Mathematics, the pioneering foundation for most modern technology today is a large science with more than 60 different branches. In the trend of industrial revolution 4.0, math is increasingly important, how to quickly grasp this increasingly massive amount of math knowledge? According to a famous former Stanford professor, in addition to the traditional way of learning from practice to rules, the human brain can also discover the meaning of a mathematical game from given axioms.
A series of abstracts translated from an article by author Keith Devlin on the American Mathematical Association (MAA) website, January 2009 issue. Author Keith Devlin is a former Stanford University math professor, director of the Stanford Mathematics Outreach Project of the Stanford School of Education, author of more than 30 books and 80 math studies.
“God makes the integers, everything else is man’s business” is the famous saying of the German mathematician Leopold Kronecker (1823-1891) that I (author Keith Devlin) used to solve the problem. I wrote last month’s article in the monthly Devlin-Devlin’s Angle section of the American Mathematical Association website. Concluding with some questions about how we teach math to new math students. I promised to write about an alternative to the popular math teaching solution in America. I will start this month’s post from those questions.
Learning math is like learning to play games where each game follows some specific logic rule.
To avoid repetition, I assume you have read what I wrote last month. Summarizing the main idea of the previous article, I present the supporting evidence for my doctoral thesis. Accordingly, while numbers and possibly other elements of basic 8th grade math (in the US) are generalized from everyday experience, the more advanced parts of the course are designed and learn according to defined rules, and often “games of symbols” that make no sense at first.
You can learn the former (generalizations of everyday experience) from the formation of a series of practice-based cognitive metaphors, where each stage brings a new understanding from what is already familiar. .
New mathematical knowledge is inferred from the given rules-axioms. (Illustration).
Learning the following way (defined rules) is similar to playing chess: at first you just follow the rules you barely understand, and then, with practice, you get to a level where meaning and understanding arise.
(The second way is to learn math like learning to play games where each game follows some specific logic rule. New math knowledge is derived from these given rules-axioms)
The first process was described by Lakoff and Nunez in Where Mathematics Comes From . Where Mathematics Comes From is a book on cognitive science of mathematics by cognitive linguist George Lakoff and psychologist Rafael E. Núñez, published in 2000.
Most of us can remember the second way as how to learn integrals . The latter seems to be an observation that contradicts the claims of Lakoff and Nunez, who argue that the metaphorical construction process they describe will yield the entirety of pure mathematics. Mathematical research because of the intrinsic development of mathematics compared to applied mathematics is the study of mathematics for application in other disciplines such as physics, economics…).
If in fact there are two fundamentally different kinds of mathematical thinking , you must learn them in very different ways. If so, the natural question is, in the traditional university curriculum, where does one end and where does the other begin? And make no mistake about it, the two forms of learning I’m talking about are very different. The first type is the meaning that gives rise to the rules, and the second is the rule that eventually gives the meaning. The learning process will change from generalization to linguistic creativity somewhere between mastering the concept of numbers (integers) and integrals.
Please note, both forms of learning can produce mathematics that makes sense in the world and can be applied to the world. The difference is that the first type is the factual connection that precedes new knowledge of mathematics, while the second is that mathematical knowledge must be ” cognitively bootstrapped” before we can have it. understand the connections between new knowledge and reality and make applications.
Two different forms of thinking in mathematics: from reality (Reality) derive abstract math rules (Maths) and play games with the rules and then return to reality. (Photo: QCAA).
Before going any further, I must point out that, since I am dealing here with human perception, strictly speaking, my simple classification of two groups is a simple, convenient one. benefits as the basis for the general points that I wish to convey. As usual, when people are connected, the world is not just black-and-white but a continuous spectrum of shades in between the two poles. If my monthly e-mail was visited by people with the names of mathematicians, they would seem inclined to try to look at things in two different ways.
In particular, it is in principle possible for a student to be instructed to learn all of mathematics in the iterated-metaphor manner described by Lakoff and Nunez. In practice, however, this is way too long to reach most contemporary mathematics. One possible way to learn advanced math relatively quickly is for the brain to be able to learn to follow a given set of rules without understanding them, and to apply them usefully and intelligently. As long as you practice enough, the brain will eventually discover (make) meaning in what has started out as a meaningless game, but to apply the rules effectively there is generally no need to. reach that stage.
As an obvious example, every year college freshmen in engineering and physics learn and apply advanced methods of differential equations without understanding them—a win that the students majoring in. math (where insight is a well-defined goal) struggled for four years to achieve.
Now, back to the university level, where the way to quickly achieve procedural competence – the second way will work for students who need to use Different math techniques, what is the best way to start math for students in the first grade? Given that young children are capable of learning to play games, which are often complex adventure games, and that they show high skill in videogames, I would guess, they can also learn elementary math along the way. how to play the game. Many video games are so complicated that it takes a lot of effort for most adults. If you don’t believe me, go ahead and try a game (I’ve played it and it’s hard to master).
But I don’t know if this method has ever been tried, and if it works, I don’t know. Actually, I suspect not. One thing we want our children to learn is how to apply math to the everyday world, and that depends on where the grounding of the subject is in that reality. A student who has learned to use differential equations in a rule-based fashion approaches work with a more mature mind and more prior knowledge and experience in applying math. In other words, the effectiveness of the rapid, rule-based approach to the rule-making ability of older children and adults may depend on the initial background, when learning student begins to abstract the basic concepts of numbers and arithmetic from her/his daily experience.
After all, as we have known so far, everyday experience is how our ancestors first set off on the mathematical path many thousands of years ago.
In my assertion above, I write “as far as we know” because, of course, all that we know is based on the archaeological evidence of the artifacts that the ancestors had discovered. first left. We don’t know how they really think about the world.
(to be continued)…