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Study Guides > Math for Liberal Arts: Co-requisite Course

Why It Matters: General Problem Solving

Critical Thinking

Thinking comes naturally. You don’t have to make it happen—it just does. But you can make it happen in different ways. For example, you can think positively or negatively. You can think with "heart" and you can think with rational judgment. You can also think strategically and analytically, and mathematically and scientifically. These are a few of multiple ways in which the mind can process thought. What are some forms of thinking you use? When do you use them, and why? As a college student, you are tasked with engaging and expanding your thinking skills. One of the most important of these skills is critical thinking. Critical thinking is important because it relates to nearly all tasks, situations, topics, careers, environments, challenges, and opportunities. It's a “domain-general” thinking skill—not a thinking skill that's reserved for a one subject alone or restricted to a particular subject area. Great leaders have highly attuned critical thinking skills, and you can, too. In fact, you probably have a lot of these skills already. Of all your thinking skills, critical thinking may have the greatest value.

What Is Critical Thinking?

Critical thinking is clear, reasonable, reflective thinking focused on deciding what to believe or do. It means asking probing questions like, "How do we know?" or "Is this true in every case or just in this instance?" It involves being skeptical and challenging assumptions, rather than simply memorizing facts or blindly accepting what you hear or read. Who are critical thinkers, and what characteristics do they have in common? Critical thinkers are usually curious and reflective people. They like to explore and probe new areas and seek knowledge, clarification, and new solutions. They ask pertinent questions, evaluate statements and arguments, and they distinguish between facts and opinion. They are also willing to examine their own beliefs, possessing a manner of humility that allows them to admit lack of knowledge or understanding when needed. They are open to changing their mind. Perhaps most of all, they actively enjoy learning, and seeking new knowledge is a lifelong pursuit. This may well be you!
Critical Thinking IS Critical Thinking is NOT
Skepticism Memorizing
Examining assumptions Group thinking
Challenging reasoning Blind acceptance of authority
Uncovering biases
The following video, from Lawrence Bland, presents the major concepts and benefits of critical thinking. https://youtu.be/WiSklIGUblo

Critical Thinking and Logic

Critical thinking is fundamentally a process of questioning information and data. You may question the information you read in a textbook, or you may question what a politician or a professor or a classmate says. You can also question a commonly-held belief or a new idea. With critical thinking, anything and everything is subject to question and examination for the purpose of logically constructing reasoned perspectives.

Questions of Logic in Critical Thinking

Let’s use a simple example of applying logic to a critical-thinking situation. In this hypothetical scenario, a man has a PhD in political science, and he works as a professor at a local college. His wife works at the college, too. They have three young children in the local school system, and their family is well known in the community. The man is now running for political office. Are his credentials and experience sufficient for entering public office? Will he be effective in the political office? Some voters might believe that his personal life and current job, on the surface, suggest he will do well in the position, and they will vote for him. In truth, the characteristics described don't guarantee that the man will do a good job. The information is somewhat irrelevant. What else might you want to know? How about whether the man had already held a political office and done a good job? In this case, we want to ask, How much information is adequate in order to make a decision based on logic instead of assumptions? The following questions are ones you may apply to formulating a logical, reasoned perspective in the above scenario or any other situation:
  1. What's happening? Gather the basic information and begin to think of questions.
  2. Why is it important? Ask yourself why it's significant and whether or not you agree.
  3. What don't I see? Is there anything important missing?
  4. How do I know? Ask yourself where the information came from and how it was constructed.
  5. Who is saying it? What's the position of the speaker and what is influencing them?
  6. What else? What if? What other ideas exist and are there other possibilities?

Problem-Solving with Critical Thinking

For most people, a typical day is filled with critical thinking and problem-solving challenges. In fact, critical thinking and problem-solving go hand-in-hand. They both refer to using knowledge, facts, and data to solve problems effectively. But with problem-solving, you are specifically identifying, selecting, and defending your solution.

Problem-Solving Action Checklist

Problem-solving can be an efficient and rewarding process, especially if you are organized and mindful of critical steps and strategies. Remember, too, to assume the attributes of a good critical thinker. If you are curious, reflective, knowledge-seeking, open to change, probing, organized, and ethical, your challenge or problem will be less of a hurdle, and you'll be in a good position to find intelligent solutions.
STRATEGIES ACTION CHECKLIST[footnote]"Student Success-Thinking Critically In Class and Online." Critical Thinking Gateway. St Petersburg College, n.d. Web. 16 Feb 2016.[/footnote]
1 Define the problem
  • Identify the problem
  • Provide as many supporting details as possible
  • Provide examples
  • Organize the information logically
2 Identify available solutions
  • Use logic to identify your most important goals
  • Identify implications and consequences
  • Identify facts
  • Compare and contrast possible solutions
3 Select your solution
  • Use gathered facts and relevant evidence
  • Support and defend solutions considered valid
  • Defend your solution

Critical Thinking, Problem Solving, and Math

In previous math courses, you’ve no doubt run into the infamous “word problems.” Unfortunately, these problems rarely resemble the type of problems we actually encounter in everyday life. In math books, you usually are told exactly which formula or procedure to use, and are given exactly the information you need to answer the question. In real life, problem solving requires identifying an appropriate formula or procedure, and determining what information you will need (and won’t need) to answer the question. In this section, we will review several basic but powerful algebraic ideas: percents, rates, and proportions. We will then focus on the problem solving process, and explore how to use these ideas to solve problems where we don’t have perfect information.

Learning Objectives

Solve problems involving percents, proportions, and rates.
  • Describing quantities and how they change
  • Write an equivalent fraction or decimal given a percent
  • Find a percent of a whole
  • Calculate absolute and relative change given two quantities
  • Express a relationship as a rate
  • Write a proportion equation given two rates or ratios, solve the proportion equation
  • Determine when two quantities don’t scale proportionally, or more information is needed to determine whether they do
Solve problems using basic geometry
  • Area
  • Volume
  • Proportions, similar triangles, ratios applied to geometric problems
Use mathematical problem solving and estimation techniques.
  • Define and implement a “solution pathway” for solving mathematical problems
  • Calculate sales tax, property tax
  • Calculate flat tax, progressive tax, and regressive tax

Licenses & Attributions

CC licensed content, Original

  • Critical Thinking Skills. Provided by: Lumen Learning Authored by: Linda Bruce. Located at: https://www.coursehero.com/study-guides/collegesuccess-lumen/critical-thinking-skills/. License: CC BY: Attribution.

CC licensed content, Shared previously

  • Critical Thinking. Authored by: Critical and Creative Thinking Program. License: CC BY: Attribution.
  • Thinking Critically. Provided by: The University of British Columbia, Vancouver Campus Authored by: UBC Learning Commons. Located at: http://www.oercommons.org/courses/learning-toolkit-critical-thinking/view. License: CC BY: Attribution.
  • Critical Thinking.wmv. . Authored by: Lawrence Bland. License: All Rights Reserved. License terms: Standard YouTube License.
  • Problem Solving. Authored by: David Lippman. Located at: http://www.opentextbookstore.com/mathinsociety/. License: CC BY-SA: Attribution-ShareAlike.