"Although professors regard improving critical thinking as the most important goal of college, tests reveal that seniors who began their studies with average critical thinking skills have progressed only from the 50th percentile of entering freshmen to about the 69th percentile."
- Derek Bok
By definition, critical thinking is the "objective analysis and evaluation of an issue in order to form a judgment". Unfortunately, this definition lacks the spirit that should be driven home when it comes to critical thinking.
To engage in critical thinking, a person must first remove from themselves from a pervasive human trait called "bias". Moreover, a significant effort must be taken to step away from "educated guesses" and "common sense" assumptions. This initial phase of looking only at the unabashed truth of a topic can be very challenging. To separate prejudice from known facts/truths - not opinions - is commonly counter to how many of us experience the world.
Once a person's biased world view has been muted, they can then make a focused effort to use meticulous, logical deductions to arrive at conclusions concerning the topic being investigated. There should be no leaps in logical arguments to rush to a conclusion (e.g., "The baby wasn't crying before you entered the room. Now the baby is crying. Therefore, you made the baby cry!"). Thinking critically is all about precision and care.
In this way, college- and university-level courses require students to participate in their education at a level that they may be unprepared for. They must think critically about problems from a perspective not encountered before. No longer can a student cram in the homework an hour before it is due - this will only fool them into thinking that they understand the material. To be successful, a student must develop a thoughtful approach to each concept. Failure to change the mindset from, "I just need to get through this class" to "I want to truly understand this subject" will also make success in college difficult.
In a mathematics course, lack of critical thinking (and true understanding of the prerequisite material) is the downfall of many students. Of all the subjects, mathematics is the ultimate exercise of critical thinking skills. Logical deductions based upon given information must be thoroughly described and adhered to.
My Style: It is imperative that you take time to understand the concepts in any and all of my courses. My exams focus on deep understanding of the material in addition to computational efficiency. When studying on your own, or while doing homework, you should be trying your best to master the material and truly understand the underlying concepts that I focus on in class.