Quoted from Oxford Languages, “to understand” is to “perceive the intended meaning of (words, a language, or a speaker.” This definition is deceptively simple. This article looks into the various forms of understanding in the context of a technical subject matter.
A word is a basic unit to facilitate the communication between individuals. Words form sentences. A sentence connect the words in a specific way to express how the concepts represented by the words are connected.
The first question one must ask is the meaning of a word. Let us examine an exmample, “Vlad told Ahmed that he did the homework just before the du date.” What is “he” referring, Vlad or Ahmed? Which homework is being referenced? Of course, the context of this sentence may help clarify and answer these questions. However, reliance of context is a source of confusion and misunderstanding.
In technical communication, communication is even more difficult because some technical terms have colloquial meanings. For example, “every member of a super class is inherited by a sub class.” Every single word of this sentence is a simple word, but the concept being represented is not (at least not to someone who is starting to learn object oriented programming).
What exactly is a “class”? We know what a class is in terms of school, or in society. But what constitute a class in this specific context?
What does “super” and “sub” mean? There is a strong implication that the words are opposites of each other, but in what way are they opposite?
What does “inherit” means in this case? We understand inherit in the context of property, belongings and money from an ancester. We may even understand “inherit” from the genetics perspective. But what does inherit mean in this specific context?
What is a “member”? We understand the word member in the context of a club, a family unit and other groups of people, but what exactly is a member in this context?
Because all the words are common, some people may read the sentence and think, “I understand this sentence.” However, it is possible for someone to “understand each word” but not to “understand the sentence.” In fact, this may be one of the reasons why some people find certain classes difficult!
A definition link words in a specific way. One potential problem with definitions is that one definition often rely on words that are defined in other definitions. In other words, there is a complex web of definitions.
However, in a technical context, the definitions do not form a loop. In other words, there is a bottom to the rabbit hole. It is essential to keep digging until the lowest level of definition is found and understood.
Having a “somewhat” or “sort of” understanding of the words in a definition may seem okay and time efficient at the time. However, a shaky understanding of the words used in a definition often cause problems later on in the form of not being able to problem solve.
As mentioned earlier, words used in technical definitions may or may not have the same meanings in colloquial contexts. It is important to make sure that a word has or does not have a specific meaning in a technical context. For example, in the definition of “a byte is a sequence of eight bits.” Does one “bit” mean a small bite, or does not it have a specific meaning?
In a definition, concepts can be connected merely sequentially. For example, an assignment operation may be described as “first compute the value of the right value, then use the computed right value to update the left value.” In other words, the definition specifically spell out the ordering of steps. Ordering can also apply to objects/nouns.
In certain cases, what may look like just a sequence may have a deeper implication. Certain kinds of ordering is “arbitrary” in the sense that in order to standardize, a specific ordering needs to be chosen. This includes the “big endian” definition, in which the most significant word of a multi-word integer has the lowest memory address.
In other cases, ordering is actually the consequence of dependency. Dependency is an important concept to grasp because it is a key ingredient to effective problem solving. For example, the description of a ternary expression may be explained as “the first expression is evaluated, if the value of the first expression is true, then the second expression is evaluated and the value of the second expression becomes the value of the ternary expression. Otherwsie, if the first expression has a value of false, then the third expression is evaluated and the value of the third expression becomes the value of the ternary expression.”
In this definition, the order is the result of dependency. The second or the third expressions cannot be evaluated first because which one becomes the value of the ternary expression depends on whether the first expression is true or not.
A sequence is a way to connect steps in a process or items in an ordered list. Dependencies is also a way to make connections, but based on wnat is necessary for what else.
Logical connections are special. On one hand, there are classes in which logical reasoning is the only topic, and there is formalization of logical reasoning using special symbols and notations. On the other hand, there is also “common sense” application of logic.
Logic is a formal method to communicate and work with conditions. A condition, in return, is something that can be either true of false. For example, “having a good crop” is a condition, it is either true or false. “Good soil is necessary for a good crop” is also a condition because it is either true or false. And finally, “having good soil” is also a condition.
Logic specifies the mechanism to work with conditions. In this example, if “having a good crop” is true (possibly by observation), and “good soil is necessary for a good crop” is also true (based on science), then logic deduces that “having good soil” must be true.
Unlike other branches of mathematics, logic proficiency has more to do with whether a person exercises reasoning than formal education. Most natural languages contain enough vocabulary to express conditions and logical operations. Furthermore, the concept of condition dependency is also naturally occuring.
Logic is important even in definitions because it is crucial to understand how a definition condition is defined. Such definitions are often used to qualify a term. A common template of such a definition is “something is blah if and only if yada.” Here, the term “if and only if” is crucial, it must be differentiated from “if” as well as “only if”.
One can view logic as a way to connect conditions.
The application of logic is a form of derivation. However, in most technical fields, there are other forms of derivation. For example, algebra is a system of symbolic derivations.
A system of derivation is a way to connect representations, often in a rigid and formal format. For example, in algebra, \(x+y=y+x\) is an axiom (addition is commutative). However this axiom does not only work when \(x\) and \(y\) are simple constants, this axiom works as long as \(x\) and \(y\) represent a numerical value. \(x\) and \(y\) can, in return, be complex expressions themselves.
An important part of derivations is when a derivation can occur. This relates to the previous section that discusses logic.
An abstraction connect a common concept to two or more instances that share the common concept. Abstraction is important in STEM because the ability to extract common concepts to model, describe, or specify specific instances is what the basis of STEM.
For example, consider two concrete items, one is a wine glass, the other is a coffee mug. Although the shape, material, and intended use are all different, the two items share many common properties. They can both contain liquid intended for human consumption.
At the core of abstraction are two questions: what is significant as a property, and what is not. Let us look at another example. This time, we look at the abstraction of “writing instrument.”
It is obvious that a traditional pencil, a ballpoint pen and a fountain pen are all examples of an “writing instrument.” The mechanism and color do not matter. But how far can we push this abstraction? Is a digital stylus considered a writing instrument? In the case of a touch screen, is a finger tip also considered a writing instrument? If not, why not?
The criteria of “membership” of an abstraction is, therefore, crucial to the understanding of an abstraction. For example, in the “writing instrument” example, perhaps the requirement of “non-electronic” is a requirement (at least in some contexts). Or, perhaps the requirement of using a hand and fingers to hold the instrument is also a requirement.
All members of an abstraction are connected by the sharing of certain traits, properties or attributes. Likewise, all non-members of an abstraction also share the lack of certain traits, properties or attributes.
Abstractions can be stacked, meaning that two or more abstractions can be further abstracted to a even more abstract concept.
Abstractions interact with other mentioned types of connections. For example, rules of transformation (algebraic rules) apply to certain types of values. The abstractions of integers and real numbers is essential to determine whether certain transformation rules can be applied.
What is referred to as “depth of understanding” is a combination of two measurements. The first one is the number of levels of connected concepts that can be tracked in the mind, and the second one is the granularity of the bottom-most layer.
The depth of understanding is important in problem solving. You can think it this way. Let’s say we are talking about a car mechanic. A car mechanic who has a “shallow” understanding looks at large groups of components of a car and has a vague idea of how they work together. A car mecanic who has a ’deep” understanding looks at the small components, understands the physics and chemistry, and piece together how a car functions from these small grains of knowledge.
A car mechanic who has a shallow understanding of a car cannot diagnose problems and propose solutions as well as a mechanic who has a deep understanding of how a car works. A hallmark of someone who has a deep understanding of a topic is the ability to explain at any level, and take into consideration a wide gamut of families of theories and sciences.