The IVA Initiative: An imperative for College Teachers of Algebra, of Developmental Mathematics, and...
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Transcript of The IVA Initiative: An imperative for College Teachers of Algebra, of Developmental Mathematics, and...
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.
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 hasa NUMERAT0R of a DENOMINATION
What K-teachers should know
“A” is for
3A = { }
A QUANTITY hasa NUMERAT0R and a DENOMINATION
What K-teachers should know
3 + 5 + 2
= 3A +5E + 2DEven the very, very younghave no trouble with
INVENTORIES
What K-teachers should know
3 + 5 + 2
= 3A +5E + 2DEven the very, very younghave no trouble with
INVENTORIES
What K-teachers should know
3 + 5 + 2
= 3A +5E + 2DEven the very, very younghave no trouble with
INVENTORIES
What K-teachers should know
3 + 5 + 2
= 3A +5E + 2DEven the very, very younghave no trouble with
INVENTORIES
What K-teachers should know
3 + 5 + 2
= 3A +5E + 2DEven the very, very younghave no trouble with
INVENTORIES
What K-teachers should know
The vector algebra of
INVENTORIES.
What K-teachers should know
The vector algebra of
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)
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 SPECIALSDenomination VALUES .
@ 2¢ @ 5¢ @ 3¢
What G-2 teachers should know
TODAY’S SPECIALSDenomination 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
MEASUREMENTSInvokes 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=100 PV=50 PV=25 PV=10 PV=5 PV=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 CONVERSIONEQUIVALENCE CLASSES
IMPROPER:6$ + 9H + 13Q + 4D + 9N + 39¢
What G-3 teachers also should know
VECTOR CONVERSIONEQUIVALENCE 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 CONVERSIONBY CARRYING:
4D + 9N + 39¢ 1$ + 2D + ØN + 4¢
IT CAN BE DONE & LEARNEDSEPARATELY
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 AREPLACE-BASED VECTORS
What G-4 teachers should also know
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
What G-4 teachers should also know
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
What G-4 teachers should also know
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
What G-4 teachers should also know
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
What G-4 teachers should also know
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
What G-4 teachers should also know
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
What G-5 teachers should know
The vector algebra of
FRACTIONS:the numbers got by
dividing whole-scaler spacesby natural numbers
What G-5 teachers should know
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!)
What G-5 teachers should know
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!)
What G-5 teachers should know
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!)
What G-5 teachers should know
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!)
What G-5 teachers should know
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!)
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 denominationscan be assigned to PLACES
0 . __ __ __ __ __ __ __ __ __ __ __ __ __
What G-5 teachers should know
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 …
What G-5 teachers should know
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 …
What G-5 teachers should know
In the English phonics, most decimal points are
pronounced asIMPROPER 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 usesall fraction denominations
The place-numbers are thefraction’s DENOMINATORS.
__. __ __ __ __ __ __ __ __ __ __ __ … 1 2 3 4 5 6 7 8 9 10 11 12
What G-5 teachers should know
The scalers are thefraction’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 conversionsAre
VECTOR CONVERSIONS(just as with measurements)
0 . __ __ __ __ __ __ __ __ __ __ __ __ __
What G-5 teachers should know
Fraction addition and subtractionare
VECTOR addition and subtraction
0 . __ __ __ __ __ __ __ __ __ __ __ __ __
What G-5 teachers should know
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