Arithmetic for Parents

A Book for Grown-Ups About Children's Mathematics

Ron Aharoni

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814602921
• Topic: Mathematics
• Subtopic: Arithmetic
• Index: Mathematics

This book is the result of a unique experience: a research mathematician teaching in an elementary school. It tells about a fascinating discovery made by the author — that elementary mathematics has a lot of depth and beauty, and that the secret to its teaching is in understanding its deep points.

The first part of the book discusses the nature of mathematics and its beauty. The second part tells about the teaching principles the author distilled from his experience. The third part is an excursion through the arithmetic studied in elementary school, accompanied by personal stories, historical anecdotes and teaching suggestions. The appendix relates the fascinating story of modern day politics of mathematical education.

The book was a bestseller in Israel, and has been translated into many languages. The extraordinary combination of mathematical and didactic insights makes it an essential guide for parents and teachers alike.

Contents:

• Elements
• The Road to Abstraction — Principles of Teaching
• Arithmetic from First to Sixth Grades
  • Meaning
  • Calculation
  • Fractions
  • Decimals
  • Ratios

Readership: Parents whose children study elementary mathematics, practioners who provide mathematical learning, and the general public.
Key Features:

• Modern day parents often wish to be involved in their children's education. This is a unique guide that can help them do it
• It is an ideal aid for parents who choose home schooling. The book will also appeal to those grownups who wish to return to their childhood experience, and understand it from a mature point of view

Part 1

Elements

What is Mathematics?

The book of nature is written in the language of mathematics.

Johannes Kepler, Astronomer

The Queen of Sciences

Mathematics is the queen of sciences and arithmetic the queen of mathematics.

Carl Friedrich Gauss, Mathematician

In second-grade classes, I try to explain the importance of numbers. I tell the children a story of a king who hated numbers so much that he forbade their use throughout his kingdom. Together, we try to imagine a world free of numbers, and discover that life in it is very limited. Since it is forbidden to mention a child’s age, children of all ages enter the first grade. You cannot pay for your groceries, nor can you set up an appointment, since you are not allowed to mention the number of hours and minutes.

This is only an illustration of the importance of mathematics in our lives. As civilization and technology advance, our lives become more and more dependent on mathematics. Steven Weinberg, a Nobel laureate in physics, dedicated two chapters of his book Dreams of a Final Theory to subjects beyond physics: mathematics and philosophy. He writes that time and again he is surprised to discover how useful mathematics is, and how futile is philosophy.

To understand why this is so, one must understand what is mathematics. This is not a simple question. Even professional mathematicians find it difficult to answer. Bertrand Russell said of mathematicians that they “don’t know what they are doing.” (His judgment of philosophers was even harsher: A philosopher in his eyes is “a blind man in a dark room looking for a black cat that isn’t there.”) This is true in at least one sense: Most mathematicians do not bother to ask themselves what it is, exactly, that they are doing.

To try answering this question, we will start with a simple example: What is the meaning of 3 + 2 = 5?

In the first grade, I ask the children to examine how many pencils there are when you add 3 pencils to 2 pencils. They know that “addition” means “joining.” Therefore they join 3 pencils and 2 pencils and count: 5 pencils. Now I ask, “How many buttons are there when you add 3 buttons and 2 buttons?” “5 buttons,” they answer, without missing a beat. “How do you know?” I insist. “We know from the previous question.” “But the previous question was about pencils. Maybe it’s different with buttons?” They laugh. But not because the question is pointless. On the contrary. It contains the secret of mathematics — abstraction. It does not matter if the objects in question are pencils, buttons or apples. The answer is the same in all cases. This is why we can abstractly say: 3 + 2 = 5.

This is an elementary, but characteristic, example: Mathematics abstracts thinking processes. Obviously, every thought is abstract to a certain degree. But mathematics is unique in that it abstracts the most elementary processes of thought. In the example of 3 + 2 = 5, the process involved is the joining of objects: 3 objects and 2 objects. One can ask many questions about these objects: Are they pencils or apples? Are they in your hand or on a table? And if they are on a table, how are they arranged? Mathematics ignores all these details, and asks a question that relates not to the various details, but only to the fact that these are objects that are joined: the resulting amount. That is, how many objects are there?

Abstract thought is the secret of man’s domination of his environment. The power of abstractions lies in the fact that they enable us to cope efficiently with the world. In other words, they save effort. They enable going beyond the boundaries of the “here and now” — something discovered here and now can be used in another place and at another time. If 3 pencils and 2 pencils equal 5 pencils, the same will be true for apples, and it will also be true tomorrow. A one-time effort provides information about an entire world.

If abstractions in general are useful, then all the more so is mathematics, which takes abstractions to their limit. Therefore, it is not surprising that mathematics is so useful and practical.

Should Everyone Learn Mathematics?

People, on learning that I am a mathematician, often react with a thin smile, barely hiding a grimace of agony: “Mathematics wasn’t one of my strong subjects.” For so many people learning mathematics is such a tormenting experience that each generation asks the same question — what for? Why is this torture necessary? Shouldn’t most people just give up on the attempt to learn mathematics? Nowadays, when a calculator can instantly perform mathematical operations, what is the point of learning the multiplication table or long division?

One answer is that mathematics is the key to all professions demanding knowledge of the exact sciences, and there are many of those. But mathematics is important not only for understanding reality. It offers much more than that — it teaches abstract thought, in an accurate and orderly way. It promotes basic habits of thought, such as the ability to distinguish between the essential and the inessential, and the ability to reach logical conclusions. These are some of the most significant assets that schooling can provide.

The question still remains unanswered — why is it so difficult? Must mathematics be a cause of suffering? A currently popular answer is “no” — the problem lies in the teaching. Common opinion is that many children considered to be “learning disabled” are actually “teaching disabled.” But it can’t be that simple. Blaming the teachers is too simplistic, and unreasonable. Anyone who claims that for hundreds and thousands of years mathematics teachers have been doing a bad job, must explain why this is so and why it isn’t so in other subjects.

The special problem in teaching mathematics lies in the difficulty of conveying abstractions. You can tell people the name of the capital of Chile, but you can’t abstract for them. This is a process each person must accomplish on his or her own. One must mentally pass through all the stages from the concrete to the abstract. The teacher’s role in this process is to guide the student so that he experiments with the principles in the correct order. This is not a simple art that is easy to come by. But neither is it impossible. One of the purposes of this book is to relay some of the principles along the path of such “midwifery” teaching, as Socrates put it.

The Three Mathematical Ways of Economy

I didn’t have time to write you a short letter, so I wrote a long one.

Blaise Pascal, Mathematician

Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself.

George P´olya, Mathematician

The true virtue of mathematics (and not many know this) is that it saves effort. This is true of any abstraction, but mathematics has turned economy of thought into an art form. It has three ways to economize: order, generalization and concise representation.

Order

Carl Friedrich Gauss was the greatest mathematician of the 19th century. One of the most famous stories in the history of mathematics tells of how his talent came to light when he was seven years old. One day, his teacher, looking for a break, gave the class the task of summing up all the numbers between 1 and 100. To his surprise, young Carl Friedrich returned after a few minutes, or perhaps even seconds, with the answer: 5050.

How did the seven-year-old accomplish this? He looked at the sum he was supposed to calculate, 1 + 2 + 3 + ··· + 98 + 99 + 100, and added the first and last terms: 1 and 100. The result was 101. Then, he added the second number to the one before last, that is, 2 and 99, and again the result was 101. Then 3 and 98, which yielded 101 again. He arranged all 100 terms in 50 pairs, the sum of each equaling 101. Their sum was thus 50 times 101, or 5050.

What little Gauss discovered here was order. He found a pattern in what seemed to be a disorganized sum of numbers, and the entire situation changed — suddenly matters became simple.

Imagine a phone-book arranged by a random order, or an unknown order. To find a phone number, you would have to go through each and every name. The order introduced into the phone-book, and the fact that we are familiar with it, saves a great deal of effort. A relatively small effort invested in alphabetical arrangement is returned many times over.

Or, think how much easier it is to live in a familiar city than in a strange one. A local knows where to find the supermarket or the laundromat. Knowing the order of the world around us provides us with orientation. Science, and mathematics in particular, has taken upon itself to discover the order of the universe, so that we may adjust our actions to it.

Generalization

There are many jokes about the nature of mathematics and mathematicians. The following is probably the best known of them all. I make a point of telling it to my students in every course I teach, since it is not only the most familiar, but also the most useful. It illustrates the principle of mathematical practice: Something once done does not require redoing.

How can you tell the difference between a mathematician and a physicist? You ask: Suppose you have a kettle in the living room. How do you boil water? The physicist answers: I take the kettle to the kitchen, fill it with water from the tap, place it on the stove and light the fire. The mathematician gives the same answer. Then you ask: Suppose you have a kettle in the kitchen. How do you boil water now? The physicist says: I fill the kettle with water from the tap, place it on the stove and light the fire. The mathematician answers: I take the kettle to the living room, and this problem has already been solved!

This brings economizing of thought ad absurdum, by placing it before true economization.

“This has already been solved” was also the answer we heard from the children who said they did not need to check how many buttons are 3 buttons and 2 buttons, since they had done the same with pencils. It appears, whether overtly or hidden, in each mathematical proof, and in every mathematical argument. “We have already done this, and now we can use it.” In fact, this idea lies behind every abstraction: What we discover now will also be valid in other situations.

Proof in Stages: Induction

There is a mathematical process that is based entirely on the principle of “this has already been done.” It is called “Mathematical Induction.” A certain point is established in stages, with each stage relying on its predecessor, that is, on the fact that the previous case “has already been solved.”

We will encounter this process several times throughout the book but will not mention it explicitly. For example, the decimal system is inductive: First, ten single units are collected to equal a new entity called a “ten.” Then ten tens are collected to equal a new entity called a “hundred,” and so forth. Yet another example is calculations: All algorithms used to calculate arithmetical operations are based on induction.

Concise Representation

The third mathematical economy is in representation. We are so used to the way numbers and mathematical propositions are represented, we forget that methods of representation were not always so sophisticated, and it was not so long ago, relatively speaking, that mathematical notation was much more cumbersome.

Let’s begin with the representation of numbers. Up until about three thousand years ago, numbers were represented directly — “4” was represented by four markings, for example, four lines. This is a good idea for small numbers, but impractical for larger ones. Using the decimal system, we can now represent huge numbers concisely: A “million” only requires seven digits.

The second type of economy is in the representation of propositions. A “mathematical proposition” is the equivalent of a sentence in spoken language. Up until a little over two thousand years ago, mathematical propositions were phrased in words, for instance: “Three and two is five.” Then, a very useful tool was invented: the formula. Its originator was probably Diophantus of Alexandria, who lived during the 3rd century B.C. Formulas are not only shorter, they are also more accurate and uniform, and allow systematic handling.

Historical Note

The notation we currently use developed slowly and gradually. Its current form was only established ...