Some of the MathPatterns taught in American ElementarySchool?s and HighSchools:
- Counting (how many do you have?)
- Mapping (you can draw a picture of it, and label it)
- 1:1 relationships
- 1:many relationships
- DatabaseNormalization (the only page on this wiki is DeNormalizationIsOk?)
- Functions
- Ordering (you can take a jumble of things or ideas, and put them in order)
- Addition
- The unknown, and ways of characterizing it:
- the concept of a variable (which can have a name, units, and a particular value in the case at hand, even if you do not know that value yet)
- GeneralSolution? (you can solve for a variable (as a function of other variables and/or known amounts), even if you do not know all the relevant amounts yet.)
- You can often clearly state an answer, preferably with formulas, numbers, and units.
- CheckBySubstitution?
- Uncertainty, Error, Precision, and SignificantFigure?s
- RoundingError?
- long division and Remainders
- Limits
- L'Hôpital's Rule
- Differentials -> Derivatives
- Running Totals -> Integrals
- Randomness, a priori odds, a postiori odds.
- MultipleSolutions? (such as PolynomialRoot?s) and reasonableness tests
- Multiple ways of describing the same quantity
- UnitConversion?
- ComplexNumbers
- Explicitly vs Implicitly (that is, as the number(s) that satisfies a problem or set of constraints)
- Truism
- ConservationLaw?s (A conservation law states that something is unchanged even when certain things may be changing - for example, when a diagram is simply magnified, directions are left unchanged, even though curvatures may change. The same concept applies to many (idealized) physical quantities.)
- Symmetry
- Identity (on wiki: ObjectIdentity)
- Additive Identity (0)
- Multiplicative Identity (1)
- UnitConversion? (multiply by a useful fraction that equals 1)
- IdentityMatrix? (A DiagonalMatrix? with 1's on the MainDiagonal?)
- Complement
- Symmetry + Complement + ConservationLaw? -> ReynoldsTransportTheorem
- Newton's LawsOfMechanics?
- InverseOperation?s
- Addition and Subtraction
- Multiplication and Division
- Exponentiation and Logarithm
- Integration and Differentiation
- LaplaceTransform and Inverse Laplace Transform
- FourierTransform and Inverse Fourier Transform
- Square and Square Root
- Polynomial and PolynomialRoot?s
- Sine, ArcSine?, and other trigonometric functions
- MatrixMultiplication? and MatrixInverses
- SpecialCases (ForbiddenOperation?s)
- DivideByZero?
- Zero (0)
- One (1)
- Infinity
- LoosenTheRequirement?s to ExtendTheDomain?. Notice that these examples invent numbers to allow inverse operations:
- Counting Numbers + Additive Identity ->
- NaturalNumbers + Subtraction ->
- Integers + Division ->
- RationalNumber?s (fractions) + PolynomialRoot?s ->
- AlgebraicNumber?s + Logarithm and TrigonometricFunction?s ->
- TranscendentalNumber?s and RealNumber?s + SquareRoot of NegativeNumber?s ->
- Imaginary numbers and ComplexNumbers
- You can use the result of one calculation as the input to another calculation
- Computer programming languages
- SpreadSheets
- CompoundFunction?s
- Polynomials
- Exponentials
- Create a meta-operation that repeats an operation an arbitrary number of times
- Addition -> Multiplication
- Multiplication -> Exponentiation (CompoundInterest?)
- Scalar -> Vector
- Vector -> Matrix
- You can systematically find the differences between a sequence of things
- Counting
- Sequence
- Series
- Induction
- Differentials
- Running Totals
- Differentiation
- Integration
Discussion
I question the appropriateness of this page in its present form even though it isn't OffTopic. All of mathematics is about patterns, so it's no surprise that this page resembles a subset of the mathematics syllabus of a school (or combination of schools) in America. However, omitting the fine detail is misleading. For example, it's unlikely sine and arcsine would be taught other than as part of at least the basics of trigonometry. Also, providing actual pages corresponding to the many of the dozens of dangling links would provide more than is appropriate for this wiki. WikiPedia is better suited for simple factual stuff that doesn't have immediate relevance for programming in general.
This page lists a group of patterns related in PatternLanguages. Most of the patterns are commonly used in computer programming. For example, an understanding of RoundingError? is needed in many programming applications.
Also, these patterns are normally taught over an extended period of time (such as from ages 2-18), in ways that often do not make the relationships between the patterns clear. And many students do not even learn about some of these concepts (such as ComplexNumbers), let alone some of the more engineering-oriented relationships between them (such as the ReynoldsTransportTheorem). By making a RoadMap of these concepts, I hope to encourage thinking about alternate ways to teach these ideas. I think most of them could be taught to children by the age of 10. That would have a profound impact on our society, as well as increasing the number and ability of our future programmers. -- JasperPaulsen
Perhaps, but the current breakdown varies enormously in degree of detail. It's hard to imagine many programmers finding useful information on pages about L'Hôpital's Rule, AdditiveIdentity?, or ArcSine?. For example, how many (commonly taught) functions would need there own page?
Some of these trailing links are near misses to pages already here. e.g. ComplexNumbers, MatrixInverse -- JohnFletcher
Those currently number about 7, whereas about 47 dangling links currently exist (at my last count).
Maybe the exposition leaves something to be desired, but the idea is good. LetsImproveOnIt? (pun intended). -- GunnarZarncke
See also: CategoryMath, MathPattern?, MathPatternLanguage, AmericanCulturalAssumption
AprilZeroSix
Math Patterns