What he says is to find something useful (well, assuming charitably that being a chess master is useful in some sense), and turn the first steps on the learning curve into a game. And then let kids play… along the learning curve… until they experience success and related social rewards, and then the motivational feedback loop is established.
Makes sense, but it’s still disappointing if this is Polgar’s main idea, since it doesn’t seem particularly novel or easy to replicate. Besides being a lot of hard work on the parent/teacher’s part as you mention, success also depends on how suitable the subject is for turning its learning curve into something fun for kids to play with, and how much the kids’ personalities fit with this style of learning (for example the kids may already have strong interests of their own and refuse to be enticed into playing the thing you want them to play, or they eventually get bored when doing similar things over and over, or the social rewards aren’t rewarding enough for them to motivate the hard work required). I’d guess that many other parents and teachers have tried something like this with various kids and various subjects, and Polgar’s children are an outlier in terms of success due to a combination of 1) Polgar being very smart and hard working, 2) it’s relatively easy to turn the learning curve of chess into something fun for kids to play with, and 3) the native intelligence and personalities of Polgar’s kids.
Makes sense, but it’s still disappointing if this is Polgar’s main idea, since it doesn’t seem particularly novel or easy to replicate.
The biggest problem with that idea is indeed that it won’t work well in a formal context like a school, where instruction necessarily takes place in a one-to-many form as opposed to one-to-one tutoring. In such a case, there’s a very real possibility that plain-vanilla “Direct Instruction” gives you the best bang for the buck, and that other “educational gimmicks” are simply misguided—including many attempts at “gamification”. Even there though, keeping individual lessons short and starting from the simplest, most appealing steps seem to be good ideas.
other “educational gimmicks” are simply misguided—including many attempts at “gamification”.
A typical example of “gamification” seems to be too much about external rewards: points, awards, visual effects and music. That may be a nice bonus, but it does not address the essence.
I think the correct approach is to explore the zone of proximal development using a playful setting). For example, you could teach addition by talking about dinosaurs walking in a forest, how groups of dinosaurs meet, and they need to know how many dinosaurs are now in the joined group. The child gets hooked on the dinosaur narrative, but you are still doing perfectly valid math.
These two things are about as different as a slot machine and a role-playing game. They may seem similar to someone who doesn’t understand the details and merely learned “make it more fun” as an aplause light.
How would this approach deal with something like mastering the basic math facts (learning to do single digit addition, subtraction, and multiplication quickly and automatically without conscious effort)? From my experience and what data I’ve been able to find (see http://mathfactspro.com/docs/MathFactsPro_Response_to_Intervention_Alignment_and_Research.pdf) it seems that a kid needs to do at least 1000 single digit addition practice problems just to master the addition facts. How to make this fun without external rewards?
1000 single digit addition problems doesn’t seem like a lot to me. The linked website suggests that 250 problems can be solved by the average student in 35 minutes with their software. That suggests that you need three hours to train 1000 single digits problems.
I think there’s a lot of software that gamifies simple addition problems well.
I agree that thousandfold repetition is necessary to increase the speed and to make the process automatic enough that the student can now focus on the larger picture. Trying to teach understanding without repetition typically has the outcome that the student has a general idea about the whole process, but is unable to actually do it, because it is impossible to pay enough attention to all levels of complexity at the same time (calculation of 5+3 competes for attention with understanding of why are we adding this 5 and this 3 in the first place).
Thinking about myself at elementary school, the joy of being able to solve a problem would be (and actually was) enough motivation to solve 1000 single-digit additions. But I don’t want to generalize from one example, and I am not opposed in principle to using gamification in such cases.
However, before gamification, I would make sure that the child understands precisely what is going on. (Just like in teaching physical movement, e.g. sport or martial art, you first want the student to do the movement correctly, and only afterwards to do it quickly. A correct but slow move can be made faster by repetition, but a mistake in an incorrect move will only get more fixed by repeating.)
So, in practice: Before moving to single-digit addition, I would make sure the child really understands the single-digit numbers per se. For example, it should be obvious to the child that five apples remain five apples regardless of whether you arrange them in a line or in a circle, or whether you count them left-to-right or right-to-left. (This can be achieved by e.g. playing stupid, asking the child to count the apples, then rearranging them, and asking the child to count them again. It should be the child who say that it doesn’t matter, because the result will be the same.) Also, make sure the child can count to 10 flawlessly. (Alternatively, if the child can only count to e.g. 7 flawlessly, limit the lesson to addition of two numbers where the result is not more than 7.)
Now we can use objects (apples? pencils? toys?), and put 2 apples on the table, then put 3 apples on the table (next heap), then count all apples on the table. Repeat a few times. Then let the child make a prediction “if we put 2 apples and 3 apples, how many apples will it be together?” and verify the prediction experimentally. Celebrate the successful predictions! If you have multiple children, let all of them make predictions, and then one of them perform the experiment. (As an adult, you never comment on the predictions until the experiment is completed. This is how you teach the children that the answer is in the “territory”, not in teacher’s head. You also show them what to do if they later forget something.)
When the children are already good at this, make them discover the commutativeness of addition by “accidentally” making them calculate “4+2″ right after they calculated “2+4”, etc. At some moment a child will notice “hey, it’s the same”. If this happens in a classroom, this is the moment when you let children debate the new discovery among themselves. Don’t tell them whether the discovery is correct or not, but perhaps suggest to make a series of further experiments (let the kids suggest the pairs of numbers to try).
And only after all of this, let them play a computer game with automatic feedback and artificial rewards, to gain greater speed. But maybe the children will be already motivated enough that you can just give them a paper sheet with thousand problems (and then let them review each other’s answers).
Experience suggests that using this method you progress a bit slowly at the beginning, but the knowledge is more solid, which allows you to save time later. (Less need to backtrack to the old lessons, when the child would e.g. fail at a more complex task because they actually made a mistake at the subproblem of addition.) Later this deeper understanding will pay off, e.g. you don’t need to teach commutativeness of addition as a separate fact later; your children will understand the concept on gut level, even if they never heard the word “commutative”.
And only after all of this, let them play a computer game with automatic feedback and artificial rewards, to gain greater speed.
It’s actually not easy to find a computer (or tablet/phone) game that is both fun and adaptive (i.e., customizes the sequence of practice problems to best fit the learner). Even filtering just for “fun”, it’s hard to find one whose fun doesn’t quickly wear off as the problems and rewards both start feeling repetitive. (Also, my kid hates time pressure so that rules out games with time limits.) If you know any good ones, please share. So far, the games that have held my kid’s interest the longest have been Mystery Math Town and Mystery Math Museum both by Artgig, but these are unfortunately not adaptive so they often waste time and game content/rewards on sub-optimal practice problems.
I think the correct approach is to explore the zone of proximal development using a playful setting.
Using a playful/intuitive narrative is indeed important, in ways that are sometimes less than obvious. It’s also something that happens already, and indeed seems to be rather emphasized in more recent textbooks (e.g. in math) - quite possibly in an excessive way! When I mentioned “gamification” I mostly meant the other gimmicks you talk about, but it seems that even having too much “narrative” can be bad. This is one reason why these methods don’t work very well in a school-based context.
I imagine a 2-dimensional graph of lifestyle suggestions, where one dimension is labeled “insights” and the other one is labeled “hard work”. Popular “life hacks” are medium or high on insights, and low on hard work. Polgár method is medium on insights, and high on hard work.
To this I would say—if you know a method for bringing up “geniuses” that is lower on hard work and perhaps higher on insight, go ahead, you have my blessing. But in absence of such methods, sometimes the only way to achieve the desirable results is to “shut up and suffer”.
how suitable the subject is for turning its learning curve into something fun for kids to play with
I always thought this was what teachers should give high priority to: researching ways how to turn specific subjects into playful learning curve. And the idea is not even new; Comenius wrote his Schola ludus in 1630. Yet somehow...
I guess this somehow rubs humans the wrong way. Perhaps we associate wisdom with high status, and play with low status, therefore our brains refuse to accept play as a legitimate way to achieve wisdom. Or perhaps from the educator’s side it feels like a huge status loss to convert their hard-won lifetime amounts of knowledge into child games.
Essentially, you need to find a human who is (1) good at the subject matter, so they see how the learning curve goes, (2) good at education and gamification in general, and (3) does not have strong feelings over converting their high-status knowledge into a low-status play. Then the human needs to publish this, and other people need to use it, i.e. overcome their own status issues. Also, a few decades later someone else has to update the games to reflect more current knowledge, otherwise they will become outdated.
the kids may already have strong interests of their own
I’d say that in that case you should either go ahead with the child’s interest (if it is something useful), or try to find some kind of intersection between what the child wants and what you want (e.g. if you want the child to be a painter, but the child is only interested in dinosaurs, let them paint dinosaurs).
it’s relatively easy to turn the learning curve of chess into something fun for kids to play with
In Czech Republic, they recently started experimenting with Hejný method of teaching mathematics, which is a way to make mathematics fun; furthermore, the method is designed to work in a classroom environment. One day I hope to Pareto-translate some of that, too, but it is much more text.
I’d say that in that case you should either go ahead with the child’s interest (if it is something useful), or try to find some kind of intersection between what the child wants and what you want (e.g. if you want the child to be a painter, but the child is only interested in dinosaurs, let them paint dinosaurs).
Related: Building Islands of Expertise in Everyday Family Activity discusses how to take a child’s intrinsic interest in something and then finding bridges between the “island of expertise” that the child develops around that, and everything else. A specific example used is a boy who’s been very interested in trains:
By the time the boy turns 3-years old, he has developed an island of expertise around trains. His vocabulary, declarative knowledge, conceptual knowledge, schemas, and personal memories related to trains are numerous, well-organized, and flexible. Perhaps more importantly, the boy and his parents have developed a relatively sophisticated conversational space for trains. Their shared knowledge and experience allow their talk to move to deeper levels than is typically possible in a domain where the boy is a relative novice. For example, as the mother is making tea one afternoon, the boy notices the steam rushing out of the kettle and says: “That’s just like a train!” The mother might laugh and then unpack the similarity to hammer the point home: “Yes it is like a train! When you boil water it turns into steam. That’s why they have boilers in locomotives. They heat up the water, turn it into steam, and then use the steam to push the drive wheels. Remember? We saw that at the museum.”
In contrast, when the family was watching football—a domain the boy does not yet know much about—he asked “Why did they knock that guy down?” The mother’s answer was short, simple, stripped of domain-specific vocabulary, and sketchy with respect to causal mechanisms—“Because that’s what you do when you play football.” Parents have a fairly good sense of what their children know and, often, they gear their answers to an appropriate level. When talking about one of the child’s islands of expertise, parents can draw on their shared knowledge base to construct a more elaborate, accurate, and meaningful explanations. This is a common characteristic of conversation in general: When we share domain-relevant experience with our audience we can use accurate terminology, construct better analogies, and rely on mutually held domain- appropriate schema as a template through which we can scribe new causal connections.
As this chapter is being written, the boy in this story is now well on his way to 4- years old. Although he still likes trains and still knows a lot about them, he is developing other islands of expertise as well. As his interests expand, the boy may engage less and less often in activities and conversations centered around trains and some of his current domain-specific knowledge will atrophy and eventually be lost. But as that occurs, the domain-general knowledge that connected the train domain to broader principles, mechanisms, and schemas will probably remain. For example, when responding to the boy’s comment about the tea kettle, the mother used the train domain as a platform to talk about the more general phenomenon of steam.
Trains were platforms for other concepts as well, in science and in other domains. Conversations about mechanisms of locomotion have served as a platform for a more general understanding of mechanical causality. Conversations about the motivation of characters in the Thomas the Tank Engine stories have served as platforms for learning about interpersonal relationships and, for that matter, about the structure of narratives. Conversations about the time when downtown Pittsburgh was threaded with train tracks and heavy-duty railroad bridges served as a platform for learning about historical time and historical change. These broader themes emerged for the boy for the first time in the context of train conversations with his parents. Even as the boy loses interest in trains and moves on to other things, these broader themes remain and expand outward to connect with other domains he encounters as he moves through his everyday life.
[...] although some of the learning may be highly planned and intentional, much of it is probably driven by opportunistic “noticing” on the part of both the parent and the child. Recent efforts to consider parent input into children’s categorization decisions, for example, have predominately been directed at developing an account for how parents structure a fixed interpretation for children. As Keil (1998) pointed out, casting parents as simple socializers who provide fixed didactic interpretations for children is unlikely to be the right model. There is nothing more annoying then someone who provides you with pedantic explanations that you do not want or that you could not make use of. In reality, however, everyday parent-child activity hinges on a dual interpretation problem. The parent needs to decide what is worth noting, based on their own knowledge and interests, their understanding of their child’s knowledge and interests, and their current goals for the interaction. Children are making the same calculation, simultaneously. Over time, the family interprets and re-interprets activity, bringing out different facets: Sometimes they highlight the science, sometimes the history, sometimes the emotion, sometimes the beauty, and so on. Thus, the family conversation changes to become more complex and nuanced as it traces the learning history of the family and extends through multiple activities.
In Czech Republic, they recently started experimenting with Hejný method of teaching mathematics, which is a way to make mathematics fun; furthermore, the method is designed to work in a classroom environment.
Many of the broader ideas in this link seem to be sensible, but the overall method still needs to be tested empirically, and the extreme emphasis on “we’ll just let kids discover the math on their own!” does not bode well, since this has been tried and failed dramatically, e.g. in the U.S.!
Sorry but math teachers need to actually know their math; they can’t get away with just being “coaches” and “facilitators”!
math teachers need to actually know their math; they can’t get away with just being “coaches” and “facilitators”!
Absolutely. If your math is not solid, and the child comes up with an unexpected idea, you can’t provide a valuable “coaching” in return.
Unfortunately, school is what it is, and teachers are what they are, including math teachers. I guess in short term you have to accept that math teachers are often incompetent in math as a fact about the world, and try to minimize the damage. :( And perhaps the long-term plan is that the next generation brought up using this method will have more math-competent math teachers.
the overall method still needs to be tested empirically
In progress, as far as I know. At least the last time I heard about it, they were testing the method on a randomly selected group of Czech elementary schools.
Makes sense, but it’s still disappointing if this is Polgar’s main idea, since it doesn’t seem particularly novel or easy to replicate. Besides being a lot of hard work on the parent/teacher’s part as you mention, success also depends on how suitable the subject is for turning its learning curve into something fun for kids to play with, and how much the kids’ personalities fit with this style of learning (for example the kids may already have strong interests of their own and refuse to be enticed into playing the thing you want them to play, or they eventually get bored when doing similar things over and over, or the social rewards aren’t rewarding enough for them to motivate the hard work required). I’d guess that many other parents and teachers have tried something like this with various kids and various subjects, and Polgar’s children are an outlier in terms of success due to a combination of 1) Polgar being very smart and hard working, 2) it’s relatively easy to turn the learning curve of chess into something fun for kids to play with, and 3) the native intelligence and personalities of Polgar’s kids.
The biggest problem with that idea is indeed that it won’t work well in a formal context like a school, where instruction necessarily takes place in a one-to-many form as opposed to one-to-one tutoring. In such a case, there’s a very real possibility that plain-vanilla “Direct Instruction” gives you the best bang for the buck, and that other “educational gimmicks” are simply misguided—including many attempts at “gamification”. Even there though, keeping individual lessons short and starting from the simplest, most appealing steps seem to be good ideas.
A typical example of “gamification” seems to be too much about external rewards: points, awards, visual effects and music. That may be a nice bonus, but it does not address the essence.
I think the correct approach is to explore the zone of proximal development using a playful setting). For example, you could teach addition by talking about dinosaurs walking in a forest, how groups of dinosaurs meet, and they need to know how many dinosaurs are now in the joined group. The child gets hooked on the dinosaur narrative, but you are still doing perfectly valid math.
These two things are about as different as a slot machine and a role-playing game. They may seem similar to someone who doesn’t understand the details and merely learned “make it more fun” as an aplause light.
How would this approach deal with something like mastering the basic math facts (learning to do single digit addition, subtraction, and multiplication quickly and automatically without conscious effort)? From my experience and what data I’ve been able to find (see http://mathfactspro.com/docs/MathFactsPro_Response_to_Intervention_Alignment_and_Research.pdf) it seems that a kid needs to do at least 1000 single digit addition practice problems just to master the addition facts. How to make this fun without external rewards?
1000 single digit addition problems doesn’t seem like a lot to me. The linked website suggests that 250 problems can be solved by the average student in 35 minutes with their software. That suggests that you need three hours to train 1000 single digits problems.
I think there’s a lot of software that gamifies simple addition problems well.
I agree that thousandfold repetition is necessary to increase the speed and to make the process automatic enough that the student can now focus on the larger picture. Trying to teach understanding without repetition typically has the outcome that the student has a general idea about the whole process, but is unable to actually do it, because it is impossible to pay enough attention to all levels of complexity at the same time (calculation of 5+3 competes for attention with understanding of why are we adding this 5 and this 3 in the first place).
Thinking about myself at elementary school, the joy of being able to solve a problem would be (and actually was) enough motivation to solve 1000 single-digit additions. But I don’t want to generalize from one example, and I am not opposed in principle to using gamification in such cases.
However, before gamification, I would make sure that the child understands precisely what is going on. (Just like in teaching physical movement, e.g. sport or martial art, you first want the student to do the movement correctly, and only afterwards to do it quickly. A correct but slow move can be made faster by repetition, but a mistake in an incorrect move will only get more fixed by repeating.)
So, in practice: Before moving to single-digit addition, I would make sure the child really understands the single-digit numbers per se. For example, it should be obvious to the child that five apples remain five apples regardless of whether you arrange them in a line or in a circle, or whether you count them left-to-right or right-to-left. (This can be achieved by e.g. playing stupid, asking the child to count the apples, then rearranging them, and asking the child to count them again. It should be the child who say that it doesn’t matter, because the result will be the same.) Also, make sure the child can count to 10 flawlessly. (Alternatively, if the child can only count to e.g. 7 flawlessly, limit the lesson to addition of two numbers where the result is not more than 7.)
Now we can use objects (apples? pencils? toys?), and put 2 apples on the table, then put 3 apples on the table (next heap), then count all apples on the table. Repeat a few times. Then let the child make a prediction “if we put 2 apples and 3 apples, how many apples will it be together?” and verify the prediction experimentally. Celebrate the successful predictions! If you have multiple children, let all of them make predictions, and then one of them perform the experiment. (As an adult, you never comment on the predictions until the experiment is completed. This is how you teach the children that the answer is in the “territory”, not in teacher’s head. You also show them what to do if they later forget something.)
When the children are already good at this, make them discover the commutativeness of addition by “accidentally” making them calculate “4+2″ right after they calculated “2+4”, etc. At some moment a child will notice “hey, it’s the same”. If this happens in a classroom, this is the moment when you let children debate the new discovery among themselves. Don’t tell them whether the discovery is correct or not, but perhaps suggest to make a series of further experiments (let the kids suggest the pairs of numbers to try).
And only after all of this, let them play a computer game with automatic feedback and artificial rewards, to gain greater speed. But maybe the children will be already motivated enough that you can just give them a paper sheet with thousand problems (and then let them review each other’s answers).
Experience suggests that using this method you progress a bit slowly at the beginning, but the knowledge is more solid, which allows you to save time later. (Less need to backtrack to the old lessons, when the child would e.g. fail at a more complex task because they actually made a mistake at the subproblem of addition.) Later this deeper understanding will pay off, e.g. you don’t need to teach commutativeness of addition as a separate fact later; your children will understand the concept on gut level, even if they never heard the word “commutative”.
It’s actually not easy to find a computer (or tablet/phone) game that is both fun and adaptive (i.e., customizes the sequence of practice problems to best fit the learner). Even filtering just for “fun”, it’s hard to find one whose fun doesn’t quickly wear off as the problems and rewards both start feeling repetitive. (Also, my kid hates time pressure so that rules out games with time limits.) If you know any good ones, please share. So far, the games that have held my kid’s interest the longest have been Mystery Math Town and Mystery Math Museum both by Artgig, but these are unfortunately not adaptive so they often waste time and game content/rewards on sub-optimal practice problems.
I think Sagaland is good at giving an intuitive understanding of numbers. If you roll a “5” and a “3″ you can visit the tree that’s 2 moves away.
A games like this, that requires you to apply the math are likely better than a game that just asks you to solve 5-3.
Using a playful/intuitive narrative is indeed important, in ways that are sometimes less than obvious. It’s also something that happens already, and indeed seems to be rather emphasized in more recent textbooks (e.g. in math) - quite possibly in an excessive way! When I mentioned “gamification” I mostly meant the other gimmicks you talk about, but it seems that even having too much “narrative” can be bad. This is one reason why these methods don’t work very well in a school-based context.
I imagine a 2-dimensional graph of lifestyle suggestions, where one dimension is labeled “insights” and the other one is labeled “hard work”. Popular “life hacks” are medium or high on insights, and low on hard work. Polgár method is medium on insights, and high on hard work.
To this I would say—if you know a method for bringing up “geniuses” that is lower on hard work and perhaps higher on insight, go ahead, you have my blessing. But in absence of such methods, sometimes the only way to achieve the desirable results is to “shut up and suffer”.
I always thought this was what teachers should give high priority to: researching ways how to turn specific subjects into playful learning curve. And the idea is not even new; Comenius wrote his Schola ludus in 1630. Yet somehow...
I guess this somehow rubs humans the wrong way. Perhaps we associate wisdom with high status, and play with low status, therefore our brains refuse to accept play as a legitimate way to achieve wisdom. Or perhaps from the educator’s side it feels like a huge status loss to convert their hard-won lifetime amounts of knowledge into child games.
Essentially, you need to find a human who is (1) good at the subject matter, so they see how the learning curve goes, (2) good at education and gamification in general, and (3) does not have strong feelings over converting their high-status knowledge into a low-status play. Then the human needs to publish this, and other people need to use it, i.e. overcome their own status issues. Also, a few decades later someone else has to update the games to reflect more current knowledge, otherwise they will become outdated.
I’d say that in that case you should either go ahead with the child’s interest (if it is something useful), or try to find some kind of intersection between what the child wants and what you want (e.g. if you want the child to be a painter, but the child is only interested in dinosaurs, let them paint dinosaurs).
In Czech Republic, they recently started experimenting with Hejný method of teaching mathematics, which is a way to make mathematics fun; furthermore, the method is designed to work in a classroom environment. One day I hope to Pareto-translate some of that, too, but it is much more text.
Related: Building Islands of Expertise in Everyday Family Activity discusses how to take a child’s intrinsic interest in something and then finding bridges between the “island of expertise” that the child develops around that, and everything else. A specific example used is a boy who’s been very interested in trains:
Many of the broader ideas in this link seem to be sensible, but the overall method still needs to be tested empirically, and the extreme emphasis on “we’ll just let kids discover the math on their own!” does not bode well, since this has been tried and failed dramatically, e.g. in the U.S.!
Sorry but math teachers need to actually know their math; they can’t get away with just being “coaches” and “facilitators”!
Absolutely. If your math is not solid, and the child comes up with an unexpected idea, you can’t provide a valuable “coaching” in return.
Unfortunately, school is what it is, and teachers are what they are, including math teachers. I guess in short term you have to accept that math teachers are often incompetent in math as a fact about the world, and try to minimize the damage. :( And perhaps the long-term plan is that the next generation brought up using this method will have more math-competent math teachers.
In progress, as far as I know. At least the last time I heard about it, they were testing the method on a randomly selected group of Czech elementary schools.