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I can only venture a guess. Many interesting mathematics results come from people simply observing patterns when doing adding, subtracting, etc. Many insights c

I can only venture a guess. Many interesting mathematics results come from people simply observing patterns when doing adding, subtracting, etc. Many insights can come when people are not afraid of getting hands dirty. It's about getting rid of a fear of computation and developing a system for exploring unknown problems. For example, consider this problem for geometric series: 1 + 2 + 4 + 8 + 16 + 32 + ... + 4096 = ?. This problem may look completely intractible and tedious without a calculator. But once you start adding: 1+2=3, 1+2+4=7, 1+2+4+8=15, 1+2+4+8=31, ..., even a child can spot the pattern. Now that they are amazed, they will be motivated to learn more about the cause of this pattern, etc, etc.

quacked 5y ago
And I in turn can only venture a guess that you have entirely forgotten the brain-liquefying tedium of modern mathematics education, as shown by the fact that you immediately revert to actually interesting patterns that a properly-educated mathematician might think about. The dollars-and-cents addition problem mapped onto the "fun" toy store narrative is not mathematically interesting until assigned the type of analytical firepower outlined in OP's blog post. It is a deliberately obtuse trick question posed as a learning tool; children labelled "gifted" will figure it out and feel the dull satisfaction of killing another "problem", and those without that benefit will simply experience more fear and boredom.
- supportlocal4h 5y ago
  The word "label" is very important here. "Smart" children are identified early. After that they are smart because everyone knows they are smart. It is very hard for a person who knows they are not smart to convince a society who knows they are not smart. Dumb people are sometimes lucky and get something right. That doesn't make them smart. Smart people sometimes make mistakes. That doesn't re-label them as dumb. The important thing is the label you start with. Smart people will recognize the grammatical "mistake" in the previous sentence and understand that this post came from a dumb person.
young_unixer 5y ago
But in the case of OP's problem, starting to sum from the first item is an arbitrary choice that "just happens" to need only one change for the sum to be exact. Where "just happens" means that it was artificially inserted by whoever wrote the problem. Would the problem work if the student started summing backwards from the last item? How many items would the student have to swap in that case?. For how many random combinations of items would the problem "work" (i.e. require exactly one change?) On the other hand, in your example you're not just trying random things until something works. Noticing that 1 + 2 = 3, etc is more analytical than adding members at random and hoping that the person who created the problem thought of the same random pattern you thought about.
- abrax3141 5y ago
  ..or down the columns instead of across the rows! It is indeed likely that the kids who got it (as claimed by the child) probably lucked into it in the afore-described way. But don’t you think that thinking about it first makes more sense than “just start summing”? And if you think about it, and know what an NP hard problem is (as this kid, shockingly! apparently did), you quickly realize that it’s impossible. Then, you need to have the meta-reflective insight that no one would give a third grader an NP hard problem, and that there must be a trick, so then, and only then, do you start to look for patterns, .. and then the bell rings and class is over and you spent the whole time meta-reflecting and looking for a trick in just one stupid problem. Sucks to be smart!
- alisonkisk 5y ago
  The sum problem is exactly the same. It's interesting as pure mathematics, but it's an artificial pattern inserted into the problem to make it easy to solve, which almost never happens in real numeric life.

quacked 5y ago

And I in turn can only venture a guess that you have entirely forgotten the brain-liquefying tedium of modern mathematics education, as shown by the fact that you immediately revert to actually interesting patterns that a properly-educated mathematician might think about. The dollars-and-cents addition problem mapped onto the "fun" toy store narrative is not mathematically interesting until assigned the type of analytical firepower outlined in OP's blog post. It is a deliberately obtuse trick question posed as a learning tool; children labelled "gifted" will figure it out and feel the dull satisfaction of killing another "problem", and those without that benefit will simply experience more fear and boredom.

supportlocal4h 5y ago
The word "label" is very important here. "Smart" children are identified early. After that they are smart because everyone knows they are smart. It is very hard for a person who knows they are not smart to convince a society who knows they are not smart. Dumb people are sometimes lucky and get something right. That doesn't make them smart. Smart people sometimes make mistakes. That doesn't re-label them as dumb. The important thing is the label you start with. Smart people will recognize the grammatical "mistake" in the previous sentence and understand that this post came from a dumb person.

young_unixer 5y ago

But in the case of OP's problem, starting to sum from the first item is an arbitrary choice that "just happens" to need only one change for the sum to be exact. Where "just happens" means that it was artificially inserted by whoever wrote the problem. Would the problem work if the student started summing backwards from the last item? How many items would the student have to swap in that case?. For how many random combinations of items would the problem "work" (i.e. require exactly one change?) On the other hand, in your example you're not just trying random things until something works. Noticing that 1 + 2 = 3, etc is more analytical than adding members at random and hoping that the person who created the problem thought of the same random pattern you thought about.

abrax3141 5y ago
..or down the columns instead of across the rows! It is indeed likely that the kids who got it (as claimed by the child) probably lucked into it in the afore-described way. But don’t you think that thinking about it first makes more sense than “just start summing”? And if you think about it, and know what an NP hard problem is (as this kid, shockingly! apparently did), you quickly realize that it’s impossible. Then, you need to have the meta-reflective insight that no one would give a third grader an NP hard problem, and that there must be a trick, so then, and only then, do you start to look for patterns, .. and then the bell rings and class is over and you spent the whole time meta-reflecting and looking for a trick in just one stupid problem. Sucks to be smart!
alisonkisk 5y ago
The sum problem is exactly the same. It's interesting as pure mathematics, but it's an artificial pattern inserted into the problem to make it easy to solve, which almost never happens in real numeric life.