There is a mathematics syllabus in Singapore. It spells out what we should teach in schools. Then there is an institute of education. It trains us how we might teach it. We do not often ask the question why we are teaching what we teach. In other words, we ask what and how. But we do not ask why.
For example, do we know how many formulas there are for finding the area of a circle? One, two, or more? There are at least three. Let A be the area of a circle with radius r, diameter d, and circumference c. Then we have
A = π r²,
A = .
A = .
The second formula says: the area A is roughly three quarters of the square that circumscribes the circle. The third formula also has a meaningful geometrical interpretation. Think of cutting an orange sideway and eating it. Then in the process we would find a rectangle with base c/2 and height r and having the same area as the circle. My question is: why do we teach to our pupils the poorest formula among the three? I am not advocating a change in the syllabus. My real question is why we do not ask such questions.
We have a new syllabus 2007. We introduced calculators as a mathematical tool at the primary level. Why? We dropped transformation geometry at the secondary level. Why? We introduced box-and-whisker plots but not scatter plots. Why? The area of a triangle is half of its base times height. Why do we need another formula ½ bc sin A? Why do we teach trigonometric identities though we no longer do so? The list can go on and on.
Do we have answers to the above questions? Do we ask such questions? Why should we ask such questions? Let us describe the historical development of mathematics from the point of view of school mathematics. What happened in the past led to what school mathematics is today.
2. Euclidean geometry
Once upon a time, mathematics in the west is nothing but Euclidean geometry. It was the geometry as expounded in Euclid’s Elements around 300 BC. For a long period of time, professor of mathematics was called professor of geometry. The changes did not come along until the sixteen century. From 1600 to 1900, it was Qing dynasty in China. In Europe, that was the time of renaissance in Italy, reformation in Germany, revolution in France, and industrial revolution in England. It culminated in colonization and finally dominance of the world by the West. This is roughly what happened to the world during the past 400 years.
As far as geometry is concerned, three major events took place. The first was the collapse of Euclidean geometry. Euclidean geometry was based on axioms including the parallel axiom. By axioms we mean something we accept to be true without questioning. Given a line and a point not on the line, we can draw only one line parallel to the given line. This is an axiom. However mathematicians in early days tried to prove it. They failed. As usual, when we failed to prove something, we like to think that maybe it is not true. Indeed, mathematicians constructed models to show that there are geometries other than Euclidean geometry.
Imagine that you and I both travel northward from two different cities on the equator. I go first and I shall end up at the North Pole. You go next. You will also end up at the North Pole. We are supposed to travel in parallel, and yet we meet at the North Pole. In the language of mathematics, we cannot draw two parallel lines on the globe. So the parallel line axiom has to remain as an axiom. We cannot prove it.
This discovery caused a severe blow to Euclidean geometry. As a result, Euclidean geometry no longer reigned supreme. Later on people found that actually Euclidean geometry describes the world of Newton and it does not describe the world of Einstein.
Spherical and other geometries spelled the fall of Euclidean geometry.
The second major event was that geometry went algebraic. René Descartes (1596 – 1650) invented coordinates, now called Cartesian coordinates. A point in geometry is a pair of numbers in the Cartesian coordinates. A line in geometry is a linear equation via the Cartesian coordinates. To find the intersection point of two lines is equivalent to finding the solution of two linear equations. In other words, Descartes provided a way to convert a problem in geometry to a problem in algebra. Solve it in algebra and convert the answer back to geometry. So a geometrical problem can now be solved algebraically. For computation, it is easier to do it in algebra than in geometry.
There are many geometries. Is there a unified approach? Felix Klein in one of his lectures gave such an approach. Thereafter it was known as Erlangen Programme. He said that what geometry is depends on what transformations we consider. For example, if we consider only transformations that do not change shapes and sizes, then we have Euclidean geometry. In Euclidean geometry, we study the properties under which certain geometrical figures do not change shapes and sizes. If we consider enlargements, under an enlargement a figure changes size but not shape. Then we have projective geometry. Any property that does not change under given transformations we call an invariant property. In the language of Felix Klein, geometry is a study of invariant properties of geometrical figures under a given set of transformations. Again it is easier to work with transformations expressed in terms of algebra, in this case, matrices. Again working with matrices we are solving a geometrical problem algebraically.
Cartesian coordinates served as a bridge for the migration to take place from geometry to algebra.
The temple of Euclidean geometry collapsed. The massive migration took place from the land of geometry to the land of algebra. It was not the end. Euclidean geometry was still alive. Then came the third event.
Euclidean geometry was not rigorous according to the way Euclid presented it. In other words, Euclidean geometry was not axiomatic as Euclid intended. By axiomatic we mean we assume certain conditions without proof and call them axioms. Based on the axioms and nothing else, we prove results. Then we use the axioms and the results, we prove further results. This is called axiomatic proof. Euclid did not give enough axioms so that he could prove theorems using only those axioms. The person who saved it was David Hilbert (1862 – 1943). He gave a series of axioms and put Euclidean geometry on a sound foundation. Hence we say he saved it. His book the Foundations of Geometry is still in print after all these years. As a result, Euclidean geometry no longer makes sense as it was. Euclidean geometry is dead. Hence we say Hilbert killed Euclidean geometry.
David Hilbert saved and killed Euclidean geometry.
Hilbert did not only kill Euclidean geometry. He killed also the teaching of Euclidean geometry in schools. Now geometry is not taught in the way I learned it when I was in schools. The approach we use in schools now was suggested by G. D. Birkhoff in his book Basic Geometry (1941). Basically, we assume various conditions for the congruence of triangles without proof. Then we proceed from there proving other results. There was an attempt in the 60s to replace classical geometry by transformation geometry or finite geometry. None worked out. We often associate proof with geometry. It is rigour that we want to impart to our students. Proof is a good way to get to rigour. Rigour is everywhere in mathematics, not just in geometry alone. So is geometry everywhere in mathematics. We should always look at algebra geometrically and geometry algebraically. In a way, Van Hiele levels no longer describe accurately the geometry we teach in our schools.
Al-jabr or algebra is an Arabic word. We can fairly say that algebra came from the Arab world. The first book on algebra was written in Arabic. It is interesting to note that Euclid’s Elements was translated into Latin also from Arabic. The presentation of algebra in those days was rhetorical. That is, everything was described in words and no formula. In those days, algebra was nothing but solving polynomial equations. Linear and quadratic equations were classified into many special cases. Then a solution was found for each case. They did so because they did not recognize the existence of negative numbers.
It is interesting to note that some Italian mathematicians Cardano and his rival Tartaglia actually made a living out of solving cubic equations in the sixteen century. Complex numbers were invented due to solving cubic equations, and not quadratic equations. When the degree of a polynomial equation increases, it becomes almost an impossible task to solve it. Some of you may remember the days when we learned elementary symmetric functions like etc. They are used to solve cubic and quartic equations. Solving cubic equations was in the textbook in the 50s.
It was a dead-end to solve polynomial equations by means of elementary symmetric functions. The best result thereafter was by Gauss (1799). He proved that there exists a solution for every polynomial equation. It is so important that the result is now called the Fundamental Theorem of Algebra. He did not show us what the solution is. He simply said there was one. This was not an isolated event. In astronomy, an Englishman and a Frenchman predicted independently that there was a planet beyond Uranus before it was found. Consequently, a German astronomer found the eighth planet Neptune in 1846. Similarly, an American astronomer Clyde Tombaugh found the ninth planet Pluto in 1930 as predicted. I met the scientist and had lunch with him in Las Cruces in 1992. There was a similar incident in the prediction of a missing chemical element in 1869. It is a powerful method. It is known as existence proof in mathematics. Nowadays polynomial equations and also differential equations are solved mainly by numerical methods. The solution of a cubic equation can be found in calculus, though not in school mathematics.
Algebra came from the Arab world and in time dominated school mathematics.
In the nineteen century, algebra took a different turn. It went structural and numerical. These are the two major events in modern algebra. There was a big story at the time. One of the three famous construction problems is how to trisect an angle by ruler and compasses. The problems are dated back to the time of Greek. They did not ask whether. Instead they asked how because they believed that it could be done. It was not until Galois (1811 – 1832) who showed that it could not be done. The tool he used was group theory in abstract algebra. It may be the first time that advanced mathematics was used to prove or disprove a simple problem, at least we thought it was simple. Definitely, it was not the last time as we can see many such examples later. Galois was also famous for dying at the age of 21 in a duel. The duel was with pistols at twenty-five paces.
The number 3 in 3 coconuts is an abstract concept. It was made abstract for good reasons. Similarly, algebra was made abstract also for good reasons, not just for solving the three construction problems. Other than group theory, there are other algebraic structures. Collectively, they are called abstract algebra. The key concept is structure, algebraic structures and different kinds of algebraic structures.
If we look at linear algebra carefully, it is nothing but geometry or geometry without pictures. Some concepts in linear algebra like eigenvalues and eigenvectors are introduced to simplify computation. Abstract algebra is not abstract at all. It is down to earth and has practical applications. A more recent example is linear codes. Telecommunication cannot do without linear codes. There is no need to say more about algebra going numerical.
In the years 1600 to 2000 geometry turned algebraic and algebra went structural and numerical.
We inherited school mathematics from the West. There was also mathematics in the East. For example, Chinese remainder theorem in abstract algebra was discovered in China 2000 years ago. In China, it was called Sunzi method. It was known to Arabs as Chinese method, and only more than 1000 years later to the West as Chinese remainder theorem.
Looking at it from Europe, there were two great dynasties in China over a period of 2000 years. They are Han Dynasty (206BC – 220 AD) and Tang Dynasty (618 – 907 AD). During Han Dynasty, it was Roman Empire in Europe. During Tang Dynasty, it was Dark Ages in Europe, though a golden age in the Arab world. Two Chinese classics in mathematics have been translated into English and other western languages. They are Jiu-zhang Suan-shu (Nine chapters of arithmetic) and Shu-shu Jiu-zhang (Book of mathematics in nine chapters). The above translation of the titles is one that is closer to the Chinese original. The books consist of collection of practical problems in farming, construction, business etc. They represented the achievement in mathematics during Han Dynasty and Tang Dynasty respectively. All problems are stated in context. No computation was given and no formula. The solution of a problem was obtained by going through a process. The emphasis is on process. Recently, the West rediscovers the Chinese approach to mathematics.
Qing Dynasty declined after the Opium War (1839 – 1842). Chinese did not really proceed from numbers to symbols. In the early twentieth century, Chinese imported school mathematics from the West. There was also mathematics, rich in content and approach, in other Asian countries in ancient time.
Chinese learn mathematics differently and they learn how before they learn why.
School mathematics is dominated by geometry and algebra. We described the major events in the past relating to geometry and algebra. It is only by going through the past we learn why we are doing what we do today. We must also look into the future to learn what we should do today.
4. Mathematics today
School mathematics that we are teaching today is mathematics of the last century. What is mathematics of this century? Here we are interested only in those having a potential of getting into school mathematics, and we shall be brief. However we make an exception for differential geometry, as this is the hottest topic in town.
As we mentioned above, equations are solved mainly by numerical methods. The use of calculators and computers makes a great impact on mathematics and mathematics teaching. At first, it was numerical computation. Lately, it was symbolic manipulation. The prediction is that we need both for doing research in mathematics that requires computation. In the 70s, there was a movement to replace calculus by finite mathematics in the first year of undergraduate study. The rationale was that students would then be better prepared for using computers. It did not succeed. At that time, it was also predicted that algorithm, a popular tool in computation, would become an important item in school mathematics. It did not happen. Computation, numerical or otherwise, requires a totally different kind of approach. The answers are approximate rather than exact as we are so used to for years. The process is iterative rather than of a finite number of steps. Sometimes being able to solve a problem is not good enough. We must solve it within a time frame or by making use of a suitable model. In short, computation is an essential component of mathematics. In time to come, some of it will get into school mathematics.
It is unlikely that differential geometry or more technically manifolds will be part of school mathematics curriculum. It is taught in the university, though not necessarily a core module. This is an area of active research in the past 50 years. Six mathematicians were awarded Fields Medal for their work in manifolds. Fields Medal is the equivalent of Nobel Prize in mathematics. In 2006 at the International Congress of Mathematicians in Madrid, Spain, a Russian mathematician Perelman was awarded Fields Medal for proving the final step of Poincaré conjecture.
What is Poincaré conjecture? If you live on a circle, locally it is a straight line. If you live on a sphere or the surface of the Earth, locally it is a flat plane. Suppose we can construct a four-dimensional sphere. It is hard to visualize it, but not so hard to express it algebraically. Note that a two-dimensional sphere is a circle, and a three-dimensional sphere is the sphere as we know it. The question asked is: if you live on a four-dimensional sphere, what is the geometrical structure locally? The conjecture says that on a four-dimensional sphere, a good local structure (called manifold) looks like a three-dimensional sphere. What I have described can be made precise mathematically. Some 50 years ago, half of the people believed that it was true and the other half not. Now most people believe it is true. The tool used to prove the conjecture is Ricci flow in partial differential equations. Finally, the long-standing conjecture has been proved. Ricci flow is now a hot topic for research.
The marriage of geometry and calculus gave birth to differential geometry.
In a sense, mathematics is nothing but modelling. We use a rectangle to model a table top. We use quadratic functions to model the free fall of an object. After modelling, we solve the corresponding mathematical problem by exact methods, approximate methods, or probabilistic methods. In schools, we teach mostly exact methods.
In the sixteen century, calculus was used to design a better steam engine. During and after World War II, many problems we encountered were discrete in nature. For example, allocation of resources for war effort cannot be solved by calculus. The method invented was later used in engineering or management after the War. For example, the design of telephone network cannot be solved by traditional mathematics. These are problems in discrete mathematics. The famous four-colour problem is also a problem in discrete mathematics. It says: we need not more than four colours to colour the countries on a given map. The problem was posed in 1852. It was proved in 1890 for five colours. It was solved for four colours in 1976 using the brutal force of computers. It is a fact that discrete mathematics is coming to schools. It is not difficult to pose a problem in context at the school level involving discrete mathematics.
Mathematical models are no longer restricted to physical sciences.
Again in 2006 at the International Congress of Mathematicians in Madrid, Spain, Kiyoshi Itô from Japan was awarded the Gauss Prize for his contribution to making progress in mathematics that has significant implication in other fields. Itô sent his daughter to receive the prize. He said what he did was pure mathematics. He had no intention to apply it elsewhere but others did it for him. The contribution is stochastic differential equations. The other fields are engineering, finance among others. In particular, Black and Scholes won the Nobel Prize for applying stochastic differential equations to derivatives in finance. Different types of problems require different kinds of tools. Some problems are so big like the design of a dam. Some problems are so fluid like derivatives in finance. Computation has to be done using probabilistic or stochastic methods. There are discrete models like telephone lines, and there are also stochastic models like derivatives in finance. Randomness is an important concept in stochastic methods. It is already in mathematics curriculum in some countries.
Mathematical tools go from exact to approximate and further to stochastic.
The ten story lines given above are the topics covered in ten lessons of a module MME802 Fundamental Concepts in Mathematics for graduate students in mathematics education at the National Institute of Education, Singapore. In each lesson, we introduce certain fundamental concepts with related skills, in addition to the story lines. For example, the first three lessons on geometry are: three theorems on triangles in hyperbolic geometry via Poincaré model, representation of translation, rotation etc by 3×3 matrices, and an axiomatic proof of (–1) ×(–1) = 1. It is a module in mathematics. The connection with school mathematics curriculum, in particular curriculum in Singapore, is also given. For more information on the school curriculum in Singapore, see the articles in References.
5. Why we teach what we teach
We taught Euclidean geometry because it was an academic pursuit. We taught algebra because it was a more efficient tool to do computation than to do it with geometry. We taught statistics because we thought it was useful.
When we review the events in the past, we note that many topics were in and out of the syllabus. For example, transformation geometry was introduced into the syllabus after Euclidean geometry was cut. Eventually transformation geometry was gone. It is not in the new syllabus 2007. Algebra was not taught in primary schools. Then it was and then it was not. Now it is in the syllabus 2007. We wonder why we teach certain topics and why we do not. I make a list below. It is by no means a complete list.
5.1. Rich in content and rich in examination questions.
Why was Mozart a prodigy in music? He was born and brought up in a rich environment, rich in music. We learn more if the topic we are learning is a rich topic. Mechanics is rich. So is Euclidean geometry. However they lost their place in school mathematics due to not satisfying other items in the current list. We shall not elaborate here. In Singapore context, the topic must also be rich in exam questions. Numerical methods was at one time in the A level syllabus. After a while, we ran out of exam questions. So the topic was dropped from the syllabus.
5.2. For computation and for rigour.
If we check the verbs used in PSLE (Primary School Leaving Examination) and in the O level exam papers. The most commonly used word is “find”. What is involved is to compute. So we teach and test computation. Another equally important, if not more important, topic that we teach and test is rigour. The value of mathematics is in rigour. Proof is part of it. If we want a baby to grow, we must feed the baby with solid food. Rigour is solid food for mathematics students. Some employers prefer mathematics students because they somehow believe that mathematics students have been trained in rigour.
5.3. For assessment though not assessment alone.
By all means, we teach for assessment. There is nothing wrong to teach for assessment. Assessment can be a negative factor in learning. It can also be positive if we play it right. It is a fashion now to talk about assessment of learning, forlearning and finally as learning. The key is not to teach for assessment only.
5.4. For knowledge and for the use of knowledge.
We always learn a piece of mathematics and learn how to use it. The difference is that now we make it explicit. If we learn a concept and there is no way to use it in the syllabus. Then we wonder why we teach it in the first place. Suppose we do use it. Then the next question is: how do we test the use of knowledge?
5.5. What we can relate to.
We do not mean solving problems in context in the sense that the problems are authentic or realistic. Sometimes it could be difficult or quite meaningless to do that. We learn better and faster through association with something we are familiar with. Hence it is a good idea to include things that we can relate to. Those things could be what students can do and can learn from. They may have nothing to do with our daily life or the environment around us. The key is to be able to relate to. Being authentic or realistic helps but it is not a major issue.
5.6. Statistics is a misfit.
What we are teaching in schools is exam statistics and, strictly speaking, not statistics. Statistics is a misfit in the mathematics syllabus. There has been suggestion that we may wish to teach statistics differently or even assess it differently. Maybe we need to ask a totally different set of questions concerning the teaching of statistics.
5.7. Certain concepts must be taught early.
It is my belief that if you want your children to eat certain kind of food, you better feed them before they were five years old. Certain things you have to learn from young. Some students learn statistics late in their school days. They forever have a problem thereafter. If we want our students to understand randomness, we better teach them when they are in the primary schools.
5.8. For workplace.
Most people get a job after schooling or university. If all they see in their workplace is computers, there is no reason why we do not use computers in schools. If team work is important, then we should start having team work in schools. After all, we educate our students for work.
I have taught mathematics at the university level for the past 45 years. Some students came back to see me after 30 years. They said they were my students. They could not remember when they were in my class. They could not recall the title of the course they took. To prove that they were my students, they told me a story I told them. I can recognize my own stories. So I know they were indeed my students. Apparently, they do not remember what I taught them. However they do remember a little bit of how I taught mathematics. The point I want to make is that teaching mathematics is not teaching content alone. Also, I am not saying that content is not important.
In the same way, we may not need to know why we teach what we teach. But it is my strong belief that some of us should know. We rely on these people to revise the syllabus perhaps another ten years from now. Without knowing why, we shall not be able to design a good syllabus.
Lee, P. Y. (2008). Sixty years of mathematics syllabus and textbooks in Singapore (1949-2005). In Z. Usiskin & E. Willmore (Eds.), Mathematics Curriculum in Pacific Rim Countries – China, Japan, Korea, and Singapore (pp.85-94). Charlotte, North Carolina: Information Age Publishing.
Lee, P.Y. (2006). What is new in the new O level mathematics syllabus? On website: newexpressmathematics.com
Lee, P. Y. (2007). Will the O level exam test what students know or will it test whether students know how to apply what they know? On website: newexpressmathematics.com