multiple intelligences approach to number systems

Multiple Multiple Intelligences Intelligences Approach to Approach to

Teaching Number Teaching Number SystemsSystems

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

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…

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?)

Criteria for inclusion Criteria for inclusion (as MI):(as MI):

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?)

Key points:Key points:

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

How can we use How can we use this?this?

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’)

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

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)

Anatomy of a LessonAnatomy of a Lesson

AttentionAttention

AttentionAttentionActivityActivity

AttentionAttentionActivityActivityAssessmentAssessment

Assertions:Assertions:

Assertions:Assertions:1. All learners can learn to some extent

with each (or almost any) approach.

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.

Assertions:Assertions:1. All learners can learn to some extent with each

(or almost any) approach.2. It is not possible to fully "understand"

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.

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

Concept: Number SystemsConcept: Number Systems

Why learn Why learn this?this?

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…

Concept: Number SystemsConcept: Number Systems

Target audience: Beginning Target audience: Beginning CSCS

How will understanding be How will understanding be demonstrated?demonstrated?

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

Getting Attention:Getting Attention:Openers... Openers...

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."

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.

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

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

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

The Odometer Analogy-1The Odometer Analogy-1 SpatialSpatial Bodily-Bodily-

KinesthetKinestheticic

012345

678901

345678

678901

1000's 100's 10's 1's

KinesthetiKinestheticc

012345

678901

345678

789012

1000's 100's 10's 1's

012345

789012

345678

789012

1000's 100's 10's 1's

012345

789012

456789

789012

1000's 100's 10's 1's

012345

789012

456789

890123

1000's 100's 10's 1's

012345

789012

456789

901234

1000's 100's 10's 1's

012345

789012

456789

012345

1000's 100's 10's 1's

012345

789012

456789

123456

1000's 100's 10's 1's

012345

789012

456789

234567

1000's 100's 10's 1's

The Odometer Analogy-The Odometer Analogy-1010

SpatialSpatial Bodily-Bodily-

012345

789012

456789

345678

1000's 100's 10's 1's

012345

789012

456789

1000's 100's 10's 1's

012345

789012

456789

567890

1000's 100's 10's 1's

012345

789012

456789

678901

1000's 100's 10's 1's

012345

890123

456789

789012

1000's 100's 10's 1's

The Base 8 OdometerThe Base 8 OdometerSame

deal – smaller wheel

512's 64's 8's 1's

0 1101 1112 112.. ...9 19910 20011 20112 202.. ...99 299100 300101 301102 302... ...109 999

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)

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

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

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

Relate octal numbers to Relate octal numbers to the musical scale.the musical scale.

MusicalMusical Spatial (patterns)Spatial (patterns)

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

Use an AbacusUse an Abacus

Bodily-KinestheticBodily-Kinesthetic SpatialSpatial Intrapersonal (leave them to play with it)Intrapersonal (leave them to play with it)

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

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)

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.

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

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)

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

Thanks!Thanks!

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