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Math | ||||

Paul Zeitz, I share with you my thoughts on the varieties of "deep structure" in mathematical "frames of mind". Your book The Art and Craft of Problem Solving has been profoundly helpful. I also share with Joanne Simpson Groaney ("Mathematics in Daily Life"), Alan Schoenfeld ("Learning to Think Mathematically..."), John Mason ("Thinking Mathematically"), Manuel Santos, and also Maria Druojkova and the Math Future online group where I am active. I have been looking for the "deep ideas" in mathematics. George Polya's book "Mathematical Discovery" documents four patterns (Two Loci, the Cartesian pattern, recursion, superposition) of the kind I'm looking for (and which bring to mind architect Christopher Alexander's pattern languages). Your book documents dozens more. I've found Joanne Groaney's book helpful and I think the other writings I mention will also be in this regard. You note in your "planet problem", pg.63, that "on the surface" it is a nasty geometrical problem but "at its core" it is an elegant logical problem. This distinction brings to mind linguist Noah Chomsky's distinction between the surface structure and the deep structure of a sentence. In general, what might that deep structure look like? George Polya ends his discussion of the pattern of "superposition" or "linear combination" to say that it imposes a vector space. In an example he gives, the problem of "finding a polynomial curve that interpolates N points in the plane" is solved by "discovering a set of particular solutions which are a basis for a vector space of linear combinations of them". The surface problem has a deep solution, and the deep solution is a mathematical structure! I list below 24 such deep structures which characterize the mathematical "frames of mind" by which we solve problems. I note in parentheses the related patterns, strategies, tactics, tools, ideas or problems. I have included every such that I have found in your book, as well as Polya's four patterns, "total order" and "weighted average" that I observed in Joanne Growney's book, and a few more that I know of. Please start with the illustrative example of constructing an equilateral triangle. 6 | ||||

Independent trials | ||||

Independent trials | ||||

Blank sheets | ||||

Blank sheets Avoid error-prone activity Get your hands dirty Knowing when to give up Mental toughness, confidence and concentration Practice Toughen up Vary the trials | ||||

Space | ||||

Space | ||||

Center | ||||

Center Simple form Axiom of the empty set The decimal point Zero is just a place holder Change your point of view The swimmer's hat Considering the simplest point in an arbitrary coordinate system Convenient notation Elegant solution Invariant with respect to permutation of some of the roots of a polynomial Invariants Euler's formula Mean Value Theorem Motel room paradox Notation | ||||

Balance | ||||

Balance Add zero creatively Completing the square by adding zero Appeal to Physical Intuition Easy invariants Homogeneous coloration Make an expression uglier Modular arithmetic Multiply cleverly by one Overcount and rectify Parity 127 people in a tennis tournament Parity problem: Dominos on a chessboard Parity of a sum or product Simplicity | ||||

Polynomials | ||||

Polynomials Addition means combine like units, list different units Designing algorithms with index cards And Combinatorial tactics Complement Complete the square Counting the Complement Division Algorithm
Eliminate radicals Extracting squares Factor Theorem Infinitely many primes Modulo n filtering Number theoretic functions Or Substitution Symmetric functions of zeroes The Fundamental Theorem of Algebra Viewing a problem modulo m x**2 + y**2 = 13 | ||||

Vector space | ||||

Vector space Indicator function Linear combination | ||||

Time | ||||

Time | ||||

Sequence | ||||

Sequence Axiom of Infinity Mathematical induction Natural numbers *0 is a natural number**Every natural number has a unique successor that is a natural number.**Every natural number except 0 has a unique predecessor among the natural numbers.**If K is a set such that 0 is in K, and for every natural number n, if n is in K, then S(n) is in K, then K contains every natural number.*
Standard induction Prove n! > 2**n where integer n >= 3 Prove that the sum of the interior angles of any n-gon is 180(n-2) degrees Transfinite induction Transfinite ordinals | ||||

Poset with maximal or minimal elements | ||||

Poset with maximal or minimal elements Four times a right triangle is the difference of two squares Right triangles are more basic than circles Arithmetic-Geometric Mean Inequality Calculus of variations Extremal arguments involving infinite sets Extreme principle
When black and white points are interspersed... Least squares Maximum Laffer Curve Monotonizing Sign of the derivative Sum of Squares Symmetry-Product Principle Tangent-line approximation Use of extreme principle with induction Well-Ordering Principle Division algorithm
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Least upper bounds, greatest lower bounds | ||||

Least upper bounds, greatest lower bounds Least-upper-bound assumes second-order logic Algorithmic proof *one number that originally was not equal to A became equal to A;**the sum of all n numbers did not change;**the product of the n numbers increased.*
Since there are finitely many numbers, this process will end when all of them are equal to A; then the product will be maximal. This proof is called "algorithmic" because the argument used describes a concrete procedure which optimizes the product in a step-by-step way, ending after a finite number of steps.
pg.195-196 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2194
Bounded monotonic sequence Convergence of upper and lower bounds Finite Injury Priority Method Growth rates of functions Limits and Colimits in Category Theory Massage Monovariant John Conway's Checker Problem Number of games in an elimination-style tournament Reversing cards in a deck Constraint optimization Noah practiced constrained optimization Branch and bound First-choice bounding functions Russian doll search | ||||

Limits | ||||

Limits Continuum hypothesis Cauchy property Convergence Infinite Injury Priority Method Integrals | ||||

Thread | ||||

Thread | ||||

Extend the Domain | ||||

Extend the Domain Apply calculus ideas to a discrete problem Appropriate new ideas Complete set of solutions Define a Function Eulerian mathematics Existence of solutions Experimentation Generalizing the scope of a problem Guess the limit Inequalities Nonexistence of solutions Substitute Convenient Values | ||||

Continuity | ||||

Continuity Continuity Intermediate-Value Theorem Interpret dynamically Stretching things Topologically equivalent | ||||

Self-superimposed sequence | ||||

Self-superimposed sequence Arithmetic Mean Auto associative memory of neurons Be on the lookout for new ideas Be wary interchanging a “limit of a limit" Catalyst Define a function Generating functions *When you multiply x**m by x**n, you get x**(m+n).**"Local" knowledge about the coefficients of a polynomial or power series f(x) often provides "global" knowledge about the behavior of f(x), and vice versa.*
The first fact is trivial, but it is the technical "motor" that makes things happen, for it relates the addition of numbers and the multiplication of polynomials. The second fact is deeper ... Given a (possibly infinite) sequence a0, a1, a2, ..., its generating function is a0 + a1 x + a2 x**2 + ... In general, we don't worry too much about convergence issues with generating functions. As long as the series converges for some values, we can usually get by ... The term "generatingfunctionology" was coined by Herbert Wilf, in his book of the same name. We urge the reader to at least browse through this beautifully written textbook, which among its many other charms, has the most poetic opening sentence we've ever read (in a math book).
pg.143-144, 149 The Art and Craft of Problem Solving, Paul Zeitz, 1999, John Wiley & Sons, Inc.2164
Geometric series tool Power series Recurrence Restate Shifted sequence Solve for the limit Telescope Telescoping tool Zeta Function | ||||

Symmetry | ||||

Symmetry Axiom schema of specification A bank of useful derivatives of "functions of a function" Algebraic symmetry Combination of techniques Complex Numbers Exploit underlying symmetry in polynomials Fixed objects Four bugs chasing each other Geometric symmetry Fetching water for Grandma Square inscribed in circle inscribed in square Harmony Invariant with respect to transformations Not quite symmetrical Completing the square by trying to symmetrize Roots of Unity Search for order Symmetrize the coefficients Symmetry The Gaussian pairing tool Tilt the picture Transformation | ||||

Truth | ||||

Truth | ||||

Truth: Whether it is true? | ||||

Truth: Whether it is true? Argument by contradiction Square root of 2 is not rational Dropping the law of excluded middle | ||||

Model: What is true? | ||||

Model: What is true? A right triangle is half a rectangle Recasting geometry/combinatorics as parity Apply algebra ideas to a calculus problem Crossover Deliberately misleading presentation e Encoding Fantasize an answer Interpreting algebraic variables as coordinates Recast a problem from one domain into another domain Recast an inequality as an optimization problem Recasting Recasting geometry as algebra Structural equivalence Brouwer's Intuitionistic logic and set theory Equivalence of geometries Galois theory String theories are maps of each other Taniyama-Shimura Theorem Two different ways Make it easier Wishful thinking | ||||

Implication: How is it true? | ||||

Implication: How is it true? Deduction Penultimate step Recast geometry as logic | ||||

Variable: Why is it true? | ||||

Variable: Why is it true? Free variables and Bound variables Bend the rules Draw a picture Draw pictures Drawing the monk problem Invent a font Loosen up Peripheral vision Without loss of generality Create notation Hard and soft constraints | ||||

Restructuring | ||||

Restructuring - 10 Evolution: Hierarchy => Sequence (for determining weights)
- 20 Atlas: Network => Hierarchy (for determining connections)
- 21 Canon: Sequence => Network (for determining priorities)
- 32 Chronicle: Sequence => Hierarchy (for determining solutions)
- 31 Catalog: Hierarchy => Network (for determining redundancies)
- 30 Tour: Network => Sequence (for determining paths)
Graph Models of Multiplication - The addition rule is at work, adding exponents. Multiplying by 10 or dividing by 10 shifts the number with regard to the decimal point, although it looks like the decimal point is moving. We may think of this as simply changing the units, the base unit.
- I think of rescaling as a product of actions that either make bigger (numerator) or make smaller (denominator). They are all multiplying against some unknown, acting upon it. I call the actions "multiplication drops", either "magnifying drops" (say, multiplying by 10) or "shrinking drops" (dividing by 10). So these are actions x actions x (object with units). Thus magnifying and shrinking can cancel out. Also, actions can be decomposed into component actions, into primes.
- Repeated addition is a recounting, a shift from larger units to smaller units. 3 x (23 x dollars) becomes (3 x 23) x dollars. Amount x (large unit) becomes action x (amount x small unit) becomes (action x amount) x small unit becomes amount x small unit.
- Multiplication can give the ways of matching units, multiple units times multiple units, as in box multiplication, accounting for all possibilities. Units times units means that conditions are satisfied, thus generating all of the solutions.
- Multiplication can be thought of as counting items that have been grouped where each group has the same number of items. For example, we can count coins by grouping together the pennies, nickels, dimes, quarter, placing them in rows or groups of 4 or 5 or 10. Number x (Value x Unit).
- Dividing out, for example, money per person. This is like multiple units. (Number of cycles) x (Number of people x Units )
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Tree of variations | ||||

Tree of variations Axiom of pairing Multiplication, division by 10 moves the number Algorithmic Proof Dividing into cases Examination of cases Fractal multiplication: Recopying the whole Repeatedly folding paper - If a rectangular paper is folded in half, and half again, and yet again, and so on, and they are all considered repeatedly applied transformations of the same whole, then that is fractal multiplication, "recopying the whole", as with your paper snowflake.
- If a paper is folded once, and then again, and again, but those actions are thought as taking place separately, and especially, if I'm focused on the labelled components (rather than the repeating whole), then it is label multiplication, "redistributing the multiple", as with your pie halves sliced in five slices each.
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Adjacency graph | ||||

Adjacency graph All parabolas have the same shape Axiom of extensionality Coloring Divisibility Graph Handshake Lemma Sleeping mathematicians Least Common Multiple Proportion multiplication: Rescaling the whole Bicycle gearing The Two Men of Tibet Triangulating, then coloring | ||||

Total order | ||||

Total order Well-ordering theorem If you can count it, it's an integer Permutations Strong induction Tally multiplication: Rescaling the multiple Cutting a stack of cheese slices - if I'm simply changing the units, so that I'm thinking now of 20 small slices rather than 10 large slices, so that 1 large slice = 2 small slices, then I'm skip counting, "rescaling the multiple".
- if I'm thinking of each large slice as consisting of a left piece and a right piece, (a first piece and a second piece, distinguishable or "labelled" with a child's ID 10x2), then I'm "redistributing the multiple", as with "sets, per each".
Bucket elimination | ||||

Powerset lattice | ||||

Powerset lattice Axiom of power set Distributing with box mathematics Box multiplication: Redistributing the set Multiply in a particular order Polya's pattern of two loci Illustrative example - Solutions to both X and Y.
- Solutions to X.
- Solutions to Y.
- Solutions to the empty set of conditions.
Reimagining the monk problem Constraint satisfaction Creative rethinking The two ropes | ||||

Decomposition | ||||

Decomposition Axiom of union Reorganizing addition to do it in your head Reorganizing multiplication to do it in your head Binomial theorem Complement PIE Conspicuous ceiling Factoring Fundamental Theorem of Arithmetic Halt or repeat Intermediate pigeonhole Label multiplication: Redistributing the multiple Partitioning Partitions Pigeonhole principle n+1 positive integers Two points a mile away Repeated application of pigeonhole principle The Factor Tactic The principle of inclusion-exclusion | ||||

Directed graph | ||||

Directed graph Axiom of regularity Connectivity and Cycles Divide out Multiplication: Redistributing the whole Eulerian Path Hamiltonian Path Handshake Lemma | ||||

Context | ||||

Context 10 + 4 = 2 Axiom schema of replacement | ||||

Other ways: Math | ||||

Other ways: Math Alternatives to mathematical induction Epsilon induction Forcing (in set theory) Thought experiments in Computer Science Thought experiments in Mathematics |