Sunday, November 26, 2023

Is there a good textbook for abstract algebra learning?

Hi there

My initial question was just to find a good textbook for learning abstract algebra. Then I decided to explain why I don't like the current one I read. And then my thoughts flew into this big text as a holly war against current education system. So the whole text is about math teaching approach which could be applied to any math area. Or even any other science area. In 2 words, it is about dilemma between 2 teaching frameworks: memorizing and problem solving. Whilst problem solving approach seems like more convenient and efficient, current math textbooks don't use it. And I don't see a reason why. Ok let's go.

I just recently started learning abstract algebra. Could you please tell what is the best textbook for learning this science for small 13 yo children? Well I know there is no such... I am not a child but I am stupid enough for adult textbooks so I really like children-teaching approach lol :) Something like "abstract algebra for dummies".

In general I mean the approach for teaching anything to children and teens. Yep, I am not so smart to study adult books.. Books for children are often based on the problem-solving approach and they are really good for understanding. That's why I am looking for the book of this type..

Here is an example of "problem solving" teaching from simple math:

We have the cubic cup and the cylinder cup of the same height and same width. In which of them can we put more peas? Let's call "cup volume" = number of peas we can put in a cup. How can we calculate the volume? Etc..

And here is an example of classical teaching ("memorizing"):

"Volume" is the measure of the 3-dimensional space occupied by matter, or enclosed by a surface, measured in cubic units. Just memorize it. Than memorize the volume formula. Then we will explain why you need it (at that time half of students usually forget what they memorized before).

Why memorizing approach is bad? Our brain needs to create new unconnected memory spaces by simply memorizing the terms. These spaces become unused due to lack of connections between them and another "trusted" spaces in our brain. "Trusted" means existing real life problems, familiar practices etc. After all memorized unconnected spaces will soon be erased as unused. That's why most students don't remember what they memorized 4 years ago (what a waste of time and college resources!). 

On the other side, problem solving approach is very natural and convenient, easy to apply in learning. Our brain makes connections between existing memories (problem -> solution). Whilst connections between exiting spaces serve for a long time in our brain, they are easy to use, easy to remember and easy to apply and develop and create new connections.

So the question is what is the best problem-solving textbook for abstract algebra? So far I tried to start with couple of them. Both start with definitions and axioms without mentioning any problems to solve. 

E. g. the operation is any act with any ordered pair of the A set yielding a unique element from A. I immediately raise numerous questions to fill missed spaces in the logic chain:

- Why exactly this set of axioms is relevant for the operation definition? 

- Why the return should belong to A?

- Why it should be unique?

- Why operation takes only 2 elements as an input?

- Why they should be ordered?

Well I understand I will get answers to all these questions somewhere 200 pages and few months later... But is there a way to start with problems not with theoretical definitions and terms?

I realize that I could ask all these questions online in forums and communities. But should I bother you asking them every time, for every new chapter I read? Or for every new page?Would it be simpler rather to get a book already having these answers in the beginning? 

To prove my position, I use a time machine. We take a ride back to the past, the year of  300 BC, Greece sea coast. The man covered with white sheet sits on a beach thinking about his future math textbook. "I am sorry for bothering you but may I ask a simple question?" - I dare to say. "Only if it is really simple. I believe I am smart enough to answer simple questions, also I am wise enough to avoid difficult ones" - Euclid responds. I continue: "When I was a child, teachers told me that a*b=b*a for any real numbers a and b. Why it is so?" Euclid smiles again and draws a rectangle on a sand. I ask again: "Why a+b=b+a?" Euclid laughs and draws a line segment divided by point in 2 parts on a sand.

I am pretty sure that with proper approach this prove could be understood by a middle school students. Instead we force our children to memorize a lot of information without explanation why they need it and why it is so. Many researches already agree that current school system is a manufacturing plant for making serial mass product of bio robots with certain set of predefined information, instructions and algorithms. The only algorithm is not developed by the system: creative and critical thinking. But science is not driven by army of robots, it is driven by critical minds.

Someone could tell me, we should memorize terms because some areas of math were developed having no practical application yet. This is not an argument to me. Let's ask young 20 yo √Čvariste Galois, why definitions of operations and groups contain exactly this set of axioms and not another? He will easily explain that different axioms would merely not be applicable to the equation solving process (though at that time people didn't know how to apply equations of certain high degrees to real life). Then he would bring couple of examples so even a person not familiar with the group theory could understand. Why don't we include this explanation in our textbooks at the beginning? Respectfully to Galois, current professors know even more than him. But the professor should first teach his middle school child from scratch with the same discipline he want to write a textbook about. And only then start writing a textbook. “If you can’t explain it to a six-year-old, you don’t understand it yourself”, the great Einstein said.

Indeed, there are plenty of books online named like "introduction to abstract algebra", "gentle introduction to abstract algebra", "elementary introduction to abstract algebra" etc. However so far I have seen this problem solving approach in none of them.. Most books just copy and paste basic terms and theorems in their own words and in their own order without changing the mainstream approach of teaching.

I remember the intro to one software engineering book. The author said: "Why you should buy this book so it is better than others? Because most others just retell official online tech reference, and you don't need to pay for money for that. You can find most tech references documentation for free on the official software website." This argument was quite reasonable to me. Since than, I am looking for the same intro in every book in every science area I learn..

In this context, physics and chemistry authors are better than mathematicians (respectfully to all of them). Physics textbooks often start with natural and familiar problem. Then they logically go to a solution. On the other hand, math authors don't intend to follow this way. The usual consequence of teaching is as follows:

1) Memorize some weird terms and axioms

2) Try to prove some proposed theorems with them

3) Repeat steps one and two recursively

4) In the end of the year we tell you why you needed it. If you ask though.

This consequence is neither logical nor convenient to me. Though Math is known as the most logical science among all. 

Am I asking for something impossible? To be math textbooks more consequent in teaching? Not at all. See my example with volume supra. The same approach could be applied to any math area I believe. E. g. for calculus basics: first tell about the problem to calculate the speed, then suggest a solution of derivative; first tell about the problem to calculate the complex figure square and then suggest a solution of integral etc.

Someone could respond: "Well you better learn and memorize terms first and don't bother teachers and textbooks authors. You problem is not a crucial problem at all. The crucial problem example is having neither a textbook nor a teacher at all." Well tell this to the 13 yo teen. And you kill any motivation to learn any science till the end of his/her life. I believe the right teaching approach is a key to multiply number of tech innovations. The right teaching approach is a key to let students love their work instead of working as bartenders and waiters after getting masters degree. (Respectfully to restaurant staff though, I personally worked as a busboy and dishwasher once). The right teaching approach is a key to let children be genius and make innovations even before they graduate. Like Evariste Galois.

The teacher opens the door and the student enters it. Let's make this door wider, so more students could pass...

So maybe someone knows any good abstract algebra textbook? It should follow the "explanation->definition" (or problem based) approach unlike "definition memorizing->explanation". I would really be grateful! 

Peace and love