What does algebra in school really teach us and what does it mean? (Part 1)

Teachers always emphasize the importance of all children needing to master algebra before graduating from high school. What exactly is algebra, is it really important as people often say no and why do many people find algebra a difficult subject? Please read to the series on algebra in school through the perspective of math doctor Keith Devlin.

Dr. Keith Devlin is currently a professor at Stanford University, has more than 80 published studies and 30 math books, among which have won prestigious American awards such as Pythagoras Prize, Carl Sagan Award, Peano Prize.

Why is math difficult for many students? How to make math easier?

Algebraic thinking, a method of thinking to handle complex problems more quickly and accurately

These secrets will help young children learn multiplication tables and make multiplication much easier

Part 1: Differences between arithmetic and algebra, generalization of algebra in schools

Algebra is not "arithmetic with characters".(Photo: Лекториум).

This paper focuses only on arithmetic and algebra in schools , unlike arithmetic and algebra of professional mathematicians with much more general meaning.

While answering the questions at the beginning of the article seems easier than answering a typical algebra question in school, surprisingly, very few people can give good answers.

It must be said from the beginning that algebra is not "arithmetic with characters" . At the most basic level, arithmetic and algebra are two different forms of thinking about numerical problems.

First , let's talk about arithmetic.

Basically, arithmetic is the use of basic calculations with the number-plus minus multiplication-to calculate the numerical value of everything. Arithmetic is the oldest subject of mathematics, dating back to the Sumerian civilization 10,000 years ago. Sumer society has reached a complex stage of using money as a means of measuring the wealth of each individual and an intermediary for the exchange of goods and services. Finally, monetary representations have paved the way for abstract imprints on clay plates that we now consider the first digital signs of humanity. Over time, these digital symbols carry their abstract meaning: numbers. In other words, the number arises first as money, and arithmetic arises as a means to use money in commerce.

We should note that the counting of numbers is preceded by the numbers and the arithmetic is thousands of years. People began to count objects (mainly family members, animals, crops, possessions .) at least 35,000 years ago. Evidence is that the skeleton engraved with the "book" calculation has been discovered. Those "books" are the basis for anthropologists to conclude that, at that time, there were records of numbers.

The ancient "books" of ancient people are a recognition, and what they directly reflect are objects in the world rather than abstract numbers.

A bone fragment dating back to 20,000 years represents the "book" calculated by one-on-one counting of the people of the time (Photo: Thier).

A second note is that, initially, arithmetic is not solved in the form of manipulating symbols as we are taught today. Modern numerical numerical representations that have been developed over the centuries, beginning in the first half of the 1st millennium AD in India, were received by Arab merchants in the second half of the 1st millennium. , and it was not until the 13th century that it was introduced to Europe. That's why today's name is arithmetic is "Arabic-Hindu arithmetic". Before receiving Arabic-Hindu arithmetic based on symbols (symbols), merchants performed calculations through a rather complex hand or abacus counting system. It was not until the 15th century that manipulation of symbols began to replace the verbal calculations described in previous arithmetic books.

A relic of Indian arithmetic: stone slabs with number 270 inscriptions in an ancient temple were built in India in 876 (Photo: Bill Casselman).

Many people find learning arithmetic a difficult task, but most of us succeed, or at least pass tests, on the condition that we practice enough. The things that help us learn arithmetic are the basic building blocks of the subject, the numbers that arise naturally in the world that surrounds us, as we count, measure, buy, make things. , use the phone, go to the bank, check the score of the baseball team I like . The numbers can be abstract because you never see, feel, hear or smell the smell of number 3, but they are inextricably linked to all the concrete things in the world we live in.

However, with algebra, you have separated the world one step further. The variables x and y that you have to learn to manage in algebra are representatives of numbers, and are always general numbers, not specific numbers. And the human brain is not suited to thinking about that level of abstraction naturally. Doing that requires a lot of effort and training.

The important thing that we need to realize is that, doing algebra exercises is a way of thinking and that way of thinking is different from arithmetic thinking . The formulas and equations include x variables and y is just a way to represent that thought on paper. They (formulas and equations) are not algebraic than a sheet of sheet music notes. We can also do algebra exercises without symbols, just like when you can play an instrument without reading the music. In fact, merchants and others need algebra that used it for 3,000 years before algebra in the symbolic form introduced by French mathematician François Viète in the 16th century. Today, the method of algebra It was originally called "eloquent algebra" or "verbal algebra" to distinguish it from the current popular algebraic notation method.

In summary, we can understand the difference between arithmetic and algebra in schools in several ways as follows:

1. Algebra is a way of thinking more logical than thinking about numbers.
2. In arithmetic, you argue (calculate) with specific numbers, and in algebra you reason (logic) about numbers. The focus of algebra is relationships, not calculations. For example, the relation a + b = c (c is constant) represents two unknown numbers a and b that are related by addition, while 3 + 5 = 8 is also a similar relationship but usually see a way to represent 8 to calculate 3 + 5 more.
3. Arithmetic is a quantitative argument with numbers, algebra is a qualitative argument about numbers.
4. In arithmetic, you calculate a number by working with the given numbers. In algebra, you use a term for an unknown number and a logical argument to determine its value.

The following example will help you understand these differences: the "I 14, my brother is 4 years older than me" problem is a simple equation x = 14 + 4, can be solved immediately by thinking numbers learning is doing addition, but the second problem is more complicated: "My brother is 2 years older than me, my younger sister is 5 years younger than me. She is 12 years old. Ask me how old I am in 3 years ". This article needs more thinking and can be solved in many ways: the first way is to analyze and represent the relationships between quantities by characters to get the answer. For example, if k is my current age, then k - 5 = 12 and my brother's age in the next 3 years is (k + 2) + 3. The second way is to show the unknowns to be found on the number line, the way The third is to put another character for the unknown number (x - 3 - 2 - 5 = 12). The algebra in this problem is to find a method to track unknown quantities with different operations on it.

Another note to be understood from the above differences is that algebra is not a numerical exercise with one or more characters representing known or unknown numbers.

For example, the following exercise is arithmetic rather than algebra: find numerical values ​​for quadratic equations of the form ax2 + bx + c = 0 by setting numerical values ​​for a, b, c in the familiar test formula attached below.

In contrast, the development of the equation from the beginning to get the experimental formula is algebraic exercise. Similarly, the exercise solves a quadratic equation by means of the quadratic complement (the initial quadratic transformation to the sum of a square and a constant) and the factorization (also called analysis Polynomial to factor, is the analysis of the equation of multiple polynomials) instead of using the experimental formula is also an algebraic exercise, not arithmetic.

Examples of polynomial transform methods to solve equations:

Humanization: 15a2b2 - 9a3b + 3a2b = 3a2b (5b - 3a - b2)

Squared compensation: x2 + 4x + 1 = (x + 2) 2 - 3

When students start learning algebra, they inevitably try to solve the problem with arithmetic thinking. It is a natural thing to do with all the efforts that you have spent on mastering arithmetic before, and this method works when the original algebra exercises are exceptionally simple.

In fact, the better a student is in arithmetic, the more likely he or she can progress in arithmetic thinking. For example, many students can solve quadratic equations x2 = 2x + 15 with basic arithmetic without completely using algebra (by guessing, numbering and taking square root).

However, it is paradoxical that those students may also find learning algebra more difficult than arithmetic. Because except for the simplest examples, in all cases, to do algebra, you must stop arithmetic thinking and learn algebraic thinking.

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