Author: dayat

Turning Grades 5-9 Students on to Mathematics

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Mathematics influences virtually everything we do, but unfortunately, many students, parents, and teachers do not connect the mathematics studied in school with the applications and uses of mathematics in everyday life. When we travel to school, pay for groceries, calculate time, follow a recipe, use a computer, play video games or participate in sports we are applying school level mathematics.

Similarly, in the workplace when we draft patterns, build models, operate machinery, calculate costs, analyze data, or plan a series of operations, we are applying mathematics and mathematical thinking.

The problem with teaching and learning mathematics both in the classroom and in a home learning situation is that mathematics is seen, and has been for many years, as difficult and of little value.

Some causes for this are,

a) too many teachers and parents do not have a strong enough knowledge and understanding of mathematics and pedagogy to create the positive learning environment necessary for 21st Century students, and

b) far too many adults claim they were never any good at math at school thus implying that success in life does not require any mathematical ability.

The net result is that many students are switched off math at an early age.

We have to find ways to switch them back on to mathematics otherwise they may leave school lacking the necessary skills to function successfully in the modern world. Students today need a much higher level of mathematical ability than 20 or 30 years ago. It is not sufficient that they know how to add, subtract, multiply and divide, know how to calculate areas and volumes, describe geometrical shapes and objects, calculate mean, median and mode or calculate probabilities. Certainly they have to know these facts but it is imperative that they also understand and are able to apply this knowledge in different ways and in different situations.

Historically mathematics has been taught mainly through rote learning and this continues today. While there is some value for a modicum of rote learning, turning learners on to mathematics requires a variety of teaching strategies including the use of hands on explorations of mathematical ideas, greater use of non routine problems that challenge students and require them to be creative, and an increased understanding of why mathematics is important.

Common sense and research tell us that children like to play, to be engaged, to investigate, and to learn and computers are an excellent tool for motivating students. Computer programs can present high quality mathematics questions combined with colourful graphics that are attractive and captivating for learners. Questions can be presented one at a time, thus avoiding the distraction or intimidation of a whole page of questions, and hints and solutions can be just a “click” away.

By interacting with a computer program students may not be cognizant of the strategies that are being employed to guide their development. They are only conscious of rising to the challenge of successfully solving the non-routine problems in an environment of make believe characters and entertaining graphical situations.

Finding quality math teaching software or good problems can be challenging but worth the effort when one sees the positive learning effects on students.

Mathematical Typesetting: A Great Way to Learn Mathematics

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Introducing LaTeX

For those of you who are unfamiliar with the idea, mathematical typesetting is the means by which mathematicians produce documents that contain complicated mathematical expressions. It’s not possible to typeset things like integral signs and summands in basic word processing systems: instead, the mathematician relies on a special typesetting language, which interprets commands as instructions to reproduce particular mathematical symbols and does so cleanly and elegantly.

LaTeX is the de facto typesetting language in modern use and is used by mathematicians and teachers around the world to write articles, create notes and print out question sheets. LaTeX usually seems quite peculiar to the uninitiated. It is not a WYSIWYG (what you see is what you get) processing system, like Microsoft Word, but rather a behind the signs document markup language: the resultant document appearance bears little resemblance to what the author actually types. Say you wanted to typeset an integral into your document. Then you would so by writing something like:

$$
\int_0^1 x^2=\frac{1}{3}.
$$

The language used consists of instructions to the typesetting system; the two dollar signs “$$” tells the typesetting system that an inline equation is about to be inserted: the “\int” command tells it that an integral sign needs to be produced and the follow-up “_0^1” informs it what bounds should be used; the “x^2” specifies the integrand; and the “\frac{1}{3}” tells it that a fraction, with numerator 1 and denominator 3, needs to be produced. The final “$$” tells it that it can stop worrying about producing mathematics, as the equation has been entirely specified.

LaTeX may seem odd at first, but it is a powerful program with a lot of features and the capability to reproduce nearly every mathematical symbol that is out there.

Why LaTeX is a great tool for learning mathematics

When I was an undergraduate mathematician, I never really got into the groove of learning my subject until I discovered LaTeX. I did what the vast majority of students do when they first go to university to study mathematics: I went to lectures, I took notes and I muddled my way through various question sheets. What I discovered was that I struggled in subjects where a full set of typeset lecture notes wasn’t provided by the lecturer. For some reason, reading my own notes that I took in class rarely helped the material to sink in. Even when I took the time to sit down and write them up properly, I still found them a difficult tool to learn with.

At the end of my first year, I’d figured out the problem. It turned out I learnt best when I had a proper set of typeset lecture notes sitting in front of me. The problem was, not all lecturers were forthcoming with their own notes, and in subjects where the lecturer decided that he wasn’t going to release them, I had to muddle through as best I could without. By the end of the first year, I decided I’d had enough, and took the problem into my own hands. I learned LaTeX and started to type up the notes for courses where they were missing. Immediately I experienced a massive upsurge in my performance as a mathematics undergraduate. My grades improved, my grasp of the material became more solid, and I enjoyed the subject much more.

Aside from the fact that I just enjoyed the actual process of typesetting itself, I attribute a large part of my eventual graduation with a first class degree to LaTeX and the notes I produced using it. LaTeX is a great tool for letting maths sink in and the major reason why is simple: when you get competent enough with it, it allows you to put all your focus and attention on understanding the material as you write it up.

This is in direct contrast to what happens when you write material up by hand. If you’re anything like me, you’re probably a bit of a perfectionist. So when it comes time to write up a definition, theorem or proof by hand, there’s a big part of your mental focus dedicated to not making a mistake. There is no feeling worse than getting to the end of your second side of notes and slipping up somewhere, putting down a “y” instead of an “x”, or something even worse. Sure, Tipp-Ex can solve the problem to a certain extent, but it is usually problematic to do so and distracts from the flow of your work. Because you’re on guard for this eventuality (and it inevitably happens, no matter how hard you try!), you can’t focus your attention fully on what is at hand, which is learning advanced mathematics.

LaTeX removes this problem, by allowing you to have as many drafts as you want. There are no distractions when working with LaTeX. In your head, you know that any error, no matter how egregious, can be fixed quickly, usually by just hitting backspace a couple of times. You can therefore focus fully on the task of understanding the mathematics, safe in the knowledge that if you make an error, you can undo it painlessly. This also makes LaTeX a great tool for revision, because if you’re reading through your notes and you stumble across something that is just catastrophically wrong, you can replace it in no time with something that makes sense.

Meaning and Nature Of Mathematics

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“Mathematics is the Science of number and Space”
“Mathematics is the Science of Measurement, quantity and Magnitude.”
These definitions clearly indicate that Mathematics is an accepted Science which deals with the quantitative aspect of our life and knowledge.

Mathematics has been recognized as one of the Central Strings of human intellectual activity throughout the centuries.
The Word “Mathematics” has been used in two distinct and different senses. i.e. One as a method used to solve the problems of quality, space, order etc. and the second as a set of laws or generalizations of truths that are discovered.
Mathematics is a tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.
In other words, as in mathematics we find results at the abstract level with the help of process of reasoning, therefore mathematics may be regarded as a science of abstract forms.
As expressed by Hindi Mathematicians, “Mathematics(Ganita) means the Science of Calculation.”
From the above opinions we include that Mathematics is a Science of Quality and Space. It deals with the questions and problems involving size, portion, area, time interval, distance etc. It is also a science of calculation involving the use of number and symbols.

It deals with the relationship between magnitudes. It deals with Numerical part of Man’s life.
In this way it is a Systematic, organized and exact branch of science, which deals with abstract concepts.

NATURE OF MATHEMATICS

Mathematics, like everything else that man has created, exists to fulfil certain human needs and desires.
The dominant intellectual interest of mankind shows that, mathematics appeals very powerfully to mankind.
It involves man’s high cognitive powers.
It has its own language-signs, symbols, terms and operations etc.
It has its own tools like intuition, logic, reasoning, analysis, construction, generality and individuality etc.
It helps in drawing conclusions and interpreting various ideas and themes.
Mathematics is the science of inductive, deductive reasoning.
Inductive-reasoning means when a particular property is true in a sufficient number of cases, then we conclude that it will true for all similar cases.

Deductive-reasoning is based on axioms, postulates, self-evident truth, undefined terms and definitions.

“Mathematics is a way to settle in the mind as habit of reasoning.”
Mathematics can be used in two distinct and different senses i.e. (i) The truth that are discovered and (ii) The methods used to learn truth.
Mathematics is of two types- Pure and Applied Mathematics
Pure Mathematics deals with theories and principles without regard to their application to concrete things.

Applied Mathematics is the practical side of pure mathematics.

That is why each and every invention and discovery in the field of physical, biological or social sciences owe much to applied mathematics.

“It is the science of number and space.”
“It is the science of measurements, quantity and magnitude.”
Rationality, accuracy, originality of thinking, certainty of results, transfer of learning in terms of similarity to the reasoning of day to day life and verification are the vital components of mathematics.
Mathematics is an expression of the human mind reflects the active will, the contemplative reason and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality.