Paul's blog on "Numbers People & Symbols People" discusses the problems of people's abilities (or inabilities) to handle abstraction. This raises two issues: first, the strong sequential dependencies in mathematical instruction; and second, the unfortunate anti-correlation between abstraction and practical reasoning of particulars.First, although professional math pedagogists might cringe at this framework, mathematical training seems to progress through at least 6 layers of increasing abstraction:1. Arithmetic math: the 4-function finger-and-toes math of small numbers2. Algorithmic math: the turn the crank math of multi-digit numbers (e.g. long division)3. Algebraic math: the unknown "x" math4. Advanced math: the appearance of triggy "scientific" functions that go beyond the original 4 functions.5. Analytic math: the limits of functions math that looks at the properties of series, derivatives, and integrals.6. Axiomatic math: the hard-core symbolic math where "+" and "*" no longer do what we're used to them doing.The point is that what makes math hard is that each step calls for a new way of thinking and lays the prerequisite foundation for the next level of math. Fail at one layer (due to a bad teacher, bad student, or bad year) and all subsequent layers are denied that person (barring costly remedial tutoring or therapy). This sequential dependency seems to afflict math more strongly than it does other school-day subjects. Failing to learn spelling doesn't prevent one from learning how to write a coherent paragraph. Failing to learn medieval history doesn't mean one can't learn the history of the industrial era. Failing to understand biology doesn't mean one can't learn chemistry. Yes, all subjects build on prior knowledge across the years of schooling, but other topics don't seem as necessarily strictly dependent on prior knowledge as math seems to be.Second, I do wonder whether the learning of abstraction doesn't come with some baggage. Paul's antimagical mathematician blog highlights one of the unfortunate characteristics of those blessed with abstract thinking abilities -- they seem rather less able to handle the unique peculiarities of situations that violate the usual assumptions of probability and math. I'm not sure if this observation says something about: 1) a problem in how abstract maths are taught; 2) the at-birth brain structures of those individuals that can learn abstract math; or 3) a more general limit of all human brains to be really good at all things at the same time -- one can train for either abstraction XOR "street smarts" in the same way that one can train for either weight-lifting XOR sprinting. Personally, I suspect that it's a bit of all three. First, training on abstraction could improve. If all classes on abstract math included word problems (and explicit discussions of the discrepancies between the world and the math), then abstract thinkers might understand the limits of abstract thought. Second, growing evidence in cognitive/neuroscience shows that we are NOT all identical tabula rasa at birth. Genetics plays big roles in many dimensions of brain chemistry and some babies are more adept at some tasks (e.g., recognizing if two 3-D rotated objects are different) than others. Third, despite the proclamations of "infinite potential" by human development optimists, gray matter is finite, class-room time is finite, and the work week is finite (even if these last two do not feel like it). Rather than insist that we all be well-rounded pegs for well-rounded societal holes, we might encourage further specialization but build explicit bridging mechanisms between different types of people.

Blame Bourbaki. QuoteAre you a numbers person or a symbols person? If this is an exclusive OR, then this question tends to pigeon-hole the person answering the question. At school, our class of 15 year olds learned that new maths, in this case some graph theory (no. 6 on your list). Understanding symbols is a different skill than cranking out a PDE. The former is kind of passive while the latter is active. One you know or don't know, the other you can learn.

QuoteOriginally posted by: CuchulainnBlame Bourbaki. Indeed! Yet the Bourbaki group's insistence on non-practical math is part of a broader pattern in humanity and all life forms. Making a great show of something absurdly impractical is a signal to mates and predators about fitness. It says "look at me, I'm so resource-rich, I can waste resources on abstract axioms, diamond-encrusted iPods, climbing Everest, pretty feathers, tweeting loudly whist hovering, etc."QuoteOriginally posted by: CuchulainnQuoteAre you a numbers person or a symbols person? If this is an exclusive OR, then this question tends to pigeon-hole the person answering the question. At school, our class of 15 year olds learned that new maths, in this case some graph theory (no. 6 on your list). Understanding symbols is a different skill than cranking out a PDE. The former is kind of passive while the latter is active. One you know or don't know, the other you can learn.Good point. And that raises the issue of the many types of intelligence. I've known people that couldn't follow the recipe for making icecubes -- they just don't have an innate sense of executing a sequence of actions. Other people (including me) have terrible memories for details. When I took organic chemistry, I did so well in the first semester that the professor tried to recruit me as a chemistry major. But then I did so badly at the second semester that I almost failed. The difference was that the first semester dealt with theory and the second semester dealt with memorizing an ungodly number of different chemical reactions (I still shudder at the words "Diels-Alder reaction"). That 6-step framework seems to require a combination of mental capacities. If these capacities are anti-correlated, then the probability of a person making it through all 6 stages is very remote. But I wonder if it possible to create different mathematical curricula for people with different capacities for rote memory facts, algorithmic sequences, pattern-matching, etc. Can everyone reach some high-level of mathematical understanding through respective and very different paths? Can we create a Brownian bridge to mathematical knowledge?

Functional analysis is a good compromise between Bourbaki and the more one-by-one solving approach because it works at two levels1. Banach/Hilbert spaces in general2. specific cases (like real numbers)And you can oscillate between the two in order to specialise and generalise. Example; Banach FPT subsumes many specific cases. So, each time you just have to check your level 2 satisfies level 1 axioms. (Ausubel theory?)

The spiral of innovation in algorithmic math?Take GCD: GCD(n,m) can be calculated by max {i, i div n and i div m}Then we observe from examples (how many, is dependent on the abilty to find abstractions) that GCD(n,m)=GCD(n-m,m) We push this into our mathematical knowledge base. And from further examples we might derive that this gives us an algorithm to calculate GCD (Euclidan) faster.(an automated theorem prover could probably find the Eucldian alg, if it is built for white box, constructive, proofs)In Axiomatic Math we do not distinguish between max_i and Euclidan, because they represent the identical i/o relation (n,m)-->GCD(n,m)To be honest, I do not see other classifications than Axiomatic and Algorithmic.(Arithmetic: if we have only Peano's Axioms in our knowledge base, it will take us some while to understand from examples that operations can be performed better when applying kind of polynomial-operations, but we can)Symbolic computation is about the manipulation of symbols, not the exploration of theorems and rules in our mathematical knowledge base?Question. Provided we have had powerful computers from the beginning of calculation. Would we have named special functions, like Sin, Gamma, Meijer G, ..; Operators, like Der, Int, ..;Structures like Groups, Fields, .. ?

QuoteOriginally posted by: exneratunriskTo be honest, I do not see other classifications than Axiomatic and Algorithmic.(Arithmetic: if we have only Peano's Axioms in our knowledge base, it will take us some while to understand from examples that operations can be performed better when applying kind of polynomial-operations, but we can)As I said, professional math pedagogists might cringe at this framework. But the question remains, what do we teach a 5 year old about math? And after they learn that, what do we teach a 6 year old? And if the X year old fails to understand the math topic of that year, will the X+1 year old have any hope of understanding the next year's topic? QuoteOriginally posted by: exneratunriskSymbolic computation is about the manipulation of symbols, not the exploration of theorems and rules in our mathematical knowledge base?Perhaps. Symbolic computation starts at the "Algebraic math" stage with the manipulations that let us solve for the unknown numerical value of x. What the full-on axiomatic stage offers is the manipulation of the more of the symbols. For example, in the ordinary introductory algebra classes (e.g., solving quadratic equations), the unknown x and y variables are ordinary numbers and the "+" and "*" retain the definitions used by the child from the first time they added and multiplied single digit numbers. By the time we get to the axiomatic stage, we've overloaded the operators to mean new abstract things and redefined the "unknowns" to be drawn from selected sets of numbers or non-numbers. We learn that our introductory algebra was just one of many possible algebras.Also, is symbol manipulation all that different from exploration of theorems? In both cases we start with one or more mathematical statements (which we assume to be true) and progressively manipulate that statement to derive other true mathematical statements. Yes, there may be vast differences in the level of generality between proving that the solution to a particular PDE is a particular result versus proving that all functions of a certain class have a certain convergence property. But the process has the same flavour -- mapping one symbolic truth to another symbolic truth for the purpose of extending knowledge of a particular mathematical form.We'll need to ask Paul about what level of symbolic knowledge he deems necessary and sufficient. QuoteOriginally posted by: exneratunriskQuestion. Provided we have had powerful computers from the beginning of calculation. Would we have named special functions, like Sin, Gamma, Meijer G, ..; Operators, like Der, Int, ..;Structures like Groups, Fields, .. ?That's a very good question! To what extent is the current level of math curriculum and general math knowledge stuck in a suboptimal state due to path dependence. Are we still teaching math in a pre-computational way? Of course, computers are still in their infancy -- should we adapt to computers or should computers adapt to us? (I'm voting for coevolution of the man-machine system). Will we ever have a "Google Math" that lets us enter a mathematical statement, hit "Search", and have it return all of the most popular/relevant solutions, theorems, source codes, applications, etc. associated with that statement?On the other hand, haven't computers lessened our understanding of symbolic/abstract solutions? Don't people rely far to much on the numerical output of computers -- not understanding the effects of the quality of the inputs (i.e., garbage-in-garbage-out), not knowing the limitations of the underlying code, and not having any intuitive sense for what the right answer might look like as a check for the correctness of the computer's output.

QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: exneratunriskTo be honest, I do not see other classifications than Axiomatic and Algorithmic.(Arithmetic: if we have only Peano's Axioms in our knowledge base, it will take us some while to understand from examples that operations can be performed better when applying kind of polynomial-operations, but we can)As I said, professional math pedagogists might cringe at this framework. But the question remains, what do we teach a 5 year old about math? And after they learn that, what do we teach a 6 year old? And if the X year old fails to understand the math topic of that year, will the X+1 year old have any hope of understanding the next year's topic? I should have mentioned that I am an (intuitive) fan of constructive, explorative learning (but I favour this without own scientific verifications and not being a teacher, I do not even have any practical experience, except application on myself. And I am a player-type ).Like create the sum-, product- and chain rule from experimenting with the Differentiation-Operator in a symbolic computation system? I think this will work?Or white-box/black-box learning. Show a higher principle even when its solver is not understood. And then explain the solver in a white box arrangement (pen-and-paper).With this one might be able to teach in quite agile leaning arrangements with short motivation-acquisition-intensification-test cycles?To the extreme, what is the use of learning descriptive geometry, if you have geometric modelers? p.s1. Symbolic computation?I include any symbols, like maps, molecular structures, circuits, .. and even programs (the functional programming paradigm makes it easier to manipulate). Unfortunately sc is often understood as simplifier a "semantic search engine" for closed form solutions. But closed form solutions are usually only valid for small "worlds". But yes, if proven, you du not need to test them with infinite many inputs . They are verified for ALL relevant input cases ( Black Scholes Formula )p.s2. Good question: is the movie (dynamic visualization) of the heat flow in a continuous casting slab as output of a finite element solver reliable?

QuoteBut the question remains, what do we teach a 5 year old about math? And after they learn that, what do we teach a 6 year old?I think geometry is not a bad place to start. It is close to analysis (in the Phytagoras sense) and they get a feeling for pi and numerical integration. For example, covering a 2d curve with rectangles give Simpson's rule. And make sure they use the crayons

QuoteOriginally posted by: CuchulainnQuoteBut the question remains, what do we teach a 5 year old about math? And after they learn that, what do we teach a 6 year old?I think geometry is not a bad place to start. It is close to analysis (in the Phytagoras sense) and they get a feeling for pi and numerical integration. For example, covering a 2d curve with rectangles give Simpson's rule. And make sure they use the crayons I like this! And if the urchins measure the volume of the crayons before and after filling in the rectangle (learning about Archimedes in the process), then they'll have a second estimate of the area of under the curve.I can see it now. Bananas and Banachs space after nap time.

QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnQuoteBut the question remains, what do we teach a 5 year old about math? And after they learn that, what do we teach a 6 year old?I think geometry is not a bad place to start. It is close to analysis (in the Phytagoras sense) and they get a feeling for pi and numerical integration. For example, covering a 2d curve with rectangles give Simpson's rule. And make sure they use the crayons I like this! And if the urchins measure the volume of the crayons before and after filling in the rectangle (learning about Archimedes in the process), then they'll have a second estimate of the area of under the curve.I can see it now. Bananas and Banachs space after nap time.Quality of maths education and pervasiveness ~ O(1/how 'developed' a country is)?

Actually, it has been a successful strategy for advancement in some places. Singapore is a great example.Has been at or very near #1 in Math and Science at 4th, 8th and 12th grade rankings for many years.Then you see a tremendous effort to move from providing a skilled labor force for multinational corps (like Hewlett Packard) to becoming an innovative (read intellectual property-driven) economy in its own right.World Economic Forum - Global Competitiveness Report

QuoteOriginally posted by: Traden4AlphaPaul's blog on "Numbers People & Symbols People" discusses the problems of people's abilities (or inabilities) to handle abstraction. Thinking by using logical rules, and operators, applied to symbols is efficient. It enables a stupid person to think he is thinking about complicated things. But really he has only become able to think about the phenomenon by discarding most of the relevant complexity - the real world mechanics - of the phenomenon being contemplated.Whether you call it abstraction or ignorance depends on whether you are deluded. Skill in using symbols to think about complicated phenomena, and imagining "abstract" patterns are predictive, has led more morons to imagine they are smart than anything else. But really you have to think through things, applying the laws of physics to all the varied and grainy details, and with all the surprises and moving parts, to come up with anything remotely approaching a useful simulation.Only simple uniform fluids and empty space can be described with coefficients and tensors. People who are good with this hammer generally see everything as a nail, and end up being wrong about the majority of the physical world and human civilization itself.

QuoteOriginally posted by: farmerQuoteOriginally posted by: Traden4AlphaPaul's blog on "Numbers People & Symbols People" discusses the problems of people's abilities (or inabilities) to handle abstraction.Thinking by using logical rules, and operators, applied to symbols is efficient. It enables a stupid person to think he is thinking about complicated things. But really he has only become able to think about the phenomenon by discarding most of the relevant complexity - the real world mechanics - of the phenomenon being contemplated.Whether you call it abstraction or ignorance depends on whether you are deluded. Skill in using symbols to think about complicated phenomena, and imagining "abstract" patterns are predictive, has led more morons to imagine they are smart than anything else. But really you have to think through things, applying the laws of physics to all the varied and grainy details, and with all the surprises and moving parts, to come up with anything remotely approaching a useful simulation.Only simple uniform fluids and empty space can be described with coefficients and tensors. People who are good with this hammer generally see everything as a nail, and end up being wrong about the majority of the physical world and human civilization itself.So very true!This is part of the fundamental human limits of only being able to think of 7 +/-2 things at the same time. Some people think they know the world because they know "the numbers" but they don't. Some people think they know the world because they know the symbols, but they don't. Both groups only know a respective 7 +/- 2 things about the world. The "relevancy" issue comes down to whether the true world resides in the neighborhood of the numerical representation (simplified to that instance of numbers) or resides in the neighborhood of the symbolic representation (simplified to that instance of symbols).Everyone understands that the numbers are wrong outside a small neighborhood. But the symbols have a high skew error distribution -- much of the time, the simplified symbolic representation works for moderately large distances from the "known" cases. But when the symbolic model fails it fails massively -- it's a widely applicable hammer that does a lot of damage if misused. Thus the symbolic solution earns a lot of nickels, but risks dollars.Overall, both approaches are clearly needed but the symbolic approach carries both extra rewards (for breadth of applicability) and extra penalties (for depth of potential error).

Err, if a number isn't a symbol what is then a symbol?? Wiki: "A symbol is something such as an object, picture, written word, sound, or particular mark that represents something else by association, resemblance, or convention."QuoteSymbols are great for showing structure, abstraction is always necessary if you are to go beyond mere arithmetic. Arithmetic is a method to 'show structure'. Everything is arithmetic if you will, because it is a formal language that can represent anything. Arithmetic is the ultimate abstraction.

QuoteOriginally posted by: macrotradeErr, if a number isn't a symbol what is then a symbol?? Wiki: "A symbol is something such as an object, picture, written word, sound, or particular mark that represents something else by association, resemblance, or convention."QuoteSymbols are great for showing structure, abstraction is always necessary if you are to go beyond mere arithmetic. Arithmetic is a method to 'show structure'. Everything is arithmetic if you will, because it is a formal language that can represent anything. Arithmetic is the ultimate abstraction.Agree. If we think of Free Algebras. Their elements are "language" representation of structures. The Free Polinomial Ring over the alphabet A:={x,y}, contains all polynomial elements constructed from (A,+,*), they are associative in +,*, abelian in +, * , distributive, have neutrals in +,*, inverse elements in +, ... All Polynomial Rings over A inherit the rules. But if we want to SolvePolynomialEquation, FindRoot, FindMinimum, .., we need to map Polynomials into Polynomial Functions: p-->p(x,y).If we cannot find exact (symbolic)) solutions, we approximate (numerically).Arithmentic works in real closed fields over {B}, we are used to B:=10?But the other way around, if we do not know any structure and special functions at all, we ALWAYS can apply SolveEquation, FindRoots, FindMinimum, ... numerically. How accurate, robust, .. depends.Goedel: a theory which contains Arithmetic has undecidable theorems.