According to educational statistics, on average, chinese students have to spend more than 12 years on mathematics and complete tens of thousands of exercises。
But a confusing phenomenon is that when the vast majority of people walk out of the school door, mathematics seems to evaporate from their memories, and few of them, except for basic additions and subtractions, are actively calling in their daily lives by means of formulas, theorems, problem solving techniques that have been rehearsed。
This huge contrast between inputs and outputs leads many to question: what is the meaning of mathematics learning?
At the beginning of the nineteenth century, german mathematician goss faced a seemingly chaotic situation, and in 1801 italian astronomer piazi discovered that the observation of asteroid valley was interrupted shortly after the disease, leaving scattered and miscalculable data. When goss took over the figures, he did not choose to give up, but to draw together all observations and errors into a chart。
The result is surprising: these seemingly obscurantious deviations present a middle-high, two-end, low-pretty curve that was later named a normal distribution, known as the goss distribution, which is one of the most central tools of modern statistics。
It is worth noting that this pattern of distribution does not exist only in distant star-space observations, where anyone who throws two dice at his home, records the sum of points hundreds of times, repeats seven times more times than two or 12 times, and the frequency map also presents a prominent medium shape。
This general pattern, which can be observed from astronomical observations to children's games, is precisely an indication that mathematical thinking is not an artificially constructed abstract symbol, but a bottom logic derived from the real world。
The problem is that traditional education tends to teach students only to calculate the parameters of a normal distribution, but rarely enables them to understand the way in which the thinking behind this curve represents, how to find order in chaos and how to deal with uncertainty in a structured way
The absence of such a way of thinking has serious consequences in real life. A medical study in the twentieth century found an anomaly in which the survival rate of the mother-smoking group was higher than that of the non-smoking group among low-weight newborns。
This finding was controversial at the time, even when it was considered that "smoking has a protective effect on lightweight babies", but an in-depth analysis revealed that it was simply an misinterpretation caused by statistical mixage。
Low-weight babies may come from two completely different causes: malnutrition due to environmental factors such as mother smoking and severe congenital diseases. The latter risk of death is much higher than the former, but researchers compare the two groups of babies and naturally draw distorted conclusions。
Without basic statistical thinking, it is difficult to discern this seemingly data-supported and logically confusing trap。
Similar cognitive deviations occurred during the second world war, with the united states military accounting for the distribution of bullet holes in returning warplanes and the discovery of a high level of ammunition in the wings and airframes, which led to plans to strengthen the armour in those parts。
But statistician ward argued against the fact that what really needed to be strengthened were the few bullet holes in the engines and cockpits, because once they had been shot, the aircraft would never be able to return, and the bullet holes in the survivors did not represent all the dangerous areas, which was the famous theory of survivor deviation。
Today, this theory is widely applied in areas such as business analysis, medical research, social investigation, and helps people avoid the deadly error of seeing only the surviving samples, a judgement that requires the capacity for reverse thinking and structured analysis that has been repeatedly emphasized in mathematics training。
From a practical point of view, mathematical thinking was more directly supportive of modern technology, and in the 1940s, mathematician uram played card games on his bed, finding that it took as much time as astronomical numbers to exhaust all possible series。
Instead, he tried to randomly extract part of the statistics and then use the samples to extrapolate the whole pattern, which was later named the monte carlo method and has now become an indispensable computing tool in the fields of nuclear physics, engineering simulation, financial risk assessment, etc。
China has made extensive use of these probability-based algorithms in its national priorities of space engineering, weather forecasting and disease modelling. Without a solid mathematical base, it can be said that today's digital transformation and intellectual development are impossible。
However, there are still serious faults in the current mathematics education, with the emphasis on quick answers in the classroom, while the real mathematical research process is full of mistakes and reruns, and the probative part of a mathematical paper may be dozens of pages, requiring dealing with numerous border situations and reverse cases。
The value of this training is not to remember a certain theory, but to develop a habit: "why" in the face of any conclusion, "where" in the face of any data。
This ability is particularly important in the era of information explosions, and when social media are filled with generalizations about "how a generation is how" for all young people, people who are logically trained are instinctive to question: is the chain of evidence for this conclusion complete? Is there a reverse? Is the sample representative?
The emphasis on mathematics education at the national level is also based on this consideration, ranging from "scientificism in the nation" to "innovation-driven development", where policymakers are well aware that the scientific literacy and logical judgement of a society directly affect the quality of public decision-making and the stability of social functioning。
When rumours can be transmitted instantaneously through social networks, citizens with basic statistical commons can effectively resist emotional misapprehension, and when technological change accelerates the iterative process, those who know mathematics can quickly adapt to new tools。
Mathematics never enter life on its own. It just waits quietly, from astrometric error curves to table dice games, from hospital statistics to battlefield survivors, from random sampling of poker cards to modern computer monte carlo algorithms, which seem unrelated, in fact point in the same direction。
Mathematics provides not a standard answer, but a framework for thinking about the world, judging information, and dealing with uncertainty. When a person truly masters it, the world becomes clearer, more manageable and more trustworthy。
