Promoting Equity in Mathematics Education

In February 2018, I visited the Mathematical Sciences Research Institute at UC Berkeley to attend a MSRI Workshop on Critical Issues in Mathematics Education 2018: Access to mathematics by opening doors for students currently excluded from mathematics. Tertiary and K-12 educators across the nation and some including myself from overseas converged at this workshop packed with group discussions and lightning talks co-organized by Francis Su who was the President of the Mathematical Association of America. The MSRI has been a prominent driving force behind math circle learning in the United States, and so for me it was rightly a pilgrimage trip to MSRI which is perched right up at the top of the Berkeley hill, offering an unbeatable panoramic view of the Golden Gate Bridge and the Bay Area.  

One striking takeaway from this workshop is equity in mathematics education. Roughly speaking, equity means giving students what they need to succeed in learning regardless of their personal or social circumstances. One of the plenary speakers, Prof. Dave Kung, gave a very powerful and fantastic talk that provoked the educators among the audience to ponder the question of "I teach math?" or "I teach students?"  and where the audience would see themselves in the spectrum between these two extremes. The former is speaking the content, while the latter is individualized instruction. Dave held polls in between his plenary and it was interesting to watch the number of hands raised in the audience when Dave asked whoever had experienced a teacher at those two extremes and where the audience thought their current department might be standing in that spectrum. 

What Dave Kung brought out is an important pedagogical question but also one that is well formulated to ask whether and how, somewhere in the spectrum, technologies can help the disaffected students and even teachers. Going by the extent in which technologies have democratized information sharing in the last ten years, it is thus necessary to consider what greater role will technology play in order to address the equity problem in mathematics education. Will it open up doors to many more students or will it shrink the equity gap between the haves and the have-nots? This one-week pilgrimage to MSRI has been extremely educational and has given me a sense on how to develop and deploy our educational software technologies going forward. 

For more than a year, we have been working on the PolyMath App, which first saw action at the Julia Robinson Mathematics Festival in Hong Kong in April 2017. In February 2018, we worked with a group of forty secondary school mathematics teachers from the English Schools Foundation (ESF) international schools in Hong Kong. There was so much to learn from the teachers at the front-line of the ever-evolving mathematics education landscape. Using the PolyMath App, we shared with the teachers how they can quickly learn binary number system and arithmetic through game-play in our PolyMath App. The binary system is the most basic and practical one, not just because we use it in computers, but also because binary number representation and its arithmetic manifest intriguingly in games and algorithms. In particular, we delved into the deep mathematical insights behind the NIM game (also called the Fish-flavored Lollipop problem set in JRMF) and the algorithm behind the Russian peasant's method of multiplication and a logical puzzle by Martin Gardner. Working with these amazing teachers was an uplifting experience!

         

Starting the Julia Robinson Mathematics Festival in Hong Kong

We are very excited to announce that the inaugural Julia Robinson Mathematics Festival in Hong Kong is going to be held on April 1st (Saturday) at the Singapore International School (Hong Kong). This is open to students in Hong Kong, ages 10 to 14.

The Julia Robinson Mathematics Festival in Hong Kong 2017 inspires students to explore the richness and beauty of mathematics through activities that encourage collaborative and creative problem-solving. This event is in joint partnership with the Julia Robinson Mathematics Festivals and the Singapore International School (Hong Kong). Details at http://www.algebragamification.com/JRMF.

We started this as a grassroots activity with a few like-minded friends -- Jian Shen (former Princeton University Math Club President who then roped in other mathematicians working in finance), Ken Shum. We all wondered what differences we can make for Hong Kong students who are new to this type of mathematics initiative. And the Julia Robinson Math Festival in Hong Kong seems perfectly in sync with the United States' 2017 National Math Festival, the grandest carnival in mathematics. We are delighted to receive full support from Mark Saul (Executive Director of Julia Robinson Mathematics Festival) and also local academics like Professor Tony Chan (HKUST President), sponsorship from the IEEE Information Theory Society and the Singapore International School (Hong Kong) whose vice-Principal Mr. Bernard Ng and I had a great time working together before.

The Julia Robinson Mathematics Festivals is a project founded by Nancy Blachman in partnership with the American Institute of Mathematics that started in 2007 at Google in the Bay Area (yes it's as old as the cool iPhone). Since then, the festivals have spread to many places worldwide. The idea of the Festivals is to allow young people to develop their talent for mathematics by providing problems, puzzles, and activities that are intriguing and accessible in a non-competitive atmosphere. A diverse audience of young people, school teachers, math lovers in the community come together to explore the joys and power of mathematics with the goal of broadening society's appreciation and support of mathematics. 

We will have a number of math-lovers from the academia and industry who will facilitate at each table of math and to give out raffle tickets! The Algebra Game Project will of course have a table of its own for students to explore the mathematics behind the game. There are also mathematical origami, puzzles and many more that we are creating and preparing right now! 

Stay tuned for April 1st! Here is the Festival poster:

Algebra Game and Algebra Maze on Google Play Store!

We recently launched our first mobile apps Algebra Game and Algebra Maze on the Google Play Store.  From day one, the Algebra Game team members basically hit the ground running to put ideas into code.  Designing mathematical games in mobile apps is a challenging experiment - binding mathematical elements with the human-computer interface is an art,  making the same piece of software work on different hardware is trial and error,  walking through the entire online app submission process is new to us!  And this whole process doesn't stop there. It continues in a loop whenever new bugs/ideas surface.  Developing software is truly a humble learning experience.  We are working hard on the iOS mobile apps - so expect to see them in App Store soon.

We are also excited to have made inroads into understanding the mathematics behind Tao's Algebra Game.  Figuring out the math to compute the fewest moves for puzzle generation thereby answering some of Terence Tao's questions is actually the ultimate endgame for us :)   This can help in cleverer puzzle generation and we will flesh out that once the math is neatly ironed out.  The Algebra Maze on the Google Play Store currently has forty-five levels altogether,  but we have embarked on newer game design for the Algebra Maze and even contemplating new functionality in these mobile app games to bring out the social element: Math is social!

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?

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.