My friend gives private mathematics lessons to young pupils, mostly high school students who need some help in their curriculum math. Here is a story about a student of him.
My friend asks him: how much is 5+6? The guy looks at the roof, then back at my friend and says: “umm..13.” My friend encourages him to think a bit more and the guy looks around and says “umm..10? no! 11! NO! 12?”. The guy was around 14 years old (going to the 8:th grade in the Finnish educational system). What was a possible reason for this behaviour? The guy didn’t know what + means? He didn’t know what 5 means? He was joking? He has ADHD? (I guessed the latter two when I heard about this; my friend was sure the student wasn’t joking.) No, dear reader, none of the above was the reason. Later, after my friend made cautious warnings to the pupil about his future and learning success, the student started to strive for his goals and a mirracle happened: after some time he had no problem solving much more advanced mathematical problems than 5+6.
In the beginning the student was lazy to think.
Not only the student was Lazy. We all are to some extent. I do recall incidents from my own postgraduate mathematical studies, in which, I confess, I behaved in an analogous manner.
Imagine that you are reading a blog post and the writer asks you to visualize the curve which is parametrized by
How likely are you to continue reading without actually putting any effort at visualizing the curve? If I then state that the curve is compact, how much effort do you put at checking it?
If you are told that Google reads all the data from your hard drive when you are logged in, do you accept this (or postpone judgement) without giving a thought to whether this statement is ridiculous or not? If you are planning a vacation trip to Paris, how much time will you spend thinking on what is the optimal way of finding a suitable hotel before starting to search? If you are lost in London and you ask someone how to get somewhere, do you blindly follow the instructions or do you try to figure out where are you on the map and where is South?
Commonly, I conjecture, people are lazy to perform such cognitive tasks, if not pressed. Eliezer Yudkowsky explains a phenomenon which he calls Guessing the Teacher’s Password: The students are tought in their physics class that heat behaves “according to heat conductivity” and other related concepts, and when they are posed with a difficult problem, they try to guess, as opposed to think which of these concepts is the correct explanation right now. Worse, Eliezer’s example shows that the students do not even understand what it means for heat to behave “according to heat conductivity”, they only learned to verbalize the sentence “heat behaves according to heat conductivity” in proper situations. (In fact, I presume that many students do not even understand what it means to understand that heat behaves “according to heat conductivity”; sorry, students.) Note: Eliezer’s point was somewhat different from mine although a parallel one; I recommend reading his post.
The same phenomenon is seen very evidently in high school math. The teacher shows a model solution to the students (“To solve a problem of this format you first take a derivative of the function, then you find its zero points, then you substitute them into the original function and calculate the answer…”). The students memorize that model solution and then carefully copy it into their exam sheet and get the better grade the carefulier they copied it. But what if the exercise is of a different format (but same content)? What if the function is denoted by h but not f? Then they are lost!
Why does this kind of teaching exist? Because it allows the students to be Intellectually Lazy. It is undoubtly very hard to teach the students to think instead of teaching them to memorize model solutions. Children learn routinely to memorize things, at least in Finnish schools, because in almost any other subject (like languages and history) the difference between memorizing and understanding is not as huge as in math, so the teachers can get away with it. Most students are not willing to adopt new ways of thinking (they are Lazy) and so they rather say “I am not talented in math, I am talented in languages!” rather than giving a math problem a chance and trying to think for five minutes.
You may ask “So why bother to be non-Intellectually-Lazy? If it is so hard, can’t we just do things the easy way?”. My responce is that the benefit of overcoming this Laziness is HHHHUUUUGGGGGEEEEE!
I noticed it first when I was 9 or 10 years old. As a homework, we had to memorize the multiplication tables. I suppose it is still a standard part of the elementary school education that the children have to memorize the multiplication tables (how horribly embarrasing!). Well, when we had the first test (there was eight of them: the n:th test tested the multiplication by n+1), I had forgotten to accomplish my homework. I totally completely forgot to do it. So I experienced a panic-like feeling, as the teacher arranged the test suddenly without a prior notification. She gave all the students empty sheets of paper and then she started pronouncing outloud questions to which we had to write answers down. And so she went: “two times four”, “two times seven” and so on. Fortunately I knew what multiplication means. So I knew that multiplying x by two is the same as x plus x. Fortunately again, the teacher gave us enough time so I could calculate x + x and then inscribe the solution into the sheet.
What delightful moments! I didn’t have to do any homework, but I still got all correct! What an indescribable pleasure I experienced! Since that first test I didn’t look at the multiplication tables out of principle! I could manage in a test without studying at all! Subsequently I passed all the eight multiplication tests with good marks (I don’t remember the exact outcomes and would be very surprised now if all of them were correct, but good outcomes they were nontheless). As a matter of fact I have never memorized the multiplication tables in my life, which sometimes shocks people who know that I go around telling that I am a professional mathematician.
Note that in the example above, being non-lazy to calculate, I actually saved time. Similarly, in high school math I barely ever looked at the model solutions. I studied the math and then I solved the problems myself. Studying the math took definitely more effort at the instant of studying, and solving the “right format” exercises might have been be easier (but certainly much less fun!) by copying a model solution rather than thinking, but what I gained in exchange to that effort is that I was able to solve a problem of any format as long as it fell into the same theory, I was quick in them, I made less conceptual mistakes (but maybe more misprints) and I also saw why math gives correct solutions. And the latter was the main reason I got inspired by math. I am not surprised the tiniest bit to see that “math is boring” to many students, if they do not understand why math works.
Finally it turned into an upside-down laziness for me. Once I realized that understanding and using multiplication saves me from doing homework, I got more and more motivated. Mind you, that wasn’t my main motivation, but a motivation nontheless. Thus I started being non-lazy in thinking (partly) because I was lazy in non-thinking. What a strange loop!
It is like buying one high quality item which lasts long instead of buying low quality items once a month. Even if the high quality shop is located five miles further (the money spent being the same in the end of the day). This habit of mine very quickly became the reason why I was labelled “mathematically talented”.
As the examples in the beginning of this post suggest, I am willing to apply the concept of Intellectual Laziness to a broad range of human activity. If this inspired also you, dear reader, next time when you find yourself in front of a non-trivial task, remember this post, pause for a moment, and think more than you otherwise would “What is the best way to accomplish this?”, “What do I gain if do not do it in a conventional way?”, “What opportunities there are?”…and share with me the journey of overcoming our Intellectual Laziness.
1. Show that the curve in the beginning of the post is compact.
2. Show that the statement about Google in the beginning of the post is not true. (Hint: the world would be a mess otherwise.)