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?'

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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.

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.
"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.