What does algebra in school really teach us and what does it mean? (Part 2)

Algebra is an important discipline in mathematics, a step farther than arithmetic in the history of mathematics. In arithmetic, you work with specific numbers, while in algebra, your research object is the relationships between numbers, abstract structures with general numbers. What is the benefit of studying algebra for us in daily life? Please read to part 2 of the two-part series on the nature of algebra in school and its meaning.

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

Why do we use "X" as an unknown in Mathematics?

Part 2: The meaning of algebra in life and some ideas about algebraic thinking

Summary of part 1: the difference between arithmetic and algebra, generalization of algebra in school

The translation series from Dr. Keith Devlin's blog, a famous Stanford math professor with over 100 published books and studies.

Is algebraic proficiency (eg, algebraic proficiency) worth the effort? Certainly, although you will have to struggle in hardship to reach that destination, based on what you will find in most algebra textbooks at school.

In today's world, most of us really need to learn how to master algebraic thinking. An example is that you need to use algebraic thinking if you want to write a macro to calculate cells in a Microsoft Excel spreadsheet. Only one example of this helps us to see why algebra, not arithmetic, is the main goal of teaching mathematics in schools. With spreadsheets, you don't need to do arithmetic exercises but computers will do it, much faster and more accurately than any other person in general. What you-humans-have to do is create that spreadsheet at the beginning. Computer can't do it for you.

The problem is not what you use the worksheet to do, to calculate points for sports tournaments, to monitor finance, to run a business / club, to find the best solution to equip your character. You in World of Warcraft, the problem is that you need algebraic thinking to set up spreadsheets to do what you want. That means thinking about numbers in general form instead of specific numbers.

According to Dr. Keith, although spreadsheets can give students today more satisfactory and meaningful applications of problems with the number of leaving trains, the number of sprinklers in his generation used to be "endure" but of course the need for algebra does not make this subject any easier. But in a world where the livelihoods of each country depend on leading the curve of technology, equipping students with the thinking skills that the world needs is important. The ability to use a computer is one of those skills. And the ability to use computers to do arithmetic exercises requires that we have algebraic thinking.

The ability to use computers to do arithmetic exercises requires that we have algebraic thinking.(Photo: Exceljet).

Sharing and questions about algebraic philosophy

The article by Dr. Keith Devlin has helped many people understand why they struggled with algebra during their school days, and the article also received lots of sharing and questions about algebraic thinking. Here are some questions, good sharing from readers and Dr. Keith's answer.

1. Compare the similarities between algebra and Microsoft Excel very well. I often teach my students algebra like running through tires (running tires, tires run) in practicing soccer. You will never run through tires in a football game, but you do it to forge your feet in a good way. Similarly, algebra is learning how to think rationally and solve problems. No boss will ask you to analyze a polynomial into a factor, but they will ask you to solve a problem logically. (Eric Blask)

(Photo: Youtube)

2. Rather than solving problems, algebra is solving problem-related problems, meaning working with non-verbal abstract symbols. This morning, when I thought about why so many students passed the geometry but had to "fight" with algebra, I thought it involved abstract symbols. Actually, a standard geometry course doesn't have many abstract symbols, except of course algebraic courses. Symbols are quite simple and often relatively specific, such as characters indicating angles (A, B, C .), or n-indexed characters (the labels of this type are a type of similar symbol. For classic, like arithmetic notation or language), things that you can put into a diagram with certainty. Even geometric proofs are relatively specific in the way you work with a diagram (which you can get into trouble!), Abstraction here is a different kind of abstraction than algebra.

As for Excel tricks, they seem to make more abstract algebraic symbols specific. Because you can change the input fields and see the changed output cells. Therefore, you will get used to visualizing the changing / temporary meaning of a symbol in many ways that without Excel you would have to imagine in your mind. So maybe it's a way of looking at differences - algebraic symbols can represent different things in different cases, due to their nature. Symbols in algebra have a trend over time / change.

In fact, that is also true for geometric proofs, except that, because of the interest in student success, we have eliminated the problem of change - we try to draw a diagram. The only typical (between variables / events without ambiguous, unclear-relational relationships) and arguments with it. Sometimes I try to help a student "see" a theorem by imagining parts of the moving diagram, showing constraints, and then observing what still has to remain.

For example, when increasing the number of degrees of an angle in a triangle, we will see other angles (or just one angle) to decrease in the same number of degrees. Intuitively, that establishes the judgment that "the sum of the angles" is a constant. What is that constant? Increase one corner until it is nearly 180 and the other two corners are close to 0, oh, that's 180 degrees. The same is true for polygons in general, and in the general case, increasing (n-2) the number of angles approaching 180, the last two angles will be 0! (David Lewis).

(Photo: Wikipedia).

But I must say, this idea often doesn't appeal to students as much as it appeals to me. If I could do that in a geometric "spreadsheet" like the Geometer's Sketchpad (an interactive software with a fee to learn Euclidean geometry, algebra, integrals .), maybe this idea would be better. . Therefore, there is more evidence that new time / change issues are new elements that make things more difficult for students to grasp. And change is something that happens in algebra - look at that world - variables! Therefore, algebra is even higher than "thinking reasonably".

Change is a common occurrence in algebra.(Photo: KNILT).

Keith's share:

I agree that higher algebra (also known as modern algebra) is inherently symbolic / abstract, but algebra in school, the focus of my writing is not so. In fact, verbal algebra was done for thousands of years until the symbolic method was introduced by Viete in the 16th century (as stated in part 1).

In my book "The Math Gene" (Gen Math) published in 2000, I investigated the cognitive problems that people encounter in the face of abstraction. If the goal is classical algebra (or algebraic thinking), then it may be unnecessary to focus on abstract symbols, and according to me the spreadsheet is a good way to establish those symbols.

Of course, the ability to manipulate abstract symbol structures is also valuable. But when using existing spreadsheets is a useful skill for everyone, mastering abstract-symbol systems is less important, because many people find it hard or impossible to capture those systems. . Therefore, we may need to delay the use of multiple symbols until students have mastered algebraic thinking.

3. "The number arises first as money, and arithmetic arises as a means to use money in commerce" (a note in part 1) . Do you think so? What about the first numbers represented by mammoth drawings (mammoth, an extinct ancient elephant nearly 5 million years ago) on cave walls?

Keith: You gave an interesting and subtle idea. What you describe is not related to numbers. It is a form of early calculation system. Computational systems are common in the early history of mankind, but they only require one-to-one correspondence (one corresponds to a representative of animals that are shaped like animals used to count and real objects needed) Counting, because there are no digits yet, the ancient people use the representative to count the numbers), like "I have this, I have many others". Numbers arise when you abstract them as intermediaries between collections of objects and related calculations. We see a similar abstraction stage when young children learn. Initially, they grasped the corresponding one-to-one idea, and only then did they grasp the concept of number.

A bone fragment dating back to 20,000 years engraved the number one of the ancient people.(Photo: CC-SA).

4. I wonder if someone can be very good at algebra but poor in arithmetic? (Jerry Lou)

Keith: Many professional mathematicians, including me, have such trends. In fact, this may be because (1) lack of interest in arithmetic and (2) mathematicians almost never use arithmetic, so all the progress they have accumulated in middle school Learning tends to die gradually due to lack of use opportunities. On the other hand, arithmetic proficiency needs a good memory, and in algebra, you can imagine everything by logic, and that can continue to happen more with mathematicians.

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