Question

How important is the self-explanation principle to your understanding of mathematics? Where the self-explanation principle refers to the providing of explanations during learning.

Answer 1

Self-explanation is the bare minimum of understanding. Changing it, trying to use it, and comparing it to other concepts is where true understanding begins to develop. I just don't believe understanding exists until there is a creative, independent push against it.

Answer 2

Let me clarify the self-explanation principle refers to the finding that people learn more deeply when they spontaneously engage in or are prompted to provide explanations during learning.

Answer 3

Yes I agree it is a bare minimum that a lot of students fail to grasp. There is also a method behind the madness. I tend to call it a running commentary. In order to be able to succeed in mathematics you are required to circulate the question in your mind and what you are expected to do with it, what principles are at work and how you will use those principles towards solving the problem. It's pretty much the planning before you begin any calculations, in the same way a general maps out of the territory and assigns his troops before heading to the battlefield..

This to me process is more important than any formula you can learn. You must know why it is you are using the formula for a particular situation, or why you are applying a particular method. A lot of students get stumped at the question, unable to even write a statement to begin. Getting stuck is ok as long as you can find the right track to reapproach the question once again. The difference between simply doing and understanding. I find having students do similar questions but with slightly different conditions helps to cement principles behind the solutions. I also emphasise the importance of mathematical definitions before anything else. You'd be surprised at how many people don't know what a function is, what a derivative is, what an absolute value is etc.

Answer 4

Psychologically speaking, such learning does involve greater learning networks and formation of 'concept nodes', thus integrating a lot of already stored information and making new ideas. I also think it's more motivating when you have a set goal in mind and you feel you've accomplished something by answering the question rather than just for the sake of spitting out something onto a page.

Answer 5

Consider below conversation between a teacher and a student :

```
Teacher :
Do you know Pythagoras theorem ?

Student :
Yes, a^2 + b^2 = c^2.
```
At this level, the teacher has no way to know with any degree of certainty that the student has learned the concept. Even google could spit that output.

If the student answers something like this :
```
Teacher :
Do you know Pythagoras theorem ?

Student :
Yes, a^2 + b^2 = c^2.
It helps in finding an unknown side in a right triangle.
```

That might impress the teacher somewhat, but there still is a possibility that the student might have simply memorized that. Its more trickier than turing test I guess. I believe the teacher must engage with the student, watch him attempt a couple variety of problems that use Pythagoras theorem to be reasonably certain.

Answer 6

yes very much, I would expect 80% of the people that pass an online math class to fail a math class from 30 years ago (from the same class). We incorporate so much technology that it is hard to know what they know.

Answer 7

Actually speaking of this, before going on to complex problem solving and mathematical concepts, I became quite shocked at how dependent students are these days on using calculators for simple arithmetic exercises. I think it is a travesty that some secondary school students haven't memorised their times and division tables. Technology has given new ways in which we can study mathematics but at the same time intuitive exercise has been forgotten somewhat. Thus mathematics can become rather a mechanical exercise.

Answer 8

The utility of times table has unfortunately been underestimated by many, including teachers. Many believe that times table is used for multiplication and division, and therefore can be replaced by calculators.

Many wonder why factoring is difficult, especially those who have not mastered their times table. Even simple tasks like integrating using the power rule requires times table skills.

When an elementary school teacher (or a parent) gives up on a student learning the times table and give him/her a calculator, the teacher has sealed the student's fate!

No wonder many students believe that math is a difficult subject.

Answer 9

I don't really know my times table. 7*9=7^2+14

I think this approach is actually way better then memorizing.

Answer 10

@zzr0ck3r that is actually how the multiplication algorithm that they teach in middle and junior high school works though it might not be mentioned explicitly that the distributive law is applied.

20 x 13 is formally written as 20 x (10 + 3) = 200 + 60 = 260.

Answer 11

lol thanks

Answer 12

my point is that you do not need to memorize things, as that is not mathematics.

Answer 13

I agree that it is not necessary to memorize anything in mathematics, but we need basic skills, whether it's the times table, or the Trachtenberg system, or anything else. In North America and most of the rest of the world, the education system has chosen times table, so we're stuck with it. So times table is the only item to be committed to memorization. In fact, in my books, it is not even to be memorized, but trained to be a reflex, together with squares of numbers possibly up to 100.

Once we get trained on the basic reflexes, we can develop other mental reflexes, such as:
1489*5=(14/2)~(8/2)~(90/2)=7445 ... (adapted from Trachtenberg system)
115^2=100(100+2*15)+15^2=13225, similarly
97^2=100(100-2*3)+3^2=9409
23*27=25^2-2^2=621
13*19=100+10(3+9)+3*9=247 (= 16^2-3^2=247)
45*11=4~(4+5)~5=495
2579*11=2~(2+5)~(5+7)~(7+9)~9=28369 ... (from Trachtenberg system)
85^2=(8*9)~25=7225, 115^2=(11*12)~25=13225, ...

All of the above are written algebraically and might appear long, but are in fact very efficient basic \(mental\) algorithms that anyone (even elementary grades) can master easily.

Yes, we can live without the times table, but we need basic math.

See also:
http://www.trachtenbergsystem.org/free-pdf-download/