A Comprehensive Roadmap for Mathematics (in progress)
Keywords: mathematics roadmap, mathematics, roadmap, mathematics study plan, mathematics references, references, mathematics books, books
This roadmap is primarily intended for students of Mathematics. This doesn't necessarily mean that students from other disciplines such as Physics and Computer Science won't benefit from it; however, looking at the roadmap could be overwhelming for them, but this is because Mathematics has many areas and the roadmap was intended to be comprehensive to include them.
The file mathematics-roadmap.jpg contains the image of the roadmap.
There are several problems with the way Mathematics is presented and taught today which causes all the confusion and struggle that the students experience. In my opinion, the main problem is the way in which Mathematics is currently written in the textbooks. Mathematics is considered to be a deductive science, i.e., starts from first principles (called Axioms or Postulates) and a set of logical rules that are used to establish results (called Theorems) from these first principles; hence, it is typically written in that systematic order to reflect its underlying logical structure. I don't mean from this that it is a "bad" way to write Mathematics in, and I would even say that this is how mathematics should be written "rigorously". However, "rigorously" doesn't imply "pedagogically effective", that is, we don't "naturally" think within the bounds of the axiomatic method. This, also, doesn't imply that we will need to get entirely rid of writing axiomatically either, but to seek somewhere between logical rigor and effective pedagogy. The lack of motivation for the axioms and definitions and the precedence of abstractions to concrete examples (or instances) make students feel that the subject could only be fully understood by an elite few (geniuses). One can hardly find a textbook on any topic that includes its history, philosophy & motivation, and to also contain all the theorems and proofs that the other textbooks contain.
I don't intend here to offer solutions to the problems mentioned above; however, using the best (pedagogically best) of available references, I wish to construct an effective and comprehensive roadmap for learning Mathematics which approximates my idea of good mathematical exposition.
I emphasize the importance of the relation of other subjects to Mathematics. Of course, Philosophy lays down the conceptual framework that encompasses the entirety of human knowledge so it relates to any field or science not just Mathematics but Philosophy always had a special relationship with Mathematics and anyone who reads Philosophy can clearly see that. Philosophy impacts one's thought and makes him independent, aware, self-reflective, critical, rigorous, and always seeking for deep understanding. Although I started the roadmap with Philosophy because of my obvious bias, you can skip it but I highly recommend reading at least one book. Also, there are many other important subjects such as Computer Science and Physics which are strongly connected to Mathematics if not sometimes regarded as subsets of Mathematics. Throughout history, Mathematics was strongly influenced by these subjects, and in turn, Mathematics also influenced them. Many ideas in Mathematics have their origins in problems in subjects elsewhere so these subjects are extremely useful for motivating these ideas.
Learning Mathematics is a tedious task that requires long periods of conscious effort and patience. I offer some tips which I consider to be of great importance when learning any subject within Mathematics (which could be applied elsewhere).
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The main goal of learning is to understand the ideas and concepts at hand as "deeply" as possible. Understanding is a mental process we go through to see how a new idea is related to previous ideas and knowledge. By "deeply" we mean to grasp as much of the ideas and relations between them as possible. A good metaphor for this is picturing knowledge as a web of ideas where everything is somehow related to everything else, and the more dense the web is, the stronger it becomes. This means that there might be no "perfect" state of understanding, and otherwise it is an on-going process. You could learn a subject and think you understand it completely, then after learning other subjects, you come back to the first subject to observe that now you understand it deeper. Here we can use a famous quote from the mathematician John V. Neumann: "Young man, in mathematics you don't understand things. You just get used to them", which I think really means that getting "used to" some subject in Mathematics might be the first step in the journey of its understanding! Understanding is the journey itself and not the final destination.
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Motivation for any new concept is a must. This includes historical development of the subject which is sometimes crucial to understanding, analogies, drawings, and many other methods. Thought is induced by problems, questions, and misconceptions; thus, knowing what questions were asked in the mind of the mathematician who developed the subject and the problems he confronted really helps guiding thought in the right direction of understanding.
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Always question the way the subject is presented. This includes questioning everything from the way terms are defined, to the way theorems are proved, even questioning whether the subject deserves the time and effort mathematicians put to it. Sometimes, there could be many different ways to define something; however, a particular definition is chosen among others for some conveniences and goals, so learning about these conveniences and goals would motivate the use of that definition. Some other times, more than one definition are studied independently so one can easily see the consequences of different definitions. We could use a good quote here from the mathematician Paul Halmos: "Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?".
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Solve as many exercises as you can to challenge your understanding and problem-solving skills. Exercises can sometimes reveal weaknesses in your understanding. Unfortunately, there is no mathematical instruction manual for problem-solving, it is rather an essential skill that requires practice and develops over time. However, it could be greatly impacted by your level of understanding of the subject. The processes of learning and problem-solving are interrelated and no one of them is dispensable in the favor of the other. There are also general techniques that could be helpful in most cases which are found in some books on problem-solving (which are included in the roadmap).
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Be metacognitive (from Metacognition which literally means "beyond cognition", i.e., "beyond knowledge"), that is, be aware of your own knowledge and thoughts and consciously think about how you think and acquire knowledge. Thought is not passive, but an active process that could reflect on itself. Metacognition and consciousness help us monitor and regulate our thought processes to increase our potential to learn. This gives us the ability to evaluate our own performance by utilizing past thought experience.
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Teach what you have learned to someone else or at least imagine that you are explaining what you learned to someone in the best possible way (which is also known as the Feynman Technique). This forces you to elaborately rethink what you have learned which could help you discover any weaknesses in your understanding.
How should one approach books? Should the reader go through every word from the first page to the last page? Should you solve every single problem? These questions are typical regarding book reading, and answering them is not a straightforward task. I will provide general guidelines, and accordingly the reader should find suitable answers for these questions.
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What is the most fundamental purpose of reading? To learn, of course. So determining what you want to learn, determines what you should read. Not only what books to read, but also what chapters within a book to read. Sometimes it suffices to read the first chapter of the book, and in other times you have to go through all the chapters. However, one isn't always sure what to read and what to leave, and in that case only read the part you are sure you will need, then after going through other books you will eventually know whether you need to return to the book to read more. Moreover, reading books is not always a "linear" process, that is, sometimes going back and forth between multiple books is necessary. The reader should be critical to himself, and he has to assess precisely what he knows and understands and what he doesn't.
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Sometimes, skimming (pass quickly through the text to note only the important points by looking for certain keywords) is possible; however, in some cases, you might arrive at a paragraph that you will need to read word by word. That is left to the assessment of your understanding. Patience is the key when dealing with books, so don't expect to go through a 100+ page book in one day and understand everything completely unless you have reasonable prior knowledge of the subject.
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When reading about a new concept, try to predict what the writer will say before you read it. What (important) questions would you ask about this concept? how would you answer them? and what would you deduce from these answers? Before you read a proof of a theorem, try to prove it yourself first. If you could carry out the proof entirely on your own, then you will become more confident of your knowledge. If you become stuck, then when you read the proof you would embrace what you didn't know and you would hardly forget the proof afterwards.
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It is possible to find repetitive exercises, in other words, you could go through several exercises that have the same idea which could be solved by the same method. In this case, solving one of them could suffice. Don't always count on your intuition, since one can think he has solved the exercise by just looking at it and at the end he finds out otherwise. Going through all the exercises of a chapter/section is up to you and your assessment of how good you did with the exercises you solved (and again, depending on the assessment of your understanding of the subject).
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Learning how and when to take notes is not easy. You don't want to waste your time copying the entire book. Most modern books have nice ways to display important information such as definitions and theorems, so it's a waste of time to write these down since you can always return to them quickly. What you should do is take notes of how you understood a difficult concept (that took you a relatively long time to understand) or anything that you would like to keep for yourself which is not included in the book, or to rewrite something in the book with your own words. Notes are subjective and they should be a backup memory that extends your own memory.
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Read critically. Books are written by people and they are not perfect. Don't take everything for granted. Think for yourself, and always ask yourself how would you write whatever you are reading. If you found out a better way to explain a concept, then write it down and keep it as a note.
The roadmap consists of topics, each represented as a labeled group of rectangles. The arrow connecting two groups (say from A to B) represents a dependency (B depends on A). Sometimes the dependency of either of the topics on the other is vague or they are interrelated so a two-sided arrow is used. Colors indicate whether the topics are essential (you can't skip them), optional but recommended (they are not essential but very beneficial), or optional (reading them or skipping them is up to you so they are just regarded as additional information). The legend indicates these colors.
Each topic has multiple books. This is since any two books in a single topic are either mostly similar but differ in a few aspects (e.g. how they explain some concepts or how the subjects are ordered) or they complement one another (so one book has subjects not discussed in the other). I recommend that the reader should look at the preface and/or introduction and the table of contents to see whether the book satisfies his/her needs and to be able to compare the differences between the books easily.
The software used to create these diagrams is draw.io. Just open the file mathematics-roadmap.html and you can start editing.