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

The IVA Initiative:An imperative for College Teachers

of Algebra, of Developmental Mathematics, and of School/College Teachers

Based on a mathematical monograph fromThe American Institute for the Improvement

of MAthematics LEarning and Instructiona.k.a The MALEI Mathematics Institute

© January, 2008 The MALEI Mathematics Institute

WHAT ALL TEACHERS OF MATHEMATICS SHOULD KNOW

IVAFImplementing 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.


“A” is for

The term, “alligator”, is the NOMEN for all things of a specific kind.


“A” is for

3A = { }


“A” is for

3A = { }

A QUANTITY hasa NUMERAT0R of a DENOMINATION


“A” is for

3A = { }

A QUANTITY hasa NUMERAT0R and a DENOMINATION


3 + 5 + 2

= 3A +5E + 2DEven the very, very younghave no trouble with

INVENTORIES


3 + 5 + 2


INVENTORIES


3 + 5 + 2


INVENTORIES


3 + 5 + 2


INVENTORIES


3 + 5 + 2


INVENTORIES


The vector algebra of

INVENTORIES.



INVENTORIESIs 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)


The SPACE of such inventories

(?)

(?) (?)

= [ 9A+7E+6D ]



VECTOR ADDITION

[ 3A+5E+2D ]

+ [ 6A+2E+4D ]

= [ 9A+7E+6D ]



VECTOR ADDITION

[ 3A+5E+2D ]

+ [ 6A+2E+4D ]

= [ 9A+7E+6D ]



VECTOR ADDITION

[ 3A+5E+2D ]

+ [ 6A+2E+4D ]

= [ 9A+7E+6D ]



For REPEATED ADDITION

The “” is read “of”

7[3A+5E+2D] = 21A+35E+14D




7[3A+5E+2D] = 21A+35E+14D




7[3A+5E+2D] = 21A+35E+14D

and its SCALER MULTIPLICATION


SCALER DIVISION of inventories

. 5A + 8E + 3D . 4 21A + 35E + 14D

. 20A + 32E + 12D .

. 1A + 3E + 2D .

Is REMAINDER division



. 5A + 8E + 3D . 4 21A + 35E + 14D

. 20A + 32E + 12D .

. 1A + 3E + 2D .




. 5A + 8E + 3D . 4 21A + 35E + 14D

. 20A + 32E + 12D .

. 1A + 3E + 2D .




. 5A + 8E + 3D . 4 21A + 35E + 14D

. 20A + 32E + 12D .

. 1A + 3E + 2D .




. 5A + 8E + 3D . 4 21A + 35E + 14D

. 20A + 32E + 12D .

. 1A + 3E + 2D .




. 5A + 8E + 3D . 4 21A + 35E + 14D

. 20A + 32E + 12D .

. 1A + 3E + 2D .




. 5A + 8E + 3D . 4 21A + 35E + 14D

. 20A + 32E + 12D .

. 1A + 3E + 2D .




. 5A + 8E + 3D . 4 21A + 35E + 14D

. 20A + 32E + 12D .

. 1A + 3E + 2D .




. 5A + 8E + 3D . 4 21A + 35E + 14D

. 20A + 32E + 12D .

. 1A + 3E + 2D .




. 5A + 8E + 3D . 4 21A + 35E + 14D

. 20A + 32E + 12D .

. 1A + 3E + 2D .


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


TODAY’S SPECIALS

@ 2¢ @ 5¢ @ 3¢


TODAY’S SPECIALSDenomination VALUES .

@ 2¢ @ 5¢ @ 3¢


TODAY’S SPECIALSDenomination VALUES .

@ 2¢ @ 5¢ @ 3¢

are VARIABLE


@ 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


@ 2¢ @ 5¢ @ 3¢


3∘(A) + 5∘(E) + 2∘(D) =

3•(2) + 5•(5) + 2•(3) = 37

LINEAR EQUATIONS

3A + 5E + 2D = 37


@ 2¢ @ 5¢ @ 3¢


3∘(A) + 5∘(E) + 2∘(D) =

3•(2) + 5•(5) + 2•(3) = 37

LINEAR EQUATIONS

3A + 5E + 2D = 37


@ 2¢ @ 5¢ @ 3¢


3∘(A) + 5∘(E) + 2∘(D) =

3•(2) + 5•(5) + 2•(3) = 37

LINEAR EQUATIONS

3A + 5E + 2D = 37


@ 2¢ @ 5¢ @ 3¢


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



MEASUREMENTSInvokes conversion tables


A fluid measurements space:Gal, Qt, Pt, C, FlOz, Tbs, Tsp

- - - - - - - - - - - -

Unit-vector conversions:1G=4Q; 1Q=2P; 1P=2C; 1C=8F; …


A fluid measurements space:Gal, Qt, Pt, C, FlOz, Tbs, Tsp

- - - - - - - - - - - -

Unit-vector conversions:1G=4Q; 1Q=2P; 1P=2C; 1C=8F; …


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


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=100 PV=50 PV=25 PV=10 PV=5 PV=1

4$ +5H +7Q +3D +7N +8¢


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


VECTOR CONVERSIONEQUIVALENCE CLASSES

IMPROPER:6$ + 9H + 13Q + 4D + 9N + 39¢


VECTOR CONVERSIONEQUIVALENCE CLASSES

IMPROPER:6$ + 9H + 13Q + 4D + 9N + 39¢

PROPER:14$ + 1H + 1Q + 2D + ØN + 4¢

LOWEST SUM OF THE NUMERATORS


VECTOR CONVERSIONBY CARRYING:

4D + 9N + 39¢ 1$ + 2D + ØN + 4¢

IT CAN BE DONE & LEARNEDSEPARATELY

FROM ADDING/MULTIPLYING


“PROPER” ADDITION =VECTOR ADDITION &

FULL CARRYING.[1Q+3D+7N+9¢] + [5Q+7D+4N+8¢] =



FULL CARRYING.[1Q+3D+7N+9¢] + [5Q+7D+4N+8¢] = 6Q+10D+11N+17¢ =



FULL CARRYING.[1Q+3D+7N+9¢] + [5Q+7D+4N+8¢] = 6Q+10D+11N+17¢ = 3$ + ØQ + 2D + ØN + 2¢


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) …




4796 = 4Y, 7H, 9T,6(S) …

short for the vector, 4Y + 7H + 9T + 6(S) = (4,7,9,6)




4796 = 4Y, 7H, 9T,6(S) …short for the vector,

4Y + 7H + 9T + 6(S) = (4,7,9,6)

ARABIC NUMERALS AREPLACE-BASED VECTORS


LINEAR ALGEBRAICPROPER 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



2487 + 3942 =(2,4,8,7) + (3,9,4,2) =

(5,13,12,9) ~(5,14,2,9) ~

(6,4,2,9) = 6429



2487 + 3942 =(2,4,8,7) + (3,9,4,2) =

(5,13,12,9) ~(5,14,2,9) ~

(6,4,2,9) = 6429



2487 + 3942 =(2,4,8,7) + (3,9,4,2) =

(5,13,12,9) ~(5,14,2,9) ~

(6,4,2,9) = 6429



2487 + 3942 =(2,4,8,7) + (3,9,4,2) =

(5,13,12,9) ~(5,14,2,9) ~

(6,4,2,9) = 6429



2487 + 3942 =(2,4,8,7) + (3,9,4,2) =

(5,13,12,9) ~(5,14,2,9) ~

(6,4,2,9) = 6429



FRACTIONS:the numbers got by

dividing whole-scaler spacesby natural numbers


Fraction DENOMINATIONS are:H(alves) or T(hirds) or F(ourths) orf(ifths) or S(ixths) or s(evenths) orE(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!)


















2F+3F = ?7E-4E = ?

1H+2T is an inventory vector.But 1H=3S & 2T = 4S;

So 1H+2T = 7S


2F+3F = ?7E-4E = ?


So 1H+2T = 7S


2F+3F = ?7E-4E = ?


So 1H+2T = 7S


2F+3F = ?7E-4E = ?


So 1H+2T = 7S


2F+3F = ?7E-4E = ?


So 1H+2T = 7S


PHONIC DENOMINATIONS:

9ths, 8ths, 7ths, 6ths, …5ths, 4ths, 3ths, 2ths, 1ths

2(4ths) +3(4ths) = 5(4ths)




2(4ths) +3(4ths) = 5(4ths)




2(4ths) +3(4ths) = 5(4ths)




2(4ths) + 3(4ths) = 5(4ths)




2(4ths) + 3(4ths) = 5(4ths)




2(4ths) + 3(4ths) = 5(4ths)

2(4) + 3(4) =5(4)


Fraction denominationscan be assigned to PLACES

0 . __ __ __ __ __ __ __ __ __ __ __ __ __


A DECIMAL POINT is a pair of place-based vectors:

Each place valueis an integer- power of ten.

Each of the placed numeratorsis a whole number

0 . 7 32 5 642 0 0 0 0 …


In a DIGITAL decimal pointall numerators are less that ten.

Those arethe PROPER decimal points.

0 . 7 3 2 5 6 4 0 0 0 0 …


In the English phonics, most decimal points are

pronounced asIMPROPER vectors.

0 . 7 3 2 is pronounced as 0. 0 0 732


0 . __ __ __ __ __ __ __ __ __ __ __ __ __


A FRACTIONAL point usesall fraction denominations

The place-numbers are thefraction’s DENOMINATORS.

__. __ __ __ __ __ __ __ __ __ __ __ … 1 2 3 4 5 6 7 8 9 10 11 12


The scalers are thefraction’s NUMERATORS

3(4) = 0 . 0 0 3 0 0 0 0 0 0 0 0 0 0 0 …


All fraction conversionsAre

VECTOR CONVERSIONS(just as with measurements)

0 . __ __ __ __ __ __ __ __ __ __ __ __ __


Fraction addition and subtractionare

VECTOR addition and subtraction

0 . __ __ __ __ __ __ __ __ __ __ __ __ __


Multiplication of fractionsby whole numbers is

SCALER MULTIPLICATIONof vectors, by scalers.

0 . __ __ __ __ __ __ __ __ __ __ __ __ __

WHERE WOULD YOULIKE 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 GETTINGTHIS GLIMPSE OF

THE IVAFINITIATIVE

Documents

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