**Document Id**: 12_04_02_1

**Date Received**: 2002-12-04 **Date Revised**: 2003-02-12 **Date
Accepted**: 2003-02-16

**Curriculum Topic Benchmarks**: 6.3.1

**Grade Level**: Middle School \[6-8\]

**Subject Keywords**: Probability, mean, average.

**Rating**: moderate

# The Mathematical Implications of Lying

**By**: Glenn Simonelli, Lakeview Elementary School, 6405 S. Main St.,
Bloomington, IN 47401

e-mail: gsimonel@indiana.edu

**From**: The PUMAS Collection http://pumas.jpl.nasa.gov

© 2002, California Institute of Technology. ALL RIGHTS RESERVED. Based
on U.S. Gov\'t sponsored research.

**Introduction.** This article explores how statistics can be
interpreted in different ways to yield different conclusions. It
describes a pair of class activities. In the first, the results are
interpreted to "show" that taking a group rather than an individual
perspective is ultimately beneficial to the individual. In the second, a
variation is added "showing" that telling the truth is better that
lying.

## 

**The Lesson.** Here is a scenario I presented to a 5^th^ grade math
class: you and a friend are trapped in a burning building. The door is
blocked and the only way out is through a window. The window is not
large enough for both of you to go through at the same time. You have
two options. You can try to cooperate with your partner to see that you
both get out quickly, or you can fight with your partner to assure that
you get out first. These choices yield four possible results: 1) you and
your friend cooperate with each other, and you both get out with minor,
first-degree burns; 2) you fight and your partner cooperates, leaving
you without any burns and your partner with severe, third-degree burns;
3) you cooperate and your partner fights, with the opposite results; or
4) you both fight, leaving the both of you with moderate, second-degree
burns. Which is the best choice?

Next I assigned number values to the different results. No burns were 0,
minor burns were 1, moderate burns were 2 and severe burns were 3.
During discussion I pointed out that higher numbers correspond with more
severe burns, and the students agreed with me that the goal was to
receive the smallest possible number. I asked students if, using this as
a guide, it was better to fight or cooperate to get out of the burning
building.

In the past I had difficulty getting the students in this class to
cooperate with each other. Although there were exceptions, as a group
these students were unusually competitive, so I was not surprised when
the consensus of the class was that, based on the possible numerical
outcomes, one would be better off fighting with their friend than
cooperating. Since the range of outcomes for cooperators is 1 (minor
burns) to 3 (severe burns), while for fighters the range is 0 (no burns)
to 2 (moderate burns), it appears that fighting is the most reasonable
choice.

**Fight vs. Cooperate.** We then played the game. I went around the
class and randomly paired up the students. When it was a pair's turn to
go, both students wrote on a piece of scrap paper either a "C," for
cooperate, or an "F," for fight. Students did not know what their
partners wrote. Then the two students showed the class what decision
they made, each student was assigned a number based on the severity of
burns they received. The decisions and scores were tallied on the board
using the following format:

[Group \#]{.underline} [Decisions]{.underline} [Results]{.underline}

> 1 F-C 0-3

2 F-F 2-2

3 C-C 1-1

4 C-F 3-0

Etc.

Each member of the class had an opportunity to participate, and we then
took time to discuss and analyze the results. The first question I asked
was, "How do we find out what was the average severity of burns received
by the people in this class?" After a brief discussion we decided that
the best way would be to add up all the numbers and divide by the number
of students. I no longer have the exact results that we recorded that
day, but there were 28 students in the class and a roughly equal number
of C and F responses, so the final tally looked something like this:

[Group \#]{.underline} [Decisions]{.underline} [Results]{.underline}

> 1 F-C 0-3

2 F-F 2-2

3 C-C 1-1

4 C-F 3-0

5 C-C 1-1

6 C-F 3-0

7 F-C 0-3

8 C-F 3-0

9 F-F 2-2

10 C-C 1-1

11 F-C 0-3

12 F-F 2-2

13 C-C 1-1

14 C-F 3-0

I will use these numbers as I relate the discussion we had after we
played the game.

**Class Discussion.** After performing the math and rounding to three
digits, we determined that the average severity of burns received was
1.46\--(forty-one total "burn" points received divided by 28 students
participating). Since this was an average (the mean), the class agreed
that, as this number went lower, an individual was less likely to
receive severe burns and as it went higher, an individual was more
likely to receive severe burns. The next step was to compare the results
of the groups with cooperating students to the groups with fighting
students. Our goal was to determine the likelihood, or probability, of
burn severity for each type of decision pairing. We made two columns,
Cooperate and Fight, and tallied pair totals under each column. We had
no difficulty assigning values when both pair members made the same
decision. If they both cooperated the result was 1 + 1, so there were
two points added to the Cooperate column, and if they both fought they
received 2 + 2 points, so a 4 was added to the Fight column. We needed
to make a decision when the pairs did not make the same choice. When the
members of a pair made different decisions their score was 0 + 3, but we
couldn't put a 3 under both columns because that would be adding their
score twice, and that wouldn't give us an accurate average. Since we
were tallying the results of each pair's decisions, I suggested that the
best thing to do would be to split that pair's score in half and add the
half score (1.5 points) to both columns. The students agreed to this
solution.

We then totaled the score for each decision and divided the total by the
number of people making that decision. Doing this using the numbers
above yields the following results:

[Group]{.underline} [Cooperate]{.underline} [Fight]{.underline}

1 1.5 1.5

2 - 4.0

3 2.0 -

4 1.5 1.5

5 2.0 -

6 1.5 1.5

7 1.5 1.5

8 1.5 1.5

9 - 4.0

10 2.0 -

11 1.5 1.5

12 - 4.0

13 2.0 -

14 1.5 1.5

The next step was to total the scores for each column and divide the
total by the number of participants in each column. These totals were,
for the Cooperate group, 18.5 divided by 15, or around 1.23, and for the
Fight group, 22.5 divided by 13, or 1.73. This meant that, if you were
in a cooperating group you were likely to have less severe burns. After
seeing these totals, most students concluded that you were better off to
be in a group that had at least one person cooperating.

**Interpretation.** It is important to realize that this result is only
apparent when you look at the probable burn severity for the pair. If
you restrict your evaluation to just an individual's burn severity, it
appears that fighting to get out is a more advantageous reaction.

Yet when considering the overall probability for the entire group, it
appears better to cooperate than to fight. The statistics show that it
is preferable to be in a group where at least one person cooperated (29
points divided by 22 students in 11 groups = 1.32) than to be in a group
where at least one person fought (34 points divided by 20 students in 10
groups = 1.70). However, to come to this conclusion you must ask the
question, "Which decision is most likely to benefit the most people?" In
order to arrive at the conclusion reached above, the students needed to
shift their perspective from a self-centered view to a group view.

One could argue that this exercise is mathematically misleading, because
if you tallied individual scores the fighters would have a lower average
than cooperators. (Cooperate: 29 points divided by 15 students = 1.93
average burn severity; Fight: 12 points divided by 13 students = 0.92
average burn severity.) But this is exactly the point of the exercise;
the best decision depends on whether students consider only their own
needs or if they think of themselves as members of a group. When taking
a group perspective, a person is more likely to make decisions that
benefit the entire group. In this game, as more people cooperate, the
average gets lower and everyone becomes less likely to receive severe
burns. As the entire group benefits, the individuals in the group also
benefit.

**Truthfulness vs. Lying.** I tried this exercise a second time with a
different class, and this time I decided to add another element to the
game. Honesty was an issue with this particular group, and I was always
looking for ways to impress upon them the value of honesty. We completed
the exercise and discussed the results, and then I told the students
that we would try it again, but with slightly different rules. Before a
student wrote down a C or F to represent their decision, they would tell
their partner (and the class) what they were going to write. The twist
was that they didn't have to tell the truth; they could lie if they
didn't want their partner to know what their decision was. We wrote down
their intentions on the board so we could tell later if they were
telling the truth or lying. After announcing what they would do, the
students wrote down their decisions and showed them to the class. On the
board, we kept track of the results of the second round this way:

[Group \#]{.underline} [Decisions]{.underline} [Results]{.underline}
[Truth or Lie?]{.underline}

> 1 F-C 0-3 T-T

2 F-F 2-2 T-L

3 C-C 1-1 T-T

4 C-F 3-0 L-L

Etc.

**Interpretation.** After everyone had an opportunity to go a second
time we totaled the scores for the honest responses and compared them
with the lies. We used a method similar to the one we used for the Fight
or Cooperate groups: if both members of the group did the same thing we
tallied the complete amount for the group, if they did different things
we split their score in half. Once again, I don't have the original
results, but these results are representative:

[Group \#]{.underline} [Decisions]{.underline} [Results]{.underline}
[Truth or Lie?]{.underline}

> 1 F-C 0-3 T-T

2 F-F 2-2 T-L

3 C-C 1-1 T-T

4 C-F 3-0 L-L

5 C-C 1-1 T-T

6 C-F 3-0 T-T

7 F-C 0-3 L-T

8 C-F 3-0 T-T

9 F-F 2-2 L-L

10 C-C 1-1 T-T

11 F-C 0-3 L-T

12 F-F 2-2 L-L

13 C-C 1-1 T-T

14 C-F 3-0 L-L

In this example 17 students told the truth and 11 lied, which closely
approximates the results that we achieved in class. Tallying the
severity of burns for members of each group yields the following:

[Group]{.underline} [Truth]{.underline} [Lie]{.underline}

1 3 -

2 2 2

3 2 -

4 - 3

5 2 -

6 3 -

7 1.5 1.5

8 3 -

9 - 4

10 2 -

11 1.5 1.5

12 - 4

13 2 -

14 - 3

The average severity of burns for the truth-tellers worked out to 22
points divided by 17 students, or 1.29, rounded off to three digits. For
the liars, the average was 19 points divided by 11 students, or 1.90.
These results were even more dramatic than the Fight vs. Cooperate
results, and initiated a lively discussion. Next we checked the results
for the different possible group outcomes (both truthful, both lying, or
mixed) and got the following:

[Group]{.underline} [Both Truthful]{.underline} [Both Lying]{.underline}
[One Truthful, One Lying]{.underline}

1 3 - -

2 - - 4

3 2 - -

4 - 3 -

5 2 - -

6 3 - -

7 - - 3

8 3 - -

9 - 4 -

10 2 - -

11 - - 3

12 - 4 -

13 2 - -

14 - 3 -

Averaging burn severity for individuals in each group worked out to:

**Both Truthful** - 17 points divided by seven groups equals 2.43 points
per group, or 1.21 per individual (the slight discrepancy is due to
rounding)

**Both Lying** - 14 points divided by 4 groups equals 3.50 points per
group, or 1.75 per individual

**One Truthful, One Lying** - 10 points divided by three groups equals
3.33 points per groups or 1.67 per individual.

**Class Discussion.** Clearly, a student that lied was more likely to
receive severe burns in this game. When asked to explain why this
happened, the students were quick to notice that they were more likely
to lie if they were planning to fight their partner. (Of the 11 students
who lied, 9 chose to fight and only 2 chose to cooperate.) When asked to
draw conclusions from this exercise, one conclusion arrived at was that
people are more likely to lie if they are planning to do something bad
to someone. Another comment was that people only lie if they are not
proud of what they are planning to do.

**Conclusion.** Honesty and cooperation were still issues in these
classrooms after the lesson, although as the year progressed the
situation improved. It is difficult to assess what impact, if any, this
exercise had on the students' behavior toward each other. There were
many discussions about honesty and cooperation in these classrooms.
Still, I think that having it displayed, however imperfectly, as part of
a math lesson made a greater impression on the students than just
telling them to cooperate with each other and to always be truthful.

**Extension**: In the example as it is, "fight" always produces the same
result. A reviewer pointed out that the exercise could be made more
realistic, though more complex, by putting probabilities on the outcomes
when fights occur, so there would be some probability that a fighter
would win.

Glenn Simonelli has taught elementary school in Bloomington, Indiana for
six years. He is currently on a leave of absence to complete a doctorate
degree in science education.