More questions: * What are some questions (based on this situation)? This is Marion Walters, advocate of Problem Posing, opening gambit. * How do you feel about the question? (Tanton's questions circle around this.) Excited? Bored? Intimidated? (I watched a great talk by math educator Zalman Usiskin at UoChicago on this topic; it was very hard to get a group of math educators to focus purely on feeling as opposed to logical analysis, but in practice feeling is an important part of mathematical intuition. People kept saying things like "I feel there might be multiple solutions".). * The flip side of asking an easier question: what would be a harder version of this question? * How would this question sound if it had been stated by <Elvis, the Mad Hatter, Bugs Bunny, Luke Skywalker>? (Change the tone of voice) • What part of the problem should we work on first? * How shall we organize or restate the problem in order to make it easier to solve? * (After working on a problem for a while) What has not worked? What have we learned from our failures? * If this problem were stated by one character to another in a crucial scene of a famous movie, what would be the subtext? What aspect of character would be revealed? * Make up a humorous nonsense version of the same question. (to break up seriousness, and explore the edge between sense and nonsense). * If you already knew the answer, what do you think the solution might look like? (guessing) * What type of problem is this? What types of techniques might work on it? (requires a list of problem solving techniques) * What branch of math is relevant to solving this problem? * Who do you know that would be good at solving this sort of problem? * What would you Google to get help solving this problem? (knowing where to look for help) * What parts DO you understand? (to combat the feeling of I don't understand anything). This question is from Tim Gallwey. Note without judgement which parts of the question you do understand, and which you don't. On Mon, Dec 4, 2017 at 8:44 PM, Brent Meeker <meekerdb@verizon.net> wrote:
Consider an extreme form of this question? Is it easier to answer?
Brent
On 12/4/2017 1:13 PM, James Propp 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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