I was waiting for conflict to be resolved and I thought I should bring a different the viewpoint about both chess and mathematics, and how it has helped towards enriching the rational thoughts in our children.

I am coming into this forum, not to criticize Frank’s work and support Arne efforts or vice-versa (which my fellow country man, Gopinath has tried) but indeed place a perspective that I had been dreaming for more than 2 decades for now (Since the days of my Scholastic Chess). I am here, bringing more perspective from, Chess-in-Schools aspect (People remember me from CISCCON 2007) and bring out what I believe from some of the cognitive science I have been exposed to in my self-made way.

First, I want this belief to be understood: “Chess is not learned out of Math Principles depicting true state of Universe” nor “Maths is learned from the Chess principles that reflect reactivity within Cultural state of Mind”. So there should be proper interface or a Cosmic gateway for chess and mathematics to synergistically operate in a child’s mind.

If this is little confusing, Let me confuse more, Math is always objective, even when counting real and imaginary numbers. Chess is subjective as in when we force ourselves to make the unforced mistakes. So combining these are like combining some sort of Analysis and Synthesis, which is an explorative and iterative process in itself. Unless, there is strong connection with the interface, there is no useful learning. The integrated methods do not scale up well with this character.

In the present context, we stand from the stand point of Science of causation or from the standpoint of the Results for positive Correlation. We indeed failed to stopped for a moment to think all these from the our own shoes. Well, I want to try to explain, what our own shoes is, and then take up this heated arguement.

Let’s look into some standing learning theories. One that hits my mind readily is the KOLB’s Learning Theory or remembered as KLT. What’s more interesting from the KLT is the stages of learning that anybody has in pursuit with learning in general – They are “Acquisition”, “Specialization“ and “Integration”. It is to be understood like, a child acquires some knowledge, works on specializing the knowledge through his advanced studies in his college time, and integrates it during the work that emerges out of his studies. (Also addressing the life cycle of today’s contemporary professional path)

So I would like to point out my hypothesis, that a child works on specialization, when it comes to chess (as acquisition is easy in chess, all he needs to understand is the Chess rules and current position) and often being tempted to integrate his/her learning, when he/she ponders to play. I believe (it is not science) that chess gives a platform to readily align to this mental framework of learning. (KLT explains a framework of how learning takes place.). I would like to talk based on these stages, and analyse both Arne’s and Frank’s views, to support Arne’s claim for further research in the Causal relationships, to make “Chess and Maths” scientific in approach.

To explain more, I am regretting the Mathematical fallacy in Chess Events, in general:
I strongly feel that a coach/players in general fails, if he just goes by guidelines in chess. I have seen many friendly games among kids stopped by chess coaches, when kids play “exchange down” or “even a piece down”, despite having some small surprises ahead. Disappointment often is observable in the child’s face, that they could have always TRIED /but no thoughts of winning yet!

Is this an action of over looking the brilliant opportunities, lack of poor education style / ethics in chess – No! It is stereotypic decision of elders generally take (based on the belief the objective of game and alignment of the kid) that children’s hate.

Why not continue the game with losing mindset? Why is losing bad in the first place? Well! I could have tricked him! Etc…(Emotional). These are imagination in general but it has high precedence over any irrational stereotypic decisions that are mistakenly mathematical decision.

We are making mistakes to associate with piece values in Chess, trying to offer trade off on a later date. That’s where we often land into value of 1-3-3-5-9 for mortal pieces–infinity for the immortal King, which is a guideline information in chess and something that does not have to be met exactly every time. Further, we force ourselves to forget this information, when we spring up with wild imagination with bizarre combination or even volley of exchanges –Such games with volley of exchanges goes to dark endgame, which are not vibrant as the middle game). As chess players, we are not guided simply by equations we are indeed misguided/ failed to sense the moment by principles that revolve around mathematical soundness, at least in Chess, which also similar in life.

Second evidence in chess is, most of the immortal games do not emphasize this material equality. They rather reflect the material imbalance and how the activity of pieces (generates the imagination) can lead better compensation for the material sacrificed. The quality of Child improves when they are exposed to the quality of pieces. If an integrated syllabus, is emphasizing the material values as a cue for learning. It does not actually use chess in the first place, but uses “X” that facilitates understanding of Chess. Such meta-data cannot be strictly taken as learning from Chess, in the pure sense /even statistically, which our friends have pointed to.

I recollect “Method in Chess”, which using a load balancing method, using four factors – “The Minor piece activity, King Safety, Positions after Exchange of queens (Bulk) and Pawn Structures” (I believe these are the factors) and weighing the Centre of gravity of the chess board to make an objective decision. This is a strange instance, that has been called as imaginative scheme but it is indeed a very formal method using the physics principles, which is mathematically exploited.

Many such examples can be related holistically, but we have to make sense from an aspect of what does inside the black box of students. Degroot’s ,Chase, Simon and Gobet have pointed out methods to analyze the mind’s eye through a process paradigm with Cognitive science. That is a scientific approach. However, let’s not get so deep to explain the appropriateness. Let’s see into what is presented as a stimulus for us to consider, the Human – Integrated System interactions, which will enable the Children’s overall performance under such systems.

One important notion within Chess, as a stimulus: The flow of thoughts, this indeed happens from “pieces to squares”. (Even Cognitively, we see the pieces first and then the squares within the chess board to localize the pieces.) Thus, the perception is from pieces to squares. There were subtle confusion in my opinion, in the example 1 in Arne’s article, whether the Square described within the example , meant a white square or a Fill in the blanks. Such cognitively overload, had not be clearly taken account into in the research programme. Simply with Chess what you see is towards a gestalt vision and you keep integrating individual sensation of the part of the position.

As in other example (Exercise 15.) that presents the probability and statistics discussions. The cognitive load on moving the bishop to its ideal squares that other bishop would meet is very high. The case is pretty simple, with opposite bishops don’t meet and hence the probability is zero.

However, Considering a similar case with the same coloured bishop. The problem is well-posed,
If it can be analytically reasoned first as follows:
- Bishops move in a diagonal. (number of squares in diagonal is a variable)
- There can be only two diagonals possible (People don’t consider knight moves as diagonal moves, relative to its opposite color, so that they have 4 diagonals with 8 squares to cover)
- Two lines can interest at a point. (Mathematical thought)
- Bishops can intersect at 2 points , if they are not on same diagonal This becomes stereotopically TWO, like Rook can given anytime cover 14 squares on a Chessboard.
- Bishops can intersect at more points if they are on the same diagonal. Then it becomes the number of squares that the bishops are move in common + 1 (The capture square, which is different for two sides). (Visual movements help you to count the number of squares)
- Even De Groot’s claim is that the visual encoding is all automatic an there is no controlled process for it. However, the perception of the information, is based on Chunking of relatively grouped information. This opens the world of information processing, that the research study has not validated with. Practically, instruction design has not used this approach for any learning in general.

However, the cognitive processes here are much aligned to mental movement of bishops and not with counting each time. So complexity of this cognitive task is much more than understand the probability by the mathematics symbol. I doubt the conventional wisdom, that this process can improve the perception on probability than learning the probability with the standard mathematical symbols. (May be there might be slight disorientation with the symbols to the students. Like what Richard Feynman used to believe in inventing, his own mathematical symbols in his young age, because he did not like log / Sine / Cosine Symbols. Feynman too did it for convenience sake, to resolve his confusion, but did not invent his language for better Standards.)

Secondly, I doubt if these methods are scalable to facilitate the understanding needed to address the acquisition (Kolb’s Stages of Learning) of probability theory ideas. Second point to consider, is that are these probabilistic reasoning helping a chess player, which in turn helps him to learn better chess and in turn better probability. Only at the deeper levels, we see the frustration of the interface. At higher levels, we see only the correlation and become reactive with it. I am sorry Frank I let you down.

There is one perspective to support Frank as well.

I am currently working with Chess in Schools Research work in India, where there are roughly around 10000+ kids participating in the research programme to study the impact of chess in academics. There has been a significant impact in education with the presence of a chess programme in their schools. Literally 99 % of the students have given a feedback stating that “Chess helped me score better marks in their subjects (Mathematics / English / Sciences /Social Science and Languages like Hindi and Sanksrit), (Study is yet to be complete to give the actual perspective to it, may be this group can help me with some Peer reviews). Assuming that this is a positive difference and is in the belief state of the students themselves, I still find it as a big challenge and I cannot still say that Chess Impacts Students, academically. What I believe in the impact of bringing chess into their educational livelihood is a new stimulus. I call this Unified Stimulus (Still no Science, my perception), they are reactive to this unified stimulus and any effects on the general outcome. They are not able to distinguish the effects of chess or effects of some specific / additional processes, which might or might not have been impacted out of chess. We humans are a kind which cannot differentiate Apples and Organes (an argument that runs into Predictably Irrational, Dan Areily)

I am seriously working towards understanding this phenomenon of where is the stimulus there is a unanimous statement in response that chess has helped them to achieve to progress? Simplest logic is the tendency promoted by chess to play a move without a pause and move out of stagnation. So in move they make, they see a progress and in every progress they see, the best is seem from the unique moves that they have played. This drive has significantly influenced the kids, not only to perform by their standards but also relate their improvement (however small it is) to chess. I don’t have any speculation that I can make out of 10,000 kids say the same Mantra and students who say no, rather associate with some development that chess helped us to look at ourselves and only that helped in shifting the focus on the performance. I can draw equal parallel’s with Frank’s work, it is only that he has not thought in those terms. The correlation with the pre and post exposure to FANTASY is positive, is my logical conclusion.

What had actually compelled me to write to this forum, was the sense to understand, how the element of FANTASY is exploited in Education, to improve the learning across the subjects in the harmonious way.

I see lot of fantasy being exploited with “Integrated Chess and Mathematics Programme”, with lot of intuitive interfaces for each question and the computational framework / rather than core mathematical framework, to entice the students to solve mathematical problems with chess symbolic interface.

Kids are blessed with Imagination, simply because they are seeing this world and its interaction as new events. However, as with experience, we all lose the fresh eyes, and the imaging mind. That’s one of the reasons, we are objective in criticizing Frank’s work with a genuine interest to further put efforts into understanding how these processes are aligned within chess and Maths framework and what value-add can this system create in general. Definitely, Arne’s thoughts have a noble cause in evolving the science within this framework. I wish I could support him in my humble means.

One thing, I see as an aspect of chess, is that it replicates the instinct of a mathematician. Let me take the simplest case of logically deducing LHS =RHS, which I regard, Mathematician’s gift to learning children.

In the process, each step that we modify in either side, involves an axiom, definition or a well known theorem. (I will provide the source for this). This can be seemly analogous to actual game of chess. We can imagine LHS as the current position and RHS as a “Checkmate position” or a Win in mind or even an objective position that you wish to go(TABIA). You seldom use the movement of pieces and threats they create (axioms), rules of chess like exchanges, promotion, Checks (definitions) and some deeper tactics and strategy , pattern recognition( Well known Theorems), which are all small theorems in practice. As such, the complexity of the chess, in relation to mathematics is in the imagination of RHS and how clear they are. They are indeed mathematically aligned applying the axioms/ theorems all require adequate quote from memory or accessible reference. But With Chess, all the axioms / definition (hidden) and Theorems are embedded within the visualization of the chess board. Once again, the triumph of representation of chess (within a finite view in a board) over mathematics is in the elegance of the interface, which mathematics cannot give due to its manifold of complexity even with representation (however, using the natural/symbolic language) and pervasiveness.

There are many other analogies with chess and maths at the fundamental process level that can help to strengthen the thought process with themselves.

If these instincts are brought to the limelight, and an instruction design emerges from it. We can best achieve a transition in Chess in Schools, but that day is still far from reality, as the efforts are discrete from both chess and mathematical front.

Chess and Mathematics framework is inherently still an ill-posed condition, that we pose a child into and we cannot expect the child to be predictable with such integrated approach, when they have to set an expectation to scale them or tune them to a specific problem.

Whereas, Children are successful with the general approach with Mathematics or with Chess. GM under age 12 to 15 is a common place today.

To answer, why Frank’s Frameworks is successful, is once again because of it richness in representation, which you use pieces than numbers. Simply because of this difference, their marriage sounds interesting.

However, it can be seen as yet another domain for Maths and certainly a new method for Chess in itself with the actual purpose of chess is different from the GOAL aspect of Checkmate.

To practically understand the debate here, Does the framework look meaningful (living the spirit of Mathematician and boosting the ego of the Chess player) or useful as it claims to be, as in Frank’s statistical experiment on Correlation / strictly not causal relations. Thanks for the Amanda / “Summariser” Psychological Statistician for pointing out, and Arne for what needs to fit in more…

There are stimuli in each of them (chess and Maths). The fields (if I may call them out of my passion) have slight overlap in the sense of appreciation and is really vast to have a sensible marriage. May be we get a NSF grant, to align to the existing K-12 Curriculum. It is not as practical as it sounds and it is more fundamental as it counts.

But there is still a difference is their pervasiveness. Mathematics is real and astronomical but Chess is finite and astronomical in imagination any day. Positioning both Chess and Math together is a new stimulus by itself and more importantly, with Frank’s Research, Chess is merely used as a language / Symbols within a implied meaning from chess and its complexity is not used in at all.

I see the analogy of ROMAN NUMERAL Systems within a Clock, as to how chess has been used in Frank’s Integrated Methodology. It is all symbol’s and perceptions. It is very important to understand the scalability and usability of the methods. Roman Numerals are fine, but their computations are non-intuitive. Invariably, I am yet to find them in a digital clock of the most bizarre design to take his arguments right,. Frank’s work is definitely is not in this flavor, if it is, it is then an individual’s perspectives or emotions, whether they are chess players or Mathematicians or Teachers.

I feel that the confidence over the methods in Chess and even Math reflects the character of person solving the problem. This is very subjective and very artistic in nature. Mathematicians are true artist with Accuracy / Chess players are artist with precision. Some times precision sounds dull, where the observable error is repeatable that where the conflict lies upon.

After going through all these discussion, confusion is from whose shoes are we looking all these aspects from. The statistics produced are from the way Children have been programmed to interface with “X”. Not the right approach indeed.

Further, I also believe that Chess/Math integrated approached for a problem is very different from Chess / Math approached by a principle. (I belong to the group, which believe in Principles first (Stimulus First) and then come to the perception of the problem. That’s where we relate to the right problem, as end of the day – It is the problem that we value more for having to solve it.

About me:

I had just wish to provide another facet of what was interesting to me with Chess (I have a fairly decent strength that I have scored more than 80% against Titled players, with the personal experience of training a GM myself) and with Mathematics. (Without its appreciation/ inspiration, I cannot have an advanced degree in Robotics, from Carnegie Mellon University).

I returned to India, to work on Chess, but I had been interrupted by all important Boeing 787 project at GE Aviation UK, which took my real time out of working with Kids. I thank many people like the NIIT / V Anand to have brought me back to the chess world of research, which I inspire and Aspire…

I hope with Chess we should actually strive to focus on significant problems that we can soften by possible marriage with Chess, and promote more of the fundamental research in the application of the game to present day psychological problems of Child, not just from education alone.

I regard Frank’s efforts as a right approach to marry Chess with a learning process/ problems but I personally take Arne stand in asking the right question for its perfect execution. Is this the burning problem to solve? And if there should be a reason as to why it works, Let’s apply it for holistic problems that World is Facing.

Thanks Arne for this opportunity!

“Don’t give me 1000 reasons why it won’t work, but give me 1 reason why it would work”

As I am likely the least formally qualified – I guess I should bow out gracefully. It saddens me that I / (we) failed miserably to have Frank Ho address the content of his method itself (which surprisingly I seem to be the only one focussed upon) but I can see how this far into the marathon it is too late to get back to the race circuit – after making a wrong turn.

However I am grateful to have had this opportunity to return to the challenge of picking up where I left off…continuing to define the ‘Math OF Chess’ – Perhaps I could submit something to your for publication: hopefully by fal. With a little expert help I could design a valid test for my ‘geometric’ introductory method of teaching Chess. Of course I’ll still be faced with the ’science versus commerce’ dilemma.

All others said (including all posting here) about the possible results or ?¢‚Ǩ?ìcausers?¢‚Ǩ? are not the direct results of their experiment but opinions and they are subject to disagreements.

I have no crystal ball to say what are or what are not the “causers” of their experiment since I have no experimental design to prove either.

I don’t think anyone here will disagree with that. But is this (a difference between pretest and posttest scores) really the only point the authors are making? Sentences like “The results of this study demonstrate that a truly integrated math and chess workbook can help significantly improve pupil?¢‚Ǩ‚Ñ¢s math scores” (as quoted in the article above), as well as Frank Ho’s contributions to the discussion, suggest that the authors are in fact making a claim regarding the *cause* of the difference, namely, that the teaching method used has caused, or at least contributed to, the difference.