Using Simple Animated Presentations (SAPs) in Teaching Elementary=20
Logic


Katarzyna Paprzycka
Department of Philosophy
University of Pittsburgh




Many of the problems that students have in grasping logic
occur at the very basic levels: of seeing patterns, of applying
derivation rules. While classical chalk-and-board and word-of-
mouth methods for presenting the material must not be replaced,
it might be helpful if they were supplemented by more animated
ways of presenting the material. This paper describes some
existing and some not yet existing SAPs (simple animated
presentations) that may be helpful in teaching Introduction to
Logic courses.

The present paper is somewhat of a mishap for at least three
reasons. First, it is hard to use exclusively the word-to-eye
method to discuss animated presentations. (Some illustrations are
included in the HTML version of this paper available at
http://www.utpb.edu/scimath/paprzyck/kp/anilogic.htm). Second,
the work that the paper reports is very much still in progress.
In fact, it would be safe to say that none of the presentations
mentioned are <I>really</I> finished, and many of them still do
not exist even in outlines. Third, I have had practically no
experience in actually using the presentations in the classroom
(except for a couple). The last two points, in particular, shape
much of the purpose of the paper. Insofar as the idea of SAPs is
both worthwhile and manageable, it would work best as a
collective rather than individual enterprise (see section 4). The
paper is as much a presentation as it is a call for help.

The paper splits into four sections. First, some existing as
well as non-existing SAPs are described, making note of some of
their more important features and uses (section 1). Section 2
explains and speculates on why SAPs ought to be conceived as
short and simple rather than long and complex presentations. In
section 3, some technical limitations of using SAPs are noted.
The final section 4 discusses some prospects for the use of SAPs.

Before proceeding, it ought to be noted that all the SAPs I
have actually developed were meant to accompany Virginia Klenk's
<I>Understanding Symbolic Logic </I>(Englewood Cliffs, NJ:
Prentice Hall, 1994). All of them were written in Microsoft
PowerPoint 4.0. (For those unfamiliar with the program, a
description of some relevant information including some terms are
included in the Appendix.) Some of the presentations were shown
to a small group of students taking "Introduction to Logic"
course taught at the University of Texas of the Permian Basin in
Spring 1996.

1. SIMPLE ANIMATED PRESENTATIONS (SAPS)

The idea for using animated presentation in teaching is at
least as old as educational television. But it is only now, with
the development of faster computers, with better graphical
capacities, and hopefully appropriate accessible software, that
the use of animation becomes a reality for an ordinary teacher.
In this section, various existing as well as non-existing SAPs
that may be used to enhance the presentation of elementary logic
are described in some detail.

A SAP is a simple and short animated presentation that uses
some representational metaphors (implemented by the use of
animation and color) to capture logical concepts and techniques.
(The meaning of this characterization ought to become clearer in
the discussion of some examples.) SAPs can be used at all stages
in an introductory course. They can enhance the students'
understanding of symbolization techniques, of the truth-table
method, and finally, and perhaps most fruitfully, of the proof
method.

Before proceeding, one point should be stressed (it will be
elaborated in section 2). One should resist the temptation to
replace the mentioned classical methods of conveying the material
with the high-tech ones. Accordingly, the presentations ought to
be brief and to-the-point, focusing but on one particular issue.
All of the SAPs actually discussed fit this format.

There is no significant order of the following discussion,
except that I begin with the area where I believe SAPs can prove
most useful, in teaching proof techniques.

The progress of a proof

SAPs may be useful in illustrating the progress of a proof.
Frequently, some students do not follow the progress on the board
as quickly as others. Even if they are able to apply the rules
correctly, they get lost in trying to see which lines in the
proof are actually used in obtaining the formula. In a SAP, at
any step of the proof, the irrelevant information may be shadowed
out and appropriate colors may be used to emphasize exactly what
(and in what way, see the discussion of inference and replacement
rule SAPs, below) is involved in making the particular step in
the proof.

The Difference between Inference and Replacement Rules

Klenk introduces both inference and replacement rules.
Moreover, the system without replacement rules is not complete.
This gives rise to the additional burden that the students have
to (rather than having an option to) learn/memorize ten
additional rules within a rather short time-span. (And the proofs
using only the inference rules get rather boring rather quickly.)
One disadvantage of this is that replacement rules are in general
harder to apply and, moreover, the differences between them and
inference rules tend to be confusing for students.

The presentations that introduce inference and replacement
rules accordingly try to emphasize the differences between these
two types of rule by using two different overarching metaphors
for them. While the overarching metaphor for inference rules is
that of a process where the premises give as a result a
conclusion, the overarching metaphor for inference rules is that
of one formula undergoing a transformation. In the case of
inference rules, a new formula is created bearing structural
affinities (emphasized by animation and color) to the original
formulas. If the application of the inference rules requires two
formulas (as does MP, for example), this means that a third
formula will be created while the other two formulas still reside
on the screen. In the case of replacement rules, since the
metaphor is that one formula is transformed into another formula,
the original formula is no longer in view during the animation
for it is being transformed. Only one formula is visible on the
screen. The proof in the latter case is a way of keeping track of
transformation of what has happened to a single formula.

Inference Rules

An inference rule allows one to infer the conclusion from
the premises. What a SAP can emphasize is the pattern involved in
such a transition. Ordinarily, a SAP will introduce a
representational metaphor that is distinctive of an inference
rule in question, and implement it using color and animation. The
<I>Modus Ponens </I> SAP can serve as an example.

Two colors are used. The formula in the antecedent is green,
the formula in the consequent is yellow. The representational
metaphor is the obvious one of separating the consequent once the
antecedent is locked in by the antecedent-formula appearing on a
separate line. The use of colors helps in keeping track of the
structural affinities of the formulas involved and is also
essential in applying MP to more complex formulas. The animation,
in turn, makes the idea of inference more vivid -- a copy of the
consequent-formula flies over to the conclusion.

The described SAP may help in one common confusion arising
in the application of  MP. It is notorious that some students
tend to disregard the significance of the line between the
premises and the conclusion in applying inference rules. Such a
student will mention MP to justify inferring A (and on a separate
line B) from =E9A =C9 B=F9. The SAP described may aid here in two
respects. First, the inference line is drawn only when (a) the
antecedent-formula appears separately on a line and (b) it is
"locked in" with the actual antecedent (by a box surrounding
both). Second, the actual application of the rule is marked by
the process of the consequent-formula <I>flying over</I> to the
conclusion.

Replacement Rules

Replacement rules differ from inference rules in two ways.
First, replacement rules can be applied in two directions.
Second, replacement rules can be applied also to subformulas. The
presentations capture these two peculiarities of the replacement
rules in two ways. The first feature is captured by the use of a
different overarching metaphor for their presentation. As
mentioned above, while the overarching metaphor for inference
rules is that of a process of creating a new formula from pre-
existing formulas, the overarching metaphor for replacement rules
is that of transformation of one formula into another (where the
actual step-by-step proof is conceived as a registration of the
transformations.) The second feature of replacement rules is
captured by the employment of a zoom-in box. The box highlights
the part of the formula where the application takes place.

As do inference rule SAPs, so replacement rule SAPs try to
catch onto a graphical metaphor that the student could think to
be distinctive of the transformation in question and recognize
later in the proofs. To take some simple examples. The <I>Double
Negation</I> SAP illustrates the application of DN in one
direction (to a formula with two tildes) with a pointed finger
appearing on the screen to shake the two tildes loose, as a
result of which they drop down and disappear from the screen. The
application of DN in the opposite direction is marked by a
"fairy-tale sparkling star" appearing on the screen and leaving
behind two tildes, which then drop to the appropriate place in
the formula (closest to where the star appeared). The <I>De
Morgan</I> SAP uses the metaphor of the initial tilde spreading
over the elements in the parenthesis: the tilde travels up and
over the parenthesis;  when it is over the major connective of
the parenthesis it splits into three tildes; the middle one drops
slowly at the same time as the dot or wedge move upward; when
they clash, they become transformed into the wedge or dot,
respectively, which then drops into place; thereupon the two
other tildes drop in their respective places. (I mention some
options for a SAP to illustrate the application of DeM in the
other direction in section 4.)

The idea of a variable

While the idea of a variable is already well-entrenched in
the conceptual structures of some students, for others variables
seem dangerously close to constants. (After all, both are
represented by small letters of the alphabet...)

One way of emphasizing both the distinctness and the
relation between the two is to employ the metaphor of a variable
as an exploding box. Individual constants then "jump" into the
variable boxes. The boxes may be marked by different colors to
mark different variables in a formula.

The Logical Structure of Formulas

A SAP can enhance the students' grasp of the idea of the
logical structure of a formula. The SAP on <I>The Logical
Structure of Formulas</I> begins first with what the students can
do in their notebooks -- the step-by-step pairing of parentheses
thus arriving at the logical structure of a formula. The
connectives are then numbered reflecting the order of the
construction of the formula. Thereupon, the subformulas (in the
order of their connectives) drop a little thus revealing the
structure of the formula. What the animation adds over and above
what can be represented in the textbook is the fact that the
student can witness the process of such deconstruction.

 The Base of a Truth-Table

A simple SAP illustrates the implementation of two
algorithms for constructing truth-tables. The use of animation
and color (different for Ts and different for Fs) helps to make
the process more vivid and clear than is usually possible using
the blackboard.

The calculation of truth-values in a truth-table

SAPs can help in teaching the truth-table method in at least
two ways. First, they can help to illustrate the calculation of
truth-values in truth-tables. In particular, the order in which
the truth-table is filled out can be emphasized by use of shading
and color. The truth-values that are no longer used are shaded;
those that are used are appropriately colored. Second, the
determination of logical properties of sentences, arguments, etc.
can be illustrated vividly with color.

Symbolization

SAPs may also be used to supplement the teaching of
symbolization. This is primarily because they allow the students
to see the progress of symbolization virtually step by step --
something that the students never see on the blackboard.
Moreover, the possibility of using color to mark different simple
sentences employed may be particularly helpful when the
schematization requires that the order of sentences be changed
(as with "<I>p</I> if <I>q</I>").

Other Uses: Visual Demonstrations

SAPs can also be used to illustrate some informal
demonstrations for which one may ordinarily use the blackboard.
One such use is illustrated in <I>Quantifier Negation Rules
</I>SAP, where a simple demonstration is used to show that the
statement "Everything is uncertain" implies the statement "It is
not the case that something is certain." The universe of
discourse is represented with a big rectangle divided into
smaller boxes representing objects. A box being filled with black
stands for an object's being uncertain, a box being filled with
red stands for an object's being certain. The demonstration
begins with the assertion that everything is uncertain, which is
illustrated with the individual boxes filling in with black. Then
the question "Is something certain?" and an appropriate
representation of what this would mean appears on the screen.
After a pause (awaiting an answer from the students), it is
answered appropriately.

Other Uses: Teaching by Fun

It is common wisdom among teachers that no matter how well-
prepared, well-thought-out and well-presented the material is,
only rarely do the cognitive aspects of the presentation keep the
students' attention. Humor is one way to break the monotony.
Voice modulations another. Unexpected role playing yet another.
And SAPs can play their part too.

While the possibilities are possibly endless, a good example
of what can be done includes the use of whimsical graphical
figures that enhance the points made in the SAP. I describe two
ways of employing such enhancements, one of which is more
ornamental serving to break the monotony, the other of which is
in addition more deeply involved in the presentation of the
material.

(All the graphics used come from the MS ClipArt Gallery.
Since MS PowerPoint allows to modify the graphics, some of the
graphics used are the result of such modifications.)

* The presentations on the inference rules include sections
on how <I>not</I> to apply the rules. Such a section may be
"spiced up" by graphical means. Some of the ones I have employed
included changing the background of the slides from blue to red
(only in the title slide, though), as well as including
thematically appropriate graphics: hell-fire and an ass.

(Incidentally, it is probably best to show the presentation
twice. For the first time, only the section on how to apply rules
ought to be shown. And only after the students have some hands-on
experience applying the rules (ideally when they committed some
of the mistakes depicted in the SAP), ought the section on how
not to apply the rule be shown.)

* The SAP on <I>Categorical Quantifier Negation Rules</I>
uses a graphical character in a more essential way. It introduces
a character named Stupido. (His appearance is the result of
transformations of the character by the generic name "Man" in the
Cartoons section of the MS ClipArt Gallery). Stupido, as his name
suggests, is prone to saying exactly the wrong things at exactly
the right times. So, when the categorical proposition "Some
students like logic" is asserted, Stupido opens his mouth and
says "No students like logic". Thereupon, the Hand of Logic
squashes him, and a tilde is added in front his claim.

Although it is hard to convey the details of SAPs, the
general idea behind them ought to have become clearer. They are
short presentations that use animation and color to illustrate
and emphasize logical concepts and techniques. I should also
mention that while the use of animation and color is inextricably
linked in SAPs, many of the advantages of SAPs can be obtained by
using color chalk or color transparencies. Moreover, many of the
metaphors that have acquired picturesque representations in SAPs
can be used in oral presentation. This is in fact the origin of
many of the metaphors I have implemented. I have been suggesting
that students think of variables as boxes, of the antecedent-
formula having to be "locked in" before MP is applied, and so on.

2. ESSENTIAL LIMITATIONS OF SAPS

One of the first limitations that ought to be noted concerns
the author. I have not actually used the presentations in the
classroom in a systematic way. Many of the comments are therefore
speculative in nature. But some of them have some base in
experience. In place of working on full-blown SAPs I have used
some color transparencies to help the students with some ideas
(primarily with truth-tables, symbolization and some with
proofs). It seems rather plausible that some of the problems I
encountered while using transparencies would have correlates if I
had used SAPs.

Perhaps the single most important limitation has already
been mentioned and is ingrained in the name. SAPs are
<I>simple</I> animated presentations. I have been tempted to
replace the use of blackboard altogether with a computer
presentation. Thus, I conceived of long presentations that
consisted not only of material for actual SAPs but also of
material that would be usually presented on the blackboard or on
regular transparencies. (Such Complicated and Long Presentations
are referred to as CLAPs below.) Here are some problems with such
an idea.

* The primary advantage of a SAP is that of emphasis as well
as a break from a certain kind of monotony of regular
instruction. If a long presentation (which included the material
>from SAPs) was used as part of regular instruction, these two
effects would be lost. The CLAP would acquire a monotony of its
own.

* Contrary to expectations perhaps, the viewing of a
presentation requires a rather focused attention. Watching a long
presentation may be simply very tiring, while switching between
regular instruction and the viewing of SAPs can be stimulating at
a very basic pre-cognitive level.

* As I mention below (section 3), SAPs are easier to view
with the lights off. This makes taking notes (as would be
inevitable in a CLAP) rather difficult.

* Yet another argument for using SAPs but not CLAPs is that
it is rather hard to make such a computer presentation
interactive. If a student asks a question, though the
presentation can be stopped, its flow is disrupted. Moreover,
there is no way of taking into account the impact of the
student's question.

* This is related to another disadvantage this time of both
SAPs and CLAPs. They are extremely inflexible. They have to be
prepared ahead of the class presentation, and they cannot be
changed on the spot. In fact, given the amount of time required
to make them, they have to be prepared a long time before the
presentation is made. This makes introducing even minor
adjustments rather time-consuming.

* SAPs can be easily repeated. MS PowerPoint, at least does
not have any kind of bookmarking that would allow the repetition
of segments of a longer presentation.

* Perhaps one of the most important advantages of using the
board for instruction is that the student actually watches
exactly what the teacher is doing down to the (seemingly)
insignificant detail of which line and what symbol is drawn
first. In a computer presentation, even if one is careful to
animate and time the presentation so as to reflect the way that
the regular demonstration would proceed on the board, something
is lost. While it would be interesting to have an understanding
of what exactly is lost (perhaps the students "empathize" with a
teacher in a way they do not with the computer?), the two ways of
presentation are significantly different. I have only anecdotal
evidence for this claim, though the particular incident described
does seem to make sense of a number of teaching episodes (from
both sides of the lectern).

The incident in question occurred when I was using
transparencies to introduce truth-tables. By then, the students
were already able to calculate truth-values of a complex sentence
given the truth-values of the simple sentences. The only
innovation was the conceptual one of taking into account all
possible truth-values of simple sentences (as well as the
corresponding representational one of constructing the base of
the truth-table) and the representational one of organizing all
this into a truth-table. After introducing the general concept
and explaining how to construct the base of a truth-table, I have
shown the students a simple transparency with a neat truth-table,
explaining all its elements. We filled it all the truth-values
row by row, together. We worked through one more example, once
again on the transparency. Then I asked the students to do
another example in groups. I got uniformly puzzled looks and the
request "Could you show us an example first?" And they did in
fact proceed to work on their own after we did one truth-table on
the board, drawing all the lines from scratch, putting in all the
sentences where they belong, etc.

This suggests that at least sometimes there is a danger of
going too far in replacing what can be done on the blackboard
with what can be more neatly and seemingly more clearly done by
means of other media. This seems to further indicate that it is
much safer to use SAPs rather than CLAPs.

* One final point worth making, which further supports the
suggestion that SAPs ought to be used only to supplement regular
presentation of the material, is that they offer optional ways of
understanding the material. The hope is that the metaphors they
present can attach themselves to the students' conceptual
structures more effectively than could regular instruction on its
own. But this may be a very individual matter. It may work for
some students but not for others. So, it should be emphasized
that SAPs are not meant to be a part of regular instruction also
in the respect that the students must not be accountable for
them. Whether they pay attention, catch on to a metaphor so that
it will become helpful to them, should be left entirely without
any formal testing procedure. They should watch SAPs for fun not
for grades.

Although this enumeration of advantages and limitations of
SAPs cannot pretend to be complete, it illustrates their value as
tool for enhancement rather than replacement of regular ways of
presenting material. The discussion seems also to indicate that
SAPs should be used only after a regular presentation of the
material, and perhaps even only after the students have had an
opportunity to try out the application of the concepts and
techniques themselves. Inevitably, a more detailed discussion on
the effective way of using SAPs will have to be postponed until
after some actual classroom experience has been gained.

TECHNICAL LIMITATIONS

I have already mentioned some limitations of a technical
nature in passing. Let me bring them together. They fall into
three categories: limitations of software, of hardware, and some
more technical ramifications of their employment in classroom
instruction.

While MS PowerPoint does have some animation capacity, it is
somewhat of an abuse (of the program and the user) to use it with
view to creating animated presentations. Its substandard
performance in this capacity should come as no surprise since it
is not a software package designed for this purpose. One of the
major problems in using it in this fashion is that it supports no
automation. Without even a simple macro language or any animation
commands, each of the slides has to be prepared by hand. One nice
and useful feature is that it includes the "Rehearse Timings"
function. The presentation is run and the time each slide is
viewed by the user (before a mouse-click) is recorded.

Especially the speed of the presentation is particularly
sensitive to the computer one uses. I have not even tried to run
any of the SAPs on a 386. A 486 33MHz can work well. But the
faster the computer the better the animation effects (potentially
at least). One problem that is worth emphasizing is that the
presentation will appear differently depending on the speed of
the computer it is run on. What is basically a transparent
animation on a 486 becomes a blur when run on a Pentium unless
the timings are changed (which has to be done by hand). Moreover,
the actual preparation and editing of a SAP is extremely slow if
a 486 machine is used.

Aside of the limitations of the software and hardware, there
are also some technical aspects of classroom use that merit
attention. Although the presentations can be run when the lights
are on, their visual appeal is significantly enhanced when the
lights are off. This means that SAPs will work effectively only
in case where the teacher has an easy access to the light switch.
Moreover, the actual technical equipment required can be rather
distracting if it is not properly placed in the classroom. The
whole apparatus I have used was placed on a rather high cart-
wheel which was either in the view of the students so that they
could barely see the presentation or else, when moved away,
distorted the shape of the projected image.

FUTURE PROSPECTS

I find the idea of enhancing the teaching of introductory
logic with SAPs very appealing. Whether it will puss master only
experience can show. But it is also exceedingly clear that it
would be very hard to develop SAPs on one's own.

If there were even a few people who would be interested in
both using and developing SAPs, it would be sensible to establish
an electronic library thereof. Such a library would have several
advantages. First, it would considerably ease the pain and time
involved in developing SAPs. At the same time, since SAPs are
revisable, each and every user could use an existing SAP to
modify it to fit his or her own taste or system. Second, the
creation of really effective SAPs is not an easy matter. This is
particularly so because we have very little insight into the way
in which forms of representation feed into the process of
understanding and, consequently, into the process of education. 
For example, it might be that there are better and worse ways of
choosing a representational metaphor for a particular replacement
rule. As an example, I might mention that I have struggled with
two ways of representing the DeMorgan transformations in the
direction other than the one already mentioned (transforming a
disjunction/conjunction into a negation). One involves the second
tilde clashing with the wedge, say, and becoming transformed into
a dot, while the first tilde jumps in front a parenthesis. The
other involves all three connectives flying over to one spot,
where they clash and become transformed into one tilde and a dot,
and then move to respective places. None of these strike me as
being particularly helpful, though I find the second preferable.)
By having some sort of forum for exchange of ideas, the chances
of finding really effective SAPs would be greatly increased.

If such a forum were to be organized, it might be reasonable
to try to find different software. The problem, however, is that
of balancing the sophistication of the software, on the one hand,
and its accessibility and cost, on the other. The enormous
advantage of MS PowerPoint is that together with the other MS
Office programs it has become something of a standard and is
available in many computer labs, offices and homes.

To summarize, I have introduced the basic idea and described
some examples of simple animated presentations which, as I have
argued, can enhance and supplement though not replace the regular
ways of presenting material in elementary logic classes. I have
discussed some of the advantages and disadvantages of their
various uses. What I have done very little of is to offer any
insight into how they work in the practice of classroom teaching.
I hope that, perhaps with the input of other interested
individuals, this can be helped.