The IVA Initiative: An imperative for College Teachers of Algebra, of Developmental Mathematics, and of School/College Teachers Based on a mathematical.

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The IVA Initiative: An imperative for College Teachers of Algebra, of Developmental Mathematics, and of School/College Teachers Based on a mathematical monograph from The American Institute for the Improvement of MAthematics LEarning and Instruction a.k.a The MALEI Mathematics Institute © January, 2008 The MALEI Mathematics Institute

WHAT ALL TEACHERS OF MATHEMATICS SHOULD KNOW IVAF Implementing the Vector-Algebraic Foundations of Curricular Mathematics

What K-teachers should know “A” is for The term, “alligator”, is the NOMEN for all things of a specific kind.

What K-teachers should know “A” is for The term, “alligator”, is the NOMEN for all things of a specific kind.

What K-teachers should know “A” is for 3A = {}

What K-teachers should know “A” is for 3A = {} A QUANTITY has a NUMERAT0R of a DENOMINATION

What K-teachers should know “A” is for 3A = {} A QUANTITY has a NUMERAT0R and a DENOMINATION

What K-teachers should know 3 + 5 + 2 = 3A +5E + 2D Even the very, very young have no trouble with INVENTORIES

What K-teachers should know 3 + 5 + 2 = 3A +5E + 2D Even the very, very young have no trouble with INVENTORIES

What K-teachers should know 3 + 5 + 2 = 3A +5E + 2D Even the very, very young have no trouble with INVENTORIES

What K-teachers should know 3 + 5 + 2 = 3A +5E + 2D Even the very, very young have no trouble with INVENTORIES

What K-teachers should know 3 + 5 + 2 = 3A +5E + 2D Even the very, very young have no trouble with INVENTORIES

What K-teachers should know The vector algebra of INVENTORIES.

What K-teachers should know The vector algebra of INVENTORIES Is about how many of each kind

What G-1 teachers should know The BASIS for the VECTOR SPACE = [ 9A+7E+6D ] & SUBTRACTION (when it works)

What G-1 teachers should know The SPACE of such inventories (?) = [ 9A+7E+6D ] & SUBTRACTION (when it works)

What G-1 teachers should know VECTOR ADDITION [ 3A+5E+2D ] + [ 6A+2E+4D ] = [ 9A+7E+6D ] & SUBTRACTION (when it works)

What G-1 teachers should know VECTOR ADDITION [ 3A+5E+2D ] + [ 6A+2E+4D ] = [ 9A+7E+6D ] & SUBTRACTION (when it works)

What G-1 teachers should know VECTOR ADDITION [ 3A+5E+2D ] + [ 6A+2E+4D ] = [ 9A+7E+6D ] & SUBTRACTION (when it works)

What G-2 teachers should know For REPEATED ADDITION The “  ” is read “of” 7  [3A+5E+2D] = 21A+35E+14D

What G-2 teachers should know For REPEATED ADDITION The “  ” is read “of” 7  [3A+5E+2D] = 21A+35E+14D

What G-2 teachers should know For REPEATED ADDITION The “  ” is read “of” 7  [3A+5E+2D] = 21A+35E+14D and its SCALER MULTIPLICATION

What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

THE CONTEXT: www.reformingalgebra.org

An E-book in Progress: THE VECTOR-ALGEBRAIC FOUNDATIONS OF ARITHMETIC Written for mathematicians who teach mathematics to teachers of mathematics, K-Collegebra

What G-2 teachers should know TODAY’S SPECIALS @ 2 ¢ @ 5 ¢ @ 3 ¢

What G-2 teachers should know TODAY’S SPECIALS Denomination VALUES. @ 2 ¢ @ 5 ¢ @ 3 ¢

What G-2 teachers should know TODAY’S SPECIALS Denomination VALUES. @ 2 ¢ @ 5 ¢ @ 3 ¢ are VARIABLE

What G-2 teachers should know @ 2 ¢ @ 5 ¢ @ 3 ¢ EVALUATION OF INVENTORY VECTORS 3 ∘ (A) + 5 ∘ (E) + 2 ∘ (D) = 3(2) + 5(5) + 2(3) = 37 LINEAR EQUATIONS 3A + 5E + 2D = 37

What G-2 teachers should know @ 2 ¢ @ 5 ¢ @ 3 ¢ EVALUATION OF INVENTORY VECTORS 3 ∘ (A) + 5 ∘ (E) + 2 ∘ (D) = 3(2) + 5(5) + 2(3) = 37 LINEAR EQUATIONS 3A + 5E + 2D = 37

What G-2 teachers should know @ 2 ¢ @ 5 ¢ @ 3 ¢ EVALUATION OF INVENTORY VECTORS 3 ∘ (A) + 5 ∘ (E) + 2 ∘ (D) = 3(2) + 5(5) + 2(3) = 37 LINEAR EQUATIONS 3A + 5E + 2D = 37

What G-2 teachers should know @ 2 ¢ @ 5 ¢ @ 3 ¢ EVALUATION OF INVENTORY VECTORS 3 ∘ (A) + 5 ∘ (E) + 2 ∘ (D) = 3(2) + 5(5) + 2(3) = 37 LINEAR EQUATIONS 3A + 5E + 2D = 37

What G-2 teachers should know @ 2 ¢ @ 5 ¢ @ 3 ¢ EVALUATION OF INVENTORY VECTORS 3 ∘ (A) + 5 ∘ (E) + 2 ∘ (D) = 3(2) + 5(5) + 2(3) = 37 LINEAR EQUATIONS 3A + 5E + 2D = 37

More abstractly : If icons have values, those are variable S Is for Spades ♠ Value = __ H is for Hearts ♥ Value = __ D is for Diamonds ♦ Value = __ C Is for Clubs ♣ Value = __ 7S+ 4H+ 6D+ 0C What G-2 teachers should also know

What G-3 teachers should know The vector algebra of MEASUREMENTS Invokes conversion tables

What G-3 teachers should know A fluid measurements space: Gal, Qt, Pt, C, FlOz, Tbs, Tsp - - - - - - Unit-vector conversions: 1G=4Q; 1Q=2P; 1P=2C; 1C=8F; …

What G-3 teachers should know A fluid measurements space: Gal, Qt, Pt, C, FlOz, Tbs, Tsp - - - - - - Unit-vector conversions: 1G=4Q; 1Q=2P; 1P=2C; 1C=8F; …

What G-3 teachers should know In fluid measurements space: [G, Q, P, C, F, T, t] Each vector is equivalent to a reduced (or proper) vector 17C+11F+10t ~ 4Q+1P+4F+1T+1t

What G-3 teachers should know In fluid measurements space: [G, Q, P, C, F, T, t] Each vector is equivalent to a reduced (or proper) vector 17C+11F+10t ~ 4Q+1P+4F+1T+1t

What G-3 teachers also should know U.S. COINS: Values are legislated PV = Place Value PV=100PV=50PV=25PV=10PV=5PV=1 4\$+5H+7Q+3D+7N +8 ¢

What G-3 teachers also should know U.S. COINS: \$(USANS), H(alves), Q(uarters), D(imes), N(ickles), ¢(ents) Vector Basis: \$, H, Q, D, N, ¢ 6\$ + 9H + 13Q + 4D + 9N + 39¢ THE VECTOR VALUE = SUM OF QUANTITY VALUES

What G-3 teachers also should know VECTOR CONVERSION EQUIVALENCE CLASSES IMPROPER: 6\$ + 9H + 13Q + 4D + 9N + 39¢

What G-3 teachers also should know VECTOR CONVERSION EQUIVALENCE CLASSES IMPROPER: 6\$ + 9H + 13Q + 4D + 9N + 39¢ PROPER: 14\$ + 1H + 1Q + 2D + ØN + 4¢ LOWEST SUM OF THE NUMERATORS

What G-3 teachers also should know VECTOR CONVERSION BY CARRYING: 4D + 9N + 39¢  1\$ + 2D + ØN + 4¢ IT CAN BE DONE & LEARNED SEPARATELY FROM ADDING/MULTIPLYING

What G-3 teachers also should know “PROPER” ADDITION = VECTOR ADDITION & FULL CARRYING. [1Q+3D+7N+9¢] + [5Q+7D+4N+8¢] =

What G-3 teachers also should know “PROPER” ADDITION = VECTOR ADDITION & FULL CARRYING. [1Q+3D+7N+9¢] + [5Q+7D+4N+8¢] = 6Q+10D+11N+17¢ =

What G-3 teachers also should know “PROPER” ADDITION = VECTOR ADDITION & FULL CARRYING. [1Q+3D+7N+9¢] + [5Q+7D+4N+8¢] = 6Q+10D+11N+17¢ = 3\$ + ØQ + 2D + ØN + 2¢

What G-4 teachers should also know In English, pronounce the following number: 4796 (In Old English, “th” was spelled “Y”; The singles, “S”, is s-ilent)

What G-4 teachers should also know In English, pronounce the following number: 4796 (In Old English, “th” was spelled “Y”; The singles, “S”, is s-ilent) 4796 = 4Y, 7H, 9T,6(S) …

What G-4 teachers should also know In English, pronounce the following number: 4796 (In Old English, “th” was spelled “Y”; The singles, “S”, is s-ilent) 4796 = 4Y, 7H, 9T,6(S) … short for the vector, 4Y + 7H + 9T + 6(S) = (4,7,9,6)

What G-4 teachers should also know In English, pronounce the following number: 4796 (In Old English, “th” was spelled “Y”; The singles, “S”, is s-ilent) 4796 = 4Y, 7H, 9T,6(S) … short for the vector, 4Y + 7H + 9T + 6(S) = (4,7,9,6) ARABIC NUMERALS ARE PLACE-BASED VECTORS

What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

What G-5 teachers should know The vector algebra of FRACTIONS: the numbers got by dividing whole-scaler spaces by natural numbers

What G-5 teachers should know Fraction DENOMINATIONS are: H(alves) or T(hirds) or F(ourths) or f(ifths) or S(ixths) or s(evenths) or E(ighths) or N(inths) or t(enths) or e(levenths) or … Vector basis: H,T,F,f,S,s,E,N,t,e,… (We’ll soon run out of letters!)

What G-5 teachers should know Fraction DENOMINATIONS are: H(alves) or T(hirds) or F(ourths) or f(ifths) or S(ixths) or s(evenths) or E(ighths) or N(inths) or t(enths) or e(levenths) or … Vector basis: H,T,F,f,S,s,E,N,t,e,… (We’ll soon run out of letters!)

What G-5 teachers should know Fraction DENOMINATIONS are: H(alves) or T(hirds) or F(ourths) or f(ifths) or S(ixths) or s(evenths) or E(ighths) or N(inths) or t(enths) or e(levenths) or … Vector basis: H,T,F,f,S,s,E,N,t,e,… (We’ll soon run out of letters!)

What G-5 teachers should know Fraction DENOMINATIONS are: H(alves) or T(hirds) or F(ourths) or f(ifths) or S(ixths) or s(evenths) or E(ighths) or N(inths) or t(enths) or e(levenths) or … Vector basis: H,T,F,f,S,s,E,N,t,e,… (We’ll soon run out of letters!)

What G-5 teachers should know Fraction DENOMINATIONS are: H(alves) or T(hirds) or F(ourths) or f(ifths) or S(ixths) or s(evenths) or E(ighths) or N(inths) or t(enths) or e(levenths) or … Vector basis: H,T,F,f,S,s,E,N,t,e,… (We’ll soon run out of letters!)

What G-5 teachers should know 2F+3F = ? 7E-4E = ? 1H+2T is an inventory vector. But 1H=3S & 2T = 4S; So 1H+2T = 7S

What G-5 teachers should know 2F+3F = ? 7E-4E = ? 1H+2T is an inventory vector. But 1H=3S & 2T = 4S; So 1H+2T = 7S

What G-5 teachers should know 2F+3F = ? 7E-4E = ? 1H+2T is an inventory vector. But 1H=3S & 2T = 4S; So 1H+2T = 7S

What G-5 teachers should know 2F+3F = ? 7E-4E = ? 1H+2T is an inventory vector. But 1H=3S & 2T = 4S; So 1H+2T = 7S

What G-5 teachers should know 2F+3F = ? 7E-4E = ? 1H+2T is an inventory vector. But 1H=3S & 2T = 4S; So 1H+2T = 7S

What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) +3(4ths) = 5(4ths)

What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) +3(4ths) = 5(4ths)

What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) +3(4ths) = 5(4ths)

What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) + 3(4ths) = 5(4ths)

What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) + 3(4ths) = 5(4ths)

What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) + 3(4ths) = 5(4ths) 2(4) + 3(4) =5(4)

What G-5 teachers should know Fraction denominations can be assigned to PLACES 0. __ __ __ __ __ __ __ __ __ __ __ __ __

What G-5 teachers should know A DECIMAL POINT is a pair of place-based vectors: Each place value is an integer- power of ten. Each of the placed numerators is a whole number 0. 7 32 5 642 0 0 0 0 …

What G-5 teachers should know In a DIGITAL decimal point all numerators are less that ten. Those are the PROPER decimal points. 0. 7 3 2 5 6 4 0 0 0 0 …

What G-5 teachers should know In the English phonics, most decimal points are pronounced as IMPROPER vectors. 0. 7 3 2 is pronounced as 0. 0 0 732

What G-5 teachers should know 0. __ __ __ __ __ __ __ __ __ __ __ __ __

What G-5 teachers should know A FRACTIONAL point uses all fraction denominations The place-numbers are the fraction’s DENOMINATORS. __. __ __ __ __ __ __ __ __ __ __ __ … 1 2 3 4 5 6 7 8 9 10 11 12

What G-5 teachers should know The scalers are the fraction’s NUMERATORS 3(4) = 0. 0 0 3 0 0 0 0 0 0 0 0 0 0 0 …

What G-5 teachers should know All fraction conversions Are VECTOR CONVERSIONS (just as with measurements) 0. __ __ __ __ __ __ __ __ __ __ __ __ __

What G-5 teachers should know Fraction addition and subtraction are VECTOR addition and subtraction 0. __ __ __ __ __ __ __ __ __ __ __ __ __

What G-5 teachers should know Multiplication of fractions by whole numbers is SCALER MULTIPLICATION of vectors, by scalers. 0. __ __ __ __ __ __ __ __ __ __ __ __ __

WHERE WOULD YOU LIKE TO RUN WITH IT? FOUNDATIONS OF MATHEAMATICS? LINEAR ALGEBRA? TEACHER EDUCATION? COLLEGE ALGEBRA? LIBERAL ARTS? DEVELOPMENTAL MATHEMATICS? PUBLIC LITERACY?

For details, visit www.reformingalgebra.org

THANKS FOR GETTING THIS GLIMPSE OF THE IVAF INITIATIVE

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