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Multiple intelligences approach to  Number Systems
 

Multiple intelligences approach to Number Systems

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An example of how number systems can be taught while incorporating Gardner's theory of multiple intelligences

An example of how number systems can be taught while incorporating Gardner's theory of multiple intelligences

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    Multiple intelligences approach to  Number Systems Multiple intelligences approach to Number Systems Presentation Transcript

    • Multiple Intelligences Approach to Teaching Number Systems
    • MI Theory:
      • First described by Howard Gardner (1983)
      • Intelligence has to do with:
        • Capacity for solving problems
        • Fashioning products in context-rich settings
    • MI Theory
      • Intelligence theory (about how we are ‘smart’)
      • not –
      • learning theory (about how we get ‘smart’)
      • The multiple intelligences are…
    • 8 Intelligences (+ 1):
        • Linguistic (words)
        • Logical-Mathematical (numbers, logic)
        • Spatial (pictures, charts, 3D)
        • Musical (music, song, sound)
        • Bodily-Kinesthetic (physical activity)
        • Interpersonal (social)
        • Intrapersonal (self, philosophy)
        • Naturalistic (living vs non-living)
        • Existential (why are we here?)
    • Criteria for inclusion (as MI):
    • Criteria for inclusion:
        • Ability to isolate (brain damage; savants; prodigies; testing; experimentation)
        • Definable set of “end-state” performances; operations (‘works’, events, rituals, etc.)
        • Susceptible to encoding (supported by symbol system – which intelligence is Braille?)
    • Key points:
    • Key points:
      • Everyone has all of them
      • We have favorites
      • Most can develop the rest
      • They often work together
      • Many ways to be intelligent within each category
    • How can we use this?
    • How can we use this?
      • If students aren't “getting it”, we may try a different approach (rather than pronouncing the student ‘not smart enough’)
    • How can we use this?
      • If students aren't “getting it”, we may try a different approach
      • A means to a fresh approach to the same old stuff
    • How can we use this?
      • If students aren't “getting it”, we may try a different approach
      • A means to a fresh approach to the same old stuff
      • Opens possibility for other ways for students to demonstrate mastery (legitimacy of different approaches)
    • Anatomy of a Lesson
    • Anatomy of a Lesson
      • Attention
    • Anatomy of a Lesson
      • Attention
      • Activity
    • Anatomy of a Lesson
      • Attention
      • Activity
      • Assessment
    • Assertions:
    • Assertions:
      • All learners can learn to some extent with each (or almost any) approach.
    • Assertions:
      • All learners can learn to some extent with each (or almost any) approach.
      • It is not possible to fully "understand" something (depth) without involving more than one "intelligence".
    • Assertions:
      • All learners can learn to some extent with each (or almost any) approach.
      • It is not possible to fully "understand" something (depth) without involving more than one "intelligence".
      • Thorough assessment (of understanding) is not possible if it is based on a single intelligence.
    • Assertions:
      • All learners can learn to some extent with each (or almost any) approach.
      • It is not possible to fully "understand" something (depth) without involving more than one "intelligence".
      • Thorough assessment (of understanding) is not possible if it is based on a single intelligence.
      • Most lessons are not "pure" in that they already address more than one intelligence.
    • Assertions:
      • All learners can learn to some extent with each (or almost any) approach.
      • It is not possible to fully "understand" something (depth) without involving more than one "intelligence".
      • Thorough assessment (of understanding) is not possible if it is based on a single intelligence.
      • Most lessons are not "pure" in that they already address more than one intelligence.
      • 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 Lesson: Number Systems defined:
      • Common elements of number bases like decimal, binary, octal, and hexadecimal
      • A way of symbolizing quantity
    • Concept: Number Systems
      • Why learn this?
    • Concept: Number Systems
      • Why learn this?
        • fundamental data form in CS is binary strings; everything else built on this
        • helps to understand many other concepts related to numbers
        • number systems are higher-level concept from binary or octal == if you get this, then binary, octal, hex, ... follows
        • an example of abstraction / symbolism
        • ‘ cause we said so…
    • Concept: Number Systems
      • Target audience: Beginning CS
      • How will understanding be demonstrated?
    • Concept: Number Systems
      • Understanding Demonstrated By:
        • ability to convert numbers between arbitrary bases [to & from base 10]
        • be able to explain an arbitrary base (such as base 5 or base 13) without having been shown that base
        • show / tell / demonstrate conversion of specific numbers from base X to base Y
        • be able to count in an arbitrary base
        • be able to perform simple arithmetic in an arbitrary base
    • Getting Attention: Openers...
    • Getting Attention: Openers... (hooks)
      • Linguistic "Aliens have landed and are starting to ask questions. They want to know about this METRIC thing."
      • Logical-Mathematical "Why do we count using base 10?"
      • Logical-Mathematical, Interpersonal "What do you suppose would be different in the world if we only had 8 fingers?"
      • Spatial "By the time we are done today, you'll know how to count to 1000 on your fingers."
    • Getting Attention:
      • Musical Play Tom Lehrer's "New Math"
      • Intrapersonal Explain to class why learning about number systems is useful.
      • Bodily-Kinesthetic Get the class to fold a piece of paper in half, then in half again, then in half again,... till they can't any more.
      • Naturalistic Explain the "6 Degrees of Separation" Theory.
    • Activities:
    • Explain general form of number systems (# symbols, powers of X, how to count)
      • Linguistic, Spatial, Logical-Mathematical
      • Do base 10, then base 8, then base 2, then base 16
      • General Rules:
        • x 0 = 1;    x 1 = x;     x 2 = 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 built
      • Logical-Mathematical, Linguistic
        • 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 * 10 4 + 7 * 10 3 + 6 * 10 2 + 9 * 10 1 + 2 * 10 0
        • = 4 * 10000 + 7 * 1000 + 6 * 100 * 9 * 10 + 2 * 1
    • The Odometer Analogy-1
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 6 7 8 9 0 1 3 4 5 6 7 8 6 7 8 9 0 1 1000's 100's 10's 1's
    • The Odometer Analogy-2
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 6 7 8 9 0 1 3 4 5 6 7 8 7 8 9 0 1 2 1000's 100's 10's 1's
    • The Odometer Analogy-3
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 3 4 5 6 7 8 7 8 9 0 1 2 1000's 100's 10's 1's
    • The Odometer Analogy-4
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 4 5 6 7 8 9 7 8 9 0 1 2 1000's 100's 10's 1's
    • The Odometer Analogy-5
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 4 5 6 7 8 9 8 9 0 1 2 3 1000's 100's 10's 1's
    • The Odometer Analogy-6
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 4 5 6 7 8 9 9 0 1 2 3 4 1000's 100's 10's 1's
    • The Odometer Analogy-7
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 4 5 6 7 8 9 0 1 2 3 4 5 1000's 100's 10's 1's
    • The Odometer Analogy-8
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 4 5 6 7 8 9 1 2 3 4 5 6 1000's 100's 10's 1's
    • The Odometer Analogy-9
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 4 5 6 7 8 9 2 3 4 5 6 7 1000's 100's 10's 1's
    • The Odometer Analogy-10
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 4 5 6 7 8 9 3 4 5 6 7 8 1000's 100's 10's 1's
    • The Odometer Analogy-11
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 4 5 6 7 8 9 4 5 6 7 8 9 1000's 100's 10's 1's
    • The Odometer Analogy-12
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 4 5 6 7 8 9 5 6 7 8 9 0 1000's 100's 10's 1's
    • The Odometer Analogy-13
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 7 8 9 0 1 2 4 5 6 7 8 9 6 7 8 9 0 1 1000's 100's 10's 1's
    • The Odometer Analogy-14
      • Spatial
      • Bodily-Kinesthetic
      0 1 2 3 4 5 8 9 0 1 2 3 4 5 6 7 8 9 7 8 9 0 1 2 1000's 100's 10's 1's
    • The Base 8 Odometer
      • Same deal – smaller wheel
      512's 64's 8's 1's 1 2 3 4 5 7 0 1 2 3 5 6 7 0 1 6 7 0 1 2
    • Look at how we count (then do the same in other bases).
      • Logical-Mathematical
      • Linguistic
      • Spatial (patterns)
      1000 999 109 ... ... 302 102 301 101 300 100 299 99 ... .. 202 12 201 11 200 10 199 9 ... .. 112 2 111 1 110 0
    • Look at how we count in different bases.
      • Logical-Mathematical
      • Linguistic
      • Spatial (patterns)
      10 20 10000 16 F 17 1111 15 E 16 1110 14 D 15 1101 13 C 14 1100 12 B 13 1011 11 A 12 1010 10 9 11 1001 9 8 10 1000 8 7 07 0111 7 6 06 0110 6 5 05 0101 5 4 04 0100 4 3 03 0011 3 2 02 0010 2 1 01 0001 1 0 00 0000 0
    • Show how to convert numbers from some base to base 10.
      • Logical-Mathematical
      • Example: 10111001 2 in expanded notation looks like:
      • = 1 * 2 7 + 0 * 2 6 + 1 * 2 5 + 1 * 2 4 + 1 * 2 3 + 0 * 2 2 + 0 * 2 1 + 1 * 2 0
      • = 1 * 128 + 0 * 64 + 1 * 32 + 1 * 16 + 1 * 8 + 0 * 4 + 0 * 2 + 1 * 1
      • = 128 + 32 + 16 + 8 + 1
      • = 185
    • Show how to convert numbers from base 10 to others.
      • Logical-Mathematical
      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
    • Relate octal numbers to the musical scale.
      • Musical
      • Spatial (patterns)
    • Show how to count in binary on your fingers. [Beware of ‘4’!]
      • Bodily-Kinesthetic
      • Spatial
    • Use an Abacus
      • Bodily-Kinesthetic
      • Spatial
      • Intrapersonal (leave them to play with it)
    • Act It Out (each person gets a wheel, list, or flip-book of numbers; have them count; when one gets to '9' they get to poke the next guy).
      • Bodily-Kinesthetic
      • Interpersonal
      9 9 9 2
    • Multiplying like Bunnies (Relate to generations of bunnies, each having 'N' babies. 'N' can be 2, 8, 10).
      • Naturalistic
      • Spatial (patterns)
    • Assessment: Musical
      • Propose a numerical code (octal mapping) for musical notes. Encode a simple song - try reading it using the numerical code.
    • Assessment: Logical-Mathematical , Linguistic
      • Explain base 'X' [using symbols, powers]
      • Explain base '5', or '13'
      • worksheets: fill in the blanks...
      647 FF 11010 32 Base 16 Base 8 Base 2 Base 10
    • Assessment: Logical-Mathematical
      • What's the next number in base 'X'?
      • Simple Additions in various bases
      • Naturalistic
      • Find examples in nature (asexual reproduction; propagation)
    • Assessment: Bodily-Kinesthetic, Spatial, Interpersonal
      • Show me n in binary using your hands.
      • Get people to be "bits" - standing = 1; sitting = 0 - do counting or arithmetic using people
    • Thanks!