What is mathematics?

Mathematics is a way of organising our experience of the world. It enriches our understanding and enables us to communicate and make sense of our experiences. It also gives us enjoyment. By doing mathematics we can solve a range of practical tasks and real-life problems. We use it in many areas of our lives.

In mathematics we use ordinary language and the special language of mathematics. We need to teach students to use both these languages. We can work on problems within mathematics and we can work on problems that use mathematics as a tool, like problems in science and geography. Mathematics can describe and explain but it can also predict what might happen. That is why mathematics is important.

For more than two thousand years some familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today the traditional place of mathematics in education is in grave danger.

These opening sentences to the preface of the classical book “What Is Mathematics?”were written by Richard Courant in 1941. It is somewhat soothing to learn that the problems that we tend to associate with the current situation were equally acute 65 years ago (and, most probably, way earlier as well). This is not to say that there are no clouds on the horizon, and by this book we hope to make a modest contribution to the continuation of the mathematical culture.

The first mathematical book that one of our mathematical heroes, Vladimir Arnold, read at the age of twelve, was “Von Zahlen und Figuren”1 by Hans Rademacher and Otto Toeplitz. In his interview to the “Kvant” magazine, published in 1990, Arnold recalls that he worked on the book slowly, a few pages a day. We cannot help hoping that our book will play a similar role in the mathematical development of some prominent mathematician of the future.

We hope that this book will be of interest to anyone who likes mathematics, from high school students to accomplished researchers.

 We do not promise an easy ride: the majority of results are proved, and it will take a considerable effort from the reader to follow the details of the arguments. We hope that, in reward, the reader, at least sometimes, will be filled with awe by the harmony of the subject (this feeling is what drives most of mathematicians in their work!) To quote from “A Mathematician’s Apology” by G. H. Hardy, The mathematician’s patterns, like the painters or the poets, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

For us too, beauty is the first test in the choice of topics for our own research,

as well as the subject for popular articles and lectures, and consequently, in the choice of material for this book. We did not restrict ourselves to any particular area (say, number theory or geometry), our emphasis is on the diversity and the unity of mathematics. If, after reading our book, the reader becomes interested in a more systematic exposition of any particular subject, (s)he can easily find good sources in the literature.

About the subtitle: the dictionary definition of the word classic, used in the

title, is “judged over a period of time to be of the highest quality and outstandingof its kind”. We tried to select mathematics satisfying this rigorous criterion. The reader will find here theorems of Isaac Newton and Leonhard Euler, Augustin Louis Cauchy and Carl Gustav Jacob Jacobi, Michel Chasles and Pafnuty Chebyshev, Max Dehn and James Alexander, and many other great mathematicians of the past.

Quite often we reach recent results of prominent contemporary mathematicians, such as Robert Connelly, John Conway and Vladimir Arnold.

We started this preface with a quotation; let us finish with another one. Max Dehn, whose theorems are mentioned here more than once, thus characterized mathematicians in his 1928 address [22]; we believe, his words apply to the subject of this essay:

At times the mathematician has the passion of a poet or a conqueror, the rigor of his arguments is that of a responsible statesman or, more simply, of a concerned father, and his tolerance and resignation are those of an old sage; he is revolutionary and

conservative, skeptical and yet faithfully optimistic.

There are four main issues in the teaching and learning of mathematics:

Teaching methods

Students learn best when the teacher uses a wide range of teaching methods. This book gives examples and ideas for using many different methods in the classroom,

Resources and teaching aids

Students learn best by doing things: constructing, touching, moving, investigating. There are many ways of using cheap and available resources in the classroom so that students can learn by doing. This book shows how to teach a lot using very few resources such as bottle tops, string, matchboxes.

The language of the learner

Language is as important as mathematics in the mathematics classroom. In addition, learning in a second language causes special difficulties. This book suggests activities to help students use language to improve their understanding of maths.

The culture of the learner

Students do all sorts of maths at home and in their communities. This is often very different from the maths they do in school. This book provides activities which link these two types of rnaths together. Examples are taken from all over the world. Helping students make this link will improve their mathematics.

Why do we learn mathematics?

Why should anyone study mathematics? Should those in high school or college be forced to take math courses, even if their intended future profession does not require higher level mathematics?

A common argument in favor of forcing math classes to be taken is that it is a necessary part of educational process to make the individual well-rounded in all aspects including science, humanities, writing classes, language, and the other basic fields of study. While this assumption is true, this is not the main reason why mathematics should be a required part of a school’s curriculum. First of all, simply removing the requirement for math classes would lower the standards for everyone either educating or being educated. Simply giving up on a generation of kids who are not good at math is definitely not the right way to go. If anything, people should be pushed harder in mathematics to do the best as possible even if that means ”failing.” Hopefully failure would lead persistence to succeed and motivation to do better.

In high school, math should be required for at least three years. This is because during the high school years, a student rarely knows what future profession he or she may have. Entering high school, one may have dreaded mathematics in previous years, but upon being exposed to higher levels of math, the person might gain curiosity or even enjoyment for mathematics. Dropping the requirement may steer students away from something that they may develop to be very good at or have a future profession in. Another reason that mathematics should be a required course is that fact that it helps to develop the problem solving process. Although one may not directly use mathematics in their everyday lives, math plays an indirect role in how they make many day to day decisions such as finances or time management. The ability to problem solve is not something that can be directly taught, but is rather a discrete skill learned over many years through trial and error, past experiences, and exercises that might help to develop such a skill.

Mathematics is certainly one these exercises. Math is one of the few fields of study where many facts are not necessarily memorized. Such a field where this is necessary is history, or a high school English class.

In math, one only has to know the basic operations, and symbolisms. With this limited knowledge, one can solve an infinite number of problems.Theorems may require a certain amount of memorizing, but even these can be derived from previously gained knowledge. In other words, gaining math skills does not just mean that one knows how to do math,

but rather opens up the brain to all kinds skills and development. Yet another reason why mathematics should be studied is the fact that basically anything that happens in the world can be related to mathematics. Gaining an understanding of this concept can greatly affect how one perceives the world around them. New enlightenment may come about with this understanding. Math is also the only true universal part of human culture.

Searching for some more insight on this project, I came across a quote said by James Caballero that caught my attention: ”I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.

Everybody a mathematician?” CAIP Quarterly 2 (fall, 1989). I believe that what was said is very apparent. Every time we stop pursuing a subject, we lose another opportunity to develop our thinking to a greater level. Mathematics isn’t just a tool for engineers, chemists, physicists, or economists. It can be used by everyone to gain problem

solving skills, take care of every day tasks, understand the world around them, and to feel a sense of accomplishment. Shutting these possibilities off by not pursuing studies in mathematics severely limits a person’s growth and ability, thus affecting everyone around them.

Reference:

1.Carson, Elizabeth and Denise Haffenden. ”NYC HOLD: Honest Open Logical Debate on math reform”. http://www.mathematicallycorrect.com/nychold.htm.
2.Mathematically Correct. ”What has happened to Mathematics Education?”. 

http://mathematicallycorrect.com/intro.htm#doyou.

3.Ross, David. ”TheMathWars”. http://www.ios.org/articles/dross mathwars.asp.

Stohr-Hunt, Patricia. ”What Good is Math?”. http://oncampus.richmond.edu/academics/19 Oct 2003.

Value of self-access e-learning in today’s education scenario

What we know about learning is an important starting point for exploring the use of technology and the design and success of online and blended learning. The basis of effective online learning is comparable to the foundation of effective learning in general. Among the many theories surrounding how people learn, this paper focuses on three aspects of learning, which in turn are tied

to the use of the online learning components integrated in the two courses of the study.

Learning theory suggests that learning is promoted or enhanced (1) when students are actively involved in the learning, (2) when assignments reflect real-life contexts and experiences, and (3) when critical thinking or deep learning is promoted through applied and reflective activities. Each of these aspects of learning are briefly reviewed, with a subsequent discussion of how the online learning components integrated in the two courses were chosen with these dimensions in mind. Numerous studies have demonstrated that a student’s active involvement in the learning process enhances learning, a process often referred to as active learning. Simply stated, active learning involves “instructional activities involving students in doing things and thinking about what they are doing”. Interactive instruction or “learning by doing” has been found to result in positive learning outcomes. Because many new technologies and web based activities are interactive, online coursework has the potential to create environments where students actively engage with material and learn by doing, refining their understanding as they build new knowledge. As we observes, “When students become active participants in the knowledge construction process, the focus of learning shifts from covering the curriculum to working with ideas. And using technology tools ‘to think with’ facilitates working with ideas and learning from that process”. In addition to active involvement, students better understand and apply material when problems

and situations are set in the context of real world issues and situations. Authentic situations and scenarios can provide a stimulus for learning, creating greater student motivation and excitement for learning, representing and simulating real-world problems and contexts, providing an important structure for student thinking. Emphasizing authentic tasks in context rather than abstract out-of-context activities creates a greater likelihood of learning. Technology and online instruction can facilitate learning by providing real-life contexts to engage learners in solving complex problems. The use of real-world situations has the potential to promote deep learning through the development of critical thinking skills. Critical thinking involves the active and skilful analysis, synthesis, and application of information to unique situations. Learning retention and performance improves as students are required to apply what they have learned and then reflect upon the learning.

Again, online instruction has the potential to provide opportunities to promote reflective thought and deep learning through realistically integrating and applying principles learned.

Online instruction, such as a simulation, thrusts learners into a learning experience, increasing engagement and providing activities that actively engage learners to analyse, synthesize, and evaluate information while constructing knowledge. Built upon a foundation of learning theory, e-learning can potentially provide many important payoffs. To learners, online instruction offers the flexibility and convenience to complete learning units when and where a learner desires. Additionally, online education has been used to reduce

costs and to provide an efficient, standardized way to deliver content. In addition to potential cost savings, e-learning has pedagogical potential beyond traditional methods related to the principles of learning discussed. For instance, multimedia capabilities can be used with learning exercises that allow learners to apply concepts realistically. Or, animation can help demonstrate concepts and events difficult to portray in traditional classes, which, in turn, can facilitate a more accurate communication of important ideas. E-learning can deliver “new”

information not contained in traditional sources, effectively reinforcing other course information through offering examples, explanations, assessments, and exercises. In this way, online instruction can potentially enhance learning compared to what can be accomplished using a classroom only approach. However, there are also potential disadvantages or limitations of online learning. For example, one study concluded that asynchronous e-learning was not effective as a standalone method to deliver technical training for information technology professionals Learners in the study commented

that e-learning eliminates classroom interaction time, where a significant amount of “real learning” takes place as users assimilate information, utilize software, apply knowledge to problem solving, and interact with the instructor and other learners.

Other potential problems of e-learning that have been identified in previous research include a sense of learner isolation; learner frustration, anxiety, and confusion; higher student attrition rates; the need for greater discipline, writing skills, and self-motivation; and the need for online users to make a time commitment to learning.

Based on these considerations, some research has stressed the importance of using a “blended learning” approach. Blended learning is a hybrid instructional

approach that combines elements of e-learning with the traditional classroom environment . It involves starting with learning objectives and then selecting the best combination of delivery methods to meet those objectives. In some situations, blended learning may involve students completing online units prior to meeting to ensure they share a common foundation of knowledge.

This allows class sessions to go into greater depth with application exercises and problem solving. Alternatively, e-learning elements can be used after class meetings to maintain an on-going dialogue among a community of participants about course-related topics through chats or discussion board postings. Other blended learning options may use a combination of pre-class and post-class e-learning components. As the use of e-learning and blended learning continues to expand significantly in higher education and business environments, we must gain an improved understanding of where, when, and under what circumstances online and blended learning can be applied most effectively as well as how it can best be implemented.

If students perceive some benefit to their learning (through either a personal interest in or an application of content), they will likely be more motivated to perform well. As one can observe, “Students who are motivated to learn will choose tasks that enhance their learning, will work hard at those tasks, and will persist in the face of difficulty in order to attain their goals.”

Mean, Median, Mode

“While the individual man is an insolvable

puzzle, in the aggregate he becomes a

mathematical certainty. You can, for

example, never foretell what any one man

will do, but you can say with precision

what an average number will be up to.”

—Arthur Conan Doyle,

The Sign of Four

Sherlock Holmes spoke these words to his colleague Dr.Watson as the two were unravelling a mystery. The detective was implying that if a single member is drawn at random from a population, we cannot

predict exactly what that member will look like. However, there are some “average” features of the entire population that an individual is likely to possess. The degree of certainty with which we would expect to observe such average features in any individual depends on our knowledge of the variation among individuals in the population. Sherlock Holmes has led us to two of the most important statistical concepts: average and variation. (While the individual man is an insolvable)

Statistics is something that surrounds us every day – we’re constantly bombarded with statistics, in the form of polls, tests, ratings, etc. Understanding those statistics can be an important thing, but unfortunately, most people have never been taught just what statistics really mean, how they’re computed, or how to distinguish the different between statistics used properly, and statistics misused to deceive.

The most basic concept in statistics in the idea of an average. An average is a single number which represents the idea of a typical value. There are three different numbers which can represent the idea of an average value, and it’s important to know which one is being used, and whether or not that is appropriate. The three values are the mean, the median, and the mode.

MEAN

The mean is what most people are taught as the average in middle school math. Given a set of values, the mean is what you get by adding up all of the values, and dividing that sum by the number of values.

The mean is a very useful number – it summarizes the properties of the group. It’s important to understand that the mean does not represent an individual – in fact, there may be no individual whose value matches the mean.

MEAN = = sum of all data values / number of data values

or, more formally,

DEFINITION: Mean

The mean is the sum of a set of values, divided by the number of values in

the set. The notation for the mean of a set of values is a horizontal bar over

the variable used to represent the set. The formula for the mean of a data

set {x1, x2, . . . , xn} is,

 How to calculate the mean?

QUESTION

What is the mean of the data set {10; 20; 30; 40; 50}?

SOLUTION

Step 1 : Calculate the sum of the data

10 + 20 + 30 + 40 + 50 = 150

Step 2 : Divide by the number of values in the data set to get the mean

Since there are 5 values in the data set, the mean is

Mean =

MEDIAN

DEFINITION: Median

The median of a data set is the value in the central position, when the data

set has been arranged from the lowest to the highest value.

Note that exactly half of the values from the data set are less than the median and the other half are greater than the median.

To calculate the median of a quantitative data set, first sort the data from the smallest to the largest value and then find the value in the middle. If there are an odd number of data, the median will be equal to one of the values in the data set. If there are an even number of data, the median will lie halfway between two values in the data set.

Example 4: Median for an odd number of values

QUESTION

What is the median of {10; 14; 86; 2; 68; 99; 1}?

SOLUTION

Step 1 : Sort the values.

The values in the data set, arranged from the smallest to the largest,

are 1; 2; 10; 14; 68; 86; 99.

Step 2 : Find the number in the middle

There are 7 values in the data set. Since there are an odd number of

values, the median will be equal to the value in the middle, namely,

in the 4th position. Therefore the median of the data set is 14.

Example 5: Median for an even number of values

QUESTION

What is the median of {11; 10; 14; 86; 2; 68; 99; 1}?

SOLUTION

Step 1 : Sort the values

The values in the data set, arranged from the smallest to the largest,

are

1; 2; 10; 11; 14; 68; 86; 99

Step 2 : Find the number in the middle

There are 8 values in the data set. Since there are an even number

of values, the median will be halfway between the two values in the

middle, namely, between the 4th and 5th positions. The value in the

4th position is 11 and the value in the 5th position is 14. The median

lies halfway between these two values and is therefore

Median =

MODE

DEFINITION: Mode

The mode of a data set is the value that occurs most often in the set. The

mode can also be described as the most frequent or most common value in

the data set.

To calculate the mode, we simply count the number of times that each value appears in the data set and then find the value that appears most often.

A data set can have more than one mode if there is more than one value with the highest count. For example, both 2 and 3 are modes in the data set {1; 2; 2; 3; 3}. If all points in a data set occur with equal frequency, it is equally accurate to describe the data set as having many modes or no mode.

Example 6: Finding the mode

QUESTION

Find the mode of the data set {2; 2; 3; 4; 4; 4; 6; 6; 7; 8; 8; 10; 10}.

SOLUTION

Step 1 : Count the number of times that each value appears in the data set

Value	Count
2	2
3	1
4	3
6	2
7	1
8	2
10	2

Step 2 : Find the value that appears most often

From the table above we can see that 4 is the only value that appears

3 times, and all the other values appear less often. Therefore the

mode of the data set is 4.

One problem with using the mode as a measure of central tendency is that we can usually not compute the mode of a continuous data set. Since continuous values can lie anywhere on the real line, any particular value will almost never repeat. This means that the frequency of each value in the data set will be 1 and that there will be no mode. We will look at one way of addressing this problem in the section on grouping data.

Example : Comparison of measures of central tendency in real life situation.

QUESTION

There are regulations in South Africa related to bread production to protect consumers. By law, if a loaf of bread is not labelled, it must weigh 800 g, with the leeway of 5 per cent under or 10 per cent over. Vishnu is interested in how a well-known, national retailer measures up to this standard. He visited his local branch of the supplier and recorded the masses of 10 different loaves of bread for one week. The results, in grams, are given below:

Monday  Tuesday  Wednesday   Thursday   Friday   Saturday   Sunday
802,4        787,8        815,7             807,4           801,5     786,6         799,0
796,8        798,9        809,7             798,7           818,3     789,1         806,0
802,5        793,6        785,4             809,3           787,7     801,5         799,4
819,6        812,6        809,1             791,1           805,3     817,8         801,0
801,2        795,9        795,2             820,4           806,6     819,5         796,7
789,0        796,3        787,9             799,8           789,5     802,1         802,2
789,0        797,7        776,7             790,7           803,2     801,2         807,3
808,8        780,4        812,6             801,8           784,7     792,2         809,8
802,4        790,8        792,4             789,2           815,6     799,4         791,2
796,2        817,6        799,1             826,0           807,9     806,7         780,2

1. Is this data set qualitative or quantitative? Explain your answer.

2. Determine the mean, median and mode of the mass of a loaf of bread for

each day of the week. Give your answer correct to 1 decimal place.

3. Based on the data, do you think that this supplier is providing bread within

the South African regulations?

SOLUTION

Step 1 : Qualitative or quantitative?

Since each mass can be represented by a number, the data set is

quantitative. Furthermore, since a mass can be any real number, the

data are continuous.

Step 2 : Calculate the mean

In each column (for each day of the week), we add up the measurements

and divide by the number of measurements, 10. For Monday,

the sum of the measured values is 8007.9 and so the mean for Monday

is

In the same way, we can compute the mean for each day of the week.

See the table below for the results.

Step 3 : Calculate the median

In each column we sort the numbers from lowest to highest and find the value in the middle. Since there are an even number of measurements (10), the median is halfway between the two numbers in the middle. For Monday, the sorted list of numbers is

789,0; 789,0; 796,2; 796,7; 801,2;

802,3; 802,3; 802,5; 808,7; 819,6

The two numbers in the middle are 801,2 and 802,3 and so the median is

In the same way, we can compute the median for each day of the week:

Day	Mean(g)	Median(g)
Monday	800.8	801.8
Tuesday	797.2	796.1
Wednesday	798.7	797.2
Thursday	803.4	800.8
Friday	802.0	804.3
Saturday	801.6	801.4
Sunday	799.3	800.2

From the above calculations we can see that the means and medians

are close to one another, but not quite equal. In the next worked

example we will see that the mean and median are not always close

to each other.

Step 4 : Determine the mode

Since the data are continuous we cannot compute the mode. In the

next section we will see how we can group data in order to make it

possible to compute an approximation for the mode.

Step 5 : Conclusion: Is the supplier reliable?

From the question, the requirements are that the mass of a loaf

of bread be between 800 g minus 5%, which is 760 g, and plus 10%,

which is 880 g. Since every one of the measurements made by Vishnu

lies within this range and since the means and medians are all close

to 800 g, we can conclude that the supplier is reliable.

Reference:

1.     Handbook on Mathematics.

2.     Book of Everything in Mathematics- Grade 9-10

3.     Wikipedia

4.     Wolfram Maths World.