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Although I have opinions regarding more than just mathematics, I’m going to focus on the meat of the question. One of the things to mention is that I believe that we should return to a track-based system for science and math (notions in parentheses are only rough ideas):MathTrack 1 (mathematicians) - Students focus on proofs and building creative skills to tackle tough problems mathematicallyTrack 2 (engineers) - Students learn some important theorems, but spend little time demonstrating them, and rather, compute with them and make use of themTrack 3 (knowledgeable citizens) - Students acquire a mathematical literacy, focusing on comprehension and understanding the meanings of various conceptsScienceTrack 1 (scientists) - Students perform experiments continuously and build their knowledge of the world using these experimentsTrack 2 (inventors) - Students learn how to apply scientific principles to build things and how to apply to various everyday things (like cooking)Track 3 (technicians) - Students acquire scientific literacy, focusing on comprehension and understanding the meanings of various conceptsStudents should not be forced into a track though. The tracks should be an option, even to the students that find them the toughest. The point is that it is based on interest. Those with little interest will spend more time learning why it should interest them in track 3 then necessary in track 1.Furthermore, every subject should:Start with a motivation (why do I want to know it)Have an intuitionHave a formal definitionInclude examples and physical experiments (where possible)Everything that follows here is roughHere, I’m only going to discuss track 1 of mathematics, and I don’t believe that the tracks should separate anyway until about 3rd grade. My fear here is not the student’s readiness, but the amount of material to teach.I also believe that philosophy is paramount to understanding both mathematics and science, and should therefore be taught alongside it. By this, I don’t just mean logic, but discussions about epistemology, identity, and value.Students should not be forced into a track though. The tracks should be an option, even to the students that find them the toughest. The point is that it is based on interest. Those with little interest will spend more time learning why it should interest them in track 3 then necessary in track 1.Furthermore, every subject should:Start with a motivation (why do I want to know it)Have an intuitionHave a formal definitionInclude examples and physical experiments (where possible)Furthermore, some hints:Wait to the last minute (just before fractions) to introduce an operation called subtraction and division, instead focus on additive and multiplicative inverses.Ensure that the main focus of algebra is to teach that: [math]\phi\ x=\phi\ y \Leftrightarrow x = y[/math]Teach tensors first, and then matrices multiplication using tensors. Use for loops to teach tensors.Kindergarten (5 to 6 yrs)While students are learning their number words, and shapes, and colors, and animals, at ages 4 and 5, they are also beginning to acquire a theory-of-mind. Although this sounds like philosophy, not math, we work through questions about what other people think they know, against what actually happened, to learn about truthlikeness and verisimilitude (without ever mentioning these words). We talk about facts, about truth, and compare them with what’s not true, and try to work through the differences between lies, mistakes, and misconceptions (all using smaller words).We also must ensure that the kindergartner understands:relative value (that 3 apples is more than 2 apples) [convert introduction to both the philosophy of value and mathematical order theory]; demonstrate how relative value can help us increase rewards (in class marshmallow experiment)various descriptions of “sameness” [covert introduction to both philosophy of identity and equivalence classes]; demonstrate how it allows us to determine whether or not we can interchange legos by color but not by shapeclassifications (tacked onto the “sameness” discussion, such as when something is a color vs a number vs an animal [covert introduction to sets]how various shapes look in a mirror, and via rotations [covert introduction to group theory]; discuss how rotating a shape can allow us to fit the shape in a holeviewing how cardboard touches the outside of a balloon at a single point (with no pressure); deforming a line drawn on a balloon, playing with silly putty and play-doh and asking about connectivity, separation, and holes [covert introduction to topology and calculus]using rubber bands, ask about whether things are connected or disconnected, to each other, and through how many rubber bands [covert introduction to graph theory]how to turn on a computer, run an application, and by the end of the year, program a “hello, world” using a visual language (replacing “world” with their name) [covert computer literacy and programming]Also, unrelated to math, but related to their ability to learn in general, understanding the emotional and philosophical notion of grit, and the ability to take the lessons outside, and to put the lessons into games, and to connect lessons between various fields together is vital for all ages.1st grade (6 to 7 yrs)Add Boolean Logic, Shape Combinations (visual set theory via Venn-like diagrams), and Geometric Actions (rotation and reflection).Allow the student to consider commutativity, associativity, and involution in regards to:The Boolean Logical operators: AND, OR, and implicationRotation, reflection, and moving (translation) of objectsAddition and multiplicationMake sure they understand the concept of logical truth being “greater than” logical falsity in Boolean Logic, and how that leads to the implication. Ask how this relates to quantity in numbers, and “insideness” in shapes. Introduce an “unknown” quantity to logic and ask where they would place it if truth had an order.Have kids demonstrate arithemetic on computers as well as on paper. Introduce branching in code.2nd grade (7 to 8 yrs)Introduce arithmetic equality and orderDiscuss how it signifies that we are discussing counts and quantityAsk if we Boolean Logic equality and number equality are the same thing (can we compare 5 = false?)Introduce direction with integersAsk about things with orientation. Ask about debt and positions. Ask about positive change and negative change, and use this to introduce negative numbers by which to add (inverses), discuss subtraction as the act of finding the distance.Discuss the number line and compare and contrast subtraction, difference, distance, and displacement.Ask how subtraction helps us figure out how much of something we need (algebraic thinking).Discuss multiplication by -1 having the meaning of “change of direction” similar to reflection of a mirrorDiscuss ordering of negative numbersDiscuss the concept of an additive inverse of a numberDetermine whether or not subtraction is commutative or associativeAddition with clock hands [covert introduction to cyclic groups]Questions about understanding “order” when we don’t know if the day is before or afterWorking from other cyclic groups, like 8 and 2Demonstrating how addition on a cyclic group of 2 works like Boolean Logic XOR and multiplication works like ANDHave kids to do proofs by exhaustion of commutativity, associativity, and involution using truth tablesDemonstrate how many boxes they need for these proofs, and how this is related to multiplication in arithmeticDiscuss contrapositive, reverse, converse, and proof that in Boolean Logic that the contrapositive is equivalent to the original statement, but that the others are not, but equal to themselves.Discuss the relationship between XNOR and equals in Boolean LogicDiscuss how there are 16 possible operations on 2 inputs, and use truth tables to demonstrate itHave kids prove that NAND and NOR can be used to for any other objectHave kids prove that [math]a \to (b \to c) = (a \wedge b) \to c[/math], and mention how this will pop back up again in sets and in arithmeticIntroduce new logic systemsDemonstrate how many boxes will be required for one proof in 3-value logic; discuss a 4-value logic, including inconsistency and unknown(both true AND false or neither true NOR false).Discuss modal logic and how it is one possible way to discuss philosophical statements, ethics, and hence in requirements for projects; ensure that they understand that there are other ways to discuss ethicsHave the students do proofs symbolically, using modal logicIntroduce 1 and 0 as being special in arithmetic; discuss how special “nothingness” and the “universe” are to the union and intersection when combining shapes; ask about similar concepts in logicDiscuss comparisons such as ancestors vs decendent, in what order you prefer things, age of each Avenger; have them graph these comparisons using a directed graphUsing a textual programming language (like Haskell) introduce functions in programming, holding actions that can be used more than once; ask them to imagine if we could write a function that rotated or mirrored an object around us, and if we could imagine functions as actions for things around us. Have them use Boolean and integer functions.Do measurements of lengths, time, and mass; playing a game outside with a number line, demonstrate how it’s impossible to add step 5 to step 8, but that we can ask the distance between them (3) and add that to 5. Make sure they understand that we are not adding steps, but adding a step to a distance [covert introduction to torsors vs groups]; ask the same question about time, why we cannot add two dates together, but we can subtract one from another3rd grade (8 to 9 yrs)Introduce quantifiersDiscuss how quantifiers relate to modal logicDiscuss set notationDiscuss how they “bind” symbols to be used laterDemonstrate how to define union, intersection, and subset using quantifiers and logic. Have them compare the boolean logic with the set logic. Ask whether or not sets make another kind of logic.Introduce lambdasDiscuss how lambdas also bind variables, like quantifiers doDiscuss how lambdas can be used to rewrite functionsDiscuss the composition of functionsTalk about state in programmingOftentimes programming requires us to hold onto information to use laterDiscuss how this idea differentiates mathematical functions from software functions (programming notion of purity)Introduce numbers in base 10, base 2, base 60, base 12, and base 16Arithmetic using these basesHow bases are a representation onlyDiscuss binary relationsIntroduce ordered pair and compare against an unordered pair (set)Discuss the notion of a relation, and how each relation can be written, how the relation can be graphed (directed graph) and put into a table (similar to how we did exhaustive proofs for boolean operations)Discuss relations to and from the same classification of objects and different objectsDiscuss the requirements of a mathematical category (and how it has little to do with the everyday notion); using this discuss how it can simplify graphing some relations, but that not every relation is a categoryDistributivitiesMultiplication over additionAND over OR and OR over ANDUnion over Intersection and Intersection over UnionSelf-distributivity of implicationIntroduce the concept of a multiplicative inverse ([math]\overbar{9}[/math])Do this before discussing division!!!!Reason why it doesn’t make since to have [math]\overbar{0}[/math]Introduce division and rational numbersUsing natural numbers, and then integersUse the number line and pie chartsDiscuss how equality for rational numbers is different, because there are different expressions that can have the same valueDiscuss how expressions and value are different thingsDiscuss whether division is commutative or associativeGeometryMeasure perimeter, area, and volume of squares and cubesHave them discuss exponentialsAsk about exponentials to higher numbers and what ranges they would expect for values if exponents were fractionalMeasure area for triangles and then other quadrilaterals4th grade (9 to 10 yrs)Binary relationsDemonstrate a few set constructions of binary relations and how it compares to the cartesian productIntroduce a formal definition of a proset and poset and compare that to a categoryIntroduce formal definitions for, and compare and contrast function, multifunction, and partial functionDiscuss bijective (invertible) functions and have them find a fewIntroduce relation composition and symbolically prove that it is associativeIntroduce function composition and prove that it is associativeDemonstrate how the cartesian product ensures the same number of items as the product: e.g. [math]|A\times B| = |A|\times |B|[/math]Demonstrate that rational numbers are not natural numbers, but that a function exists to make natural numbers into rational numbers, and that a partial function exists to bring them backUse unordered pairs in constructionsconstruct the rules for addition, multiplication, division, and subtraction of rational numbersconstruct the rules for addition, multiplication, and subtraction of integersDiscuss divisibility and primenessDiscuss two numbers being coprimeDiscuss the GCD and LCM, MAX and MIN, and introduce all of them as lattices, compared with logic operators use this to introduce the union and intersection of a setFundamental Theorem of ArithemeticIntroduce Combinators with recursion and partial applicationDemonstrate the equivalence to lambdasBooleans constructed using lambdas and combinatorsNatural numbers constructed with lambdas and combinatorsIntroduce dependent typing of functions and how a function acts like a logicIntroduce the concept of a monoidDescribe it axiomaticallyDiscuss all of the monoids that we have used up to this point (AND, OR, XOR, ADD, MULT, rotations, etc.)Function composition as a monoidUse axioms to defineSets and use those to prove theorems from setsEuclidean geometry (using Tarski’s axioms, since we don’t have real numbers) and use those to prove some triangle theorems from thereDescribe philosophically how axiom systems act more like a contract for structures to adhere to or not5th GradeExponentiation and polynomialsCompare and contrast exponents of a variable and repeated iteration of a functionDiscuss the theorems of exponentiation. and how those theorems relate to logic (Heyting-like)Discuss how many unique functions can be created from one set to anotherDiscuss the multiple solutions to a quadratic and the principal rootSolve a quadratic without the quadratic formulaIntroduce some simplistic Galois Theory using quadraticsDemonstrate how [math]a^{\frac{1}{n}}[/math] cancels out [math]a^{n}[/math] when multiplied together, using the laws of exponentiationHave the student construct integers and rational numbers using IdrisIntroduce the students to typeclasses and proof burdensHelp them recognize the built-in Uint8 and Uint16 are implemenations of cyclic numbers, and that BigInt is an implementation of NatSI units of measure and differences between differential pressure, gauge pressure, and absolute pressure and how this corresponds to the notion of affine spaces and fields (both defined axiomatically here)Basic probability using sets and fractionsProofs about geometric shapes, rectanglesDiscuss how membership in the rational numbers is an all-or-none (classical) case for logic, and therefore how proof-by-contradiction can prove that [math]\sqrt{2}[/math] is not rationalIntroduction to decimals using base-2, base-10, and base-12; describe how this relates to the problem of [math]\sqrt{2}[/math] not being rational; ask how decimal representations of base-7 would look for fractions with a denominator divisible by 7Graphing functionsDecimals as it pertains toPercentagesMoneyUse set-complement and Boolean negation to introduce Boolean AlgebrasDrawing Hasse diagrams for ordersDoing Boolean Logic with electronic circuitsType Theoretical notionsComparing [math]A \times (A \times A)[/math], [math](A \times A) \times A[/math], and [math]A^3[/math] and using isomorphisms to discuss univalenceDescribing functional notation [math]f:A\to B[/math] and connections with logic and exponentiation (a logical version of “modus ponens”)An introduction to Per Martin Löf Type Theory alongside categoriesIntroduction to the disjoint union [math]A+B[/math]Discussions of constructors, deconstructors (pattern-matching), and containersRecursive functionsGreatest and least fixed points (by experimentation)Recursively built sequences (corecursion)Equational ReasoningMonotonicity of functions: discussion of how [math]a < b \Leftrightarrow -a > -b[/math]File reading in programming to count how many of some character as a stream (character container)6th GradeUsing excel to create histogramsComputing the mean from a data setComputing the standard deviation from a data setDoing the same, but from Idris/HaskellComputing cost of fencing materials to do a construction job using IdrisPolygons, vertices, and computing edges, vertices, perimiters, and areasThe point of algebra, that: [math]x=y\Leftrightarrow \phi\ x = \phi\ y[/math]Equational reasoningIntroduction to roots as right-inverse functions of exponentiationIntroduction to logarithms as left-inverse functions of exponentiationLaws of logarithms, proved by the inverse relation to exponentiationNotational syntaxesPolish notation and reverse polish notation vs infix notationAbutment of variables as either function application, group product, or rig productMultisetsIntroduction to 2D vectorsDistance using Pythagorean TheoremScalingTranslationAxioms for a vector spaceHow Types are a vector spaceMultisets as a vector spacePolynomials as a vector spaceIntroduction to complex numbersPlotting as 2D vectorsRotationFundamental Theorem of AlgebraProbability TheoryDependent probabilitiesAs a logicTensors of rank-2Transformations of vectorsComplex multiplication represented as tensorsRank-2 tensors of Boolean valuesRelationsComposing relation