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 Chris Brown. His topic, from Geometry: using a truth table.
How can this topic be used in your students’ future courses in mathematics or science?
Truth tables apply directly to the field in Computer Science, as in its essence, it runs on Boolean logic. Boolean logic simply means that everything has a result of True or False. This can be seen explicitly when dealing with logic gates, which are different paths that a computer program follows as it tests whether inputs are true or false based on given conditions. Based on the results, the program will continue to run, testing different cases, based on each result in a complex chain of tests. For example, for a simple program, let’s say you may input any integer, n, between 10 and 20 inclusive. If the number is divisible by 2, then it will compute n divided by 2. If the number is not divisible by 2, then it will return the original number. Then, if the resulting number is divisible by 2 as well, it will once again compute n divided by 2. If the resulting number is not divisible by 2, then it will return the resulting number. This sequence of tests follows the conditional statement, “If an integer between 10 and 20 inclusive is divisible by 2, and it’s resulting value is also divisible by 2, then the chosen integer has 22 within its prime factorization.” For the “and” truth table: if the integer chosen was 10, we see the True & False = False case; if the integer 16 was chosen, we see the True & True = True case; if the integer 19 was chosen, we see the False & False = False case. With variations and chains of logic gates, Computer Science has every single type of truth table embedded within the Boolean logic it uses.
How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?
Logical humor has often been used in the more intelligent based humor of popular culture, and truth tables and arguments are even more so apart of this. In the movie “Get Smart,” released in the year 2008, features a quirky, and humorous data analyst named Maxwell Smart who by an odd turn of events was promoted to field agent. On one of Smart’s missions to infiltrate the enemy base, he, Siegfried, and Shtarker wittingly enters into a logical argument that is a beautifully crafted logical argument. I have written the lines below.
Smart: I understand that you are the man to see if someone is interested in acquiring items of a nuclear nature
Siegfried: How do I know you are not Control
Smart: If I were Control, you would already be dead
Siegfried: If you were Control, you would already be dead
Smart: Since Neither of us are dead, so I guess I am not Control
Shtarker: That actually makes sense!
While this is not an example of a truth table per say, truth tables and propositional logic was the foundation of how this argument was created. What we see in lines 3-5 is the following propositional formula:
((p → q) ∧ (p → s))
p = Smart being Control
q = Siegfried Being Dead
s = Smart Being Dead
By viewing the truth table, we see that when q and s are false, then p must be false; as stated in Line 5 of the movie.
How can technology be used to effectively engage students with this topic?
The technology tool that I found was listed on the Stanford University website and is one that the students can easily use to check over their work. The website, attached below, allows students to enter in their propositional logic formulas for any complex length and has functionality for all necessary, binary logical operators. The site also allows for the usage of many logical expressions, not just 2. Inputting the formulas is very user friendly and allows for multiple representations of each logical operator. For instance, “or” can be represented by “\/” and also “or,” and can even both be used within the same formula chain. If a character or statement is used that the system does not recognize, the system will highlight the symbol in red and say, “illegal character,” which I personally find easily understandable for all ages. What I love most about this website is that as the formula is being entered, the student is able to see the table being created as it is being entered.