DinS Written on 2018/4/26
Let’s try something fun. I’ll use calculus to show that why you should work at a 30-minute interval.
First see this graph.
This is a sketch of efficiency function. It describes your working efficiency η with the pass of time t.
In fact psychologists have studied this function for quite some time and there should a real representation for it. Here, however, I’ll be lazy and won’t dig deep into it. All you need to know is that it decreases slowly at the beginning, then round 30 minutes it sharply drops and decreases slowly again(Assumption ①). Maybe sigmoid function can be tweaked to represent it? I hope this matches your common sense.
The x-axis is the time, in minutes. The y-axis is your efficiency, in unit/minutes. Unit is an abstract concept, depends on what particular work you’re doing. Anyway, you can multiply efficiency and time to get how much work you have done in given time t. That’s exactly the area under the curve. In other words we can use integrals to represent it.
Now I need to make two more assumptions. They are meant to make the problem easier and can be discussed later.
Assumption ②: Everyone’s efficiency function is the same.
Assumption ③: Once you rest for certain time, say 5 minutes, you recover instantly, as if you drink some potion and max HP. In math this means you’re back to η(0) again.
With all three assumptions I can prove that why you should work at a 30 minutes interval, hopefully.
Step 1. We can calculate the amount of work without rest. Take an hour as the standard.
Step 2. Draw a graph that represents the case when you work 25minites and rest 5 minutes.
By assumption ③ we know that it looks like this. Now we can write the amount of work with rest.
Step 3. Compare the amount of work you’ve done.
then the rest-strategy wins because you’ve actually done more work during an hour. Using the integral sum rule we can simply a bit.
So the problem becomes whether the first 25 minutes work amount dwarfs the rest amount of work. By assumption ① we know that efficiency drops sharply around 30 minutes. If we are sloppy enough we can conclude that
Which means we should work at a 30-minute interval. To be more precise, work 25 minutes and rest 5 minutes. In fact that’s my working schedule. I have a timer used especially for this. Adopting this strategy will boost your performance without getting backache, neckache and shoulderache.
You may ask: where’s the math? Well, it’s all about math. Math is a way of thinking. Let’s look back the whole procedure. We first build a model for the question, using graphs and notations to make it concise. Next we determine our goal and make some assumptions. After that we gradually approach the goal with reasoning. Finally we arrive at a conclusion. That’s math.
If I have persuaded you to adopt this working style, I’m glad. This shows that thinking mathematically can achieve its effect without crazy formula and notations that scares off the readers.
If I haven’t persuaded you, I can make it more complicated by changing the assumptions. Right now assumption ① is a natural language description. I can use a tweaked version of sigmoid function instead. Then I can actually calculate the number out and do crazy anti-differentiation.
I can also change assumption ③ to make it more close to reality. Say there’s a recovery-rate. How much you recover depends on how much time you’ve rested. This is also a function. Now you have two functions to work with. Quite complicated.
I can still make it more complicated. Assumption ② says everyone behaves the same. We can make each individual a specific efficiency function. If that’s the case we’ve to use matrix when calculating, if you know the trick of linear algebra.
So you see, I can make it hard enough as long as I move closer to reality. But remember math is about abstraction. It’s because reality is complicated that we use math to simplify our reason. Complexity is not the nature of math. We first get simple cases right, and if need calls we gradually move to harder cases. It’s the way of think that makes math fun.