Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.I haven't watched the video, but I hope one of the approaches the two students consider involves asking the teacher what the problem means by "the chairs."
This video shows an excerpt of a conversation between two students comparing approaches to solving a problem and trying to understand why they got different answers and where one of them made an error.
Three halls contained 9,876 chairs altogether. One-fifth of the chairs were transferred from the first hall to the second hall. Then, one-third of the chairs were transferred from the second hall to the third hall and the number of chairs in the third hall doubled. In the end, the number of chairs in the three halls became the same. How many chairs were in the second hall at first.
By happenstance, my students and I read the first half of G.K. Chesterton's essay on fairy tales in class today. When I say "read," I mean that my students and I read each sentence individually and out loud and then stopped so I could explain what the sentence meant and why after first asking the person who had just read to take a crack at it.
Chesterton's opening lines:
Some solemn and superficial people (for nearly all very superficial people are solemn) have declared that the fairy-tales are immoral; they base this upon some accidental circumstances or regrettable incidents in the war between giants and boys, some cases in which the latter indulged in unsympathetic deceptions or even in practical jokes. The objection, however, is not only false, but very much the reverse of the facts.I asked my students what the words "the objection" meant. When nobody knew, I pointed out that the words "the objection" function as an anaphora: the definite determinative (the) tells you that you already know which (or what) objection because you've seen it before, in the text. It's the objection, not "an objection."
So, if you've already seen "the objection," and you've only read one other sentence, what is the objection? It's got to be inside that one other sentence.
At that point, my student who was educated outside the United States in his early elementary years (and who speaks very lightly accented English) figured it out.
My answer to the Illustrated Math problem, which I arrived at on my own and without a lot of conjecturing and solution-pathway-planning and special-case mongering and the like is: 4,938.
The Common Core era is going to be an unhappy one for mathematically gifted children.