Critical Thinking

(Background pic is a statuethat is actually on my bookcase in my home office)

I want my students to think about a problem and ask themselves a series of questions as they work to find a solution: What question is being asked? What do I already know? What do I need to know? Have I solved similar problems before?  Could I modify the problem to solve a simpler version first? Does my answer make sense?  

Since I am preparing students for the future, I must help them to think critically and apply what they know by offering interesting examples and applications in order to motivate them to learn. In upper-level courses, students engage more with the language of mathematics by writing proofs or explanations; in lower-level courses students focus on computations and the concepts behind those calculations. Many of the activities I prepare for my courses involve using concepts in a relevant application. For example, students in linear algebra explain the application of solving systems of linear equations to determine the stoichiometric coefficients in a chemical reaction. I usually witness many students grow in their writing abilities by the end of the semester.

Next, I describe three examples: one from calculus 1, one from finance, and one from statistics. After I have gone over the derivative rules in calculus 1, I divide a graph and its derivative into several pieces. I purposefully provide each student with only a partial description of the graphs or the function or its derivative. Students work in groups to piece together the information and produce the original graph and its derivative. In this activity, students are reviewing definitions, concepts, and applying critical thinking by arranging the pieces into a coherent whole. There are also process skills involved such teamwork and time management. 

I co-taught an experimental class with a finance faculty at Florida Southern College, James Farrell, called Math Behind Finance (MAT/FIN 3998). Students spent several class periods analyzing steps and deriving the formulas shown in their text. Students used critical thinking in checking each step and measuring whether the assumptions being made were valid. These steps involved knowledge of non-trivial calculus and statistics. Critical thinking is also used when linking the mathematical steps and assumptions to real world implications, such as what the calculations imply about possible trading strategies. 

In statistics, students explore how sample data can indicate information about a population. I often have students run simulations to collect random samples, then use that data to make calculations. For example, I create a small population of 16 fish in a tank, and collect random samples of size 4. The question is “what is the mean length of the fish in the tank?”. To answer this question, students could only use the information collected from the random samples. Each student used a random number generator to select a simple random sample of size 4, then calculated the mean length of their sample. Each student then came to the board and drew a dot on a number line corresponding to their results. During this process, students are asked to explain patterns they recognize and to ponder the reason behind it. Once we describe concepts of probability, I can return to this graph and link it to the theoretical notions that are describing the behavior of sample means. Below are the drawings that were on the board.

Student Evaluations of Critical Thinking

From my student evaluations, my students overall feel they engage in critical thinking in my classes. While my statistics courses have been a separate story of engagement, all courses rate above 6 after 2018.  Below are my overall student ratings for the question, “The professor challenged me to think critically and/or creatively about material in the course.”

The question is rated from 1 - 7, with 7 being strongly agree, 4 being neutral, and 1 being strongly disagree. 

The line graph shows that students in Elementary Statistics felt they were engaging less in critical thinking during the COVID semesters than before COVID. This is likely due to the lack of group work and activities I would have normally used to incorporate critical thinking. After the fall 2021 semester, the averages began to increase once again as we transitioned to in-person teaching.

Below is a table of the student responses to the prompt, “The professor challenged me to think critically and/or creatively about material in the course”  across each class and semester taught. The question is rated from 1 - 7, with 7 being strongly agree, 4 being neutral, and 1 being strongly disagree. 

I end this section with some comments I have received in my evaluations that concern student's ability to think critically.