standardized Testing" on the traditional standardized testing in Asia and how it affected mathematics proficiency and even the lifetime success of a student. The article's author started a First-in-Math venture to host math tournaments in the United States to play 24 Game - in fact a card game that originates from China in the 1960's. Have a go at the 24 Game to test your reflex and sharpen your mind (The game at is free)! This is big business indeed - going by the 17 billions (and still counting) of problems solved as advertised on the First-in-Math's website.

One interesting takeaway from the Edweek article is that:
Testing has its place as long as it doesn’t push kids away from a sense of wonder and fascination for the world around them....  A far more worthy goal would be to create a system wherein the whole individual is addressed, developed, and encouraged to thrive in the pursuit of a better life. 

I think this pretty much sums up personalized learning as the next frontier in education. What truly amazing technological innovations can we envision in this quest to advance personalized learning? Will that be an ecosystem of mobile digital tutors, an Internet of peer learning, automated software to individualize learning, learning analytics at scale, smarter interactive computer tests driven by computational neuroscience, AI, cloud computing and machine learning? Or will it be more powerful surveillance drones?

Monday, June 1, 2015

Diophantus Puzzle

Diophantus was a Greek mathematician whose work laid down important foundation for the development of algebra. Listed on Wikipedia is an interesting puzzle to guess the age of Diophantus:

'Here lies Diophantus,' the wonder behold.
Through art algebraic, the stone tells how old:
'God gave him his boyhood one-sixth of his life,

One twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun;
In five years there came a bouncing new son.

Alas, the dear child of master and sage
After attaining half the measure of his father's life chill fate took him. 
After consoling his fate by the science of numbers for four years, he ended his life.'

This Diophantus puzzle has a similar flavor to the Carroll's puzzle in my previous post. Here, however, a finite sequence of moves (in fact, events in the timeline) is applied to an unknown - Diophantus's age. 
How long did Diophantus live?

Friday, May 29, 2015

Carrollian Algebra

Alice talking to White and Red Queens
Alice's Adventures in Wonderland and its sequel Through the Looking Glass are some of the well-known children's classic tales written by Lewis Carroll, whose real name is Charles Lutwidge Dodgson, a mathematics lecturer at Oxford University. 

There are various algebra games designs embedded in Carroll's work. Let's use a few examples to illustrate. 

The first example is quiz-based design that comes from an excerpt in Through the Looking Glass, by Lewis Carroll:

'Can you do Addition?' the White Queen asked. 'What's one and one and one and one and one and one and one and one and one and one?'

'I don't know,' said Alice. 'I lost count.'

'She can't do Addition,' the Red Queen interrupted, 'Can you do Subtraction? Take nine from eight.'

'Nine from eight I can't, you know,' Alice replied very readily: 'but—'

'She can't do Subtraction,' said the White Queen. 'Can you do Division? Divide a loaf by a knife—what's the answer to that?'


Both Alice and the reader are entertainingly quizzed by the Queens. Try arguing with arithmetic!

The second example is puzzle-based design that comes from an excerpt in The Hunting of the Snark, by Lewis Carroll: 

Taking Three as the subject to reason about --
     A convenient number to state--
We add Seven, and Ten, and then multiply out 
    By One Thousand diminished by Eight.

The result we proceed to divide, as you see, 
    By Nine Hundred and Ninety and Two:
Then subtract Seventeen, and the answer must be 
    Exactly and perfectly true.


In a way, the Carroll's puzzle-based design is similar to the Jinx Puzzle introduced by Harold Jacob in his book Elementary Algebra (1979)

In Carroll's puzzle game, an initial number (can be any and need not be Three) first gets manipulated through a finite sequence of arithmetic moves (the above first stanza) to become another numerical result. A second finite sequence of arithmetic moves (the above second stanza) is then carefully chosen to go from this numerical result back to the initial number. Thus, the two finite sequences are essentially inverse of each other.

In Jacob's Jinx Puzzle game, an initial number (can be chosen from a few possibilities) goes through a carefully-crafted finite sequence of arithmetic moves to become a particular desired numerical result, giving the impression that All roads lead to Rome.

Saturday, May 16, 2015

Exploring Algebra Games

Maria Andersen had a nice blog article with a nicely-made Prezi video showing an overview of some of the existing games out there in the market for learning elementary algebra.

Many of these existing algebra games are commercial and they seem to have good and catchy graphics. The game design however seems to be essentially quiz-based. There are also a few puzzle-like ones that require players to develop a playing strategy. It will be interesting to think of different game design for learning algebra. 

It is important to observe that many of these games are mobile apps (that run on smartphones or tablets). Imagine the ease of purchasing them off-the-shelf from Google Play and Apple stores. Mobile app platforms have thus made it so much easier for these games to reach out to young players. 

One wonders the possibility of deploying these apps in real classroom setting. In this case, how often and how do such games actually get deployed in real primary schools for teaching elementary algebra?

I'd imagine that for these games to penetrate the schools, the teachers have to vet and select some eventually. However, it is kind of hard to get real statistics on how effective these selected games can be used as teaching aids in real classroom setting.

Quiz-based games in a certain sense "digitize" the paper-and-pencil approach to working on a quiz, and thus perhaps emphasize more on the drilling aspect. 

Then, what game design can be more efficient in motivating young children to learn algebra as compared to quiz-based games? 

Sunday, May 3, 2015

Explorer - The First Step

Yes, that's you - you're the explorer.
[ From the Preface: Games & Puzzles, Discovering the Art of Mathematics by V. Ecke and C. von Renesse with J. F. Fleron and P. K. Hotchkiss, 2015 ]

In 2008, the National Academy of Engineering (NAE) made Advance Personalized Learning as one of the fourteen grand challenges for engineering in the 21st century. Since then we have witnessed the emergence of amazing technologies for learning such as mobile apps, MOOC, Social Learning Networks, Youtube, Gamification, ... .Everyone learns differently so how to cater to the different learning styles of each person to learn mathematics?

Let's start by thinking how to advance personalized learning to teach children, K12 students and even adults to learn mathematics such as arithmetic and algebra. This is the journey to explore the wonders of learning mathematics through playing.