How We Learn: Generalizing Special Cases
How many of you remember the first time that you took your car to the empty parking lot and practiced shifting gears? One by one, you progressed from going at 5 mph to 15 mph. Then you practiced your parking by slowly turning the wheel hand over hand as you squeezed in the car between the lines. And then, you slowly eased into the nearby street and practiced accelerating and decelerating by the stop sign. And that’s it! After that, you are ready to drive in the real world with a bunch of other cars on the road.
Of course, as it turns out, driving is not actually that simple, but the process of starting in a simple situation and gradually being exposed to more complex situations is one that we are quite familiar with.
In this post, I seek to demonstrate examples of where this process shows up in our day to day lives.
mathematics
Let’s start with the basics, shall we? 1,2,3,4,5, etc. Most of us started learning math by learning how to count with natural numbers. There is one pencil over there. There are two people in this room. There are three dogs here, etc.
Of course, fast forward a year or two, and all of a sudden, zero!? Negative numbers?! What does that even mean? How can we point to something and be like there is zero of that thing? And negative objects? Don’t even get me started there.
As we progress through math, we start to understand integers, fractions, and all real numbers in general. Yet, when we first start with learning operations such as addition and subtraction, we are first exposed to only a special subset of real numbers. In this case, natural numbers. As we progress and get more comfortable with natural numbers, only then does it make sense to learn how to add negative numbers and fractions.
Of course, by the time that we get to algebra and calculus, it is more convenient for us to use our general knowledge of numbers to do operations than it is for us to remember the distinct differences between natural numbers and fractions. Or as our math teachers will tell us, it is better to understand the process than it is for us to remember a bunch of different rules for different situations.
music
Whole notes. Half notes. Quarter notes. Treble clef. Bass clef. For many of us, when we first learned how to read music, we learned it in distinct cases like this. We learned how to play the right-hand part on the piano by finding where middle C is. After starting out learning to hold a whole note out for 4 beats, we press the key down and count. And we repeat for half notes, counting to 2. And quarter notes.
By the time that we learn these notes, it is likely ingrained into our memory. But wait, what are dotted half notes? And how can we have notes that have more than 4 beats? This is when understanding the process starts to come into play.
As we progress through our understanding of music, we start to pick out patterns and understand the underlying mechanisms that determine the rhythm of a piece. When we understand why a whole note has the value that it does, then we are no longer constrained by our ability to remember the values or our ability to retrieve them from our memory every time we encounter a differently valued note.
So the next time that you catch yourself struggling to determine where to place the left-hand in relation to the right-hand in Chopin’s Fantaisie Impromptu, just remember to break it down into its components. Look at the measure and see how the values fit into each measure. That way, you won’t have to remember what every little note value is worth again.
language
Remember when you learned how to form the past tense in English? You know, -ed? As it turns out, the same process of generalizing special cases applies again here.
When we first learn how to form the past tense of words, we begin by looking at a few specific cases that align with the rule. We begin by memorizing how to spell words such as talked and walked. After a while, we start to pick up the patterns inherent in the language. Next thing you know, generalizing the past tense becomes second nature.
Of course, there are exceptions to the rule. By and large though, it is far easier for us to remember how the rule works of changing a word into the past tense, than it is for us to remember every single word’s spelling.
conclusion
Remember the driving example that I had at the beginning of the post? Well, as many of you know, learning to drive involves learning how to navigate driving on icy roads, as well as other surfaces.
One interesting pattern to note is that, when it comes to learnability, it is easier for us to begin with a few special cases. As we progress, when we begin to complicate things, we start to recognize patterns and rules that guide us. In that sense, for applicability, it is easier for us to use our general knowledge to figure out what to do in specific situations.
Of course, I could cover a bunch of other subjects where you could note these same patterns, but if you only took one thing from this article, it should be this: Learning is a continual process, one where the situations keep on getting more complex. If you want to learn best, try to understand the underlying processes and use them to simplify concepts.