Mathematical doodles.
- Thread starter Veho
- Start date
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I saw this a few days ago after watching a vlogbrother vid where they recomended her.
Her mobius strip videos are quite nifty too.
Her mobius strip videos are quite nifty too.
That quote was from <a href="http://www.maa.org/devlin/LockhartsLament.pdf" target="_blank">A Mathematician's Lament by Paul Lockhart</a>
I'd definitely recommend reading it, here are some more quotes:
I'd definitely recommend reading it, here are some more quotes:
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->The most striking thing about this so-called mathematics curriculum is its rigidity. This is
especially true in the later grades. From school to school, city to city, and state to state, the same
exact things are being said and done in the same exact way and in the same exact order. Far
from being disturbed and upset by this Orwellian state of affairs, most people have simply
accepted this “standard model” math curriculum as being synonymous with math itself.
This is intimately connected to what I call the “ladder myth”— the idea that mathematics can
be arranged as a sequence of “subjects” each being in some way more advanced, or “higher”
than the previous. The effect is to make school mathematics into a race— some students are
“ahead” of others, and parents worry that their child is “falling behind.” And where exactly does
this race lead? What is waiting at the finish line? It’s a sad race to nowhere. In the end you’ve
been cheated out of a mathematical education, and you don’t even know it.
Real mathematics doesn’t come in a can— there is no such thing as an Algebra II idea.
Problems lead you to where they take you. Art is not a race. The ladder myth is a false image of
the subject, and a teacher’s own path through the standard curriculum reinforces this myth and
prevents him or her from seeing mathematics as an organic whole. As a result, we have a math
curriculum with no historical perspective or thematic coherence, a fragmented collection of
assorted topics and techniques, united only by the ease in which they can be reduced to step-bystep procedures.<!--QuoteEnd--></div><!--QuoteEEnd--><!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->It is not at all uncommon to find second-year algebra students being asked to calculate
[ f(x + h) – f(x) ] / h for various functions f, so that they will have “seen” this when they take
calculus a few years later. Naturally no motivation is given (nor expected) for why such a
seemingly random combination of operations would be of interest, although I’m sure there are
many teachers who try to explain what such a thing might mean, and think they are doing their
students a favor, when in fact to them it is just one more boring math problem to be gotten over
with. “What do they want me to do? Oh, just plug it in? OK.”<!--QuoteEnd--></div><!--QuoteEEnd--><!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->SIMPLICIO: But aren’t you asking an awful lot from our math teachers? You
expect them to provide individual attention to dozens of students,
guiding them on their own paths toward discovery and enlightenment,
and to be up on recent mathematical history as well?
SALVIATI: Do you expect your art teacher to be able to give you individualized,
knowledgeable advice about your painting? Do you expect her to
know anything about the last three hundred years of art history? But
seriously, I don’t expect anything of the kind, I only wish it were so.
SIMPLICIO: So you blame the math teachers?
SALVIATI: No, I blame the culture that produces them. The poor devils are
trying their best, and are only doing what they’ve been trained to do.
I’m sure most of them love their students and hate what they are being
forced to put them through. They know in their hearts that it is
meaningless and degrading. They can sense that they have been made
cogs in a great soul-crushing machine, but they lack the perspective
needed to understand it, or to fight against it. They only know they
have to get the students “ready for next year.”
SIMPLICIO: Do you really think that most students are capable of operating on
such a high level as to create their own mathematics?
SALVIATI: If we honestly believe that creative reasoning is too “high” for our
students, and that they can’t handle it, why do we allow them to write
history papers or essays about Shakespeare? The problem is not that
the students can’t handle it, it’s that none of the teachers can. They’ve
never proved anything themselves, so how could they possibly advise a
student? In any case, there would obviously be a range of student
interest and ability, as there is in any subject, but at least students
would like or dislike mathematics for what it really is, and not for this
perverse mockery of it.
SIMPLICIO: But surely we want all of our students to learn a basic set of facts and
skills. That’s what a curriculum is for, and that’s why it is so
uniform— there are certain timeless, cold hard facts we need our
students to know: one plus one is two, and the angles of a triangle add
up to 180 degrees. These are not opinions, or mushy artistic feelings.
SALVIATI: On the contrary. Mathematical structures, useful or not, are invented
and developed within a problem context, and derive their meaning
from that context. Sometimes we want one plus one to equal zero (as
in so-called ‘mod 2’ arithmetic) and on the surface of a sphere the
angles of a triangle add up to more than 180 degrees. There are no
“facts” per se; everything is relative and relational. It is the story that
matters, not just the ending.
SIMPLICIO: I’m getting tired of all your mystical mumbo-jumbo! Basic arithmetic,
all right? Do you or do you not agree that students should learn it?
SALVIATI: That depends on what you mean by “it.” If you mean having an
appreciation for the problems of counting and arranging, the
advantages of grouping and naming, the distinction between a
representation and the thing itself, and some idea of the historical
development of number systems, then yes, I do think our students
should be exposed to such things. If you mean the rote memorization
of arithmetic facts without any underlying conceptual framework, then
no. If you mean exploring the not at all obvious fact that five groups
of seven is the same as seven groups of five, then yes. If you mean
making a rule that 5 x 7 = 7 x 5, then no. Doing mathematics should
always mean discovering patterns and crafting beautiful and
meaningful explanations.<!--QuoteEnd--></div><!--QuoteEEnd--><!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->What is happening [in high school geometry] is the systematic undermining of the student’s intuition. A proof, that is,
a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof
should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted
argument should feel like a splash of cool water, and be a beacon of light— it should refresh the
spirit and illuminate the mind. And it should be charming.
There is nothing charming about what passes for proof in geometry class. Students are
presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a
format as unnecessary and inappropriate as insisting that children who wish to plant a garden
refer to their flowers by genus and species.
...
not only are most kids utterly confused by this pedantry— nothing is more mystifying
than a proof of the obvious— but even those few whose intuition remains intact must then
retranslate their excellent, beautiful ideas back into this absurd hieroglyphic framework in order
for their teacher to call it “correct.” The teacher then flatters himself that he is somehow
sharpening his students’ minds.
...
The problem with the standard geometry curriculum is that the private, personal experience
of being a struggling artist has virtually been eliminated. The art of proof has been replaced by a
rigid step-by step pattern of uninspired formal deductions. The textbook presents a set of
definitions, theorems, and proofs, the teacher copies them onto the blackboard, and the students
copy them into their notebooks. They are then asked to mimic them in the exercises. Those that
catch on to the pattern quickly are the “good” students.
The result is that the student becomes a passive participant in the creative act. Students are
making statements to fit a preexisting proof-pattern, not because they mean them. They are
being trained to ape arguments, not to intend them. So not only do they have no idea what their
teacher is saying, <i>they have no idea what they themselves are saying.</i><!--QuoteEnd--></div><!--QuoteEEnd-->
especially true in the later grades. From school to school, city to city, and state to state, the same
exact things are being said and done in the same exact way and in the same exact order. Far
from being disturbed and upset by this Orwellian state of affairs, most people have simply
accepted this “standard model” math curriculum as being synonymous with math itself.
This is intimately connected to what I call the “ladder myth”— the idea that mathematics can
be arranged as a sequence of “subjects” each being in some way more advanced, or “higher”
than the previous. The effect is to make school mathematics into a race— some students are
“ahead” of others, and parents worry that their child is “falling behind.” And where exactly does
this race lead? What is waiting at the finish line? It’s a sad race to nowhere. In the end you’ve
been cheated out of a mathematical education, and you don’t even know it.
Real mathematics doesn’t come in a can— there is no such thing as an Algebra II idea.
Problems lead you to where they take you. Art is not a race. The ladder myth is a false image of
the subject, and a teacher’s own path through the standard curriculum reinforces this myth and
prevents him or her from seeing mathematics as an organic whole. As a result, we have a math
curriculum with no historical perspective or thematic coherence, a fragmented collection of
assorted topics and techniques, united only by the ease in which they can be reduced to step-bystep procedures.<!--QuoteEnd--></div><!--QuoteEEnd--><!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->It is not at all uncommon to find second-year algebra students being asked to calculate
[ f(x + h) – f(x) ] / h for various functions f, so that they will have “seen” this when they take
calculus a few years later. Naturally no motivation is given (nor expected) for why such a
seemingly random combination of operations would be of interest, although I’m sure there are
many teachers who try to explain what such a thing might mean, and think they are doing their
students a favor, when in fact to them it is just one more boring math problem to be gotten over
with. “What do they want me to do? Oh, just plug it in? OK.”<!--QuoteEnd--></div><!--QuoteEEnd--><!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->SIMPLICIO: But aren’t you asking an awful lot from our math teachers? You
expect them to provide individual attention to dozens of students,
guiding them on their own paths toward discovery and enlightenment,
and to be up on recent mathematical history as well?
SALVIATI: Do you expect your art teacher to be able to give you individualized,
knowledgeable advice about your painting? Do you expect her to
know anything about the last three hundred years of art history? But
seriously, I don’t expect anything of the kind, I only wish it were so.
SIMPLICIO: So you blame the math teachers?
SALVIATI: No, I blame the culture that produces them. The poor devils are
trying their best, and are only doing what they’ve been trained to do.
I’m sure most of them love their students and hate what they are being
forced to put them through. They know in their hearts that it is
meaningless and degrading. They can sense that they have been made
cogs in a great soul-crushing machine, but they lack the perspective
needed to understand it, or to fight against it. They only know they
have to get the students “ready for next year.”
SIMPLICIO: Do you really think that most students are capable of operating on
such a high level as to create their own mathematics?
SALVIATI: If we honestly believe that creative reasoning is too “high” for our
students, and that they can’t handle it, why do we allow them to write
history papers or essays about Shakespeare? The problem is not that
the students can’t handle it, it’s that none of the teachers can. They’ve
never proved anything themselves, so how could they possibly advise a
student? In any case, there would obviously be a range of student
interest and ability, as there is in any subject, but at least students
would like or dislike mathematics for what it really is, and not for this
perverse mockery of it.
SIMPLICIO: But surely we want all of our students to learn a basic set of facts and
skills. That’s what a curriculum is for, and that’s why it is so
uniform— there are certain timeless, cold hard facts we need our
students to know: one plus one is two, and the angles of a triangle add
up to 180 degrees. These are not opinions, or mushy artistic feelings.
SALVIATI: On the contrary. Mathematical structures, useful or not, are invented
and developed within a problem context, and derive their meaning
from that context. Sometimes we want one plus one to equal zero (as
in so-called ‘mod 2’ arithmetic) and on the surface of a sphere the
angles of a triangle add up to more than 180 degrees. There are no
“facts” per se; everything is relative and relational. It is the story that
matters, not just the ending.
SIMPLICIO: I’m getting tired of all your mystical mumbo-jumbo! Basic arithmetic,
all right? Do you or do you not agree that students should learn it?
SALVIATI: That depends on what you mean by “it.” If you mean having an
appreciation for the problems of counting and arranging, the
advantages of grouping and naming, the distinction between a
representation and the thing itself, and some idea of the historical
development of number systems, then yes, I do think our students
should be exposed to such things. If you mean the rote memorization
of arithmetic facts without any underlying conceptual framework, then
no. If you mean exploring the not at all obvious fact that five groups
of seven is the same as seven groups of five, then yes. If you mean
making a rule that 5 x 7 = 7 x 5, then no. Doing mathematics should
always mean discovering patterns and crafting beautiful and
meaningful explanations.<!--QuoteEnd--></div><!--QuoteEEnd--><!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->What is happening [in high school geometry] is the systematic undermining of the student’s intuition. A proof, that is,
a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof
should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted
argument should feel like a splash of cool water, and be a beacon of light— it should refresh the
spirit and illuminate the mind. And it should be charming.
There is nothing charming about what passes for proof in geometry class. Students are
presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a
format as unnecessary and inappropriate as insisting that children who wish to plant a garden
refer to their flowers by genus and species.
...
not only are most kids utterly confused by this pedantry— nothing is more mystifying
than a proof of the obvious— but even those few whose intuition remains intact must then
retranslate their excellent, beautiful ideas back into this absurd hieroglyphic framework in order
for their teacher to call it “correct.” The teacher then flatters himself that he is somehow
sharpening his students’ minds.
...
The problem with the standard geometry curriculum is that the private, personal experience
of being a struggling artist has virtually been eliminated. The art of proof has been replaced by a
rigid step-by step pattern of uninspired formal deductions. The textbook presents a set of
definitions, theorems, and proofs, the teacher copies them onto the blackboard, and the students
copy them into their notebooks. They are then asked to mimic them in the exercises. Those that
catch on to the pattern quickly are the “good” students.
The result is that the student becomes a passive participant in the creative act. Students are
making statements to fit a preexisting proof-pattern, not because they mean them. They are
being trained to ape arguments, not to intend them. So not only do they have no idea what their
teacher is saying, <i>they have no idea what they themselves are saying.</i><!--QuoteEnd--></div><!--QuoteEEnd-->
The Standard School Mathematics Curriculum
LOWER SCHOOL MATH.
The indoctrination begins. Students learn that mathematics is not
something you do, but something that is done to you. Emphasis is placed on sitting still, filling
out worksheets, and following directions. Children are expected to master a complex set of
algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part,
and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables
are stressed, as are parents, teachers, and the kids themselves.
MIDDLE SCHOOL MATH.
Students are taught to view mathematics as a set of procedures,
akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are
handed out, and the students learn to address the church elders as “they” (as in “What do they
want here? Do they want me to divide?”) Contrived and artificial “word problems” will be
introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison.
Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’
and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent
preparation for Algebra I.
ALGEBRA I.
So as not to waste valuable time thinking about numbers and their patterns, this
course instead focuses on symbols and rules for their manipulation. The smooth narrative thread
that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance
algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no
characters, plot, or theme. The insistence that all numbers and expressions be put into various
standard forms will provide additional confusion as to the meaning of identity and equality.
Students must also memorize the quadratic formula for some reason.
GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of
students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy
and distracting notation will be introduced, and no pains will be spared to make the simple seem
complicated. This goal of this course is to eradicate any last remaining vestiges of natural
mathematical intuition, in preparation for Algebra II.
ALGEBRA II.
The subject of this course is the unmotivated and inappropriate use of coordinate
geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic
simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety
of standard formats for no reason whatsoever. Exponential and logarithmic functions are also
introduced in Algebra II, despite not being algebraic objects, simply because they have to be
stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder
mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.
TRIGONOMETRY.
Two weeks of content are stretched to semester length by masturbatory
definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of
a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and
obsolete notational conventions, in order to prevent students from forming any clear idea as to
what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All
Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and
symmetry. The measurement of triangles will be discussed without mention of the
transcendental nature of the trigonometric functions, or the consequent linguistic and
philosophical problems inherent in making such measurements. Calculator required, so as to
further blur these issues.
PRE-CALCULUS.
A senseless bouillabaisse of disconnected topics. Mostly a half-baked
attempt to introduce late nineteenth-century analytic methods into settings where they are neither
necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to
obscure the intuitively clear notion of smooth change. As the name suggests, this course
prepares the student for Calculus, where the final phase in the systematic obfuscation of any
natural ideas related to shape and motion will be completed.
CALCULUS.
This course will explore the mathematics of motion, and the best ways to bury it
under a mountain of unnecessary formalism. Despite being an introduction to both the
differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be
discarded in favor of the more sophisticated function-based approach developed as a response to
various analytic crises which do not really apply in this setting, and which will of course not be
mentioned. To be taken again in college, verbatim.
***
And there you have it. A complete prescription for permanently disabling young minds— a
proven cure for curiosity. What have they done to mathematics!
There is such breathtaking depth and heartbreaking beauty in this ancient art form. How
ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an
art form older than any book, more profound than any poem, and more abstract than any abstract.
And it is school that has done this! What a sad endless cycle of innocent teachers inflicting
damage upon innocent students. We could all be having so much more fun.
LOWER SCHOOL MATH.
The indoctrination begins. Students learn that mathematics is not
something you do, but something that is done to you. Emphasis is placed on sitting still, filling
out worksheets, and following directions. Children are expected to master a complex set of
algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part,
and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables
are stressed, as are parents, teachers, and the kids themselves.
MIDDLE SCHOOL MATH.
Students are taught to view mathematics as a set of procedures,
akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are
handed out, and the students learn to address the church elders as “they” (as in “What do they
want here? Do they want me to divide?”) Contrived and artificial “word problems” will be
introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison.
Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’
and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent
preparation for Algebra I.
ALGEBRA I.
So as not to waste valuable time thinking about numbers and their patterns, this
course instead focuses on symbols and rules for their manipulation. The smooth narrative thread
that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance
algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no
characters, plot, or theme. The insistence that all numbers and expressions be put into various
standard forms will provide additional confusion as to the meaning of identity and equality.
Students must also memorize the quadratic formula for some reason.
GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of
students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy
and distracting notation will be introduced, and no pains will be spared to make the simple seem
complicated. This goal of this course is to eradicate any last remaining vestiges of natural
mathematical intuition, in preparation for Algebra II.
ALGEBRA II.
The subject of this course is the unmotivated and inappropriate use of coordinate
geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic
simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety
of standard formats for no reason whatsoever. Exponential and logarithmic functions are also
introduced in Algebra II, despite not being algebraic objects, simply because they have to be
stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder
mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.
TRIGONOMETRY.
Two weeks of content are stretched to semester length by masturbatory
definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of
a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and
obsolete notational conventions, in order to prevent students from forming any clear idea as to
what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All
Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and
symmetry. The measurement of triangles will be discussed without mention of the
transcendental nature of the trigonometric functions, or the consequent linguistic and
philosophical problems inherent in making such measurements. Calculator required, so as to
further blur these issues.
PRE-CALCULUS.
A senseless bouillabaisse of disconnected topics. Mostly a half-baked
attempt to introduce late nineteenth-century analytic methods into settings where they are neither
necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to
obscure the intuitively clear notion of smooth change. As the name suggests, this course
prepares the student for Calculus, where the final phase in the systematic obfuscation of any
natural ideas related to shape and motion will be completed.
CALCULUS.
This course will explore the mathematics of motion, and the best ways to bury it
under a mountain of unnecessary formalism. Despite being an introduction to both the
differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be
discarded in favor of the more sophisticated function-based approach developed as a response to
various analytic crises which do not really apply in this setting, and which will of course not be
mentioned. To be taken again in college, verbatim.
***
And there you have it. A complete prescription for permanently disabling young minds— a
proven cure for curiosity. What have they done to mathematics!
There is such breathtaking depth and heartbreaking beauty in this ancient art form. How
ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an
art form older than any book, more profound than any poem, and more abstract than any abstract.
And it is school that has done this! What a sad endless cycle of innocent teachers inflicting
damage upon innocent students. We could all be having so much more fun.
While I have seen a great many videos* on this and similar subjects these last few years (and probably before that as well) I shall certain add "+1" to what came before me in this thread.
* http://www.ted.com/talks/tags/id/70/page/7 probably being as good a start as any.
My personal favourite boredom type activity- trying to wedge/connect two or more unconnected (at least in my head at this point in time) areas of science/maths together.
Looking at all this it seems I was somewhat fortunate- while I recognise most of the things being mentioned in this thread it seems I was for the most part shielded from it all (proud would perhaps be the wrong word but it certainly causes no concern that to this day I do not know times tables and for that matter never have done). What I truly detest though is having to think in a curriculum- "is this method allowed?" (something that often appears when helping others with revising), "there is a far "simpler" method but I have to use this one", being told "you can not solve this..... yet*" and so forth.
*quadratic equations before imaginary/complex numbers "appear" would be a good example.
* http://www.ted.com/talks/tags/id/70/page/7 probably being as good a start as any.
My personal favourite boredom type activity- trying to wedge/connect two or more unconnected (at least in my head at this point in time) areas of science/maths together.
Looking at all this it seems I was somewhat fortunate- while I recognise most of the things being mentioned in this thread it seems I was for the most part shielded from it all (proud would perhaps be the wrong word but it certainly causes no concern that to this day I do not know times tables and for that matter never have done). What I truly detest though is having to think in a curriculum- "is this method allowed?" (something that often appears when helping others with revising), "there is a far "simpler" method but I have to use this one", being told "you can not solve this..... yet*" and so forth.
*quadratic equations before imaginary/complex numbers "appear" would be a good example.
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shlong said:Awesome
Also kinda awesome, but not doodly is
How does that guy not have tons of women all over him?
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