Middle schools spend all their time teaching math content, focusing on how to learn and apply different routines to solve math problems, and spend little (if any) time trying to convey to students what math is. It's a bit like describing soccer in terms of executing a series of passes to get the ball into the goal. Both precisely describe different key features, but they both ignore what the whole is and its ins and outs.
After looking at the course requirements, I can understand why this is, but I think it's wrong. Especially today, it is useful for any citizen to have a general awareness of the nature, reach, capabilities, and limitations of mathematics.
Over the years, I've met many people with graduate certificates in closely related disciplines such as engineering, physics, computer science, or even mathematics. These people told me that until they had completed all their secondary and college education, they did not have a good grasp of what constitutes modern mathematics . It was not until later, when they caught glimpses of the true nature of the subject from time to time in their lives, that they began to appreciate that mathematics had permeated every aspect of modern life.
01
more than arithmetic
Most of the mathematics used in science and engineering today is no more than three or four hundred years old, and many are less than a hundred years old. However, the mathematics included in the regular high school compulsory courses are all at least that old, some even more than two thousand years old!
There's nothing wrong with teaching something that old. As the saying goes, if it ain't broken, don't fix it. Algebra (the word "algebra" comes from the Arabic "al-jabr", meaning "to reset" or "to put pieces together again") was developed by Arab merchants in the 8th and 9th centuries to improve the efficiency of their business transactions.
Algebra is as important and useful as it was then (8th and 9th centuries), even though we apply it today in spreadsheet macros rather than counting with our fingers like in the Middle Ages. However, times are passing and society is also developing. In the process, a need for new mathematics arose and was met in a timely manner. Education also needs to keep pace.
It can be said that mathematics began with the invention of numbers and arithmetic. It is believed to have been around 10,000 years ago, with the creation of money. (Yes, its origins apparently have to do with money!)
Over the next few centuries, the ancient Egyptians and Babylonians expanded the discipline to include geometry and trigonometry. In those civilizations, mathematics was mostly practical, much like a "cookbook". (“Do this to a number or a geometric figure, then do this, and you will have your answer.”)
The period from 500 BC to 300 AD is the period of ancient Greek mathematics. Ancient Greek mathematicians paid a lot of attention to geometry. In fact, they handled numbers geometrically, viewing them as measurements of length. And when they discovered that there were some lengths that their numbers could not correspond to (in essence, they discovered irrational numbers), their research on numbers basically came to an end.
In effect, the ancient Greeks made mathematics a field of study rather than just a series of techniques for measuring, counting, and accounting. Around 500 BC, Thales of Miletus (now part of Turkey) introduced the idea that precisely expressed mathematical claims could be logically proved by formal arguments. This innovative idea marked the birth of the theorem, which is the cornerstone of today's mathematics. The publication of Euclid's "Elements of Geometry" brought the formalization method of the ancient Greeks to its peak. It is said that this book is the most widely circulated book after the Bible all the time.
In general, middle school mathematics is based on all the developments I've listed above, plus two more from the 17th century: calculus and probability theory . In fact, mathematics in the last three hundred years has not entered the middle school classroom at all. However, most of the mathematics used in the world today was developed in the last two hundred years, not even the last three hundred years!
Therefore, anyone whose views on mathematics are locked in by typical secondary school teaching is unlikely to realize that the study of mathematics is a thriving worldwide activity, nor to accept that mathematics has penetrated to a large extent Most industries in today's society and life.
For example, it is impossible for them to know which institutions in the United States employ the most PhDs in mathematics. (Although the exact number is an official secret, the answer is almost certainly the NSA. Most of these mathematicians work in code-breaking, intercepting encrypted messages through surveillance systems for the authorities to read. Although the intelligence authorities still won’t admit that This, but at least that's usually what people think.
While most Americans probably know that the NSA does code-breaking, many don't realize that code-breaking requires mathematics, and don't perceive the NSA as an agency that employs large numbers of advanced mathematicians. )
Over the past hundred or so years, mathematical activity has increased dramatically, and development has been particularly rapid. At the beginning of the 20th century, mathematics could reasonably be seen as consisting of about a dozen distinct disciplines: arithmetic, geometry, calculus , and a few others. Today, the number of these categories is about sixty or seventy, depending on how you count them. Some disciplines, like algebra or topology, have split into different subfields; others, such as complexity theory or dynamical systems theory, are entirely new fields of study.
The remarkable development of mathematics led to the emergence of a new definition of mathematics in the 1980s: the science of patterns. From this description, mathematicians define and analyze abstract patterns—value patterns, shape patterns, motion patterns, behavior patterns, crowd voting patterns, repeating probability event patterns, and so on.
These patterns may be real or imagined; visible or mental; static or dynamic; qualitative or quantitative; practical , and possibly for recreation. They may come from all around us, from the pursuit of science, or from the inner workings of the human brain. Different models make different branches of mathematics. For example,
Arithmetic and number theory study patterns of number and computation.
Patterns for geometric studies of shapes.
Calculus allows us to deal with patterns of motion.
Logic studies patterns of reasoning.
Probability theory deals with patterns of probability.
Topology studies patterns of closure and position.
Fractal geometry studies the self-similarities found in the natural world.
02
Mathematics Symbol
One of the hallmarks of modern mathematics that is recognizable even to ordinary people at a glance is the use of abstract symbols: algebraic expressions, complex-looking formulas, and geometric diagrams. Mathematicians' reliance on abstract notation reflects the abstract nature of the patterns they study.
Different aspects of reality need to be described in different forms. For example, when studying terrain or describing to someone how to find your way in an unfamiliar town, drawing a map is most appropriate. It is far less appropriate to do so in words. Similarly, annotated line drawings (blueprints) are best used to represent the structure of buildings, while musical notation is best used to depict music on paper. For different kinds of abstract and formal patterns and abstract structures, the most appropriate means of description and analysis is to use mathematical symbols, concepts and algorithms.
For example, the commutative law of addition can be stated in everyday language like this:
When adding two numbers, their order doesn't matter.
However, it is usually written in symbolic form:
Although for simple examples like the above, the symbolic form has no obvious advantage, the complexity and abstraction of most mathematical models makes it too cumbersome to apply any tool other than formal symbols. Thus, the development of mathematics also included a steady increase in the use of abstract symbols.
Although symbolic mathematics in its modern form is generally believed to have been introduced by the 16th-century French mathematician François Véctor, algebraic notation appears to have first appeared in the writings of Diophantus of Alexandria (lived around 250 AD) . His 13-volume treatise on Arithmetic (of which only 6 volumes survive) is generally considered the first algebra textbook. It is worth mentioning that Diophantus used special symbols for unknowns and powers of unknowns in equations, and symbols for subtraction and equality.
Math books these days have a tendency to overflow with symbols. However, just as musical notation is not music, mathematical notation is not mathematics. A page of notes represents a musical composition, but it is only when the notes on the page are sung or played by an instrument that what you hear is the music itself. Music comes alive through performance and becomes part of our experience. Music does not exist on paper, but in our heads.
The same goes for math. Symbols on paper are merely representations of mathematics, and printed symbols come to life only when they are read by competent performers (in mathematics, some mathematically trained people)— Mathematics lives and breathes in the reader's mind like an abstract symphony.
Again, abstract notation is used because the patterns that mathematics helps us identify and study are abstract. For example, mathematics has played a vital role in helping us understand invisible patterns in the universe. In 1623, Galileo wrote:
"The great book of nature can only be read by those who know what language it is written in, and that language is mathematics."
In fact, physics can be precisely described as seeing the universe through a mathematical lens.
To give an example, it is precisely because of the mathematical systematization and understanding of the laws of physics that we have air travel today. When a plane flies overhead, you can't see anything supporting it. Only through mathematics can we "see" the invisible forces that keep it aloft. The forces in this case were identified by Newton in the 17th century, who also developed the mathematics needed to study them.
Although it wasn't until a few centuries later that technology advanced to a point where we could actually use Newton's mathematics (enhanced by a host of other mathematics developed during this time) to build an airplane. This example is a good example of one of my favorite memes used to describe what mathematics is: mathematics makes the invisible visible.
03
Modern University Mathematics
Having briefly outlined the historical development of mathematics, I can begin to explain why modern university mathematics is fundamentally different from the mathematics taught in secondary schools.
Although mathematicians long ago expanded their field of study beyond numbers (and the algebraic notation for representing them), until a hundred and fifty years ago they viewed mathematics as primarily a science of computation. In other words, proficiency in mathematics actually means being able to calculate or use symbolic expressions to solve problems. In general, secondary mathematics is still largely based on this earlier tradition.
However, during the 19th century, as mathematicians tackled ever more complex problems, they began to find that sometimes those earlier intuitions about mathematics were not enough to guide their work. Counterintuitive (sometimes even paradoxical) results lead them to realize that some of the methods they develop to solve important practical problems lead to results they cannot explain.
For example the Banach-Tarski paradox. The paradox says that, theoretically, taking a ball, you could somehow cut it into parts and recombine them to get two identical balls, each the same size as the original. Because the mathematics is correct, the Banach-Tarski result must be accepted as a fact even though it defies our imagination.
Thus, it is understood that mathematics can lead to areas that can only be understood through mathematics itself. To ensure that we can trust the discoveries made using mathematical methods without having to verify them in other ways, mathematicians turn to methods within mathematics and use them to test the discipline itself.
In the mid-19th century, this introspection led to the adoption of a new and different idea of mathematics, one that focused less on calculus or calculating answers than on formulating and understanding abstract concepts and relationships. This is a shift from emphasizing to emphasizing understanding. Mathematical objects are no longer considered primarily given by formulas, but rather as carriers of conceptual properties. Proof is no longer the transformation of items according to the rules, but the logical reasoning process starting from the concept.
This revolution (and it was enough to call it a revolution) completely changed the way mathematicians saw their subject. In other parts of the world, however, this change has not yet happened. Except for professional mathematicians, people first noticed a change in the situation when the new perspective emerged from the required undergraduate courses. As a college student studying mathematics, if you feel like your head is spinning when you first encounter this "new mathematics", you can blame them on Dirichlet, Dedekind, Riemann, and all the others who helped introduce this new mathematics. Methods of mathematicians head on.
As a preview of what's to come, I'll give an example of this change. Before the nineteenth century, mathematicians were accustomed to the fact that 
a formula such as this gave a function such that, from any given number 
, a new number could be obtained 
. Then came the revolutionary Dirichlet. Forget those formulas, he said, and just focus on what the function does in terms of input-output behavior. According to Dirichlet, a function is any rule that yields new numbers from old numbers. This rule cannot necessarily be expressed by an algebraic formula. In fact, there is no reason to limit your attention to numbers. A function can be any rule that starts from an object to get a new object.
With this definition, the function on the real numbers is legalized by the following rules:
If 
it is a rational number, let 
; if 
it is an irrational number, let 
.
Try plotting this monster function!
Mathematicians began to study the properties of this abstract function. Such functions are not given by some formula, but by their behavior. For example, does the function have properties such that it always gives different answers when you give it different initial values? (This property is called injectivity.)
This abstract, conceptual approach bore fruit in the development of the new discipline known as real analysis. In their own right, mathematicians have studied abstract concepts such as continuity and differentiability of functions. French and German mathematicians invented definitions of continuity and differentiability 
. To this day, each generation of students who take calculus-based mathematics courses struggle to master it.
Also, in the 1850s, Riemann used differentiability to define a complex function, and the definition of the function given by the formula was regarded by him as the second definition.
The famous German mathematician Gauss (1777-1855) came up with residue classes (you're likely to come across them in algebra class), a precursor to what we now consider standard. This approach defines mathematical structures as sets with specific operations whose behavior is specified by axioms.
Following Gauss, Dedekind studied new concepts such as rings, fields, and ideals, each defined as a family of objects with specific operations. (Again, you'll probably run into these concepts pretty quickly after taking calculus.)
There are more changes to follow.
Like most revolutions, the changes that took place in the 19th century had sprouts very early. The ancient Greeks were undoubtedly interested in mathematics as a conceptual exploration, not just as calculation. Leibniz, the co-inventor of calculus in the 17th century, also thought deeply about these two approaches. But until the 19th century, mathematics was largely viewed as a series of algorithms for solving problems.
However, for today's mathematicians who grew up learning the revolutionary mathematical concepts, mathematics is just a product of the revolution in the 19th century. The revolution may not have been spectacular and largely forgotten, but it was done and had far-reaching consequences. Moreover, it sets the stage for this book, whose main purpose, after all, is to provide the basic mental tools needed to enter the new world of modern mathematics (or, at least, to learn to think mathematically).
At present, although post-19th century mathematical concepts have become a staple of college mathematics courses after calculus, it has not had much influence in middle school mathematics, which is why you need such a book to help you complete this time. The reason for the transition. There was an attempt to introduce this new method into the middle school classroom, but it went terribly wrong and was quickly abandoned.
This was the so-called "New Math" movement of the 1960s. What went wrong then was that when messages of innovation passed from prestigious universities to secondary schools, they were seriously misinterpreted.
For mathematicians around the mid-nineteenth century, computation and understanding have always been important. The revolution in the 19th century was just a shift in the view of mathematics: calculation and understanding, which is the essence of mathematics, and which only plays a derivative or supporting role.
Unfortunately, in the 1960s, the message to middle school teachers across the country was often, "Forget about the calculus tricks and just focus on the concepts." Such an absurd and terribly bad strategy led satirist (and mathematician) Tom Lehrer to write in his song "New Math": "It's the method that counts, never mind Got the right answer." A few years later, most of the "new math" (mind you, it's actually well over a hundred years old) was dropped from the middle school syllabus.
The nature of educational policymaking in a free society makes it unlikely that such a change will happen again in the foreseeable future, even if it might do better the second time around. Nor is it clear (at least to me) whether such a change is desirable in itself. There are some educational arguments (although this is debated due to lack of conclusive evidence) that the human mind requires a certain level of mastery over the computation of abstract mathematical objects before it can think about the properties of these objects.
043
Why do you need to learn this?
You should now understand that this shift in the 19th century was a change in professional mathematics circles, where mathematicians went from seeing mathematics as computational to seeing it as conceptual. As professionals, they are more interested in the nature of mathematics. But for most scientists, engineers, and others who use mathematics in their daily work, the situation remains largely the same, and remains so today. Computing (and getting it right) remains as important as ever, and its use is more widespread than at any time in history.
Thus, to anyone not in the mathematical community, the shift looks more like an expansion of mathematical activity than a change of focus. College students studying mathematics today are not only taught problem-solving routines, they are also (additionally) required to grasp the concepts behind them and be able to justify the methods they use.
Is such a request reasonable? Professional mathematicians need this conceptual understanding because their job is to develop new mathematics and test its validity. But why is this required of those students, whose future careers (such as engineers) will only use mathematics as a tool?
There are two answers, both quite reasonable. (Spoiler: There are only two answers on the surface, but when you dig deeper, they are actually the same.)
First, education is not all about acquiring specific tools for a future career. Mathematics, one of the greatest creations of human civilization, should be taught alongside science, literature, history, and the arts in order to pass on our cultural treasures from generation to generation. Living is not just about jobs and careers. Education is preparation for life, and job-specific skills are only part of it.
The first answer certainly needs no further explanation. The second answer is aimed at the issue of "as a tool needed for work".
Needless to say , many jobs require mathematical skills. Many people, when looking for a job, find that they lack a mathematical background. In fact, in most industries, the demand for mathematics at almost any level is actually higher than usually predicted.
Over the years, we have become accustomed to the fact that progress in industrial society requires a workforce with mathematical skills. However, if you look more closely, these people fall into two categories. One class consists of people who can find a mathematical solution to a given mathematical problem (that is, a problem that has been expressed in mathematical terms). The other category consists of people who, after getting a new problem, such as manufacturing, can use mathematical methods to identify and describe the key features of the problem, and use mathematical descriptions to accurately analyze the problem.
In the past, there was a lot of demand for employees with Type 1 skills and little demand for people with Type 2 skills. Our mathematics education process is largely able to meet these two needs. Although mathematics education has been concerned with producing the first type of workers, some of them must also be good at the second type of activity. So everything is fine.
But in today's world, where companies must constantly innovate to remain competitive in business, the need has shifted to a second type of people: mathematically minded people who can think outside the box, rather than inside the box. Think inside. Now, all of a sudden, there's the problem.
There has always been a need, and our education system should support their development, for people with a range of mathematical skills who can work alone for long periods of time, with a deep focus on a particular mathematical problem. But in the 21st century, there is a greater need for the second type of talent. Since we don't have a name for such individuals ("mathematically capable people" or even "mathematicians" in public perception usually refer to the first category of people), I propose a name for them: innovative mathematical thinking Those (innovative mathematical thinkers).
This type of new individual (well, it's not really new, I just don't think anyone noticed them before), first needs to have a good conceptual understanding of mathematics, know its power, scope, when and How it is applied, and its limitations. They also need to have a solid grasp of some basic math skills, but they don't need to be particularly advanced. More importantly, they are able to function in team work (often interdisciplinary), see things in new ways, learn quickly and master new skills that may be needed, and apply old ways in the new situation.
How can we educate such individuals? We want to work on the education of conceptual thinking. This kind of thinking is hidden behind all concrete mathematical skills. Remember that old saying? "It is better to teach a man to fish than to give him a fish." The same is true for mathematics education in the 21st century.
There are so many different math skills out there, and new skills are being developed all the time, it's impossible to fully include them in K-16 education. By the time a college freshman graduates to work, many of the specific skills learned in college are likely no longer relevant, and new skills are all the rage. The focus of education must be on learning how to learn.
The growing complexity in mathematics led 19th-century mathematicians to shift (or expand, if you prefer) the focus of computational skills to the underlying, fundamental, conceptual ability to think.
One hundred and fifty years later, society has changed again with the aid of more complex mathematics. This shift in focus is no longer just important to mathematicians, but to everyone, if they approach mathematics with the mindset that they want to apply it to reality.
So, now you not only know why mathematicians in the 19th century shifted the focus of mathematical research, but also why, starting in the 1950s, university mathematics students were also required to master conceptual mathematical thinking.
In other words, now you know why your university wants you to take this bridging course. Hopefully, you also now realize why this is so important to your life, and not just for the immediate need to pass college math classes.
recommended reading
By: Keith Devlin
Translator: Lynn
A Primer on Mathematical Thinking for High School Students, College Students, and Anyone Who Wants to Improve Their Analytical Thinking Skills
