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.
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.