Chess and Math

The effect of math and chess
The Chess Academy, Chicago, USA
Research studies have shown that chess can be used as an effective game-based teaching method. However, all the past studies used chess as a separate instructional tool. There were no math contents in chess instruction provided and there was no math and chess integrated workbook used. This study examined the effect on pupils’ math scores when a truly integrated math and chess workbook was used as an instructional practice workbook. The results show that the integrated math and chess workbook significantly increased pupils’ math scores between pre-tests and post-tests among grade 1 to grade 8 pupils.
Introduction
Research papers have demonstrated that chess instruction improves analytical reasoning, problem solving skills, and academic achievement (Chrisiaen & Verholfstadt (1978); Frank & D’Hondt (1979); Smith & Cage (2000)). Research conducted by Gaudreau (1992) shows no significant differences among the groups on basic calculations. These research studies point to the direction that chess has strong effect on improving children’s cognitive ability than their arithmetic computation ability. By teaching math and chess as two separate subjects, children do not have opportunities to work on basic arithmetic operations using acquired chess knowledge, this may explain why by playing chess, it may not statistically significant improve children’s basic arithmetic computation ability.
How to maximize the benefits of chess instruction in such a way that not only chess benefits children’s cognitive development, but also their computation ability? All the past chess instruction research studies have used chess instruction as an independent teaching tool and it is not truly integrated with math instruction. The author Frank Ho created a math and chess integrated workbook. The theoretical basis of how math and chess are integrated has been published by Ho (2006). We believe that with the creation of truly integrated math and chess workbooks, pupils will be able to increase their computation ability by working on these math and chess integrated workbooks. This is particularly important for those children who have no interest in playing chess, but they could still get benefit of chess instruction by working on math and chess integrated workbooks.
No research has been done before on the effects of using math and chess integrated workbook, this study will compare the effect of pupils’ math computation ability before using the math and chess integrated workbook and after using it to see if there is a significant difference.
Method
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. Tests of TONF (The Compass Learning Explorer Online Diagnostic Tool was used for both the pre-test and post-test. The Compass Learning Explorer Assessment meets the requirements as a true valid and reliable criterion-referenced assessment tool.) were given to all pupils for both tests. Each lesson consisted of lecturing, practice on math and chess integrated worksheets and chess playing.
Results
Paired t test was used to analyze the data. The results of this study shows significantly different on their math scores for all grade 1 to grade 8 pupils between pre-test and post-test at level of p is less than 0.01.
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
Discussion
The results of this study demonstrate that a truly integrated math and chess workbook can help significantly improve pupil’s math scores. Our observations show that the effect of using a truly integrated math and chess workbook also provides mental entertainment and thought by pupils as more fun than traditional computation practices. Pupils were able to sit longer when working on math and chess integrated workbook than working on traditional computation worksheets.
The result of this research is particularly interesting for children who do not have a high interest in playing chess since the math and chess integrated workbook involves visualization, analyzing, spatial relation and data processing, these types of problems provide high order cognitive skills. Without spending substantial time on playing chess, we believe that children can get the similar benefits of playing chess on cognitive effects by working on math and chess integrated workbooks. This may require further study.
Why children like to work on math and chess integrated workbook than on the traditional computation worksheets? 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. This also gives children ample opportunities to think visually. Most of the time, the computation questions themselves are not written for children to work on immediately but for children to "create" themselves and these questions have to be actually "mapped" out by following directions and children love them. Children learn best while having fun.




Number of positions in chess after n moves
Chess mathematics can be fascinating. At first sight chess seems to be easy to calculate. It has logical patterns and finite board space. However, the simplest questions may require serious mathematical skills.
A good example is the number of possible positions after n moves, n being 1, 2, 3, etc. After the first move there are exactly 20 positions, after the second, there are 400. White has a choice of 20 first moves, Black the same number of replies, making 400 different possible positions after one move for each color. From here on it is difficult to keep on counting since the number is rapidly growing. After the third move we have 5362 positions, and after the fourth the number is 71852. A really large number for 8x8 board!
These numbers are a good back up of the complexity of The number of Shannon. In 1889 Cunningham got close to the number of moves after the 4th move, stating they are 71782. Fabel got even closer in 1895, he calculated 71870 possible moves. The first one to find the correct number, 71852, was C. Flye St. Marie in 1903.
As far as the Chessdom team knows, there are estimations of the number of positions after the 5th and the 6th moves. They are 809798 and 9132484 respectively. However, we would like to receive confirmation or a more correct information from our mathematician readers. Do not forget to include special moves like en passant .

The number of Shannon
a simple proof how deep chess can be
Claude Shanno
Claude Elwood Shannon (1916-2001) was a famous electrical engineer and mathematician, remembered as "the father of information theory". He was fascinated by chess and was the first one to calculate with precision the game tree complexity of chess i.e. the number of possible chess games. He based his calculation on a logical approximation that each game has an average of 40 moves and each move a player chooses between 30 possible moves. That makes a total of 10120 possible games. This number is known as the number of Shannon.
To a similar conclusion came Peterson in 1996. An interesting comparison is the estimation of the total numbers of atoms in the universe 1081 . The number of legal positions in chess according to him, however, is about 1050 .
All these calculations will suffer slight changes when we apply new rules to chess, such as the Sofia rule or further estimation of the effect of en-passant. However, the numbers are close enough to show you how deep chess can be.
Other game tree complexities (log game tree):
Tic tac toe 5 Connect Four 21 Othello 58 Chess 120 Backgammon 140 Connect six 140 Go 766