Thanks to you all who took the time to listen to my talk on Radio 702 with Azania from 2-3 pm today (Wednesday 27 Jan 2016).
There were a large number of callers and some very interesting questions raised. I would like to respond to these individually and to attempt to summarise the topics.
What is mathematics?
The first question concerned what is mathematics? Is it a language?
It is a language, and it is a symbolic language, in which the symbols represent things. The important element of mathematics as a language is that it can be manipulated, and these manipulations are also a part of the mathematical language. Part of learning mathematics is learning this language. If you cannot read you cannot understand, and thus the important first step is how to read and understand mathematical statements, in the language of mathematics.
Mathematics helps us to understand our world, and is the basis for almost every discipline of knowledge. Mathematics is not a single topic, but a range of components which help in different ways. Numbers help us to quantify the world – to count and to measure in units. Geometry helps us to understand the spatial structure of our world. Calculus helps us to understand the infinite. Logic helps us to model the reasoning processes. And mathematics goes on and on into so many exciting areas, such as topology, in which a doughnut and a coffee cup are seen as the same; and fractals, which help us to build computer models of natural features of the physical world such as mountains and trees.
Loving and hating mathematics
Azania asked whether there are people who can do maths and others who cannot. In essence, this is question of whether there is a “maths gene” which we are born with.
We do not actually know the answer to this question, but I take the position that anyone can learn mathematics and to become and expert, since the human brain can accommodate this knowledge. I thus argue that it is environment of the learner – what they are exposed to, including the teachers and the materials – which is the largest impact on whether a learner will become proficient in mathematics, rather than their innate ability to learn mathematics and to retain this knowledge.
Everyone I speak to tells me that there was a time in their life when they loved mathematics, and this was often in the early grades. However, something happened and by the time they left school they were convinced that they could not do mathematics. I blame the system, not the individual learners.
I see the Senior Phase, Grades 7-9, as the problem years for mathematics. This has been referred to as the “transition years” by international researchers who have identified the challenges in these grades. The learners are taken from the comfort of simple mathematics, which they were engaged in from Grades R-6 and now have to confront a range of new mathematics concepts, which are more general, more powerful, and which requires more abstract thinking skills. However, this does not make them necessarily more difficult.
Is mental mathematics still needed?
Within this web site I have introduced a range of articles on the topic of mental arithmetic, and over the next few weeks I will be completing this set of articles. Each article has some practical work, and useful documents which you can download to help you.
I place a high value on learners developing a memory of common operations, such as the “times table”. However, mental arithmetic is more important than just multiplication, and this is needed by all learners. For example, the ability to quickly find the prime factors of a number is essential in helping to cancel fractions (oops, I think I have already lost you).
Mental mathematics is that which we can do without having to use paper and pencil (who still uses pencils?) and without using a calculator. There are two types of mental mathematics: firstly that which is learned by rote and which can be recalled at will, such as the times-table; secondly the calculations which we can do in our head without using a calculator, making use of the rote learned knowledge.
In all cases it is quicker to use our head than to spend the time typing out the calculations on the calculator. Think about it – you want the answer to 11 + 19 – which you immediately work out as 30 without any effort at all (since the last digits add to 10, and there are two other 10s, which adds up to 30 – easy peezy lemon squeezy). To use the calculator for this same sum we first need to find the calculator in our bag, and then switch it on, and then type in 1 1 + 1 9 = and finally we have our answer. This took you about 10-15 seconds. The mental answer was available to you in less than one second.
The question I raise in the articles on Mental Arithmetic in this site how much should be mental arithmetic and when should you rather use the calculator. Also what should you learn by rote and memorise, and what should be done mentally by simply calculations.
Check it out here: Mental Arithmetic #1
Is Mathematics Man-Made?
This was one of the philosophical questions which I received today, and this is as deep into pure philosophy as you can get. The core question is “Was mathematics discovered or invented?”
I have a book by Stephen Hawking called “God Created the Integers”, which is the first part of a sentence which continues “…and man created everything else”. This is a collection of very important mathematical works from Euclid (around 300 BCE) up to Alan Turing (1950s), each of which is a major contribution to the history of mathematical thought. An excellent book, but not for the meek. This is mostly advanced in nature, but if you have the will you can work your work way through many of the articles in this 1100 (yes, one thousand one hundred!) page tome.
This is a question for which we will never have an answer. We see elements of nature which exhibit mathematical properties, but this does not mean that these were built using mathematics. Many refer to mathematics as the “language of God” in the sense that when God created the universe he (she?) would have needed a team of engineers and plans, and would have needed mathematics to describe his creation. Whereas this is a cynical approach to creation, I can rather temper this by saying that IF God used a language to express his creation, then that language would have been mathematics.
This is a question which is ontological in nature, meaning that it concerns the existence of reality – what we consider to be real and existing in the world which is outside of ourselves, as distinct from what is internal and created by our minds.
I believe that mathematics is a bit of both. Firstly that there are univerals in the world, which are mathematical in nature, but for which there is no language to express this. Secondly, that we humans created the languages to express these. Take the number 2, it is not the symbol “2” which is important but rather than this represents anything which occurs in pairs, and the Romans used the symbol II for this, and both “2” and “II” mean the same thing. Every historical culture has developed a different conception for the number two, and it is only relatively recently that these have been unified, largely due to the planet which has shrunk to a unified knowledge culture by the modern communications and information systems – so we now largely all speak the same mathematical language.
Books on Mathematics
A number of callers made reference to books on mathematics which have inspired them. Let me go through some of these and I am also providing links for you to purchase these from loot.co.za. I receive a small commission from purchases from loot.co.za and this helps to finance the ongoing work on this site.
I mentioned the book “Elements” by Euclid, which is from around 300 BCE, around 2300 years ago, and which was the standard text book on mathematics for at least 1800 years. This is the largest non-fiction book of all time outside of the religious texts. It is still sold in a variety of forms, and one version I have has colour-coded graphics which make the topics far easier to understand.
The Elements forms the basis for geometry in the Grade 10-12 mathematics, which is still called Euclidean Geometry. This has not changed in 2300 years, and perhaps longer, since it is postulated the Euclid merely documented the knowledge of prior mathematics, and in particular the Pythagorians.
Mathematics for the Million. Lancelot Hogden.
Taking only the most elementary knowledge for granted, Lancelot Hogben leads readers of this famous book through the whole course from simple arithmetic to calculus. His illuminating explanation is addressed to the person who wants to understand the place of mathematics in modern civilization but who has been intimidated by its supposed dificulty. Mathematics is the language of size, shape, and order—a language Hogben shows one can both master and enjoy.
“A great book, a book of first-class importance.”—H. G. Wells
The Magic of Math, Arthur Benjamin
This book was referred to, but was not one which I have in my collection, and I will be purchasing this to include into the Museum of Mathematics. There are a host of new books on mathematics which are written for the general public, and these help to turn people towards the Light Side of the Force for mathematics, and away from the Dark Side.
Short-Cut methods for arithmetic
There have been many recommendations for how to perform arithmetic calculations quickly and without the help of calculating devices.
The Trachtenberg method was referred to by two of the callers, and I will also include Vedic Mathematics as another alternative.
Each of these methods is not magic, but is based on changing the way in which the calculatoions are performed, and especially seeing how the decimal numbers, such as 123, are really a short-hand notation for 100 + 20 + 3.
The Trachtenberg Method
Jakow Trachtenberg developed this system during WW2, before computers were developed. This starts with simple approach to multiplying by various numbers, and then builds up a more comprehensive approach to performing calculations with very little need to write intermediate calculations (such as with long multiplication and long division), and can be done as fast as it takes to enter the values into a calculator.
Vedic Mathenatics is another short-cut method, using principles similar to Trachtenberg, with the difference that this is claimed to be far older, and can be traced to the Vedas. However, this claim is disputed with Vedic scholars stating that this knowledge does not exist in the Vedas. Even ignoring the claims and counter-claims on the origins, these are useful techniques which are comparable to Trachtenberg, and which serve largely the same purpose.
The Role of Mathematical Literacy
Mathematical Literacy was included into the Grade 10-12 curriculum in 2003. From that time all learners in Grades 10-12 are required to take either Mathematics or Mathematical Literacy.
The school subject of “Mathematics” is pure mathematics, which was previously structured into Higher Grade and Standard Grade. This has a focus on more conceptual knowledge about mathematics, with some practical examples which use this conceptual basis. However, there is far less real-world examples and application of mathematics.
“Mathematical Literacy” addresses the need for all learners to become mathematically proficient for the real-world, such as how to handle money when shopping, how to read an electricity account, and how to measure quantities and areas. This is important for all people in society, and in my opinion should be better placed into the Life Orientation syllabus, being common knowledge which is needed for this modern life and its challenges. However, from Grade 10-12 there is little new mathematics actually added, and most of the mathematics is derived from the groundwork conducted in Grades 7-9.
Many schools will suggest that learners take Mathematical Literacy if they believe the learner has a greater chance of success, but the problem for the learner is that a lack of Mathematics in Grade 12 will deny access to many of the new and exciting areas for further study, including the entire breadth of the science and engineering disciplines.
I strongly advocate using Grades 7-9 to help to improve learners’ mathematical knowledge so that they can take Mathematics from Grade 10 onwards, and my approach is to understand the conceptual difficulties to help to remediate these and to allow the learners to progress faster. I fully realise that conceptual difficulties are only one of many challenges for learners at school, but even with all of these other difficulties addressed, there is no sure way to ensure that learners are able to build on their prior knowledge to advance their mathematics.
What is Infinity?
This is quite a difficult question to answer simply, and yet this has been under continual discussion for at least 2500 years.
There are two ways to conceptualise infinity which are important as an introduction to the calculus, as developed jointly by Issac Newton and Gottfried Leibniz. Whereas they built on top of the work of others, it was how they integrated this into a theory which them worth of the title of the developers of the Calculus.
The first conceptualisation is to determine the value at a particular place or time, such as to determine the speed of a car when it is accelerating. The challenge is that the car is moving all of the time, and so how doe we determine the speed at a specific point in time. Well one way is to find the speed at a time (t), and then to measure it again at time (t+dt) which is a short time (dt) afterwards, and then to work out the time using these two speeds. But this is an estimation and is not the true speed. So rather we need a way to determine how to break down time into smaller and smaller units, until these are infinitely small. We cannot actually see these times, but we can deal with these mathematically, and we can get an answer. This is called the Differential Calculus and is part of the Grade 12 Mathematics curriculum.
The second conceptualisation concerns how to measure the area of volume of a shape, such as working out the area of a circle, which has been approximated many times and in many ways over the past 2500 years. This can be addressed by dividing up the shape into smaller and smaller units and then adding the area of these smaller shapes. This is also an approximation, and to be accurate we need to make these smaller shapes infinitely small. This is the Integral Calculus and is not included into the school Mathematics curriculum. This is an essential element of much science and engineering.
Does Zero Exist?
Whereas the understanding of inifinity is actually quite a practical problem, the definition of zero is far more troublesome.
I used a simple example during the talk in which you own two cows, and you sell one and you then have one, and then you sell the other and you have none. But can you actually own zero cows? And since this means that you own nothing, is this then the same as owning zero sheep, or zero dogs? So I have no elephants, and you have no tigers, and thus we own the same! But the challenge is that owning no cows actually means something for the person who previously had two and sold them and is planning to buy two more. This is where mathematics hits reality.
The zero symbol has been introduced for a number of purposes, but two in particular are essential for our understanding of mathematics. The first is as a place-holder for decimal numbers, so that when we write 102 to represent “one hundred and two” we put in the zero to indicate that there are no tens. This was an exceptionally important part of the Arabic number system, in the 12th century, and which was introduced and which now forms the core of how we write numbers.
The second usage is to help us deal with the calculations such as (4 minus 4) and to be able to represent this as a number, which is the same as the cows I mentioned above.
Building Blocks of Mathematics
One of the most important elements of learning mathematics is that each new topic in mathematics builds on prior work in mathematics
When the building blocks of a new topic are already in place, then this new topic is seen as a small increment to the prior knowledge and it relatively easy to understand to to conceptualise.
However, when any of these building blocks is missing, or is misunderstood, or is only partially developed (a misconception), then learning the new topic is almost impossible.
My own research is concerned with the problem of how to discover what a learner knows, and in particular where they have misconceptions, or a lack of knowledge, which causes difficulties in the new topic. I argue that this can be detected by automated methods, and can be computerised as a tool to assist teachers. When a teacher has a class of 40 learners, they cannot possibly understand the needs of each learner, and we need to help the teachers by providing better tools to help them do their teaching. This is part of my ongoing work – to try to develop artifical intelligences as support for mathematics teachers.
The Role of Teachers
This leads onto another question concerning the roles of the teachers, and is was stated by one caller that bad teachers can kill mathematics, whereas a good teacher can instil a life-long love of learning mathematics. This applies to all subjects, and the choice of a career for many learners is influenced heavily by the inspiration which a good teacher provided in “selling” their subject to the learners – making their subject exciting as something to do in the future.
My own background is computer science and in particular artificial intelligence, which requires some quite difficult mathematics, since we need to model the complexities of the human mind, and use this to build better devices to help humanity. Many new inventions have AI built into them, and this promises to be one of the key driving forces of the next generation of technologies, such as self-driving cars, and robots to set up new habitats on Mars prior to human occupation. Yes learners – this is the new generation which you are moving into.
To inspire learners into the mathematical disciplines we need to provide examples which they find exciting, so that learning becomes something which they want to do, rather than something they have do to before they get on with life.
The Museum of Mathematics
I did not address each of the questions individually, but I have combined these and I hope that you have found this interesting and valuable.
Beyond anything else, we need to create a mathematically-expert generation of learners, since these will move the country into an advanced state for the next generation. Currently South Africa is the worst country in the world, by almost every international measure, on maths and science learning and education, and this is a crisis which needs to be fixed.
My contribution to helping address this crisis is firstly by providing this web site – mathematics.co.za – with a range of free articles and tools to help learning, and also to build up a Museum of Mathematics, which is a sustainable and long-term repository of trusted knowledge of mathematics, which all teachers, parents, administrators, and learners can use to help to learner mathematics better and to instill a love of mathematics.
I welcome discussion on this and please send me an email to discuss your issues. At present I do not have online comments, and I will include good questions in for the benefit of all.
My Approach to Teaching Mathematics
I have developed a method, which is based on the work of Georg Polya, which I called APEX.
I have written about this in another page on this site concerning the need to prepare for mathematics examinations and you should read this to understand my method. If you are interested I will be happy to expand on this and I hope to hear from you.