PROBLEM-SOLVING - DEGREES OF UNDERSTANDING

"Before solving a problem, you must first understand it."

A.                        A.    Defining Understanding

B.    Levels of Understanding

C.    More Examples of Understanding from First Principles

D.    Conclusions

A. Defining Understanding.

Understanding is the ability to grasp the meaning of something—whether it's words, concepts, processes, or even the thoughts and emotions of others. For our purposes here, it is to understand the problem. It involves knowing how something works, why it happens, and how to apply that knowledge effectively. Next, before any problems can be solved, it is necessary to understand what is asked or posed. This means understanding the problem as posed, not some variation or misinterpretation of the problem, not some problem you can understand, but understanding what is given.

Students all too often interpret the problem in terms they do understand, and this can be quite different from the actual problem. Students are not unique, as the same mistake is made by us all at some time or other. Thus, we need to consider various levels of understanding from the basic to the most advanced. As in previous chapters, we discuss all problems in general, and although they do apply to technical problems, our discussion is toward problems of all types. Most particularly, we are well beyond math class.

B. Levels of Understanding.

The following hierarchy outlines different degrees of understanding, beginning with a complete lack of comprehension and progressing toward deeper insight and innovation. Note the similarity to Maslow's hierarchy of needs the DIKW pyramid, a model that illustrates the progression of data into knowledge and wisdom (Data-Information-Knowledge-Wisdom), and of course Bloom’s taxonomy of thinking skills. The stages below apply to comprehension of the problem itself, not necessarily about the full package of problem and solution. Comprehension here includes also possibilities toward a solution, such as ideas or pathways.

1.     Misunderstands the problem - any and all reasons, including terms, language, the question asked.

2.     Understands the terms in the problem, but lacks clarity of what is asked.

3.     Understands the terms and the question but doesn’t know what to do – Can restate the problem but lacks a method for solving it.

4. Understands the terms, the question, and possible components:
• Can apply a single, often memorized, method to solve the problem.
• Can combine two or more methods and relevant facts.
• Can recognize transformations, reductions, or simplifications to approach the solution more effectively.
• Understands what parts of the problem are analytical and other parts are social, psychological, and even political.
5. Applies diverse, seemingly unrelated knowledge – Brings insights from different fields to enhance problem-solving.
6. Recognizes broader connections – Sees how this problem relates to other concepts within the same discipline. Understands underlying principles of the problem.
7. Innovates and devises new methods – Creates novel solutions, discovers deeper connections, and pioneers new ways of thinking about the problem.

We can connect here these levels with the traditional topics of knowledge, understanding, and wisdom as follows. Knowledge aligns with levels 1–3: recognizing terms but struggling with application. Understanding emerges in levels 4–5: the ability to apply multiple concepts and integrate insights. Wisdom characterizes levels 6–7: recognizing broader implications, drawing meaningful connections, and innovating solutions. Achieving the seventh level is essential for all of the big problems in every area.

By viewing these levels from another angle or perspective we can also consider levels of problem understanding in the following schemata.

• Understanding as a Process – More than a static state, understanding continuously integrates the unknown into the known.
• Different Degrees of Understanding – It can be thorough, substantial, partial, or even incorrect (misunderstandings or misconceptions).
• Explaining from First Principles – True understanding means being able to explain a concept or problem in terms of its fundamental principles. First-principles thinking allows for deeper problem-solving and innovation because it avoids assumptions and rebuilds understanding from core truths.
• Example: One person may know how to repair a bicycle, while another can explain the physics behind its gear system and leverage mechanics.
• Example, One person may know the procedures of double-entry bookkeeping while another understands the full merits of why it is so effective.

In the problem-solving literature, fundamental understanding is not often mentioned, let alone treated. However, great problem solvers have fundamental understandings as starting points for finding solutions.

Implicit in this discussion is the language of the problem. Beware of vague language and particularly vague terms such as “best”, “optimal”, “minimal”, and the like. The problem statement must make clear what these mean. For example, the words “safest” often works in opposition to the word “fastest”. Nothing can defeat problem understanding like a vague problem description or goals for the solution.

The Tarot Card Analogy

Much like a branching tree, the Tarot offers multiple perspectives, principles, and degrees of understanding. It does not dictate absolute truths but instead provides fragmented pieces of insight. Individuals must interpret and apply these insights in ways that align with their personal growth and comprehension. The Tarot card analogy recognizes varied levels of understanding, and even fundamental principles, particularly in non-analytical problem-solving.

C. More Examples of Understanding from First Principles.

To be an excellent problem solver, it is of great value to thoroughly understand the problem from first principles. That is, more than merely understanding the problem, you understand where it came from and the physical or social concepts underlying the problem. Explaining from first principles means breaking down a concept to its most fundamental truths and then building up understanding logically from there. Here are several examples from an assortment of subjects. The format is the subject, surface-level knowledge, and fundamental knowledge. Sometimes a broad surface knowledge is called “expert” knowledge. For example, your local computer repair man may ably fix your computer, never understanding how it really works. Everyone may be acquainted with the auto mechanic whose technique is to replace parts until the car works again.  In the examples below, we sketch out the first principles’ knowledge, with no attempt to be fully complete. BTW, we take up problems concerning Artificial Intelligence (AI) in #24 below.

Remember, when confronting a new problem at work or school, it is always of relevance, importance, and often of considerable benefit to fully understand the problem, its genre, and principles within which a solution may be found.

1. Electricity and Circuits

• Surface-level knowledge: "Electricity flows through wires to power devices."
• First-principles explanation:
• Electrons move through conductive materials due to a difference in electric potential (voltage).
• This movement of charge is called electric current, and it follows Ohm’s Law: V=IR, where V is the voltage, I is the current, and R is the resistance in a simple circuit.
• Components like resistors, capacitors, and transistors control the flow of electricity by manipulating voltage and current.

2. Rocket Propulsion

• Surface-level knowledge: "Rockets work by pushing fuel out of the engine to move forward."
• First-principles explanation:
• Newton’s Third Law states that for every action, there is an equal and opposite reaction.
• A rocket burns fuel, rapidly expelling high-speed gas downward, creating an equal force pushing the rocket upward.
• The Tsiolkovsky rocket equation (Δv=veln⁡(m0/mf)\Delta v = v_e \ln(m_0/m_f)Δv=ve​ln(m0​/mf​)) explains how velocity change depends on exhaust velocity and mass ratio.

3. Cooking and Heat Transfer

• Surface-level knowledge: "Food cooks faster at higher temperatures."
• First-principles explanation:
• Heat is energy transfer, which occurs via conduction, convection, and radiation.
• Higher temperatures increase molecular motion, accelerating chemical reactions that change food’s structure (e.g., protein denaturation, Maillard reactions).
• Water-based foods can’t exceed 100°C at sea level due to boiling point limitations, whereas frying in oil allows much higher temperatures for different cooking effects.

4. Bicycle Gearing System (from above)

• Surface-level knowledge: "Lower gears make pedaling easier on hills."
• First-principles explanation:
• A bicycle operates on the principle of torque and mechanical advantage.
• A lower gear increases the number of rotations of the pedals per wheel rotation, reducing the force required.
• The chain transfers force to the rear wheel using a combination of gear ratios, optimizing efficiency depending on terrain and effort.

5. Airplane Lift and Flight

• Surface-level knowledge: "Planes fly because air moves over their wings."
• First-principles explanation:
• Wings are shaped to create a pressure difference: air moves faster over the top, reducing pressure (Bernoulli’s Principle).
• The angle of attack directs air downward, generating lift (Newton’s Third Law).
• Additional factors like drag, thrust, and weight must balance for stable flight.

6. Music and Sound Waves

• Surface-level knowledge: "Higher notes come from tighter strings."
• First-principles explanation:
• Sound is a longitudinal wave that travels through air by compressing and expanding molecules.
• A string’s vibration frequency depends on its tension, length, and mass per unit length (wave equation f=12LTμf = \frac{1}{2L} \sqrt{\frac{T}{\mu}}f=2L1​μT​​).
• Increasing tension or shortening the string raises frequency, creating a higher pitch.

7. Computer Processors and Logic Gates

• Surface-level knowledge: "A computer processes information using a CPU."
• First-principles explanation:
• At the fundamental level, computers use transistors, which act as switches controlling electrical signals.
• These transistors form logic gates (AND, OR, NOT) that combine to perform binary operations.
• Complex circuits execute arithmetic, store data, and process instructions using binary code (1s and 0s).

8.     Why Do Bridges Have Arches?

• Surface-level knowledge: "Arches make bridges stronger."
• First-principles explanation:
• A bridge must support weight by distributing force.
• An arch shape naturally redirects compressive forces along its curve, reducing bending stress.
• This allows materials like stone (which withstands compression but not tension) to support heavy loads efficiently.

9. How Do Gyroscopes Help Stabilize Vehicles?

• Surface-level knowledge: "Gyroscopes keep things balanced."
• First-principles explanation:
• A spinning gyroscope resists changes in its axis due to angular momentum.
• When external forces try to tilt it, the change occurs perpendicular to the applied force (precession).
• This property stabilizes motorcycles, drones, and even spacecraft.

10. Why Do Airbags Inflate in a Crash?

• Surface-level knowledge: "Airbags protect passengers."
• First-principles explanation:
• When a crash occurs, the car experiences rapid deceleration (high negative acceleration).
• Sensors detect this and trigger a chemical reaction (often sodium azide decomposing) to rapidly produce gas, inflating the airbag.
• The airbag absorbs kinetic energy, reducing the force on the passenger by increasing the time over which deceleration occurs (impulse-momentum theorem.

11. Why Do We Need Sleep?

• Surface-level knowledge: "Sleep helps the body recover."
• First-principles explanation:
• The brain uses sleep to clear metabolic waste via the glymphatic system.
• Neurons undergo synaptic plasticity, strengthening important neural connections and pruning unnecessary ones.
• Sleep cycles (REM and non-REM) regulate memory consolidation, hormonal balance, and immune system function.

12. Why Do We Shiver When Cold?

• Surface-level knowledge: "Shivering helps warm the body."
• First-principles explanation:
• The body maintains homeostasis, keeping core temperature around 37°C.
• Cold temperatures reduce enzymatic efficiency and slow biological reactions.
• The hypothalamus detects the drop in temperature and triggers involuntary muscle contractions (shivering), generating heat via ATP breakdown (thermogenesis).

13. Why Do Onions Make You Cry?

• Surface-level knowledge: "Onions release chemicals that irritate the eyes."
• First-principles explanation:
• Cutting an onion breaks its cells, releasing sulfur-containing compounds.
• Enzymes convert these compounds into syn-propanethial-S-oxide, a volatile substance.
• When it reaches the eyes, it reacts with water to form mild sulfuric acid, which irritates nerve endings and triggers tear production to dilute and wash it away.

14. Why Do Hard Drives Fail Over Time?

• Surface-level knowledge: "Hard drives wear out eventually."
• First-principles explanation:
• Traditional hard drives use spinning magnetic disks (platters) and mechanical read/write heads.
• Moving parts experience friction and mechanical stress, leading to wear.
• Magnetic fields degrade over time, causing data corruption.
• Solid-state drives (SSDs) have no moving parts, but their flash memory cells degrade after a limited number of write cycles due to electron leakage.

15. How Does Internet Data Travel?

• Surface-level knowledge: "Data moves through cables and wireless signals."
• First-principles explanation:
• Digital data is encoded into binary (1s and 0s).
• It is transmitted as electrical signals (fiber-optic cables use light pulses, while wireless signals use electromagnetic waves).
• Data packets are broken down, sent across networks using TCP/IP protocols, and reassembled at the destination.
• Routing algorithms determine the most efficient path for data transfer, optimizing speed and reliability.

16. Why Does Inflation Happen?

• Surface-level knowledge: "Prices go up over time."
• First-principles explanation:
• Inflation occurs when the supply of money increases faster than the supply of goods and services.
• More money in circulation reduces its purchasing power (supply-demand imbalance).
• Factors like increased production costs, supply chain disruptions, and consumer demand shifts also contribute.
• Governments control inflation via monetary policy (interest rates, money supply regulation).

17. Why Are Some Products Cheaper in Bulk?

• Surface-level knowledge: "Buying in bulk saves money."
• First-principles explanation:
• Fixed costs (packaging, transportation, marketing) are spread over more units, reducing per-unit cost.
• Bulk purchases reduce transaction costs for suppliers.
• Businesses encourage bulk sales to increase inventory turnover and predict demand more reliably.

18. Why Do Ice Cubes Float?

• Surface-level knowledge: "Ice is lighter than water."
• First-principles explanation:
• Water expands when it freezes due to hydrogen bonding, forming a crystalline structure.
• This structure has lower density than liquid water (9% less dense), making ice buoyant.

19. Why Does Salt Melt Ice?

• Surface-level knowledge: "Salt lowers the freezing point of water."
• First-principles explanation:
• Salt dissolves in water, breaking into ions (Na⁺ and Cl⁻).
• These ions disrupt the formation of ice crystals, preventing the water molecules from bonding into a solid structure.
• This lowers the freezing point below 0°C, keeping water in a liquid state longer.

20. Why Do People Procrastinate?

• Surface-level knowledge: "People avoid tasks they don’t like."
• First-principles explanation:
• The brain weighs short-term rewards higher than long-term benefits (present bias).
• Difficult tasks trigger anxiety, leading to avoidance behavior.
• The prefrontal cortex (rational thinking) competes with the limbic system (impulses), often leading to delay in action.
• Breaking tasks into smaller, manageable steps reduces cognitive resistance and increases motivation.

21. Why Do Optical Illusions Trick Us?

• Surface-level knowledge: "Our brain misinterprets images."
• First-principles explanation:
• The brain processes visual information using assumptions based on past experiences and context.
• Optical illusions exploit these shortcuts by presenting conflicting depth cues, motion cues, or contrast effects.
• The brain resolves ambiguity by filling in missing information, sometimes creating incorrect perceptions.

22. Why Do Democracies Tend to Have Political Gridlock?

• Surface-level knowledge: "Politicians can’t agree, so things don’t get done."
• First-principles explanation:
• Competing Interests: Democracies are designed to balance different viewpoints, ensuring no single group dominates. This means policies must accommodate diverse interests, slowing decision-making.
• Checks and Balances: Many democratic systems (especially those with separation of powers) deliberately introduce friction to prevent hasty decisions and abuses of power.
• Incentive Structures: Politicians respond to their voters, interest groups, and party leaders, sometimes prioritizing reelection over compromise.
• Majority vs. Minority Tension: Majority rule ensures decisions reflect the largest group’s interests, but minority rights must also be protected, which leads to negotiation and delays.
• Polarization & Media Influence: Modern media and social platforms amplify ideological divisions, making compromise politically risky for elected officials.

This first-principles approach shows that gridlock isn’t just about individual politicians failing to agree—it’s an intentional feature of democratic systems designed to balance power, prevent authoritarianism, and reflect diverse perspectives. However, excessive gridlock can also lead to inefficiency and public frustration.

23. Why Do Social Movements Rise and Fall?

·        Surface-level knowledge: "People protest when they are unhappy, but movements eventually lose momentum."

·        First-principles explanation:

o   Grievances and Collective Identity: Social movements emerge when a group perceives injustice or inequality. Shared identity (e.g., racial, economic, or ideological) unites participants.

o   Resource Mobilization: Movements require leadership, funding, media attention, and organizational structures to sustain momentum. Without resources, even passionate movements struggle.

o   Political Opportunity: Movements grow when political conditions allow for change (e.g., weak opposition, sympathetic leadership, or crises that expose systemic issues).

o   Framing and Narrative: Successful movements shape public perception by framing issues in ways that resonate emotionally and logically.

o   State and Institutional Response: Governments may accommodate, co-opt, or repress movements. Excessive repression can radicalize activists, while partial concessions may demobilize them.

o   Internal Divisions and Burnout: Over time, disagreements on strategy, leadership conflicts, or activist fatigue can weaken movements.

o   Normalization or Co-optation: If a movement’s ideas become mainstream, it may dissolve as its goals are achieved—or be absorbed by political parties and institutions, losing its original radical edge.

This first-principles approach explains why movements aren’t just about protests in the streets—they are shaped by deeper forces like political structure, organizational strength, and public perception. This, you could say, is the political signature of a people, and it differs from society to society. This, in turn, explains why some societies are not suitable for Marxism while others require a strong-man control.

24. How to Use Artificial Intelligence (AI)?

·        Surface-level knowledge: "AI engines carry vast knowledge and can serve to answer my questions."

·        First-principles explanation: AI engines actually report a the first of preponderance of information of compiled from searching various websites, and sometime books on a given subject. The tools include Bayesian statistical methods, and do not always discriminate what is correct from not correct. Moreover, it is mostly historical and missing the recent developments.

o   Bias. An issue of algorithms used to compile information

o   Laziness. Reliance or worse, dependence, on output from AI without due diligence of checking and understanding what has been reported.

o   Inaccurate data analysis. Data compiled by AI engines is not always accurate, primarily due to the algorithms and sources used.

o   Vague. Asking questions using generalized or vague language often results in answers off the mark from good intentions.

o   Ethical absence. The two most crucial aspects of human nature are ethics and morality, but it is difficult to combine both of these into artificial intelligence.

By understanding these first principles of AI and a few others, you can construct problems that stump the AI engine into giving incorrect answers and possibly identify that incorrect answers have been previously generated.

D. Conclusions.

The bane of all problem-solvers is certainly lack of knowledge and methods, but just as important are the abilities to cut through misconceptions, misunderstandings, disinformation, misinformation, red herrings, vagueness, what is analytical, what is missing, and the like. Understanding precisely what the problem is must precede all solution methods. Explaining from first principles allows us to go beyond surface-level explanations and understand how things work at a fundamental level. This method is used in science, engineering, economics, and everyday problem-solving to break down complex concepts into their most basic truths, component, to set the problems read to be solved.

© 2025 G Donald Allen