Speaking from my education and experience in engineering, I would like to share some general approaches to problem solving. These should help you with solving problems in math, science, or engineering. The key topics are
A – break a problem into smaller pieces
B – consider a simpler version of the problem
C – consider available information and tools
I will discuss these in detail below.
Break a Problem Into Smaller Pieces
One of the first, and most important, lessons I learned studying engineering was the approach of splitting a problem into smaller pieces, solving each piece, and combining it all together to get the final solution. I would see this lesson repeated during my engineering career.
A typical electrical engineering circuit lab involves being given a circuit diagram, solving the circuit to figure out how the circuit should theoretically perform, and then going into the lab to build the circuit for real and seeing if your experimental measurements match your theoretical calculations. If you did it right, they should be close. They will almost never match exactly, but they should be close.
In my first circuit lab, we were assigned an amplifier circuit and told to find the gain. An amplifier takes an input signal and makes it bigger. The gain is the multiplier by which the signal gets bigger. For example, if you put in a 2 volt signal and get out a 10 volt signal, then the gain is 5, because 2 times 5 is 10. I tried and I tried, but I simply could not figure out the gain of this amplifier circuit. So I went to the TA (teaching assistant) and asked for help. He looked at my work and said something like “hey man, you’re doing this way too hard.” Then he started drawing boxes around parts of the circuit. He pointed to the first box and asked “what’s that?” I answered “that’s an inverting amplifier.” He answers “right, so the gain is -R1/R2.” He pointed to the other boxes and asked similar questions. Then, he explained “now that you know the gain of each individual stage, you simply multiply the individual stage gains together to get the gain of the complete overall circuit.”
He taught me the very important lesson of breaking a problem down into smaller pieces, solving the pieces, and combining it all to get the final solution. I would see this lesson later in my engineering career. In one instance that I recall vividly, I went with a senior engineer to a customer site to show them an RF test setup and how to use it properly. The setup at first was not working properly. The senior engineer wisely stripped it all down to bare bones, and started adding parts one by one, testing after adding each new part – until he found which part was not functioning properly and knew to replace that part.
And in fact, this is how REAL engineering is done. For example, when the military wants a new F35 jet fighter, they don’t turn to one egghead engineer to design the whole thing. No, they turn to a large defense contractor like Lockheed Martin and say “we will pay you this large sum of money to design us a new jet fighter that meets these desired specifications (weapons payload, maximum speed, range, stealth capability, and so on).” Then Lockheed turns around and assembles a huge group of engineers. The person in charge will say something like “you 10, design the airframe; you 10, design the radar; you 10, design the avionics; …” and so on. Within each team, it gets broken down further. For example, the radar team leader might say “you, design the oscillator; you, design the amplifier; you design the matching networks; you, design the digital phase shifters” and so on. The big task gets broken down into small bite sized pieces, and as each engineer finishes their part, it all gets slowly integrated and combined together to produce the brand new jet fighter.
Consider the following excerpt from p.40 of December 2024 IEEE Spectrum magazine discussing microchip layout
The first step is to partition the chip into high-level functional blocks, such as CPU cores, memory blocks, and so on. These large partitions are then subdivided into smaller ones, called macros and standard cells. An average system-on-chip (SoC) has about 100 high-level blocks made up of hundreds to thousands of macros and thousands to hundreds of thousands of standard cells.
Next comes floorplanning, in which functional blocks are arranged to meet certain design goals, including high performance, low power consumption, and cost efficiency … [falling under] a branch of mathematical programming known as combinatorial optimization.
Again, notice the engineering practice of breaking a large problem into smaller, more tractable problems.
Consider A Simpler Version of the Problem
If the problem seems complex and intractable, start by simplifying. Consider the simplest version of the problem that you can think of. How would you solve that simple problem? Can you apply the same or similar methodology to solving a more complex problem? What can you learn from the simple example that can be applied to the full, complex, problem? Start from simple, and slowly add complexity until you are able to solve the full problem which initially seemed too complicated to solve.
Scientists and engineers do this all the time. For example, often a physics problem will stipulate “ignore air resistance” or “ignore friction” in order to direct you to solve a simpler problem with less complexity, in order to practice and gain confidence with the underlying theory being illustrated in the problem. In research, this same approach is often applied. Scientists will seek to control the variables, keep things as simple as possible in initial problem solving stages.
An engineer designing an RF circuit might start by initially ignoring things like smaller parasitics, or effects of heating during the initial design process. Once a rough design has been set, then they would add into the model those smaller parasitics or heat effects and fine tune the design to accommodate those additional complexities.
Consider Available Information and Tools
Especially when dealing with word problems, start by considering what information is available, and what tools you have in terms of known laws or formulas. Read through the problem description, parse it and extract the key pieces of information. For example, perhaps you read the problem and see that the object’s mass and speed are given. Then consider what formulas or laws you have that involve or take as inputs those pieces of information. What about the formula for momentum? The formula for kinetic energy? Both of those formulas would use the mass and speed of an object, which, in this example, was given by the problem statement. Then start building step by step from these elements, using your known formulas and laws to manipulate and process the given information, and move forward to solving the problem to answer the specific question(s) that it posed.
What if you are given that the satellite effectively floats above a fixed point on the Earth’s surface? From that, you know the orbital period must be exactly 24 hours. Then consider your formulas for orbital mechanics that would require knowing the orbital period, and go from there. What if you are given a spring constant and a constant force? Consider those pieces of information, and what formula or formulas use those pieces of information.
Admittedly, the full path to solving the problem will not always immediately be clear, but you can at least start by considering available information from the problem statement, and your available tools i.e. formulas and laws, and start taking a step forward.
If you’d like personalized instruction to help with academic problem solving, try a NO COST one-hour lesson today!