Across disciplines and professions, math competence is highly desirable. According to a Gallup poll in 2013, mathematics is considered the most valuable school subject in America [1]. Studying mathematics is known to develop logical thinking, analytical skill, and the ability to solve complex problems. But what exactly is it in mathematics that helps math learners develop these qualities?
Mathematics is a very big subject that comprises of many different branches, among which are the familiar ones such as: arithmetics, algebra, geometry. Mathematics is so broad and so deep that there exists a gap between the public’s perception of mathematics and what mathematicians actually study. For example, in school mathematics, a major focus is on number operations: addition, subtraction, multiplication, division, starting from integers to rational, to real, and complex numbers. Questions that mathematicians deeply care about such as can we count rational numbers the way we count natural numbers are often not discussed in a typical curriculum. In many cases, the questions that mathematicians ask can easily be understood by a middle school student, such as, can we write any even number as a sum of two prime numbers [2], but the answers are so complex or even unknown. It is in search of questions like this that mathematicians continue to build the structures of mathematics, discover new connections between existing branches and develop new branches of mathematics. In so doing, they adhere strictly to the rigorous standard of mathematics that few outsiders could follow. In other words, school mathematics teaches us how useful mathematics can be but the inner workings of mathematics are often left out. It is this inner workings that make up the true power of mathematics.
Let’s look at the most important elements of mathematics and why we should teach these to young children.
Logic
Logic is the systematic study of inference, the process of deriving a conclusion from a set of premises [3]. Mathematics- being built on a logical foundation- is an ideal place to learn logic. In learning mathematics, we constantly learn to check for consistency, learn to make valid arguments, learn to examine the soundness of each assumption, or premises in the language of logic. In fact, early history of mathematics and logic were so intertwined. One of the most important works in mathematics, Euclid’s Elements [4] c. 300 BC, not only constitutes to the glory of ancient Greek mathematics but is also instrumental in the development of logic [5].
Logic is most obvious in mathematical proofs, but that should not limit us from introducing logic early to children. Logic puzzles are wonderful for teaching logic to elementary school students. Let’s look at an example of a logic puzzle.
Problem: In each of the three drawers, there is one book that belongs to the following subjects: English, Art, and Math. One mischievous student labeled the drawers as: Math, English, English or Art. Knowing that all the labels are wrong, can you determine which drawer contains which book?
In contrast to problems that only require recalling facts such as what is the sum of two numbers, problems like this require students to reason, to try different hypotheses, potentially run into contradictions, and learn to resolve them. They are also interesting and are likely to pique students’ interest in mathematics early on.
Rigor
Mathematicians have a very high standard when it comes to the truth. They won’t believe any mathematical statement until it is proved to be true, not even an obvious one like if you draw a line connecting two points- one inside, one outside a circle- then the line would intersect the circle [6]. This level of rigor may sound too daunting, even to college students. Yet there are a lot of opportunities to teach proof to elementary school children.
Consider the following example.
Problem: Prove that there is no biggest number.
Solution: Suppose there is a biggest number. Add one to it, we have a bigger number. So we run into a contradiction with our assumption that there is a biggest number. Therefore, there is no biggest number.
This is an example of proof by contradiction, a powerful and popular method of proof in mathematics. Most importantly, the simple line of reasoning here is understandable, and can be constructed by an elementary school student.
Studies on the teaching of proof to elementary school students show that they are capable of forming conjectures, explaining their findings and convincing others of the validity of their ideas using popular methods of proof such as contradiction, recursion, proof by cases [7] . In one example, a group of 3rd graders were able to recognize the pattern that the sum of two even numbers is always an even number. One of them started to “appeal to authority” by saying that he was told of this fact by his older sister. But then the other students started to examine more cases, had suspicion about the validity of the statement when the numbers got really large, and eventually found a way to construct a proof that apply to all even numbers.
Proof, therefore, should not be shunned or delayed in math education. By introducing the concept of proof, we encourage students to be skeptical of any mathematical statements. By requiring them to prove mathematical conjectures, we give them the opportunities to defend their ideas using logical reasoning, and the feeling of empowerment in the absolute truth that they found. Proof really teaches children critical thinking. Isn’t it one of the most important reasons why we make our children learn mathematics?
Generality
Mathematicians always strive for generality as the goal is to study patterns and changes rather than one specific case. A well known example is the extension from natural numbers, to negative numbers, to rational numbers, to real numbers, then to complex numbers [8]. Generality makes mathematics, the so-called language of science, so powerful in its ubiquitous applications. Vastly different phenomena across different disciplines can be described by the same mathematical concept. For example, population growth in biology, acceleration in physics, compound interest in finance all can be described by the same type of equations called differential equations.
Obviously, teaching generality at the school level is difficult and not always possible, but the important thing is to build this mindset for young math learners. Here are two concrete examples .
Problem 1: You can pick 10 crayons in two colors red and blue. How many different combinations can you get?
After we let the students do this problems for a few cases: 5, 7, 10 crayons, we should ask them to generalize it to the case of an arbitrarily large number crayons. They will then quickly realize how many different combinations they will get with 100 crayons, instead of having to do it all over again.
Problem 2: This is a well known problem called cutting pizzas. You are to cut a pizza using only straight cuts. The goal is to have as many pieces as possible. With 4 cuts, what is the largest number of pieces you can make?
In helping students to solve this problem, it is a good idea to ask them to start from 1 cut and increase the number of cuts. If we cut the pizzas the usual way, we will get 2 pieces with 1 cut, 4 pieces with 2 cuts, and 6 pieces with 3 cuts. However, if you tinker with it a little bit, you’ll realize that we can get 7 pieces with 3 cuts. At this point, the students will feel motivated to find out how many pieces he can get with 4 cuts. We can then encourage them to continue with 5 pieces. Over time, we can expect them to ask questions such as: what is the largest number of pieces with a certain number of cuts, or what is the minimum number of cuts to have a certain number of pieces. They may not know the answers but it is an important part of their math training.
Connectedness of different concepts
Although mathematics is extremely broad, it is very common to find connectedness between seemingly very different concepts. In elementary school, students think of a circle as a geometrical object. But in high school they will learn that a circle can be represented by an equation. By writing all geometrical objects from a line, to a circle, to a sphere as equations, we learn to look at these objects in a different light. That will also will allow us to solve much more complicated problems that are intractable if we are to use purely geometrical tools.
To make mathematics truly interesting and enlightening to young children, it is important to show them the connectedness of different concepts in mathematics whenever possible. Let’s look at the following two problems.
Problem 1: 5 friends want to catch up with one another over the telephone. How many phone calls do they have to make, assuming that each person only talks once to another person.
Problem 2: Emma wants to make beaded earrings using 5 beads having the same shape and size as shown in the picture on the right. If 2 of the beads are in blue color and the other 3 are in white color, how many different earrings can she make?
If we give these two problems a week or two apart, at first glance, most likely the students will think that they are two different problems. Yet, with some guidance, they may be able to draw a connection between the two. The two problems can be shown to be equivalent to the following one. There are 5 points on a piece of paper. Draw all connections that connect every pair of points. There are 10 connections in total and that is the answer to both the problems above. Each connection can be seen as
a link between two blue beads in problem 1 or a phone call between two friends in problem 2.
In my experience, this aspect of mathematics is one of the most intriguing for young children. They rejoice in the aha moment when they find out how a new problem is connected to an old one. Making the connection between different problems, different concepts whenever possible will help students avoid being overwhelmed by new concepts. Rather than looking at mathematics as a bunch of unrelated problems that need different toolboxes to solve, students will see it as a connected web of knowledge and are more prepared to build complex ideas on a solid foundation.
In conclusion, the above mentioned essences of mathematics are what make it a super intellectual subject. Unfortunately, these same properties also make mathematics seem daunting to many. Nevertheless, incorporating these elements in the teaching of mathematics early on is crucial in increasing children’s understanding of, and therefore their performance in the subject. It also shows the students the inner beauty of mathematics. As mathematics gets increasingly difficult and not every one will continue it at a higher level, grade school might well be the only time for them to experience the true beauty of this special subject.
References:
[1] Gallup poll
https://news.gallup.com/poll/164249/americans-grade-math-valuable-school-subject.aspx
[2] Goldbach’s conjecture wikipedia
https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
[3] Logic wikipedia
https://en.wikipedia.org/wiki/Logic
[4] All 13 books of Euclid’s Elements
https://www.claymath.org/library/historical/euclid/
[5] Euclid’s Elements wikipedia
https://en.wikipedia.org/wiki/Euclid%27s_Elements
[6] Jordan’s curve theorem wikipedia
https://en.wikipedia.org/wiki/Jordan_curve_theorem
[7] “Teaching and Learning Proof Across the Grades” Rutledge; 1st edition, 2009. ISBN-10: 0415887313 https://www.amazon.com/dp/0415887313
[8] “What is mathematics. An Elementary Approach” by Richard Courant and Herbert Robbins.Oxford University Press, 1978. ISBN-10: 0195025172