Mathematics is sometimes seen as a separate world altogether from the humanities. Questions from the humanities have gray areas and room for debate, whereas math is a cold crystal ball that deems answers "right" or "wrong" and has students memorizing convoluted methods to learn its ways.
But math has gray areas! Consider these questions:
- If there are three candidates running for office, how should voters choose a winner?
- What is 8 ÷ 2(2 + 2)?
- If you earn $50,000 per year and pay $1,000 per month in rent, what is the best strategy for saving money?
Let's dig into each of these.
1. If there are three candidates running for office, how should voters choose a winner?
This one seems so straightforward, right? Everyone votes for a winner, and whoever gets the most votes win. This is, of course, not how all US government candidates are elected. Moreover, the idea is fraught with mathematical paradox! To see why, let's imagine there are exactly 100 voters and three candidates: Alex, Betty, and Carlos. Consider two voting scenarios.
- SCENARIO A: Everyone votes for one person. Alex gets 40 votes, and Betty and Carlos get 30 votes each.
- SCENARIO B: Everyone votes for two people and numbers them 1 and 2 for first and second choice. Just like Scenario A, 40 people put '1' for Alex, 30 people put '1' for Betty, and 30 people put '1' for Carlos. But then EVERYONE who votes '1' for Alex or Carlos votes '2' for Betty. (The people who vote '1' for Betty are split between Alex and Carlos.)
Is it fair that in Scenario A, Alex wins, but if you include the first- and second-place votes in Scenario B, Betty would be overall more popular and, arguably, should win? As much as we want voting to be perfectly fair, by changing the voting process, the exact same people end up electing different candidates. (Classroom activity: task students with deciding how to vote and have them compare the pros and cons of each other's methods.)
2. What is 8 ÷ 2(2 + 2)?
This math problem has gone viral before. It seems so clear cut! The challenge is that people either forget or misuse the standard order of operations, which requires evaluating parentheses first, followed by multiplication and division in order from left to right. Thus the "correct" answer of 16 (which is 8 ÷ 2 • 4) is obscured by the sloppy notation. However, order of operations is something learned, not immutable mathematical law! The proper order of operations has evolved throughout history and varies geographically. Students and mathematicians alike ignore it sometimes and with good reason, for example by leaving out parentheses when they're implied or writing an expression as it would be entered into a four-operation calculator. (Classroom activity: ask students to invent their own algorithms for evaluating expressions and explore how those algorithms would affect specific formulas.)
3. If you earn $50,000 per year and pay $1,000 per month in rent, what is the best strategy for saving money?
Here, math flirts closely with the humanities. Figuring out a strategy for saving money absolutely involves mathematics, and the more details one considers, the more sophisticated it can get.
- How long are you trying to save for?
- What are your other expenses?
- How will inflation affect the amount you have saved?
Yet exploring the question fully requires asking questions whose answers are not strictly mathematical.
- What is your personal risk tolerance?
- How compulsively do you spend?
- How were you taught about money growing up?
Despite what many students (and some teachers) think, math problems rarely have a single method for solving them and frequently have multiple solutions! Students should be encouraged to explore new ways to solve the same problems.
Let's take a look at a straightforward math problem that will look familiar to any fourth grader:
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