Simple answer? No. While his teaching style may work for him, this kind of Common Core approach has proven to confuse students. A tutor can spend one-on-one time, but incorrect answers only help if you have the time to walk a student through their process and help them find where they went wrong. This is not possible in most teaching situations. But no, having more than one way to arrive at a correct answer does not qualify as an "interpretation" and definitely not hermeneutics.
Like in programming, there are rules that must be followed, in order. But as long as you follow those rules, you are free to create any kind of program you can imagine. It is similar with math. You can create just about any self-consistent mathematics you like, but there are strict rules to how you can solve equations.
(Oct 6, 2016 02:36 AM)Leigha Wrote: [ -> ]Is there room for hermeneutics in math?
From wikipedia:
Hermeneutics is the theory and methodology of interpretation, especially the interpretation of biblical texts, wisdom literature, and philosophical texts. It started out as a theory of text interpretation but has been later broadened to questions of general interpretation.
The formalism of mathematics and logic are strings of symbols linked by syntactic operators. In order for a string of symbols to acquire a meaning, in theoretical physics or whatever it is, the symbols have to be interpreted. The symbols have to stand for something, in other words. The truth-values of logical formulae will be dependent on their interpretations.
There's a whole sub-field of relatively advanced logic that addresses these interpretations and their various properties, called formal semantics.
https://en.wikipedia.org/wiki/Interpretation_(logic)
That's not exactly what hermeneutics means in Biblical and literary interpretation, and in areas like interpreting historical events where these methods have been extended. But it's probably the closest mathematical analogue that I'm aware of.
Quote:I have a good friend who tutors kids in math. He feels that even within mathematics, there is ''room for interpretation,'' as he puts it. It got me to thinking - could hermeneutics offer an approach to describing the nature of developing and understanding mathematical equations and processes? I've always viewed math as a very cut and dry type of subject, with no room for individual interpretation. My friend tried to explain this to me a bit further in that when he is tutoring his students in math, he isn't only concerned with helping them to arrive at the correct answers, but he wants them to explore different ways of arriving at the correct answers.
I don't think that what is being described there is hermeneutics so much as it's
heuristics. A heuristic is an approach that facilitates problem solving, learning or discovery. A heuristic needn't be a well-defined method or an algorithm, it might be more intuitive and creative than that. (Rules-of-thumb, analogies, educated guesses...) Heuristics are of great interest in psychology, artificial intelligence and even in the philosophy of science (hypothesis formation and originality in laboratory technique). To some extent, intelligence is a function of one's ability to recognize, create and exploit heuristics.
https://en.wikipedia.org/wiki/Heuristic
(Oct 6, 2016 02:36 AM)Leigha Wrote: [ -> ][...] I have a good friend who tutors kids in math. He feels that even within mathematics, there is ''room for interpretation,'' as he puts it. It got me to thinking - could hermeneutics offer an approach to describing the nature of developing and understanding mathematical equations and processes? I've always viewed math as a very cut and dry type of subject, with no room for individual interpretation.[...]
To use Stephen J. Gould as a particular example below[#1], mathematics is popularly(?) regarded as an abstract field which can produce "proofs" for its most valued or celebrated affairs, due to its foundation of axioms and floating on its own independently of concrete world contingencies. Relativism-free "certainties" and "absolutes" may thus be entertained within its boundaries, where such quantitative entities and constructs thereby resist being modified in themselves via relationships to others (i.e., yielding interpretations).
Whether or not advocates of embodied cognition (like George Lakoff below [#2]), or other interlopers can come along, and disturb any potentially idealized and fixed views of mathematics is ultimately up to the High Priests who regulate its mainstream appearance to either its internal practicing members or the public and academic institutions beyond its borders.
[#1] Stephen J. Gould:
Facts are the world's data. Theories are structures of ideas that explain and interpret facts. Facts don't go away when scientists debate rival theories to explain them. Einstein's theory of gravitation replaced Newton's in this century, but apples didn't suspend themselves in midair, pending the outcome. And humans evolved from ape-like ancestors whether they did so by Darwin's proposed mechanism or by some other yet to be discovered. Moreover, "fact" doesn't mean "absolute certainty"; there ain't no such animal in an exciting and complex world. The final proofs of logic and mathematics flow deductively from stated premises and achieve certainty only because they are not about the empirical world. Evolutionists make no claim for perpetual truth, though creationists often do (and then attack us falsely for a style of argument that they themselves favor). In science "fact" can only mean "confirmed to such a degree that it would be perverse to withhold provisional consent." I suppose that apples might start to rise tomorrow, but the possibility does not merit equal time in physics classrooms. --Evolution as Fact and Theory; Discover, May 1981
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[#2] George Lakoff:
Our answer is that the ordinary embodied mind, with its image schemas, conceptual metaphors, and mental spaces, has the capacity to create the most sophisticated of mathematics via using everyday conceptual mechanisms. Dehaene stopped with simple arithmetic.
We go on to show that set theory, symbolic logic, algebra, analytic geometry, trigonometry, calculus, and complex numbers can all be accounted for using those everyday conceptual mechanisms. Moreover, we show that conceptual metaphor is at the heart of the development of complex mathematics.
This is not hard to see. Think of the number line. It is the result of a metaphor that Numbers Are Points on a Line. Numbers don't have to be thought of as points on a line. Arithmetic works perfectly well without being thought of in terms of geometry. But if you use that metaphor, much more interesting mathematics results.
Or take the idea, in set-theoretical foundations for arithmetic, that Numbers Are Sets, with zero as the empty set, one as the set containing the empty set, and so on. That's a metaphor too. Numbers don't have to be thought of as being sets. Arithmetic went on perfectly well for 2000 years without numbers being conceptualized as sets. But if you use that metaphor, then interesting mathematics results.
There is a third less well-known metaphor for numbers, that Numbers Are Values of Strategies in combinatorial game theory. So which is it? Are numbers points? Are they sets? Are numbers fundamentally just values of strategies in combinatorial games?
These metaphors for numbers are part of the mathematics, and you make a choice each time depending on the kind of mathematics you want to be doing. The moral is simple: Conceptual metaphor is central to conceptualization of number in mathematics of any complexity at all. It's a perfectly sensible idea. Conceptual metaphors are cross-domain mappings that preserve inferential structure. Mathematical metaphors are what provide the links across different branches of mathematics.
One of our most interesting results concerns the conceptualization of infinity. There are many concepts that involve infinity: points at infinity in projective and inversive geometry, infinite sets, infinite unions, mathematical induction, transfinite numbers, infinite sequences, infinite decimals, infinite sums, limits, least upper bounds, and infinitesimals. Núñez and I have found that all of these concepts can be conceptualized as special cases of one simple Basic Metaphor of Infinity. The idea of "actual infinity"-of infinity not just as going on and on, but as a thing- is metaphorical, but the metaphor, as we show turns out to quite simple and exists outside of mathematics. What mathematicians have done is to provide elaborate carefully devised special cases of this basic metaphorical idea.
What we conclude is that mathematics as we know it is a product of the human body and brain; it is not part of the objective structure of the universe - this or any other. What our results appear to disprove is what we call the Romance of Mathematics, the idea that mathematics exists independently of beings with bodies and brains and that mathematics structures the universe independently of any embodied beings to create the mathematics. This does not, of course, result in the idea that mathematics is an arbitrary product of culture as some postmodern theorists would have it. It simply says that it is a stable product of our brains, our bodies, our experience in the world, and aspects of culture.
The explanation of why mathematics "works so well" is simple: it is the result of tens of thousands of very smart people observing the world carefully and adapting or creating mathematics to fit their observations. It is also the result of a mathematical evolution: a lot of mathematics invented to fit the world turned out not to. The forms of mathematics that work in the world are the result of such an evolutionary process. [The
pure mathematics POV, however, doesn't necessarily care about practical applications. It would be the un-tethered, "float on its own" abstract territory.]
It is important to know that we create mathematics and to understand just what mechanisms of the embodied mind make mathematics possible. It gives us a more realistic appreciation of our role in the universe. We, with our physical bodies and brains, are the source of reason, the source of mathematics, the source of ideas. We are not mere vehicles for disembodied concepts, disembodied reason, and disembodied mathematics floating out there in the universe. That makes each embodied human being (the only kind) infinitely valuable - a source not a vessel. It makes bodies infinitely valuable - the source of all concepts, reason, and mathematics. --"PHILOSOPHY IN THE FLESH" - A Talk with George Lakoff; EDGE 51— March 9, 1999
(Oct 6, 2016 06:49 AM)Yazata Wrote: [ -> ] (Oct 6, 2016 02:36 AM)Leigha Wrote: [ -> ]Is there room for hermeneutics in math?
From wikipedia:
Hermeneutics is the theory and methodology of interpretation, especially the interpretation of biblical texts, wisdom literature, and philosophical texts. It started out as a theory of text interpretation but has been later broadened to questions of general interpretation.
The formalism of mathematics and logic are strings of symbols linked by syntactic operators. In order for a string of symbols to acquire a meaning, in theoretical physics or whatever it is, the symbols have to be interpreted. The symbols have to stand for something, in other words. The truth-values of logical formulae will be dependent on their interpretations.
There's a whole sub-field of relatively advanced logic that addresses these interpretations and their various properties, called formal semantics.
https://en.wikipedia.org/wiki/Interpretation_(logic)
That's not exactly what hermeneutics means in Biblical and literary interpretation, and in areas like interpreting historical events where these methods have been extended. But it's probably the closest mathematical analogue that I'm aware of.
Quote:I have a good friend who tutors kids in math. He feels that even within mathematics, there is ''room for interpretation,'' as he puts it. It got me to thinking - could hermeneutics offer an approach to describing the nature of developing and understanding mathematical equations and processes? I've always viewed math as a very cut and dry type of subject, with no room for individual interpretation. My friend tried to explain this to me a bit further in that when he is tutoring his students in math, he isn't only concerned with helping them to arrive at the correct answers, but he wants them to explore different ways of arriving at the correct answers.
I don't think that what is being described there is hermeneutics so much as it's heuristics. A heuristic is an approach that facilitates problem solving, learning or discovery. A heuristic needn't be a well-defined method or an algorithm, it might be more intuitive and creative than that. (Rules-of-thumb, analogies, educated guesses...) Heuristics are of great interest in psychology, artificial intelligence and even in the philosophy of science (hypothesis formation and originality in laboratory technique). To some extent, intelligence is a function of one's ability to recognize, create and exploit heuristics.
https://en.wikipedia.org/wiki/Heuristic
Yazata, I saw your comment over at SF, and answered you there. I should copy and paste my response here, I will do so later. I think that this term does sum it up nicely, but if you look at the broadened definition of hermeneutics, then it fits.
The main thing that interests me on this is that it can greatly help students who aren't ''naturally'' good at math due to being taught a very rigid way of problem solving. This helps them to tailor their problem solving skills to what is most comfortable for them, as opposed to forcing students to learn the same procedural paths in problem solving. As long as the student arrives at the correct answer, then the path shouldn't matter. I think this concept is most appealing for teachers who can explore different ways of getting through to kids who struggle in math.