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?'

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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.

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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.

Exploring Algebra Games

Maria Andersen recently wrote a blog article with a nicely-made Prezi video giving an overview of existing games out there in the market to learn elementary school algebra.

Many of these games are commercial and they seem to have good and catchy graphics. It seems that most of the game design are primarily 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. 

Another observation is that many of these games are mobile apps that run on smartphones or tablets. Imagine the super duper ease of getting the games off-the-shelf from Google Play and Apple app stores and playing them instantly. Mobile app platforms have thus made mathematical games more accessible for many people. One wonders the possibility of a math teacher deploying these apps in a real classroom, and how would such math games complement the teaching curriculum? 

In the first place, teachers have to vet these games and find one that suits the curriculum. Given that teachers are busy, it is not easy to get real statistics or data to show how useful the mobile games are as a teaching aid and whether the students actually learn. But suppose a teacher picks a mobile game, then what game design might attract the teacher and the students?  

Quiz-based games in some sense "digitize" the paper-and-pencil approach to working on a fixed set of questions, possibly providing hints in the form of multiple choice. Quiz-based games can certainly be useful for drilling. On the other hand, puzzle-type games are typically harder to design, because it involves the element of strategic thinking. If we can instil in students the idea that learning mathematics is to think strategically, then this can be a more efficient way for students to build self-motivation and take on challenging problems as compared to the quiz-based game design.

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.