Critical Thinking

Critical thinking is an essential skill for any mathematician. In our quest to understand the world through numbers, we often encounter claims or statements that demand scrutiny. Let’s explore how to assess the validity and reasonableness of these claims. When confronted with a claim, always start by asking the right questions: What is the evidence supporting this claim? Is the reasoning logical and consistent with what we know from established mathematical principles? Don’t accept a claim at face value. Instead, delve deeper, challenge assumptions, and evaluate the evidence.

To better understand this, consider the claim: “In a city, the average person has [latex]3.5[/latex] friends.” We know that it’s impossible for a person to have [latex]0.5[/latex] of a friend, so we should be skeptical. Perhaps the statement could be more accurately presented as “On average, a person in this city has [latex]3[/latex] or [latex]4[/latex] friends.”

In our quest to assess the validity and reasonableness of claims, context is critical. The same statement can have different implications depending on the conditions surrounding it.

Consider a statement like, “The average household income in Town A is [latex]$80,000[/latex].” While this figure might seem high or low depending on your point of view, understanding the broader context is key. What’s the cost of living in Town A? How does this average compare to other towns in the same region or with similar living conditions? By asking such questions, you not only assess the reasonableness of the claim but also gain a deeper understanding of its implications.

You are watching or reading news and see the following headline:

U.S. Weekly Jobless Claims Double to [latex]6.6[/latex] million

1. You do not see the name of the organization that published the headline. What groups might have wanted to publish this statement? What are some social issues or political ideas that this statement might support?
2. What additional quantitative or math literacy questions could be asked to more fully understand the statement?

Show Solution

1. Answers will vary. Sample answer: The U.S. Department of Labor for informational purposes, a political party interested in passing legislation, etc.
2. Answers will vary. Sample answer: What constitutes a “jobless claim”? Who is gathering this data? How many of the claims are new? How many people are employed in the U.S.?

It can be challenging to identify statements in the media that may be somewhat true but also misleading. You may not be able to locate the necessary information to make a full determination, but you can start by asking if the statement is reasonable. This headline above is taken from an article published April 2, 2020, by CNBC describing that the number of jobless claims had increased from [latex]3.3[/latex] million to [latex]6.6[/latex] million over the course of a week.1 Do you think this headline was reasonable? An evidence-based approach doesn’t just allow you to assess the validity of a claim—it also helps you decide on the next steps. If the evidence supporting a claim is weak or nonexistent, further investigation may be necessary. Remember, it’s your job to turn hunches into reasoned arguments. This means not only looking for evidence that supports a claim but also being willing to ask challenging questions. As you practice this skill, you’ll become more proficient at separating fact from fiction in the world of mathematical claims.

Recognizing and Challenging Assumptions

At the core of every claim or argument lie certain assumptions – often unspoken – that shape the final conclusion. Assumptions can be as straightforward as believing the sun will rise tomorrow, or as complex as assuming a certain factor is the leading cause of a specific event. In mathematics, assumptions might take the form of accepted properties or axioms, such as “parallel lines never intersect.” Once we’ve identified the assumptions underlying a claim, we can challenge them. This isn’t about disproving the claim, but rather about deepening our understanding and ensuring we have the full picture. Asking questions like “What if this assumption is not true?” or “Are there exceptions to this assumption?” can lead to insightful discussions and a more nuanced understanding of the claim.

1. Consider the claim, “The city’s new park will increase nearby property values.” What assumptions underlie this claim? How might you challenge these assumptions?
2. Evaluate the statement, “Implementing a four-day school week will improve student performance.” Identify and challenge the assumptions in this claim.
3. Think about the assumption, “People are more productive when they work from home.” How could you test the validity of this assumption?

Show Solution

1. The claim, “The city’s new park will increase nearby property values,” makes several assumptions. It assumes that people value proximity to a park when buying property, that the park will be well-maintained and safe, and that it won’t bring any negatives (like increased noise or traffic). To challenge these assumptions, we could research whether parks have boosted property values in other areas, or whether other factors—like schools or transport links—are more influential.
2. The statement, “Implementing a four-day school week will improve student performance,” assumes that fewer days in school will lead to more rest, less stress, and thus better academic performance. It also assumes that the same amount of learning can occur in four days instead of five. Challenging these assumptions could involve looking at studies from places that have tried a four-day school week, or considering other factors that affect student performance, such as teaching quality, class size, or home environment.
3. The assumption, “People are more productive when they work from home,” implies that home environments are conducive to work, that people have the discipline to work effectively outside a traditional office, and that productivity can be accurately measured across different job roles and industries. To test this assumption, we could compare studies on productivity levels before and after the COVID-19 pandemic forced many people to work from home. We could also consider whether the nature of the work or the worker’s personal circumstances (like whether they have young children or a suitable workspace) affect this assumption’s validity.

Remember, the goal is not necessarily to prove or disprove the assumptions but to develop a nuanced understanding of the claims by recognizing and critically evaluating underlying assumptions. This process helps in making well-informed decisions and forming robust arguments.

Improving Quantitative Statements: Clarity through Details and Structure

Details serve as the building blocks of clear quantitative statements. They provide the necessary context and precision, transforming bare numbers into meaningful insights. For instance, a statement such as, “Company [latex]X[/latex]‘s revenues have grown” lacks critical details. How much growth are we talking about? Over what period? By adding details—”Company [latex]X[/latex]‘s revenues have grown by [latex]25\%[/latex] over the past fiscal year”—we’ve made the statement far more informative and meaningful. However, be careful not to drown your audience in details. The aim is to provide enough context to understand the statement, but not so much that it becomes confusing. A balance must be struck between clarity and complexity. In mathematics, the structure of your quantitative statement can be as important as the numbers themselves. Consider the statement, “Out of every [latex]100[/latex] students, [latex]61[/latex] prefer online classes.” While this is clear and precise, restructuring the statement can make the same information more relatable: “A little more than half of students prefer online classes.” Though both statements convey the same information, the second is structured in a way that feels more intuitive to many people.

1. Add essential details to improve the clarity of this statement: “Car emissions have reduced.”
2. Restructure the following quantitative statement to make it more relatable: “[latex]7[/latex] out of [latex]10[/latex] dentists recommend this toothpaste.”
3. The statement “The company’s profits have tripled” lacks context. How can you add details to enhance its clarity?

Show Solution

1. The statement, “Car emissions have reduced,” lacks specificity. To improve its clarity, we could specify the extent of the reduction, the type of car emissions being referred to, and the time period over which the reduction occurred. For instance, “Carbon dioxide emissions from passenger vehicles in the U.S. have reduced by [latex]15\%[/latex] over the past decade.”
2. The quantitative statement, “[latex]7[/latex] out of [latex]10[/latex] dentists recommend this toothpaste,” might be more relatable if we convert the fraction to a percentage and add a touch of context. For instance, “[latex]70\%[/latex] of dentists, which is a significant majority, recommend this toothpaste.”
3. “The company’s profits have tripled” lacks context. To enhance its clarity, we could add information about the time frame and base level from which the profits tripled. For instance, “Over the past fiscal year, the company’s profits have tripled from [latex]$1[/latex] million to [latex]$3[/latex] million.”

In each case, by adding relevant details or restructuring the statement, we’ve made the information more precise and comprehensible. This helps in making the numbers more relatable and easier to grasp for anyone engaging with the statement, thereby enhancing communication and understanding.