multiple intelligences approach to number systems
TRANSCRIPT
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 1
Multiple Multiple Intelligences Intelligences Approach to Approach to
Teaching Number Teaching Number SystemsSystems
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 2
MI Theory:MI Theory: First described by Howard First described by Howard
Gardner (1983)Gardner (1983)
Intelligence has to do with:Intelligence has to do with:1.1. Capacity for solving problemsCapacity for solving problems2.2. Fashioning products in context-Fashioning products in context-
rich settingsrich settings
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 3
MI TheoryMI TheoryIntelligence theory (about how Intelligence theory (about how
we we areare ‘smart’) ‘smart’)
not –not –learning theory (about how we learning theory (about how we getget ‘smart’) ‘smart’)
The multiple intelligences are…The multiple intelligences are…
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 4
8 Intelligences (+ 1):8 Intelligences (+ 1):1.1. Linguistic (words)Linguistic (words)2.2. Logical-Mathematical (numbers, Logical-Mathematical (numbers,
logic)logic)3.3. Spatial (pictures, charts, 3D)Spatial (pictures, charts, 3D)4.4. Musical (music, song, sound)Musical (music, song, sound)5.5. Bodily-Kinesthetic (physical Bodily-Kinesthetic (physical
activity)activity)6.6. Interpersonal (social)Interpersonal (social)7.7. Intrapersonal (self, philosophy)Intrapersonal (self, philosophy)8.8. Naturalistic (living vs non-living)Naturalistic (living vs non-living)9.9. Existential (why are we here?)Existential (why are we here?)
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 5
Criteria for inclusion Criteria for inclusion (as MI):(as MI):
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 6
Criteria for inclusion:Criteria for inclusion:1. Ability to isolate (brain damage; savants;
prodigies; testing; experimentation)
2. Definable set of “end-state” performances; operations (‘works’, events, rituals, etc.)
3. Susceptible to encoding (supported by symbol system – which intelligence is Braille?)
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 7
Key points:Key points:
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 8
Key points:Key points: Everyone has all of themEveryone has all of them We have favoritesWe have favorites Most can develop the restMost can develop the rest They often work togetherThey often work together Many ways to be intelligent Many ways to be intelligent
within each categorywithin each category
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 9
How can we use How can we use this?this?
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 10
How can we use this?How can we use this? If students aren't “getting it”, If students aren't “getting it”,
we may try a different approach we may try a different approach (rather than pronouncing the (rather than pronouncing the student ‘not smart enough’)student ‘not smart enough’)
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 11
How can we use this?How can we use this?• If students aren't “getting it”, If students aren't “getting it”,
we may try a different we may try a different approach approach
• A means to a fresh approach to A means to a fresh approach to the same old stuffthe same old stuff
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 12
How can we use this?How can we use this? If students aren't “getting it”, If students aren't “getting it”,
we may try a different approachwe may try a different approach A means to a fresh approach to A means to a fresh approach to
the same old stuffthe same old stuff Opens possibility for other ways Opens possibility for other ways
for students to demonstrate for students to demonstrate mastery (legitimacy of different mastery (legitimacy of different approaches)approaches)
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 13
Anatomy of a LessonAnatomy of a Lesson
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 14
Anatomy of a LessonAnatomy of a Lesson
AttentionAttention
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 15
Anatomy of a LessonAnatomy of a Lesson
AttentionAttentionActivityActivity
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 16
Anatomy of a LessonAnatomy of a Lesson
AttentionAttentionActivityActivityAssessmentAssessment
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 17
Assertions:Assertions:
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 18
Assertions:Assertions:1. All learners can learn to some extent
with each (or almost any) approach.
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 19
Assertions:Assertions:1. All learners can learn to some extent
with each (or almost any) approach.2. It is not possible to fully "understand"
something (depth) without involving more than one "intelligence".
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 20
Assertions:Assertions:1. All learners can learn to some extent
with each (or almost any) approach.2. It is not possible to fully "understand"
something (depth) without involving more than one "intelligence".
3. Thorough assessment (of understanding) is not possible if it is based on a single intelligence.
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 21
Assertions:Assertions:1. All learners can learn to some extent
with each (or almost any) approach.2. It is not possible to fully "understand"
something (depth) without involving more than one "intelligence".
3. Thorough assessment (of understanding) is not possible if it is based on a single intelligence.
4. Most lessons are not "pure" in that they already address more than one intelligence.
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 22
Assertions:Assertions:1. All learners can learn to some extent with each
(or almost any) approach.2. It is not possible to fully "understand"
something (depth) without involving more than one "intelligence".
3. Thorough assessment (of understanding) is not possible if it is based on a single intelligence.
4. Most lessons are not "pure" in that they already address more than one intelligence.
5. Many aspects of a lesson are also not pure : attention-getting can help learning; activities can gain attention or be used to assess; people can learn from assessments.
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 23
Concept for this Concept for this Lesson: Lesson:
Number SystemsNumber Systems
defined:defined: Common elements of number Common elements of number
bases like decimal, binary, bases like decimal, binary, octal, and hexadecimaloctal, and hexadecimal
A way of symbolizing quantityA way of symbolizing quantity
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 24
Concept: Number SystemsConcept: Number Systems
Why learn Why learn this?this?
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 25
Concept: Number SystemsConcept: Number Systems Why learn this?Why learn this?
fundamental data form in CS is binary fundamental data form in CS is binary strings; everything else built on thisstrings; everything else built on this
helps to understand many other helps to understand many other concepts related to numbersconcepts related to numbers
number systems are higher-level number systems are higher-level concept from binary or octal == if you concept from binary or octal == if you get this, then binary, octal, hex, ... get this, then binary, octal, hex, ... followsfollows
an example of abstraction / symbolisman example of abstraction / symbolism ‘‘cause we said so…cause we said so…
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 26
Concept: Number SystemsConcept: Number Systems
Target audience: Beginning Target audience: Beginning CSCS
How will understanding be How will understanding be demonstrated?demonstrated?
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 27
Concept: Number SystemsConcept: Number Systems Understanding Demonstrated By:Understanding Demonstrated By:
ability to convert numbers between ability to convert numbers between arbitrary bases [to & from base 10]arbitrary bases [to & from base 10]
be able to explain an arbitrary base be able to explain an arbitrary base (such as base 5 or base 13) without (such as base 5 or base 13) without having been shown that basehaving been shown that base
show / tell / demonstrate conversion of show / tell / demonstrate conversion of specific numbers from base X to base specific numbers from base X to base YY
be able to count in an arbitrary basebe able to count in an arbitrary base be able to perform simple arithmetic in be able to perform simple arithmetic in
an arbitrary basean arbitrary base
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 28
Getting Attention:Getting Attention:Openers... Openers...
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 29
Getting Attention:Getting Attention:Openers... (hooks)Openers... (hooks) LinguisticLinguistic "Aliens have landed and are "Aliens have landed and are
starting to ask questions. They want to starting to ask questions. They want to know about this know about this METRICMETRIC thing." thing."
Logical-MathematicalLogical-Mathematical "Why do we "Why do we count using base 10?" count using base 10?"
Logical-Mathematical,Logical-Mathematical, InterpersonalInterpersonal "What do you suppose would be different "What do you suppose would be different in the world if we only had 8 fingers?" in the world if we only had 8 fingers?"
Spatial Spatial "By the time we are done today, "By the time we are done today, you'll know how to count to 1000 on your you'll know how to count to 1000 on your fingers."fingers."
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 30
Getting Attention:Getting Attention: MusicalMusical Play Tom Lehrer's "New Math"Play Tom Lehrer's "New Math" IntrapersonalIntrapersonal Explain to class why learning Explain to class why learning
about number systems is useful.about number systems is useful. Bodily-KinestheticBodily-Kinesthetic Get the class to fold a Get the class to fold a
piece of paper in half, then in half again, piece of paper in half, then in half again, then in half again,... till they can't any more.then in half again,... till they can't any more.
NaturalisticNaturalistic Explain the "6 Degrees of Explain the "6 Degrees of Separation" Theory.Separation" Theory.
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 31
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 32
Explain general form of number Explain general form of number systems (# symbols, powers of X, systems (# symbols, powers of X,
how to count)how to count) Linguistic,Linguistic, Spatial, Spatial, Logical-Mathematical Logical-Mathematical Do base 10, then base 8, then base 2, then base 16General Rules:
x0 = 1; x1 = x; x2 = x * x; x-1 = 1/x; x-2 = 1/ (x*x); leading zeros are not significant, and unless they appear to
the right of a decimal place have no effect on the value of the number
when adding and subtracting the decimal points of real numbers must be vertically aligned
when dividing two real numbers they must both be adjusted (multiplied by their base) until the divisor is an integer
for real number addition and subtraction the exponents must be the same
for real number multiplication one must multiply the mantissas and add the exponents
for real number division one must divide the mantissas and subtract the exponents
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 33
Explain how numbers are Explain how numbers are builtbuilt
Logical-Mathematical,Logical-Mathematical, LinguisticLinguistic represented by 10 distinct symbols: 0,1,2,3,4,5,6,7,8,9 based on powers of 10 each place to the left of a digit in a string increases by
a power of 10; each place to the right of a digit in a string decreases by a power
of 10
Example: 4769210 in expanded notation looks like: = 4 * 104 + 7 * 103 + 6 * 102 + 9 * 101 + 2 * 100
= 4 * 10000 + 7 * 1000 + 6 * 100 * 9 * 10 + 2 * 1
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 34
The Odometer Analogy-1The Odometer Analogy-1 SpatialSpatial Bodily-Bodily-
KinesthetKinestheticic
012345
678901
345678
678901
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 35
The Odometer Analogy-2The Odometer Analogy-2 SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
678901
345678
789012
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 36
The Odometer Analogy-3The Odometer Analogy-3 SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
345678
789012
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 37
The Odometer Analogy-4The Odometer Analogy-4 SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
456789
789012
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 38
The Odometer Analogy-5The Odometer Analogy-5 SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
456789
890123
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 39
The Odometer Analogy-6The Odometer Analogy-6 SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
456789
901234
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 40
The Odometer Analogy-7The Odometer Analogy-7 SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
456789
012345
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 41
The Odometer Analogy-8The Odometer Analogy-8 SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
456789
123456
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 42
The Odometer Analogy-9The Odometer Analogy-9 SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
456789
234567
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 43
The Odometer Analogy-The Odometer Analogy-1010
SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
456789
345678
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 44
The Odometer Analogy-The Odometer Analogy-1111
SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
456789
456789
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 45
The Odometer Analogy-The Odometer Analogy-1212
SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
456789
567890
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 46
The Odometer Analogy-The Odometer Analogy-1313
SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
789012
456789
678901
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 47
The Odometer Analogy-The Odometer Analogy-1414
SpatialSpatial Bodily-Bodily-
KinesthetiKinestheticc
012345
890123
456789
789012
1000's 100's 10's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 48
The Base 8 OdometerThe Base 8 OdometerSame
deal – smaller wheel
12345
70123
56701
67012
512's 64's 8's 1's
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 49
0 1101 1112 112.. ...9 19910 20011 20112 202.. ...99 299100 300101 301102 302... ...109 999
1000
Look at how we Look at how we count count
(then do the (then do the same in other same in other
bases).bases). Logical-Logical-
MathematicalMathematical LinguisticLinguistic Spatial (patterns)Spatial (patterns)
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 50
Look at how Look at how we count in we count in
different different bases.bases.
Logical-Logical-MathematicalMathematical
LinguisticLinguistic Spatial Spatial
(patterns)(patterns)
00 00000000 0000 00
11 00010001 0101 11
22 00100010 0202 22
33 00110011 0303 33
44 01000100 0404 44
55 01010101 0505 55
66 01100110 0606 66
77 01110111 0707 77
88 10001000 1010 88
99 10011001 1111 99
1010 10101010 1212 AA
1111 10111011 1313 BB
1212 11001100 1414 CC
1313 11011101 1515 DD
1414 11101110 1616 EE
1515 11111111 1717 FF
1616 1000010000 2020 1010
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 51
Show how to convert Show how to convert numbers from some base to numbers from some base to
base 10.base 10. Logical-MathematicalLogical-MathematicalExample: 10111001Example: 1011100122 in in expanded notationexpanded notation
looks like:looks like:= 1 * 2= 1 * 277 + 0 * 2 + 0 * 266 + 1 * 2 + 1 * 255 + 1 * 2 + 1 * 244 + 1 * 2 + 1 * 233 + 0 * + 0 *
2222 + 0 * 2 + 0 * 211 + 1 * 2 + 1 * 200 = 1 * 128 + 0 * 64 + 1 * 32 + 1 * 16 + 1 * 8 + 0 = 1 * 128 + 0 * 64 + 1 * 32 + 1 * 16 + 1 * 8 + 0
* 4 + 0 * 2 + 1 * 1* 4 + 0 * 2 + 1 * 1 = 128 + 32 + 16 + 8 + 1= 128 + 32 + 16 + 8 + 1 = 185 = 185
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 52
Show Show how to how to convert convert numbers numbers
from base from base 10 to 10 to
others.others.
Division Quotient Remainder Binary Number
2671 / 2 1335 1 1
1335 / 2 667 1 11
667 / 2 333 1 111
333 / 2 166 1 1111
166 / 2 83 0 0 1111
83 / 2 41 1 10 1111
41 / 2 20 1 110 1111
20 / 2 10 0 0110 1111
10 / 2 5 0 0 0110 1111
5 / 2 2 1 10 0110 1111
2 / 2 1 0 010 0110 1111
1 / 2 0 1 1010 0110 1111
Logical-Logical-MathematicaMathematicall
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 53
Relate octal numbers to Relate octal numbers to the musical scale.the musical scale.
MusicalMusical Spatial (patterns)Spatial (patterns)
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 54
Show how to count in Show how to count in binary on your fingers. binary on your fingers. [Beware of ‘4’!][Beware of ‘4’!] Bodily-KinestheticBodily-Kinesthetic SpatialSpatial
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 55
Use an AbacusUse an Abacus
Bodily-KinestheticBodily-Kinesthetic SpatialSpatial Intrapersonal (leave them to play with it)Intrapersonal (leave them to play with it)
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 56
Act It OutAct It Out(each person gets a wheel, list, or flip-(each person gets a wheel, list, or flip-
book of numbers; have them count; when book of numbers; have them count; when one gets to '9' they get to poke the next one gets to '9' they get to poke the next
guy).guy).
Bodily-KinestheticBodily-Kinesthetic InterpersonalInterpersonal 99999922
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 57
Multiplying like Multiplying like BunniesBunnies
(Relate to generations of (Relate to generations of bunnies, each having 'N' bunnies, each having 'N' babies. 'N' can be 2, 8, babies. 'N' can be 2, 8,
10).10).
Naturalistic Naturalistic Spatial Spatial
(patterns)(patterns)
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 58
Assessment: Assessment: MusicalMusical
Propose a numerical code Propose a numerical code (octal mapping) for musical (octal mapping) for musical notes. Encode a simple song - notes. Encode a simple song - try reading it using the try reading it using the numerical code.numerical code.
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 59
Assessment:Assessment: Logical-MathematicalLogical-Mathematical, , LinguisticLinguistic
Explain base 'X' [using symbols, Explain base 'X' [using symbols, powers]powers]
Explain base '5', or '13'Explain base '5', or '13' worksheets: fill in the blanks...worksheets: fill in the blanks...
Base 10Base 10 Base 2Base 2 Base 8Base 8 Base 16Base 163232
1101011010FFFF
647647
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 60
Assessment:Assessment: Logical-MathematicalLogical-Mathematical
What's the next number in base What's the next number in base 'X'?'X'?
Simple Additions in various basesSimple Additions in various bases
NaturalisticNaturalistic Find examples in nature (asexual Find examples in nature (asexual
reproduction; propagation)reproduction; propagation)
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 61
Assessment:Assessment: Bodily-Kinesthetic, Bodily-Kinesthetic, Spatial, Spatial,
InterpersonalInterpersonal
Show me Show me nn in binary using your in binary using your hands.hands.
Get people to be "bits" - standing = Get people to be "bits" - standing = 1; sitting = 0 - do counting or 1; sitting = 0 - do counting or arithmetic using peoplearithmetic using people
University ofUniversity of CalgaryCalgary © 2003 K.Becker MI Number Systems 21-Sep-03 62
Thanks!Thanks!