James Tanton added a few items to my list: * Now that we see what the answer is <zero, a symmetrical rectangle aka a square, etc> could we have seen that more swiftly? * Was that approach to answering the question enjoyable? Could we have avoided doing XX? * Do you think this solution would be a pleasure for someone else to read? Have we made it easy to read and understand? Is what we have presented inviting to read? * What don't you like about the question? Shall we just change that and answer an easier question first? * What do you wish was also in the picture/equation/...? Can we just make it appear? (My "If there is something in life ..." maxim.) * Oh bother. I don't see what we need in order to proceed. I think we should weep. (To spur the class on to do something with the issue/question at hand) * Who says you have to do the questions in the order presented to you? (If part e is easier for you, do that part first!) * Why would anyone want to answer this question? * What would make this question interesting? * Is there a three-dimensional version of this? * What is it exactly that made this question hard? * What did we actually do here? What is the general thing we've established? On Mon, Dec 4, 2017 at 5:48 PM, Scott Kim <scottekim1@gmail.com> wrote:
Love it! This list of questions is analogous to the sorts of questions an English teacher asks students after they read a book/essay…to get the most learning out of a mathematical problem it helps to have discussion around the puzzle. (In general I think math education can learn a lot from common practices in English education).
There is exactly one problem solving curriculum for math at the elementary level in the US; it's called Cognitively Guided Instruction and it's gaining in popularity. Teachers using CGI indeed are coached to ask these sorts of questions to their students. https://blog.heinemann.com/what-is-cgi/
On Mon, Dec 4, 2017 at 1:13 PM, James Propp <jamespropp@gmail.com> wrote:
Here's a personal list of types of questions I like to ask in the classroom:
What is the answer? Does this answer make sense? Is there another way we could arrive at this answer? Does this remind you of something else we've done? What do these things have in common? What question might this lead us to ask? Is there a pattern here? What mistake did I just make? How am I fooling you? Is this wrong answer the right answer to a different question? Are we using the right definition? Have I given you enough inform to answer the question? What other information might you need? Can we think about this a different way? Is that a rigorous argument, or is there a subtle point that we're glossing over? How convinced are you? Can someone give a concrete example? Can we generalize? In plain English, what is this equation telling us? What kinds of mistakes do you think people are most prone to make when using this procedure? Does anybody have a question? (I'm still learning how to ask this one; some subtlety is required so as not to make students feel dumb.)
Has anyone published a longer list of this kind?
Jim
On Mon, Dec 4, 2017 at 11:55 AM, Andy Latto <andy.latto@pobox.com> wrote:
Are you asking for a terminology for boolean questions, or more general ones?
For boolean questions (those admitting only two possible answers, "true" and "false"), all that needs to be done is to add a single symbol, typically a question mark, to our formal notation. You can place it after a statement to turn it into the question "is the following statement true?". You can place it in other places (immediately after a quantifier, on top of an equals sign or inequality, etc.) but these are just simple syntactic transformations that add no expressive power.
But the more interesting questions are the more open-ended ones.
How fast does this function grow as x gets large? (answer is a function of x, along with a piece of terminology like O() or o()) Can we say anything interesting about ___? (answer is a mathematical statement) How can we precisely define what it means for a transformation to be natural? (answer is the invention of the field of category theory)
I'm not sure how you would define a formal syntax for this sort of question, much less a formal semantics. And I'm skeptical about how useful it would be even if you could define it.
Andy
On Sun, Dec 3, 2017 at 2:06 AM, Scott Kim <scottekim1@gmail.com> wrote:
I'm thinking about teaching the problem solving process in mathematics, and have run into a curious question: can one ask a mathematical question purely in mathematical notation? I believe the answer is no — mathematical questions always require human language in addition to mathematical notation. Problem statements are therefore always extramathematical in nature.
In practice, school kids frequently see problems stated in forms like: 13+78 = ___, which means "what is the sum of thirteen and seventy eight?" But that involves a nonmathematical symbol (the blank), AND ends up misleading kids to think that the equals sign means "the answer is". Very sloppy. We can do better.
This is an extension of something that has always bothered me: if mathematics is so rigorous, then how come it is conducted in a mish mosh of English and formal notation? The practical reason for mixing in human language is clear…like a computer program, a formal mathematical proof is more readable if it is annotated with comments. But too much reliance on human language means that formal proofs are not checkable by computer — a weird situation at best. I suppose that makes me a formalist, or a computer scientist…I've certainly got the latter bias because I don't trust anything I can't program.
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