Here’s another comment on that New York Times piece:

“mystery number game …. ‘I’m thinking of a mystery number, and when I multiply it by 2 and add 7, I get 29; what’s the mystery number?’ “

See, that’s what I mean, the ubiquitous Common Core approach to math teaching these days wouldn’t allow for either “games” or “mystery”: they would insist that your son provide a descriptive narrative of his thought process that explains how he got his answer, they would insist on him drawing some matrix or diagram to show who that process is represented pictorially.

And your son would be graded on his ability to provide this narrative and draw this diagram of his thought process, not on his ability to get the right answer (which in child prodigies and genius, by definition, is out of the ordinary, probably indescribable).

Actually, I do often ask CJ to talk out his process after we do a mystery number. I share with the commenter the worry of slipping into a classroom regime where students are graded on their ability to recite the “correct” process. But in general, I think asking about process is great. For one thing, I learn a lot about how arithmetic facility develops in the mind. I asked CJ the other night how many candies he could buy if each one cost 7 cents and he had a dollar. He got the right answer, 14, not instantly but after a little thought. I asked him how he got 14 and he said, “Three 7s is 21, and five 21s is a dollar and five cents, so 15 candies is a little too much, so it must be 14.”

How would you have done it?

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Interestingly, I have no idea how I do arithmetic; my brain produced a “14” as I was reading the problem without any concious intermediate steps. I then verified that this is correct by asking “14*7=?” and getting 98.

Maybe you remember enough digits of 1/7 that this was a problem of recollection rather than arithmetic procedure.

7×7 is 49, so double that

6*15 is less than 100. 7*15 is more than 100. Must be 14.

I disagree with the commenter that the common core process doesn’t allow for “games”. For me, the fun of a problem isn’t just plucking out an answer from nowhere, but instead retracing the process by which I got there, figuring out what principles at work, which ones

generalize, and whether it’s applicable to other kinds of problems. Part of the fun is finding the tools to use elsewhere.

There are two kinds of “mysteries” here. There is the mystery of the problem, where we wonder what the right answer is, why a certain phenomenon occurs, and then there is the mystery of the solution process. I think there is nothing wrong with taking away the mystery of the solution process. Math is not magic as so many of our students tend to think. I think making math seem like some abracadabra we do in our heads only serves to keep more people away from it.

I also don’t believe that removing the mystery of the solution process removes the mystery of the problem. Would Sherlock Holmes stories be more compelling if Holmes just pointed at the murderer, exclaimed that he did it and never explained the process with which he arrived at that conclusion?

I think the danger with the reasoning component of common core (which I actually like a lot) is in asking young children to write their process. Kids are still learning to write, so math reasoning needs to be done in a discussion rather than homework format at least through elementary, and honestly probably until we are asking them to write proofs (in geometry).

Something like “Ten times seven is 70… another five-times-seven takes you to 105, which is too much, so try four-times-seven…” I know that the ten-times part of the estimate comes instantly, and the five-times part only slightly less so.

Had it been “how many can you buy with TWO dollars” I think the “twenty times seven” would have come almost instantly, not as fast as the ten-times-seven part did, but more like how the five-times-seven bit did. The algorithm for Y/X may be “Find ten times X; if that doesn’t get you close to Y, find ten times Z times X (where Z is a whole number greater than one) until you get the closest multiple of 10(X); then add or subtract 5(X) and see if that gets you close enough; and if not, add or subtract X as necessary until you get a plausible answer OR until you have trouble remembering where 10X and 10(Z)(X) got you and you give up and just do the division.”

Although, reading my answer above, it’s interesting (for the hypothetical two-dollar version of the problem) that I really did start with “twenty times seven and then start adding” rather than “thirty times seven which is obviously closer and would actually get you to the answer in fewer steps, but would require subtracting rather than adding.”