Sexy maths: Skills of a Chess Grandmaster

For a while, the chess Olympiad this year looked like producing a surprise winner but closer inspection of Israel’s team sheet revealed that it was pretty much business as usual: half the players were named Boris!

Other than a brief blip in the 1970s, the biennial event has produced remarkably consistent results. From 1952 to 1990, the Soviet Union ruled the contest, and after the superstate’s fragmentation either Russia or one of its former union satellites struck gold every time. As it turned out this year, the Soviet diaspora’s turn in the spotlight was short-lived and Armenia triumphed for its second successive Olympiad.

Despite being connected by being born under the red flag, those that dominate the game are better categorised by their membership of a different club: the mathematical mafia. Legend has it that the game was invented by a mathematician in India who elicited a huge reward for its creation. The King of India was so impressed with the game that he asked the mathematician to name a prize as reward. Not wishing to appear greedy, the mathematician asked for one grain of rice to be placed on the first square of the chess board, two grains on the second, four on the third and so on. The number of grains of rice should be doubled each time.

The King thought that he’d got away lightly, but little did he realise the power of doubling to make things big very quickly. By the sixteenth square there was already a kilo of rice on the chess board. By the twentieth square his servant needed to bring in a wheelbarrow of rice. He never reached the 64th and last square on the board. By that point the rice on the board would have totalled a staggering 18,446,744,073,709,551,615 grains.

Playing chess has strong resonances with doing mathematics. There are simple rules for the way each chess piece moves but beyond these basic constraints, the pieces can roam freely across the board. Mathematics also proceeds by taking self-evident truths (called axioms) about properties of numbers and geometry and then by applying basic rules of logic you proceed to move mathematics from its starting point to deduce new statements about numbers and geometry. For example, using the moves allowed by mathematics the 18th-century mathematician Lagrange reached an endgame that showed that every number can be written as the sum of four square numbers, a far from obvious fact. For example, 310 = 172 +42 + 22 + 12.

Some mathematicians have turned their analytic skills on the game of chess itself. A classic problem called the Knight’s Tour asks whether it is possible to use a knight to jump around the chess board visiting each square once only. The first examples were documented in a 9th-century Arabic manuscript. It is only within the past decade that mathematical techniques have been developed to count exactly how many such tours are possible.

It isn’t just mathematicians and chess players who have been fascinated by the Knight’s Tour. The highly styled Sanskrit poem Kavyalankara presents the Knight’s Tour in verse form. And in the 20th century, the French author Georges Perec’s novel Life: A User’s Manual describes an apartment with 100 rooms arranged in a 10×10 grid. In the novel the order that the author visits the rooms is determined by a Knight’s Tour on a 10×10 chessboard.

Mathematicians have also analysed just how many games of chess are possible. If you were to line up chessboards side by side, the number of them you would need to reach from one side of the observable universe to the other would require only 28 digits. Yet Claude Shannon, the mathematician credited as the father of the digital age, estimated that the number of unique games you could play was of the order of 10120 (a 1 followed by 120 0s). It’s this level of complexity that makes chess such an attractive game and ensures that at the Olympiad in Russia in 2010, local spectators will witness games of chess never before seen by the human eye, even if the winning team turns out to have familiar names.

 Article Source : Times Online
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Chess and Math: a happy couple?

Chess and math have always slept side by side. But are they a happy couple? I think every chess player has had the experience of someone asking you, in high school, if your math grades were as good as your chess results. Sadly, for me the answer was often ‘no’.      By Arne Moll

In fact, the reason I did so badly in high school math was probably … my chess addiction. I spent so much time on chess that I completely neglected math (and other subjects.) But, as the song goes, old habits die hard. And, of course, it’s not unreasonable to suppose math and chess are related, or that results in both could be correlated. In fact, several peer-reviewed studies have pointed out the advantages of using chess as a teaching method [1]. Recently, a paper by John Buky and Frank Ho was published on the effect on pupil’s math scores using an integrated math and chess workbook [2]. In this article, I will take a look at the results from a chess players perspective, discuss some problems and give some suggestions for further research.

A critical look at the curriculum

The curriculum and the workbook are described in detail on the site of the Chess and Math Academy based in Chicago [3]. Examples are also mentioned in other peer-reviewed articles by Ho [4]. The idea is that learning about, e.g., algebraic notation and pattern recognition can be transferred to math concepts:

“By working on mathematical chess puzzles, students get training on how to transfer chess knowledge to improve math ability. Since chess is a whole number based strategy game so it is important for students to get exposure to computational mathematical chess puzzles. […] Algebraic notation learned in chess could be transferred to the concepts of coordinates […]. The King’s triangular shape of movement to create opposition in chess is an example on how the use of a geometrical shape would take a special meaning in chess. […] One notable math knowledge learning in playing chess but not widely taught is the set theory. Chess players constantly use the concept of Venn diagram to look for interaction among pieces.”

The first thing to note is that these relations between chess elements and their supposed mathematical mirror-elements are not intuitive for the chess player. Chess is not mathematics, and they work by different type of rules. Mathematics is rigorous, chess is not. In his classic Secrets of Modern Chess Strategy, John Watson remarks: “In chess, general rules will never have universal application […]” [5]. In chess, unlike in mathematics, there are no absolute truths, as anyone who has ever tried to calculate a ‘book’ bishop sac on h7 will know. The famous chess grandmaster Richard Reti makes the point in his Modern Ideas in Chess (1922) [6]:

“What is really a rule in chess? Surely not a rule arrived at with mathematical precision, but rather an attempt to formulate a method of winning in a given position or of reaching an ultimate object, and to apply that method to silimar positions.”

Although making a link between chess and math teaching is very creative and interesting, it is in a way also slightly suspect. Haven’t we all seen stereotyped commercials in which chess or chess pieces stand for ’strategy’ or ‘cleverness’ (or ‘nerdiness’)? The link is also a bit forced. For instance, although I am familiar with the concept of Venn diagrams, I have, as a chess player, never realized that it could be linked to the interaction among pieces. In fact, even now that I know of a possible relation between the two concepts, I’m not entirely sure how exactly the two can be linked. Should I draw Venn diagrams inside my head next time I’m trying to play a game? Do my chess skills help me understand complex Venn problems better?

I also doubt the author’s assertion that chess is a “whole number based game”. The idea probably stems from the supposed absolute value of the pieces (a Rook is worth 5 pawns, a Knight 3 pawns, etc.). However, it has long been known that this approach is too simplistic. Most strong chess players will probably agree that a bishop is, on average, worth not 3, but 3,5 pawns. A knight is slightly less, perhaps 3, but it all depends on the circumstances. And as every chess player knows, the value of the King in chess is, in a way, “infinite”. In my opinion, the idea that chess is a whole number game is, at best, simplistic. It is certainly confusing.

Buky and Ho are themselves aware of the non-obvious relation between chess and math and the possible effect on pupils. As they say:

“The effect of transferring math knowledge learned in chess will be less significant if the chess teacher does not take the efforts of emphatically point out the math concepts. The task of transferring math knowledge learned in playing chess would be much easier if students are offered the opportunities to work on mathematical chess puzzles.”

In other words, although the two concepts may not be grasped intuitively, having a good teacher and by doing ‘mathematical chess puzzles’, the non-intuitiveness of the two concepts could be overcome. The first condition is so obvious it’s strange it’s even mentioned at all, the second sounds plausible, but of course, the puzzles have to be clear enough for pupils to understand the implicit relations between the two. Let’s have a look at an example from the workbook.

Even disregarding the question whether one can simply attach values to chess pieces (in the example, a queen is probably worth 9 ‘pawns’ and a bishop 3), I find this example quite confusing. A first practical question is if should we fill in chess figurines or numbers in the blank boxes? And how are we supposed to interpret the mathematical operations? 12 minus a bishop equals a queen, but what is a queen plus a bishop? These may sound like trivial questions, which can be solved by proper instruction, but if for me, a skilled chess player and a professional IT developer, these puzzles are not clear, then how must these puzzles appear to 4th graders?

Here’s another example where the same question arises:

To be honest, this example baffles me. I just don’t understand what is going, but I realize it could be just me. So I asked IM Maarten Solleveld, who is a professional mathematician at Goettingen University, German, if he understood these examples. He didn’t. Moreover, he thought they were confusing to pupils, especially young ones. He writes:

“I don’t see the use of most of these examples. In fact, I find many of the excercises weak from an educational point of view, or even counter productive. The authors clearly cannot imagine themselves that it’s possible to confuse kids with strange puzzles.”

Before moving on to the research done by Buky and Ho, let me give one last example of one of their puzzles.

I think this example is especially noteworthy. In itself, making a link between chess and statistics is very original, although personally, I would rather be interested in a question like ‘What’s the chance my opponent is going to find this accurate reply to my bluff move?” than in the outcome of the probability of two pieces meeting within a certain number of moves. More to the point, it’s hard to imagine what use the excercise is given that fact that in order to get the solution to the problem, the concept of probability has to be taught to pupils anyway – so why involve chess in it? After all, the answer (0) only makes sense if you know the difference between probability 1 and probability 0.5 and probability 0. In other words, what is, in alle these excercises, the added value of adding chess concepts to the puzzles?

The results of the research

This is also my main criticism on the research mentioned in the author’s article. Buky and Ho tested their ideas on 119 pupils using a paried t-test:

“One hundred and nineteen pupils, in grade 1 to grade 8, from five public elementary schools in Chicago, Illinois, USA, participated in the after-school program for 120 minutes, twice a week, for a total of 60 hours of instruction. None of the students has possessed any substantial knowledge in chess. The study began by administering pre-tests in the first week of this study at the beginning of the program on 10/23/06 and a post-test was conducted at the end of the program on 3/28/07.

The test results were encouraging:

Group	Group One	Group Two
Mean	36.46	55.45
SD	15.82	19.37
SEM	1.45	1.78
N	119	119
t = 12.8729

The results show that pupils performed much better on the second (post-) test, showing they learned a lot (36.46 vs. 55.45) in the period between the two tests (roughly 5 months). The authors conclude:

The results of this study demonstrate that a truly integrated math and chess workbook can help significantly improve pupil’s math scores.

Discussion

This is, of course, good news for the Chess Academy’s teaching method, and for the pupils making use of it. Teaching childeren with the chess and math integrated workbook actually improves their mathematical skills. So far, so good. There is a problem, however. Since Buky and Ho’s emphasis is on the difference between their own method and “traditional computation practices”, it would, in my view, have been much more relevant (and interesting) to do a control test where one group of pupils gets their math lessons by the chess and math method, and the other group gets “normal” after-school math lessons. After all, only by doing a pre- and post-test for both groups, one can establish whether the chess and math method actually works better than the tradional method. This cannot be established with the research done by Buky and Ho.

As far as I can see, we’re still left with several possible explanations for the results they got:

Learning how to play chess improves the pupil’s mathematical skills anyway, regardless of the method or the workbook.

Since even without learning chess, pupil’s mathematical skills are likely to improve over time, it’s possible that pupils always perform better on the post-test conducted several months later.

Since the research was done in an after-school program, the pupils have in the mean time been learning a lot of math during school, making any progress in this area not more than natural.
–
What should we make of this? Because the traditional method was not compared to the chess and math method, I think it’s too early to conclude that the new method is actually better than the traditional method. And this ultimately has to be the 64,000 dollar question for the chess and math method. In theory, it’s even possible that while the chess and math method scores well on the test, the traditional method (or no method at all!) would score even better. By that rationale, the current research does not yet show that the method actually does anything at all. [7]
Also, while it may be true that, according to Buky and Ho, “math and chess integrated work has visual images, chess symbols, directions, spatial relation, and tables; all these are stimuli to kids and keep their interests high while working on computation problems”, it is not clear to me that the stimuli used in the curriculum actually contribute to improving mathematical skills. (Perhaps it was simply the teacher’s explanations that improved their results, not the puzzles themselves.) A small, non-representative poll among chess players and professional scientists does indicate that adults find the puzzles as presented in the curriculum confusing and/or unclear. Perhaps this is simply a matter of insufficient explanation, but in any case, it’s something the authors do not seem to acknowledge in their article. A final point of doubt concerns the benefits to the pupil’s chess abilities, as opposed to the benefits of their math skills. The authors are silent on this. In my opinion, it certainly won’t help childeren applying simplistic rules like ‘a bishop equals 3 pawns’ in chess. Such an approach will most probably backfire. The concept of probabilities, too, can hardly be of use in a practical game. There may be advantages also, but unfortunately, these are not indicated by the authors.
In the end, we should credit the authors for mentioning the most important aspect of all education: “Children learn best while having fun“. This is definitely true, and when the authors say that the children they work with enjoy their method, there’s no reason to doubt them. Indeed, if we agree that chess is fun (and I hope all readers of ChessVibes will agree on that!), we should perhaps be able to integrate chess in a useful way into any curriculum, be it math or biology or English literature. But chess is surely not unique in stimulating children to perform better in maths. Who knows, draughts may work even better. Or poker. In any case, if they’re confused by exercises, puzzles or chess figurines – even though they may enjoy them visually – it’s perhaps too early to write tradional methods off. In any case, more research is needed.
And perhaps, math should just be math, and chess should just be chess.
References
[1] For a list of popular and scientific publications on chess and teaching, see the Chess Academy website.
[2] Ho, F., Buky, J. (2008). The Effect of Math and Chess Integrated Instruction on Math Scores. The Chess Academy
[3] Ho, F. (2006). Chess for Math Curriculum. The Chess Academy
[4] Ho (2006). Enriching math using chess. Journal of the British Columbia Association of Mathematics Teachers, British Columbia, Canada, Vector. Volume 47, Issue 2.
[5] Watson, J. (1998). Secrets of Modern Chess Strategy. Gambit Publications (p. 11)
[6] Reti, R. (1922). Modern Ideas in Chess
[7]See http://www.encyclopedia.com/doc/1O87-onegrouppretestpsttstdsgn.html for a discussion on the one group pre-test post-test design.

This article was orginally written by Arne Moll , one of the authors of Chessvibes .

You can read the original article here .

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