In almost every STEM subject, precise symbols and rigorous application of definitions are extensively used. By comparison, human-human interaction is often fuzzy, flexible, verbose, and ambiguous.
Students who have started studying STEM early on are already used to working with precise symbols and rigorous use of definitions and reasoning. However, for students who first encounter a STEM college-level subject, especially one that is at the transfer level, these common elements of STEM subjects can present a challenge.
A symbol is a concise and yet precise way to represent a concept. For example, the digit 0 represents “zero,” “none,” “nada,” etc. But in some contexts, it can also represent “false.” Yet, in some other contexts, it simply represents “the identity of addition.” As a result, it is important to establish the context in which a symbol is defined. In this example, “0” represents different concepts in algebra, boolean algebra, and abstract algebra.
What is conciseness? It means “no simpler way to denote” something. What about precision? It means “no two ways to interpret” something. Both are important in the use of symbols and names in any STEM subject.
What a symbol represents is seldom construed out of thin air. Symbols relate to each other by definitions. If the knowledge of a STEM subject is a fabric, then symbols are buttons on the fabric, and definitions are the mesh of threads that connect the buttons.
For example, the commutative axiom of algebra (as well as boolean algebra and abstract algebra!) has a definition of
\[x+y = y+x\]This definition makes use of $x$ and $y$ as symbols of variables, $+$ as the symbol of addition, and $=$ as the symbol of equality.
In studying a STEM topic, the first step is to spot definitions that involve symbols. For example, in physics, a statement may say, “Let $v$ be the velocity.” This definition is crucial to understanding other definitions in the same context.
In more complex STEM subjects, there may be many symbols in a context. The human mind can track about 5 to 7 items in short-term memory. Until the definitions of symbols are fully understood and become a part of long-term memory, it can be challenging to keep track of many new symbols that are all being introduced at the same time.
This is where note-taking becomes important. Some authors compare note-taking to developing a second brain. Note-taking is a mechanism to expand the capacity of short-term memory.
In the study of a complex STEM topic, a list of symbols and their definitions and meanings can be very helpful. The creation of the list is just a form of studying. Reading, writing, listening, and speaking all involve different neural pathways. The more neural pathways are involved, the more likely we are to remember, to recall, and to apply something that we learn. This is also why handwriting is considered better than typing for people who are tactile sensitive.
Does a list from the text do the same? Can’t the professor provide a list of definitions?
The effect of reading, spotting definitions, and developing a list as a process is different from that of reading a prepared list. This is because the former is an active process, and the latter is more passive.
Ideally, it is best to read the material to be learned and integrate note-taking into the reading process. This involves reading and writing as two pathways to connect concepts.
A face-to-face lecture is an opportunity to engage the other two pathways in the learning process: listening and talking. Any questions in the notes from reading can be asked in class. The formulation of a meaningful question involves a significant amount of thought. Formulating a question exercises the representation of a concept in one’s mind and how to connect that to the notations and words. The benefits of asking questions are not limited to asking professors questions in a lecture, but also among peers.
A student who reads ahead and understands everything can still benefit from a lecture. A confirmation of one’s understanding is also important because it reinforces the neural pathways involved in the understanding of concepts. Depending on the individual, this confirmation can also trigger the brain’s reward system, which makes learning “fun.”
The converse is someone who did not read ahead before a lecture. The sensation of being overwhelmed by the number of symbols and definitions can trigger anxiety. Anxiety itself triggers the brain to prepare for fight-or-flight. That preparation involves brain chemistry that reduces cognitive capacity. Some individuals may simply “feel bad,” but others can spiral into panic. In panic mode, essentially all cognitive capability is suppressed. Repeated episodes can quickly cause a long-term spiral that affects academic performance and mental health.
Yes. Depending on the individual, it can be a significant amount of time.
Time budgeting, management, and prioritization are all crucial in the learning process. This module focuses on how to use the resource (time); read the linked module on how to plan to make sure the resource is available and used efficiently.
In the context of understanding complex STEM topics, “rigor” means:
In the context of STEM, rigor correlates with precision. By practicing rigor in the learning process, precision in communication is almost automatic. Precision means “expressing to and understanding from an external source without ambiguation.” This is the ability to explain a concept to another person without the other person needing clarification. This is the ability to listen to someone and notice “but there are multiple ways to interpret your sentence.”
The nature of STEM careers requires rigor and precision. Not noticing the ambiguity of units can crash a Mars Orbiter. The lack of rigor makes learning new concepts harder. Many STEM careers involve continuous self-supported lifelong learning, rigor makes this possible.
Rigor and precision in studying STEM subjects do not need to spill over to “life.” One can continue to be easygoing, chill, and tolerant in all other aspects of life.
When people think of rigor, it is usually thought of as the opposite of being curious and having fun.
Scientific rigor often leads to evidence that shows existing theories may not be entirely correct. For example, the phenomenon of starlight bending around the sun is evidence that Newtonian physics is not entirely correct. This would not have been discovered without rigor.
Curios scientists would get energized because they would have thought, “Fascinating, the current theories do not explain this; what can possibly cause this?” The ensuing journey can be gratifying. This example is fictional because looking for light bending was an experiment to confirm Einstein’s theory of relativity.
Another example, this time non-fictional, involves debugging computer program code. A rigorous developer would test code more methodically and extensively than one who uses a few test cases and declares “It works!” Upon the discovery of the computer program failing at least one test case, what are the possible reactions?
The irony is that the rigorous developer is more likely to think, “I was very careful and rigorous when I wrote the code and the test cases, what could I have missed?” This is a curious response. The process of discovering and correcting the “bug” (defect in code) is enjoyable. The reward here is the understanding of why the program does not work. With an understanding of how this program does not work, the developer is likely to be more effective in the future.
The developer who was in a hurry to declare, “It works!”, on the other hand, is more likely to be frustrated and feel negatively about the discovery of at least one defect. This is because the reward is likely to be associated with “getting it done.” There is also a chance that the ego is involved, the initial pride of “look what I accomplished” is partially crushed by knowing the program does not work 100%.
Rigor can refer to the education process leading to the application of rigor, and it can also refer to just the application of rigor.
The latter can be an equity issue because some students have been practicing and applying rigor for years, while others have little or no exposure to rigor. To equitize, some students may benefit from additional support to develop rigor. This leads back to the former kind of rigor in the previous paragraph.
UMich publishes a paper on rigor and equity. All the points are agreeable and valid in this paper.
If we consider student rigor as the y-value in a graph, the UMich article provides ways to change the first derivative, $\frac{dy}{dx}$. However, there is still a lot of reliance on the instructor/professor. A self-sufficient approach is like improving the second derivative, $\frac{d^2y}{dx^2}$. Even a small amount of non-negative second derivative eventually leads to an accelerating increase of the y-value, the ability to apply rigor.
Starting with the verbal-only passing of knowledge before there was writing, technology (and inventions) has been gradually empowering individuals to be self-sufficient learners. Paper, the press, digital medium, the Internet, and now (in 2024) generative AI have all contributed, in extreme acceleration, to the self-sufficiency of learning.
Specifically, LLM Gen-AI (large language model generative artificial intelligence) empowers students to be self-sufficient learners who practice rigor, and learning how to use LLM Gen-AI becomes a crucial component in STEM education. Can LLM Gen-AI replace human professors? Perhaps not yet for some time.
At this point (2024), LLM Gen-AI can take on the role of a personal teaching assistant who meets students where they are. LLM Gen-AI is not without its limitations, and as a result, a professor still needs to guide students on how to make the most of the LLM Gen-AI and address questions and doubts when the needs arise.