[math-fun] How do you ask a question in mathematical notation?
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. — Scott
Scott, I had to reread your email several times to make sure I wasn't missing something. Usually I agree with your take on things, but this posting seems wrong to me, both pedagogically and historically. Maybe if I had a better understanding of where you're coming from I could write a different reply, but anyway, here goes: 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 could write Fermat's Last Theorem as a quantified proposition that uses only mathematical symbols and then stick a question mark at the end of it: (\exists n \in N) (\exists x,y,z \in Q) (n > 2 \and xyz \neq 0 \and x^n + y^n = z^n) ? Would that count? I guess your answer would be "no". Mine would be "yes", since I'm pretty sure I've seen researchers do it in seminars. It seems to me that your question hinges on how broadly one defines "mathematical notation". People with different definitions will have different definitions, so unless there's a right definition of the phrase "mathematical notation" there's no right answer to your question. Some definitional issues have right answers; for instance, some students are puzzled by the fact that 0 is an even number, but one can usually convince them that they're using the wrong definition of "even number". Ditto for "Is a square a rectangle?" But it's unclear to me whether "mathematical notation" is like "even number" and "rectangle" in that respect. One difference is that "even number" and "rectangle" are inherently mathematical concepts, whereas the term "mathematical notation" is part mathematical, part historical-social. Mathematical notation is an interface (and a storage device) that was designed by, and for, people, so it's a social construct. Sometimes we have to draw a definitional line even when there's no God-given line, in questions like "When does human life begin and end?" In such cases, it's helpful to respond with the counter-question "For what purpose?" Which brings me back to the question of why you're asking the question in the first place. Is it about how we teach math, or is it about how we can be sure mathematical assertions are true?
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.
This seems to conflate mathematics with notational (formulaic) mathematics. But most of mathematical notation is a fairly recent innovation. Math, even algebra, was rhetorical until a few centuries ago. Fermat's statement of FLT was made up entirely of words (including the very human word "however").
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.
I agree that the way we teach the meaning of the equal sign is often deficient. Teachers talk a lot about this. (Good teachers like James Tanton, anyway.)
We can do better.
What might "doing better" look like? Do you want a typology of questions, for starters? Would it include a punctuation mark for open-ended questions that begin "Is there anything interesting to be said about ..." or "Are there any patterns governing ..."? I would argue that if we want to engage students, we need fewer symbols and more words. Symbols are great for stripping away distracting aspects of a problem so we can focus on pure form, but part of what gets stripped away is why we might care about the problem in the first place. Incidentally: In the classroom, my favorite questions are the ones my students ask, and my favorite subterfuges are the ones that lead them to ask the very questions I want to answer! 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.
It's definitely a problem that so little of mathematics has been checked by computer. We agree there! But would a system for having computers check a human-designed proof require a question mark? All we need is a command that says "Check this proof for errors!" But even a computer with superhuman proof-checking powers would have no curiosity and hence no need for question marks; we humans are the ones that ask the questions. So when you write "Mathematical questions always require human language", part of me wants to say "Yes, and that's a good thing!" Thanks for your question, Scott, and I apologize for misunderstanding you, as I'm sure I must have done in several spots. Jim Propp
Of course the main reason for using natural language is that we apply mathematics and so an important part of learning is the interpretation of the symbolic notation for application. You can't ask questions in symbols just because you haven't introduced the necessary meta-mathematical symbolic language. But you easily could and write something like ?x[13+78=x]. But then you would still need to teach the natural language interpretation. Brent On 12/2/2017 11:06 PM, Scott Kim 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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Mathematics is the study of form. We define formal systems and then ask questions about them. The questions are not part of the systems, and I don't feel like they ought to be. One could define a formal system that contained something question-like embedded in its structure, but doing mathematics with this system would still involve asking external questions about that system. I do not feel that this is a flaw. Geology is the study of rocks, but geological questions, the things that geologists investigate all the time, are not themselves rocks. Nobody complains about that. On Sun, Dec 3, 2017 at 12:31 PM, Brent Meeker <meekerdb@verizon.net> wrote:
Of course the main reason for using natural language is that we apply mathematics and so an important part of learning is the interpretation of the symbolic notation for application. You can't ask questions in symbols just because you haven't introduced the necessary meta-mathematical symbolic language. But you easily could and write something like ?x[13+78=x]. But then you would still need to teach the natural language interpretation.
Brent
On 12/2/2017 11:06 PM, Scott Kim 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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Warren — Thanks for pointing me to Mizar. I didn't know about it, and will investigate. If it's syntax is annoying, then it's a call to arms to make a better language! Designing a good formal proof language that is easy for humans to read is similar to the problem of designing a good programming language that is easy for humans to read. Brent — Of course language is important, and we shouldn't operate just in formal language. The practical problem is that the gulf between English and mathematical notation is so wide that many young students fail to cross it (I'm developing math games for kids ages 6-10). I'm therefore designing intermediate forms to bridge the gap. These will not be formal notations, but will likely to visual templates for expressing mathematical ideas — a better user interface for math, so to speak. Allan — Loved your hilarious geological analogy. Arguing that the content and the form of math should necessarily be alike is clearly a losing battle. Nonetheless I would like to see more of the elegance of thought that is so core to practice of math be applied to the presentation of mathematical ideas. On Sun, Dec 3, 2017 at 9:57 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Mathematics is the study of form.
We define formal systems and then ask questions about them. The questions are not part of the systems, and I don't feel like they ought to be. One could define a formal system that contained something question-like embedded in its structure, but doing mathematics with this system would still involve asking external questions about that system.
I do not feel that this is a flaw. Geology is the study of rocks, but geological questions, the things that geologists investigate all the time, are not themselves rocks. Nobody complains about that.
On Sun, Dec 3, 2017 at 12:31 PM, Brent Meeker <meekerdb@verizon.net> wrote:
Of course the main reason for using natural language is that we apply mathematics and so an important part of learning is the interpretation of the symbolic notation for application. You can't ask questions in symbols just because you haven't introduced the necessary meta-mathematical symbolic language. But you easily could and write something like ?x[13+78=x]. But then you would still need to teach the natural language interpretation.
Brent
On 12/2/2017 11:06 PM, Scott Kim 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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Regarding the pedagogy of = : a bad teacher would say that 32/12 = 16/6 is “wrong” because it isn’t the simplest form. A good teacher would say “that’s true: and can you say more that’s true, expressing 32/12 even more simply?" In passing, note that Khan Academy is a bad teacher in this sense. - Cris
On Dec 3, 2017, at 12:02 PM, Scott Kim <scottekim1@gmail.com> wrote:
Warren — Thanks for pointing me to Mizar. I didn't know about it, and will investigate. If it's syntax is annoying, then it's a call to arms to make a better language! Designing a good formal proof language that is easy for humans to read is similar to the problem of designing a good programming language that is easy for humans to read.
Brent — Of course language is important, and we shouldn't operate just in formal language. The practical problem is that the gulf between English and mathematical notation is so wide that many young students fail to cross it (I'm developing math games for kids ages 6-10). I'm therefore designing intermediate forms to bridge the gap. These will not be formal notations, but will likely to visual templates for expressing mathematical ideas — a better user interface for math, so to speak.
Allan — Loved your hilarious geological analogy. Arguing that the content and the form of math should necessarily be alike is clearly a losing battle. Nonetheless I would like to see more of the elegance of thought that is so core to practice of math be applied to the presentation of mathematical ideas.
On Sun, Dec 3, 2017 at 9:57 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Mathematics is the study of form.
We define formal systems and then ask questions about them. The questions are not part of the systems, and I don't feel like they ought to be. One could define a formal system that contained something question-like embedded in its structure, but doing mathematics with this system would still involve asking external questions about that system.
I do not feel that this is a flaw. Geology is the study of rocks, but geological questions, the things that geologists investigate all the time, are not themselves rocks. Nobody complains about that.
On Sun, Dec 3, 2017 at 12:31 PM, Brent Meeker <meekerdb@verizon.net> wrote:
Of course the main reason for using natural language is that we apply mathematics and so an important part of learning is the interpretation of the symbolic notation for application. You can't ask questions in symbols just because you haven't introduced the necessary meta-mathematical symbolic language. But you easily could and write something like ?x[13+78=x]. But then you would still need to teach the natural language interpretation.
Brent
On 12/2/2017 11:06 PM, Scott Kim 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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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The only interesting equalities are those that have different left- and right-hand sides. They are witnesses of a situation where the two sides are actually *isomorphic*, but we've modded out by the equivalence. Categorification is a creative process that removes such equivalences. It involving turning elements into objects, equations into isomorphisms, and adding new coherence laws on the isomorphisms. It's creative, because there's often no canonical way to do that. See, for instance, http://math.ucr.edu/home/baez/week147.html, where Baez categorifies division of natural numbers into the weak quotient of a groupoid acting on a set and gets a notion of groupoid cardinality. On Sun, Dec 3, 2017 at 1:01 PM, Cris Moore <moore@santafe.edu> wrote:
Regarding the pedagogy of = : a bad teacher would say that 32/12 = 16/6 is “wrong” because it isn’t the simplest form. A good teacher would say “that’s true: and can you say more that’s true, expressing 32/12 even more simply?"
In passing, note that Khan Academy is a bad teacher in this sense.
- Cris
On Dec 3, 2017, at 12:02 PM, Scott Kim <scottekim1@gmail.com> wrote:
Warren — Thanks for pointing me to Mizar. I didn't know about it, and will investigate. If it's syntax is annoying, then it's a call to arms to make a better language! Designing a good formal proof language that is easy for humans to read is similar to the problem of designing a good programming language that is easy for humans to read.
Brent — Of course language is important, and we shouldn't operate just in formal language. The practical problem is that the gulf between English and mathematical notation is so wide that many young students fail to cross it (I'm developing math games for kids ages 6-10). I'm therefore designing intermediate forms to bridge the gap. These will not be formal notations, but will likely to visual templates for expressing mathematical ideas — a better user interface for math, so to speak.
Allan — Loved your hilarious geological analogy. Arguing that the content and the form of math should necessarily be alike is clearly a losing battle. Nonetheless I would like to see more of the elegance of thought that is so core to practice of math be applied to the presentation of mathematical ideas.
On Sun, Dec 3, 2017 at 9:57 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Mathematics is the study of form.
We define formal systems and then ask questions about them. The questions are not part of the systems, and I don't feel like they ought to be. One could define a formal system that contained something question-like embedded in its structure, but doing mathematics with this system would still involve asking external questions about that system.
I do not feel that this is a flaw. Geology is the study of rocks, but geological questions, the things that geologists investigate all the time, are not themselves rocks. Nobody complains about that.
On Sun, Dec 3, 2017 at 12:31 PM, Brent Meeker <meekerdb@verizon.net> wrote:
Of course the main reason for using natural language is that we apply mathematics and so an important part of learning is the interpretation of the symbolic notation for application. You can't ask questions in symbols just because you haven't introduced the necessary meta-mathematical symbolic language. But you easily could and write something like ?x[13+78=x]. But then you would still need to teach the natural language interpretation.
Brent
On 12/2/2017 11:06 PM, Scott Kim 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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
I don't recall if I've ever shared this photo of Scott with Scott, but some others here may enjoy it (as well): http://chesswanks.com/pix/ScottKim.jpg It was taken 1986 December x at 3 PM (PST, taken at the Craft and Folk Art Museum in Los Angeles), where x is an integer less than 14. I'm representing it as an unknown because I haven't been able to figure it out (which is more annoying than usual because I happen to know the exact time of day). Noteworthy is the Mac Plus running an 8 MHz Motorola 68000, which could accommodate a phenomenal 1 MB (expandable, up to 4 MB) RAM.
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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
Incidentally, Unicode has the following mathematical symbols with a question mark: ≟ U+225F QUESTIONED EQUAL TO ⩻ U+2A7B LESS-THAN WITH QUESTION MARK ABOVE ⩼ U+2A7C GREATER-THAN WITH QUESTION MARK ABOVE To be included in Unicode, they had to be used in print. Leo On Mon, Dec 4, 2017 at 8:55 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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
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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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
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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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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James, Getting off topic but, I recently heard a few of these questions at "escape rooms" from actors while trying to solve puzzles and realized how frustrating some are. Maybe our students feel the same way:
What is the answer? I'm aware that I'm trying to get an answer - this is an infuriating question.
Does this answer make sense? Folks either do their own standard sanity checks or need you to point them toward a check that they haven't thought of. I'd prefer "What's the limiting case of ... " or "WRONG".
What question might this lead us to ask? I think the goal here is to get someone to verbalize their thoughts. Maybe a better way to achieve that is a less interrogative "Talk me through your thoughts on X so I can help"
Is there a pattern here? "I OBVIOUSLY don't know, or I would have solved it". Give me a hint or leave me alone.
How convinced are you? AKA "are you sure?" Maybe I'm poorly socialized, but I take this as epistemological. Probably not the discussion anyone wants to have.
James, I think you're missing the context of these sorts of questions, which is open-ended group exploration of a topic in a classroom, rather than students tackling work-sheets, or people trying to solve escape-rooms. It's a very different context! I agree that most of these questions would be very annoying in the latter situation.
What is the answer?
I included that one just to be complete, so as not to imply that ALL questions need to be of non-traditional type. Here "What is the answer?" is intended as a placeholder for the typical sort of question a teacher asks students in the midst of a class. I'm aware that I'm trying to get an answer - this is an infuriating
question.
You can't be aware that you're trying to get an answer if I haven't even asked the question yet. :-)
Does this answer make sense? Folks either do their own standard sanity checks
Most students don't have the habit of doing sanity checks. The mindset is write down an answer --- any answer --- and move on. or need you to point
them toward a check that they haven't thought of. I'd prefer "What's the limiting case of ... " or "WRONG".
But what if the answer is right? I'm just trying to get the students to use common sense as well as the algorithms that they may or may not have learned correctly. In any case, I try to avoid saying "wrong", to say nothing of "WRONG".
What question might this lead us to ask? I think the goal here is to get someone to verbalize their thoughts. Maybe a better way to achieve that is a less interrogative "Talk me through your thoughts on X so I can help"
"Help" is the wrong verb here, since the class is just exploring some ideas together.
Is there a pattern here? "I OBVIOUSLY don't know, or I would have solved it". Give me a hint or leave me alone.
Again, James is assuming a different context for these questions than the one I had in mind. If the class has just figured out the first four terms of a sequence, it's natural to pause to say "Does anyone see a pattern here?"
How convinced are you? AKA "are you sure?" Maybe I'm poorly socialized, but I take this as epistemological. Probably not the discussion anyone wants to have.
After I present a proof, I want the students to think about it critically. Sometimes I'll put in a mistake on purpose. In that case, the answer I'm hoping for is "No, I'm not convinced, and here's why." I should stress that these sorts of questions only make sense in a classroom culture that encourages trying out ideas even if they turn out to be wrong, and figuring things out together. Setting up the social dynamics for this is a tricky business! I do my best, but I still find that fewer than half of my students participate in discussions; the rest just lurk. Work in progress. I measure my success this semester in part by the fearlessness with which one of my favorite students participates. He usually disagrees with the other students about what the right answer is (until they convince him), and he's nearly always wrong, but he still keeps at it. But let me say again that I think I see where James is coming from, and that I agree that in certain contexts (such as helping a student with homework during an office hour), some of these questions would probably be really irritating! Jim
I think you're missing the context of these sorts of questions, which is open-ended group exploration of a topic in a classroom, rather than students tackling work-sheets
All good retorts - I was assuming a one on one student-teacher exchange. Natural language questions might be even more dependent on context than other sentence constructions. Even things like an http GET request only have definite meaning in context of the questioner and entity being questioned. I'm more convinced of Allan's point that most questions belong in meta-(whatever field).
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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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James — Hey that's a great idea. A list of such questions might be something that Escape rooms could give to players. On Tue, Dec 5, 2017 at 8:23 AM, Scott Kim <scottekim1@gmail.com> wrote:
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.
— Scott _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (10)
-
Allan Wechsler -
Andy Latto -
Brent Meeker -
Cris Moore -
Hans Havermann -
James Davis -
James Propp -
Leo Broukhis -
Mike Stay -
Scott Kim