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, mobile app 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, as we launch some of the mobile app software revolving around the Algebra Game, I hope that this is one small step towards developing learning technologies that make an impact no matter how small. To make Advance Personalized Learning at scale possible, we want to make learning mathematics and computer science a cool thing for everyone. We will post pictures of our learning endeavours on our Algebra Game Facebook Page and hope to release more fun software in the future  — stay tuned!

Terence Tao's Algebra Game

Terence Tao, a famous 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?   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 accessible only through a web browser.  If this were a Facebook game, can it reach K12 students 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?

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.

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?

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.

Explorer - The First Step

Yes, that's you - you're the explorer.
"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 tech, MOOC, flipped classroom, social learning networks, Youtube, gamification, .... Everyone learns mathematics differently. Can a math teacher cater to the different learning style of each and every student?

Perhaps, we can start by teaching children, K12 students and even adults to see mathematics as not just arithmetic or algebra. Mathematical skills and knowledge can be shaped through playing. This is the journey to explore the less trodden paths of personalized learning through innovations in computer science.