A Multiple-Intelligences Approach to Teaching Number Systems
Katrin Becker
Computer Science Education Group
Department of Computer Science
University of Calgary
2400 University Drive NW
Calgary, Alberta Canada
T2N 1N4
(403)220-5769
email: becker@cpsc.ucalgary.ca

Abstract
Gardener's Theory of Multiple Intelligences [Gardner 1997] has implications for
how we can develop lessons to cover various concepts and topics. The concept of number
systems is presented, framed within the context of multiple intelligences, and various
approaches to teaching this lesson are described. The underlying implication is that
students are likely to learn "better" when content is presented in more than one format.
The Theory of Multiple Intelligences provides us with a convenient framework within
which to create these various approaches.
1.0 Introduction
Knowledge of machine representation of data is fundamental to CS. Since all
information represented by computer is ultimately represented in binary, an
understanding of binary number systems is crucial, and almost all CS programs include
this topic somewhere in their elementary curriculum. Often this topic is treated in a rather
matter-of-fact manner. Instructors view the material as elementary, and many students
view the material as a tedious, necessary evil. It is possible to use the Theory of Multiple
Intelligences first developed by Gardner [Gardner 1997] as a basis for inspiration that
will allow instructors to vary the approach used, and perhaps even make it interesting.
Current research in education [Taylor, Gilmer, and Tobin, Eds., 2002][Bransford,
2000] suggests that students are better served in the long run when instructors teach with
the goal of understanding rather than simple retention. Understanding is more difficult to
achieve than remembering. [Wiggins & McTighe 1998] If the primary goal is that learners
understand number systems, then we must determine how this understanding can be
demonstrated. In this case, understanding will be demonstrated as follows. Having
mastered the number systems lesson, learners will be able to:
• convert numbers between arbitrary bases [to & from base 10].
• explain an arbitrary base (such as base 5 or 13) without first having been shown
that base.
• show / tell / demonstrate conversion of specific numbers from base X to base Y.
• count in an arbitrary base.
• perform simple arithmetic in an arbitrary base.

2.0 Multiple Intelligences & Lesson Design
The Theory of Multiple Intelligences was first proposed by Gardner [Gardner
1997] . As a theory of mind, it claims that intelligence can take multiple forms and that
most people excel in just a few, rather than equally in all. Gardner describes eight
"intelligences": Linguistic (which deals with words & language); Logical-Mathematical
(having to do with numbers, logic); Spatial (visual aspects, such as pictures, charts, and
3D); Musical (music, song, sound); Bodily-Kinesthetic (physical activity); Interpersonal
(social awareness & interaction); Intrapersonal (self, introspective, philosophy); and most
recently described: Naturalistic (dealing with nature; life sciences).
By acknowledging this theory, we open the possibility for development of various
forms of approach to a lesson, which at the very least provides for variety. Given that we
accept that this theory has at least some merit, a few assertions can be postulated:
1. 1. All learners can learn to some extent with each (or almost any) approach. How
effectively or efficiently they are able to learn using an approach that does not match
their strength remains to be shown.
2. 2. It is not possible to fully "understand" something (depth) without involving more
than one "intelligence". For example, being able to recite some explanation does not
prove mastery. If on the other hand, the learner can demonstrate that their
understanding can be transferred from one form of expression to another, it would
seem reasonable to assume that they have achieved at least some level of mastery.
3. 3. Thorough assessment (of understanding) is not possible if it is based on a single
intelligence. If information can only be repeated in the same context as it was
presented, the information may be inert, and not transferable [Bransford, 1989]. Such
information is not particularly useful.
4. 4. Most lessons are not "pure" in that they already address more than one
intelligence. Many lessons are presented verbally or augmented with verbal or written
explanation, even if the kernel of the lesson is visual, or mathematical.
The implications this has for instructional design is that if we present material that
has been explicitly framed within the context of this theory, we are likely to reach more
learners than we would if we simply used well-known, standard approaches. With a bit of
effort, most topics can indeed be framed within the context of this theory.
If we further divide a typical "lesson" into three main components, or aspects, we
can use this as a template for new approaches that might not otherwise have presented
themselves. The three aspects of a typical lesson are: Attention, where the instructor
attempts to engage the class and garner interest; the Activity itself, which is at the heart
of the lesson; and Assessment, how we determine if the lesson has been learned. These
components serve as a convenient template, but it must be remembered that 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.
Current thinking in Education would dictate that the goals of a lesson and its
assessment are intimately linked [Bransford, 2000]. That is, a clear specification of the

objectives of a particular teaching exercise would also suggest particular assessment
measures. Given that is the case, then, we must answer the question, "Why teach people
about Number Systems?"
3.0 Why Teach Number Systems
The ACM/IEEE Computing Curricula 2001 [ACM CC2001] lists low-level
representation as being core to a CS curriculum. If we acknowledge the validity of this
curriculum document, then this in itself is reason enough. A further rationale is possible.
ACM/IEEE CC2001 Topics
AR2. Machine level representation of data [core hours: 3]
• •
Bits, bytes, and words
• •
Numeric data representation and number bases
• •
Fixed- and floating-point systems
• •
Signed and twos-complement representations
• •
Representation of nonnumeric data (character codes, graphical data)
• •
Representation of records and arrays
Learning objectives:
• •
Explain the reasons for using different formats to represent numerical data.
• •
Explain how negative integers are stored in sign-magnitude and twos-complement representation.
• •
Convert numerical data from one format to another.
• •
Discuss how fixed-length number representations affect accuracy and precision.
• •
Describe the internal representation of nonnumeric data.
• • Describe the internal representation of characters, strings, records, and arrays.

The rationale behind teaching this unit is fairly clear. The fundamental data form
in CS is binary strings and everything else is built on this. An understanding of binary
representation helps to understand many other concepts related to numbers. Also, number
systems are a higher-level concept from binary or octal. It would stand to reason that if
students can be made to understand the higher-level concept, learning any specific
number system (like binary, octal, or hex) is simplified. Finally, a deep understanding of
abstraction and symbolism is at the very heart of CS and algorithm design. Familiarity of
number systems and how they relate to each other as well as how they can be used to
represent other things is closely tied to this.
4.0 Openers – Getting Attention
The "opener" is typically the beginning of a lesson, lecture, or unit that is
designed to capture the learner's attention and inspire them to pay attention to the rest.
The "opener" often consists of the instructor stating the topic ("Today we're going to talk
about ..."). The Theory of Multiple Intelligences gives us a basis from which to vary this
approach.
A linguistic approach to an opener might involve a story, for instance: "Aliens
have landed and are starting to ask questions. They want to know about this METRIC
thing." The idea here is that the rest of the lesson can then be framed within the context of
attempting to explain our counting system to aliens who are completely unfamiliar with
it. Sometimes an opener can be as simple as a question like: "Why do we count using base

10?" This would count as a logical-mathematical approach. This question could be used
to begin a discussion around the abstract nature of our current counting system [Bucci
2001]. There is, in fact no natural basis (aside from the number of fingers we possess) for
using base 10.
Some questions can be posed as queries to ponder: "What do you suppose would
be different in the world if we only had 8 fingers?" (Logical-Mathematical,
Interpersonal/Social). An opener that takes advantage of spatial intelligence would be "By
the time we are done today, you'll know how to count to 1000 on your fingers." [Selvia,
1995]
If the facilities permit it, a musical approach could include playing the song "New
Math" by Tom Lehrer [Lehrer 1965]. This song, written by a Mathematician runs through
a subtraction problem first in base 10, and then in base 8.
For some learners, an intrapersonal (somewhat philosophical) approach works
best. This would involve explaining to class why learning about number systems is
useful. This approach is one of the ones that is almost always helpful, at least to some
extent.
While dancing around the lecture theatre is an uncommon activity in a CS class,
there are still a number of physical activities that can be acted out to address the needs of
those learners who have a strong bent towards learning through Bodily-Kinesthetic
Intelligence. Get the class to fold a piece of paper in half, then in half again, then in half
again,... until they can't any more. This provides a means of introducing the concept of
binary numbers.
Perhaps one of the most difficult facets to address in a CS course is the
naturalistic one. Often when struggling for a natural analogy of some concept, we end up
in mathematics, which may seem very natural for experts, but is often highly un-natural
to novices. A common example of this would be the use of Fibonacci numbers to
demonstrate recursion. It has been shown that connecting a new concept to something the
learner is already familiar with helps them to construct new meaning and make new
connections [Mintzes 1997]. One of the problems with the Fibonacci numbers example of
recursion is that we often end up having to introduce a new concept in order to introduce
a new concept. There may be a certain poetic irony in this approach, but it rarely helps
the students understand. Perhaps a more appropriate opener that engages the naturalistic
intelligence might be to explain the "6 Degrees of Separation" Theory. [Guare, 1990]
5.0 The Core of the "Lesson"
Many instructors teach number systems using primarily mathematical approaches.
While there is nothing wrong with this, the author is suggesting alternate approaches that
can be used to augment, rather than replace them. In fact, the mathematical approach is
ultimately how students must be able to work with various number bases and
representations, and as such can be considered foundational. However, it may not be the
most "user-friendly" approach to take. Many novice students are still somewhat fearful of

mathematics and introducing this topic from a purely mathematical perspective may
leave many students lost precisely when they should be encouraged and inspired.
The following brief sections explain various approaches to teaching about number
systems, some well known and quite traditional, others novel. Each is identified as to
which "intelligences" it engages.
5.1 Explain how numbers are built [Engages: Logical-Mathematical, Linguistic]
One approach is to start by explaining base 10, as that is the one base we can
assume everyone is already familiar with. This typically feels rather awkward, both for
the instructor and the class as the decimal system is so familiar that we have long since
stopped examining it in this way. This is where the attention-getting technique of trying
to explain our number system to aliens can help us frame this exercise. Describing the
decimal number system allows us to establish the terminology and approach we are
using. Having done that, we can then use an identical approach to explain base 8, then
base 2, and then base 16. Approaching the explanation in this order gives us the ability to
isolate the "variables" and point out the pattern. The initial description of decimal
numbers draws on principles and concepts learned in primary school. Decimal numbers
are:
• • 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
Once the idea of describing numbers in this way has been introduced, we can
expand on this by generalizing the rules. The general rules that follow have been
described in a way that allows them to be applied to any arbitrary number base.
General 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.

0
1
2
3
4
5

3
4
5
6
7
8

6
7
8
9
0
1

6
7
8
9
0
1

1000's

100's

10's

1's

Odometer

This approach integrates three forms of
intelligence, namely: linguistic, logical, and spatial.
Some people are weaker in one or more of these
areas, and as a result some students have difficulty
following this explanation because of the
integration required. Augmenting this approach
with others that engage other forms of intelligence
are likely to facilitate retention and transfer.
Someone who is very visual for example, may feel
more comfortable with the next approach. Having
acquired a "sense" of the concept, they will then be
able to return to the more formal approach and try it
again.

5.2 The Odometer Analogy [Engages: Spatial; Bodily-Kinesthetic]
Another approach that draws on spatial abilities is to describe and show how
number systems and counting can be represented using an odometer. [Heller, S., 1997]
Again, it is recommended that the explanation begins with base 10 as this is the form of
odometer with which most students will have prior experience. A different base can be
represented by simply changing the "size" of the wheels in the odometer. If we make the
wheel a bit smaller, we only have room for 8 numbers, and we end up with octal. The
mechanism remains unchanged. Binary can also be done, using a very small "wheel". As
a diversion from the usual exercises, students can actually build a paper version of an
odometer and use it to explore counting. One great advantage of this approach is that it
allows students to see first hand how two's compliment arithmetic operates. We can count
both forwards and backwards. It also allows us to look at representational limits, overand underflow, and of we take it far enough we can also use it to begin to explain some of
the problems encountered when working with floating point
numbers. Although they are hard to find these days, old
0
110
mechanical adding machines operate in much the same fashion
1
111
and make excellent demonstration tools.

2
..
9
10
11
12
..
99
100
101
102
...
109

112
...
199
200
201
202
...
299
300
301
302
...
999
1000

How We Count

5.3 Look at how we count (then do the same in other bases).
[Engages: Logical-Mathematical; Linguistic; Spatial
(patterns)]
An alternative to the odometer approach examines
"counting". [Petzold, C., 1999] We look at how numbers grow in
size and magnitude and then apply the same technique using
other bases. Once we have demonstrated several other bases, it
becomes possible to identify similar elements and thereby
establish the pattern used.
5.4 Show How to Convert Numbers from Some Base to Base
10. [Engages: Logical-Mathematical]

The ability to convert numbers from base 10 to some other base as well as from
some arbitrary base back to base 10 is often required when working with low-level
representations. It is also sometimes helpful when deciphering certain classes of program
errors. As in the previous
approaches, starting with decimal
Division Quotient
Remainder
Binary Number
sets the "scene".
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

Example: 101110012 in expanded
notation looks like:
= 1 * 27 + 0 * 26 + 1 * 25 + 1 * 24
+ 1 * 23 + 0 * 22 + 0 * 21 + 1 * 20
= 1 * 128 + 0 * 64 + 1 * 32 + 1 *
16 + 1 * 8 + 0 * 4 + 0 * 2 + 1 * 1
= 128 + 32 + 16 + 8 + 1
= 185
5.5 Show how to convert
numbers from base 10 to others.
[Engages: Logical-Mathematical]

Converting from some
arbitrary base is often viewed by
novices as being simpler than the
2 / 2
1
0
010 0110 1111
reverse (converting to some
1 / 2
0
1
1010 0110 1111
arbitrary base), perhaps because the
former involves multiplication and
Converting to Base 10
the latter division. As a result these
two conversion techniques can be
tackled separately, and it is recommended that they be tackled in the order listed.
5 / 2

2

1

10 0110 1111

5.6 Relate octal numbers to the musical scale.
[Engages: Musical; Spatial (patterns)]
This approach is as yet untried in a classroom
environment, although the idea has been presented to
various students and other teachers with encouraging
reviews. An understanding of octal number systems
can be constructed by mapping octal numbers onto
the musical scale, and using them to encode various
A Musical Approach
tunes. This exercise would lend itself best to
implementation in a small class or lab environment.
It could even be adapted as an on-line, interactive exercise.

5.7 Show How to Count in Binary on Your Fingers. [Engages: Bodily-Kinesthetic;
Spatial]
One of the un-acknowledged strengths of
this approach is the kinesthetic nature of
this activity. The tactile nature of this
B exercise helps some students internalize the
Finger Counting
patterns involved in binary counting. They
Image Credit: [ivan]
can recall the physical movements, which
in turn can trigger their logical recollection
of the concept. Also, it's fun (watch out for the number four).
5.8 Use an Abacus [Engages: BodilyKinesthetic; Spatial; Intrapersonal
(leave them to play with it)]
This approach has definite
constructive overtones as the exercise
will typically involve the learners trying
to solve various problems using the
abacus with the hope that one side-effect
Abacus
of this activity will be a sort of 'epiphany'
involving the relationships of numbers,
counting, and arithmetic, that can be applied to various number systems. If resources
permit, devices can be built that behave just like an abacus, but that employ various
number bases – which allows for comparison.
5.9 Act It Out [Engages: Bodily-Kinesthetic;
Interpersonal]
For this activity, each person gets a wheel with
numbers on it, a list, or a flip-book of numbers. The 'action'
is that we have the participants count out loud, and when one
gets to '9' (s)he gets to poke the next guy beside him(her).
The original 'counter' reverts to '0' (zero), and the next guy
(the one who was poked) remembers the next number. The
key element here is that each participant remembers only
Act It Out
their own 'behaviours'. The audience gets to see this
"counter" in action so the entire class can be involved. In a
university or college environment such "performance art" is rare in the sciences, but often
leaves a lasting impact.

2999

5.10 Multiplying like Bunnies [Engages: Naturalistic;
Spatial (patterns)]
As stated previously, opportunities for a naturalistic approach
are sometimes difficult to create in CS. When dealing with
number systems, we can relate a number system to generations

Base 10

Base 2

Base 8

Base 16

32
11010
FF
647

A Worksheet

of bunnies, each having 'N'
babies. In this way 'N' can
be 2, 8, 10, or even 16.
6.0 Assessment

No teaching unit can
rightfully be considered complete if it lacks a means of assessment. The success or
viability of any given approach cannot be determined if there is no way to assess the
learner's mastery of the subject matter. [Resnick & Resnick 1992] In this aspect as well as
the others, it is possible to focus on various intelligences. This gives us a means of
determining whether the material the students were to master can be transferred from one
arena to another.
A musical assessment might be to propose a numerical code (octal mapping) for
musical notes. Encode a simple song - try reading it using the numerical code. Requiring
students to explain some given base 'X' (such as 5 or 13) using symbols and/or powers is
another means of determining mastery. An instrument that can be used both for practice
and assessment is the well-known "worksheet", where students can fill in the blanks. This
makes use of the Logical-Mathematical, as well as the Linguistic aspects of intelligence.
Variations would include questions such as, "What's the next number in base 'X'?" or
simple addition exercises in various bases. A Naturalistic approach might include a
requirement to find examples in nature (such as asexual reproduction or propagation).
Bodily-Kinesthetic, Spatial, Interpersonal intelligences can be combined with each other
and/or verbal skills with tasks such as, "Show me n in binary using your hands". The idea
of performance art can also be used to effect here by getting individuals to be "bits":
standing = 1; sitting = 0. An entire class can be made to do counting or arithmetic using
people. While a more light-hearted approach may become tiresome if used to excess, its
occasional implementation can be very effective.
7.0 Conclusion
The Theory of Multiple Intelligences is still classified as a theory, but this need
not prevent us from using it a tool to help us create a rich and varied learning
environment. The fact that the learners we must reach are technically adults should also
not prevent us from trying a more light-hearted approach from time to time. When used
to augment sound, well-tested approaches the result can be that some students who were
traditionally lost by a primarily linguistic or mathematical approach may be able to use
these other approaches to form the necessary connections and build the required
understandings.
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