HOW WE TEACH
FIRST PRINCIPLES
The goal of learning includes finding simple ways to perform our tasks. That is why math teachers spend considerable time finding and teaching efficient methods to their students. The good intention is for their students to solve the problems with not just accuracy but also with speed.
There is one caveat, though. Many students end up only learning the short method. They start to think that math is all about knowing how to perform the shortest procedure but not the search for it.
Multiplication vs Repeated Addition
Here is the simplest scenario to illustrate this point. We gave a student a problem which was ahead of what she was doing in school:
Jane bought 8 pencils. Henry bought three times as many pencils as Jane. How many pencils did Henry buy?
She realised that the working "8 times 3" is required, so she checked her ruler which has the times tables printed on it.
Teacher: Hey, don't look at that.
Student: But I haven't memorised the 8 times table.
Teacher: Then you do 8+8+8. Multiplication just means repeated addition.
Student: But that way is so difficult...
Teacher: Do the addition first, then memorise it later.
Student: (after a while) 24.
Teacher: Ok. Now memorise it. 8 times 3 is 24.
It would be easier if this student already knew the 8 times table. But since she didn't, it became a good opportunity to teach her how to be resourceful: Do we know the times table? No. But multiplication just means adding repeatedly. It is harder. Yes. But do we have any better idea? No. So we do the repeated addition...
This mindset of using just the basic principles - in other words, first principles - will become a useful habit in learning math. It means that whenever we solve a problem, we are always trying to make the best use of our limited prior knowledge. We may not come up with the best solution, but it is definitely the best we can think of at that moment. Good things come naturally from doing that. We learn new things faster because our brain is all warmed up. We can also make complicated concepts simple by tracing them back to their simplest building blocks...
However, it generally requires more effort to use first principles, so some children may resist it at first. But we must teach them the value of grit in problem-solving, and get them prepared before they face harder challenges later. (Other than helping in math, grit is also "the secret to outstanding achievement", argues psychologist Angela Duckworth.)
High-ability Students Who Underachieve
Your child may not find 8+8+8 difficult. But "difficult" is a relative concept. How does your child respond when they encounter a problem that is "difficult" to them?
It is not uncommon to see even students from the Mensa club underachieve. As problems become harder, and children learn from teachers who only teach shortcuts, even a high-ability child can lose the ability to think from first principles - they only want to work on a problem if they know the shortest way.
Then, there is also a small group of high-ability students who underachieve because of the opposite reason - they neglect to learn efficient methods and over-rely on first principles.
So, how do we juggle between using first principles, and learning the shortcuts?
P5 Boy from Gifted Education Programme
Firstly, there are children who can do both, and they thrive in math. Let us see what we can learn from this student, a P5 boy in the Gifted Education Programme.
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This is a homework question from the gifted programme. Like most children faced with a challenging problem, he was unable to immediately come up with a nice method. Nonetheless he did whatever he could and found the correct answer - a common feat among children who think from first principles. But he didn't just stop there. He also wanted to learn the "proper working". Here is his next message about half an hour later. He already worked out a better working on his own - and still asking to see if there is a better way.
Preparing for Math Olympiad
When this boy was preparing for math competitions, he continued to learn this way - use first principles, then find a better way later. In the end, he managed to get into the national top 30 in the National Mathematical Olympiad of Singapore (NMOS).
Rather than just telling you his achievements, it is more meaningful to tell you how an 11-year-old in the national top 30 learns. Here is his working paper when he was learning this Annual Mathlympics question (Q28 from year 2016 Final Vault Section):
A very long walkalator moves at a constant speed of 1.5 m/s. Grandma step onto the walkalator and stands. Grandpa steps onto the walkalator two seconds later and continues to walk on it at 1 m/s. Two seconds after that, Grandson reaches the start of the walkalator. He did not get on the walkalator but jogs beside it at 2 m/s. After a certain time, Grandpa is ahead of Grandma who is exactly halfway between Grandpa and Grandson. Find the distance between the start of the walkalator and Grandma at that moment.
As usual, he tried as much as he could until he got really stuck. Then only he asked for help. From his working paper, we could tell what was his thought process. As much as possible, we will teach a new thing in the context of what our students already know. So we gave him a hint related to what he had done, and he could use it to solve the problem correctly. Then, (refer to the arrow and equation in red) we showed him the place which he could have come up with a shorter method. He verified that it works and that concluded our learning for that problem. By showing him how the short method is linked to his natural thought process, it is easier for him to recall and apply it in other problems.
All now agree that the mind can learn only what is related to other things learned before, and that we must start from the knowledge that the children really have and develop this as germs, otherwise we are ... talking to the blind about colour.
(G.S. Hall)
Beyond First-Principles
The events above happened within a short period of half a year. Within that time, we are glad to see the boy was keen to go beyond first-principles and learn other methods.
From his work displayed here, you may observe he uses a wide range of methods to solve problems - from models to algebra, and from diagrams to numerical analysis (also known as guess and check). Often, he tried multiple methods to solve a single problem, showing us clearly that even gifted problem solvers like him do not always know what to do when they first see a problem. Over time, they eventually learn to recognize different types of problems and know the best method to solve each of them.
It is a rare opportunity to observe how gifted problem-solvers learn. Most of us only get to see what they can do and achieve. Because it definitely seems like they recognize the problems and know the best way to solve them, we can be misguided into thinking that problem-solving is all about learning how to recognize a problem and associate it with the best method. But the actual learning process is hardly as simplistic as that.
When students take their skills beyond the first-principles, it can mask the first-principles thinking which is still there. Next is an update of the boy's work one year later. He was in Primary 6, learning a problem from RIPMWC. Again, he could solve the problem with first-principles - in this case, by investigating simpler problems. You may notice he was now able to use of abstract notations to generalize (and prove) the central concept of the problem. It is not hard to imagine how abstract his workings will become in a few more years.
Such abstract abilities at his age (12 years old) is not common. For your information, that year he was in the top 40 of APMOPS and achieved high-distinction in RIPMWC.
In Closing
Without the habit of using first principles, students have little chance to warm up their brain before learning an advanced technique. Naturally, they will find the technique harder to master and use flexibly. They can "learn" a lot but still unable to apply correctly. It is like the blind trying to learn about colour.
Using first-principles not just reflect your child's level of grit, it also improves their ability to learn. It is always easier to help a child to excel when they have this habit.