# Engaging students: Using a truth table

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Saundra Francis. Her topic, from Geometry: using a truth table.

How could you as a teacher create an activity or project that involves your topic?

An activity that can be done with truth tables is notice and wonder. You could give students an example of a truth table, say finding the results of p and q, and have them write down what they notice and what they wonder about the table. This can lead to a great discussion about what they notice about the truth table and how they think it works. Some students might be able to figure out what the table represents or it could help display misconceptions before the lesson starts. Students can then present their wonders, which you can make sure to answers by the end of the lesson. This activity is a good way to discover if students have prior knowledge of the concept and you can see what they hope to have learned by the end of the lesson.

How has this topic appeared in pop culture?

In this Alice in Wonderland clip where Alice attends the Mad Hatter’s tea party we notice the importance of truth tables and logic to distinguish what the Mad Hatter, March Hare, and the Dormouse are discussing. Alice is using logic to discover what the Mad Hatter means and argue with him about his logic. Once you have students watch this video you can ask students to share examples of where logic is displayed. You can choose one example to transfer into a truth table later in the lesson to show students that we can use the logic displayed outside of the classroom. This will engage students and display the importance of using logic to distinguish what others mean.

What interesting things can you say about the people who contributed to the discovery and/or development of this topic?

Bertrand Russell, one of the contributors that made truth tables as we use them today, worked within the philosophical branch that combined mathematics and logic. Russell was born in Trelleck, United Kingdom in 1872. He attended Trinity College in Cambridge achieved a First Class distinction in philosophy. In 1903 Russell wrote The Principles of Mathematics with Dr. Allan Whitehead. This book extended the work of Peano and Frege on mathematical logic. This was the book where he contributed to truth tables. In 1918 Russell was imprisoned for six months due to a pacifist article he wrote. While incarcerated he wrote Introduction to Mathematical Philosophy. Russell received the Nobel Prize in literature in 1950 in recognition of his thoughts on freedom of thought and humanitarian ideals. He is known for his humorous and controversial writing.

# Engaging students: Writing if-then statements in conditional form

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Sarah McCall. Her topic, from Geometry: writing if-then statements in conditional form.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

For this exercise, I would like to kill two bird with one stone (engage students as well as deepen their understanding) by choosing word problems that are funny and have to do with topics that students are familiar with. For this activity, students could work individually or in groups to construct if then statements out of two given clauses, and then construct two if then statements out of their own clauses: one true and one false. This activity would encourage students to be as funny and outrageous as possible. For example:

1. Construct a logical if then statement using the following:
I do not take a nap today.
I will cry about my homework.
2. Construct a logical if then statement using the following:
Miss. McCall’s dog is the cutest dog on the planet. Pigs can fly.
3. Construct a false if then statement using any two clauses you can think of.
4. Construct a true if then statement using any two clauses you can think of.

Questions 3 and 4 are important because they engage students by requiring them to use their creativity and humor, as well as their logical skills. Students should ask themselves, if I want my entire statement to be false, should my “if” and “then” clauses be true or false? This could also be used as a class project where students present their creations to the class afterwards, which would further facilitate student understanding and involvement.

How could you as a teacher create an activity or project that involves your topic?

To get students engaged in this material, I would like to switch it up from the usual mathematics lesson by introducing students to a whole new side of mathematics: logic! For this activity, students would work in groups to solve the following riddle (citations listed below):

Somewhere on an island far far away, there are two large families quite different from you and I. The people in one of these families, the Truthtellers, tell the truth all the time, even when they’d rather lie. They can’t say something false even by mistake! And the other family, the Liars, always lies, even when they wish they could tell truth. Everything they say is a lie. Now suppose you meet two people on this island: Ashley and Amanda, and Ashley tells you “We are both liars”. What family is Ashley from? What about Amanda?

The goal of this exercise is to get students to use their logical intuition and eventually conclude that Ashley must be a Liar, because if she is a Truthteller, then she cannot say she is a Liar. Hopefully students will be introduced to how to communicate their ideas in a simple if then format and realize that logic is fairly intuitive and natural.

How can this topic be used in your students’ future courses in mathematics or science?

Many students often worry (or grumble) that they may never use the topics they are learning in geometry/math in the real world. However, as more and more students are considering jobs in STEM fields, I think it is important to acknowledge that these if then statements are not going away! Not only are if then statements used to develop hypotheses for science experiments, they will also be used in upper level math courses like calculus. Learning how to apply if then logic to everyday situations as well as theorems is a skill that will be helpful to all students who plan to go on to any STEM, medical, scientific or research fields. I believe starting off a lesson on if then logic with these reminders would be helpful in getting students to be engaged in their own learning, because they will see that their learning now will affect them later on.

# Predicate Logic and Popular Culture (Part 172): Clement Clarke Moore

Let $C$ be the set of all creatures, let $H(x)$ be the proposition “$x$ is in the house,” and let $S(x)$ be the proposition “$x$ is stirring.” Translate the logical statement

$\forall x \in C (H(x) \Rightarrow \lnot S(x))$.

Of course, this matches the first two lines of one of the most popular poems in the English language.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

# Predicate Logic and Popular Culture (Part 171): Hunter Hayes

Let $P$ be the set of all people, and let $H(x,y)$ be the proposition “$x$ has $y$.” Translate the logical statement

$\forall x \in P (x \ne I \Rightarrow \exists y \in P(H(x,y)) \land \forall y \in P (\lnot H(I,y))$.

This matches the chorus of this song by Hunter Hayes.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

# Predicate Logic and Popular Culture (Part 170): Tim McGraw

Let $T$ be the set of all times, let $P$ be the set of all places, and let $S(x,t)$ be the proposition “I will see you at place $x$ at time $t$.” Translate the logical statement

$\forall x \in P \forall t \in T (S(x,t))$.

This matches the last line of the chorus from this classic song by Tim McGraw.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

# Predicate Logic and Popular Culture (Part 169): Thomas Rhett

Let $P$ be the set of all people, and let $M(x)$ be the proposition “She wants to marry $x$.” Translate the logical statement

$\exists x \in P (M(x)) \land \lnot M(me)$.

This is the last line of the chorus from the canonical country song “Marry Me.”

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

# Predicate Logic and Popular Culture (Part 168): Halsey

Let $P$ be the set of all people, let $L(x)$ be the proposition “$x$ will love you.” Translate the logical statement

$\exists x \in P (L(x)) \land \lnot L(I)$.

This matches the lines from “Sorry” by Halsey.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

# Predicate Logic and Popular Culture (Part 167): Taylor Swift

Let $P$ be the set of all people, and let $T(x,y)$ be the proposition “$x$ trusts $y$. Translate the logical statement

$\forall x \in P \lnot( T(I,x) \lor T(x,I))$.

This matches the chorus from one of Taylor Swift’s recent hits.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

# Predicate Logic and Popular Culture (Part 166): Ed Sheeran

Let $T$ be the set of all things, let $S(x)$ be the proposition “I see $x$,” and let $F(x)$ be the proposition “I have faith in $x$.” Translate the logical statement

$\forall x \in T (S(x) \Rightarrow F(x))$.

This is one of lines from the recent smash hit by Ed Sheeran.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

# Predicate Logic and Popular Culture (Part 165): Eleanor Roosevelt

Let $P$ be the set of all people, let $C(x)$ be the proposition “You consent to let $x$ make you feel inferior,” and let $F(x)$ be the proposition “$x$ makes you feel inferior.” Translate the logical statement

$\forall x \in P (F(x) \Rightarrow C(x))$.

This is the contrapositive of a famous quote by Eleanor Roosevelt.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.