Mathematics and Its Reality

Mathematics is the abstract study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. (Wikipedia)

Mathematics is not an invention. Discoveries and laws of science are not considered inventions. Inventions are material things and processes. However, there is a history of mathematics, a relationship between mathematics and inventions, and mathematical instruments are considered inventions.

According to Mathematical Thought from Ancient to Modern Times, mathematics as an organized science did not exist before the classical Greeks of the period from 600 to 300 BC entered upon the scene. There were, however, prior civilizations in which the beginnings or rudiments of mathematics were created.

When civilization began to trade, a need to count was created. When humans traded goods, they needed a way to count the goods and to calculate the cost of those goods. The very first device for counting numbers was the human hand, counting on fingers. To count beyond ten fingers, mankind used natural markers, rocks or shells. From that point, counting boards and the abacus were invented.  (inventors.about.com)

Refer these to your siblings/children/younger friends:

HOMEPAGE of Free NAT Reviewers by OurHappySchool.com (Online e-Learning Automated Format)

 

HOMEPAGE of Free UPCAT & other College Entrance Test Reviewers by OurHappySchool.com (Online e-Learning Automated Format)

To see how our MODERN ELearning Reviewers work, please try this 5-item sample:

 

Importance of mathematics

Mathematics expresses itself everywhere, in almost every facet of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering - describing our understanding of all that we observe.

The Mathematics Everywhere & Everyday Exhibition explores the many wonders and uses of mathematics in our lives. This exhibition is divided into nine areas focusing on different aspects of mathematics.

In Counting Counting various quantities is one of the activities that people engage in from young. However, sometimes we wonder, just how big is one million? See how much space one million saga seeds occupy and be awed by numbers that you can relate to in your daily lives like the number of heartbeats in a typical lifetime.

In Nature 1, 1, 2, 3, 5, 8, 13... This is the Fibonacci Sequence, where each number is derived from adding the previous two numbers. This sequence of numbers can be found in many natural patterns like in pineapples, sunflowers, nautilus and pine cones. Our eyes are usually drawn to objects that are symmetrical. Leonardo Da Vinci’s Vitruvian Man is often used as a representation of symmetry in the human body.

In Shapes, Curves & Patterns Circles, squares and triangles are just a few of the shapes that are familiar to us through our daily lives. Discover the usefulness, beauty and 'cleanness' of the round shape, and learn more about the other shapes that can be found around us.

In Games & Puzzles Many, if not all puzzles and games require mathematical logic and deduction. This section uses the fun and excitement of various popular games and puzzles, and the exhilaration of solving them, to attract and engage the visitors to realise the mathematics in fun and games. .

In Time & The Heavens Mathematics was developed to understand the cycles of nature as observed in the seasons. Ancient people understood the need to define time in relation to celestial movements for agricultural, astronomical, astrological and navigational reasons. This section looks at the mathematics of astronomy, its relationship to the inventions of various cultural-historical calendars, and the division of time into units of hours, minutes and seconds.

Challenges & Controversies When we talk about mathematics, equations and formulas will pop into our mind. There are hundreds of equations in mathematics, but which is the Most ‘Beautiful’ Equation of all? You are invited to vote for your favourite. What do modern mathematicians actually do? What problems are of interest today? These are some of the issues explored in this section. Some outstanding challenges that remain unresolved are still intriguing many mathematicians. Discover what some of these challenges are.

In Real Applications Mathematic is used in our everyday lives; from figuring out the amount needed to buy your lunch to calculating the bank’s interest. This section explores some of the real life applications of mathematics. For example, internet banking is getting more and more common these days, and we depend on cryptology – the study of protecting information using codes – to keep our transactions safe. Learn more about how it is done in this section.

Awesome, Fearsome Calculus Calculus is the study of change and it is one of the most important fields in mathematics. Isaac Newton and Gottfried Leibniz are usually credited with the invention of calculus. Newton used calculus in his laws of motion and gravitational attraction. (http://www.science.edu.sg/exhibitions/Pages/Mathematics.aspx)

Two perspective on Mathematics teaching

Practical perspective

Students learn the mathematics adequate for general employment and functioning in society, drawing on the mathematics used by various professional and industry groups. He included in this perspective the types of calculations one does as part of everyday living includingbest buy comparisons, time management, budgeting, planning home maintenance projects,choosing routes to travel, interpreting data in the newspapers, and so on.

The aims of emphasizing the practical aspects of the mathematics curriculum as being:
… to educate students to be active, thinking citizens, interpreting the world
mathematically, and using mathematics to help form their predictions and
decisions about personal and financial priorities.

Specialized perspective

The specialised perspective as the mathematical understanding which forms the basis of university studies in science, technology and engineering. He argued that this includes an ability to pose and solve problems, appreciate the contribution of mathematics to culture, the nature of reasoning and intuitive appreciation of mathematical ideas

The aims of the specialised aspects are described as being that:
… mathematics has its own value and beauty and it is intended that students will
appreciate the elegance and power of mathematical thinking, [and] experience
mathematics as enjoyable.

Five strands of desirable mathematical actions for students

Both perspectives need to incorporate a sense of ‘doing’, that the focus should be on the mathematical actions being undertaken during the learning.

1.      Conceptual understanding

In describing actions and tasks relevant for teacher learning, explained that conceptual understanding includes the comprehension of mathematical concepts, basic notion was that well-constructed knowledge is interconnected, so that when one part of a network of ideas is recalled for use at some future time, the other parts are also recalled. Students to understand how to perform various mathematical tasks they must also appreciate why each of the ideas and relationships work the way that they do.

2.     Procedural fluency

They defined this as including skill in carrying out procedures flexibly, accurately, efficiently, and appropriately, and, in addition to these procedures, having factual knowledge and concepts that come to mind readily. Pegg explained that initial processing of information happens in working memory, which is of limited capacity. He focused on the need for teachers to develop fluency in calculation in their students, as a way of reducing the load on working memory, so allowing more capacity for other mathematical actions. An example of the way this works is in mathematical language and definitions. If students do not know what is meant by terms such as ‘parallel’, ‘right angle’, ‘index’, ‘remainder’, ‘average’, then instruction using those terms will be confusing and ineffective since so much of students’ working memory will be utilised trying to seek clues for the meaning of the relevant terminology. On the other hand, if students can readily recall key definitions and facts, these facts can facilitate problem solving and other actions.

3. Strategic competence

Describe strategic competence as the ability to formulate, represent and solve mathematical problems.

-a set of critical control processes that guide an individual to effectivelyrecognise, formulate and solve problems. This skill is characterised as selecting or devising a plan or strategy to use mathematics to solve problems arising from a task or context, as well as guiding its implementation.

4.  Adaptive reasoning

Describe adaptive reasoning as the capacity for logical thought, reflection, explanation and justification.

Some mathematics texts did pay some attention to proofs and reasoning, but in a way which seemed:

… to be to derive a rule in preparation for using it in the exercises, rather than togive explanations that might be used as a thinking tool in subsequent problems.

The most importance things that an educator must possess in teaching mathematics are havingenough knowledge of  content and teaching, knowledge of content and students, and knowledge on the curriculum.

5.     Productive disposition

Describe productive disposition as a habitual inclination to see mathematics as sensible, useful and worthwhile, coupled with a belief in diligence and one’s own efficacy. As the name of this strand suggests, this is less a student action than the other strands, but it remains one of the key issues for teaching mathematics, because positive disposition can be fostered by teachers, and possessing them does make a difference to learning.

Math Anxiety

Math anxiety or fear of math is actually quite common. Math anxiety is quite similar to stagefright. Why does someone suffer stagefright? Fear of something going wrong in front of a crowd? Fear of forgetting the lines? Fear of being judged poorly? Fear of going completely blank? Math anxiety conjures up fear of some type. The fear that one won't be able to do the math or the fear that it's too hard or the fear of failure which often stems from having a lack of confidence. For the most part, math anxiety is the fear about doing the math right, our minds draw a blank and we think we'll fail and of course the more frustrated and anxious our minds become, the greater the chance for drawing blanks. Added pressure of having time limits on math tests and exams also cause the levels of anxiety grow for many students.

Where Does Math Anxiety Come From?

Usually math anxiety stems from unpleasant experiences in mathematics. Typically math phobics have had math presented in such a fashion that it led to limited understanding. Unfortunately, math anxiety is often due to poor teaching and poor experiences in math which typically leads to math anxiety. Many of the students I've encountered with math anxiety have demonstrated an over reliance on procedures in math as opposed to actually understanding the math. When one tries to memorize procedures, rules and routines without much understanding, the math is quickly forgotten and panic soons sets in. Think about your experiences with one concept - the division of fractions. You probably learned about reciprocals and inverses. In other words, 'It's not yours to reason why, just invert and multiply'. Well, you memorized the rule and it works. Why does it work? Do you really understand why it works? Did anyone every use pizzas or math manipulatives to show you why it works? If not, you simply memorized the procedure and that was that. Think of math as memorizing all the procedures - what if you forget a few? Therefore, with this type of strategy, a good memory will help, but, what if you dont' have a good memory. Understanding the math is critical. Once students realize they can do the math, the whole notion of math anxiety can be overcome. Teachers and parents have an important role to ensure students understand the math being presented to them.

Myths and Misconceptions about Mathematics

None of the following are true!

You're born with a math gene, either you get it or you don't.

Math is for males, females never get math!

It's hopeless, and much too hard for average people.

If the logical side of your brain isn't your strenght, you'll never do well in math.

Math is a cultural thing, my culture never got it!

There's only one right way to do math.

Overcoming Math Anxiety

1.       A positive attitude will help. However, positive attitudes come with quality teaching for understanding which often isn't the case with many traditional approaches to teaching mathematics.

2.      Ask questions, be determined to 'understand the math'. Don't settle for anything less during instruction. Ask for clear illustrations and or demonstrations or simulations.

3.      Practice regularly, especially when you're having difficulty.

4.      When total understanding escapes you, hire a tutor or work with peers that understand the math. You can do the math, sometimes it just take a different approach for you to understand some of the concepts.

5.      Don't just read over your notes - do the math. Practice the math and make sure you can honestly state that you understand what you are doing.

6. Be persistent and don't over emphasize the fact that we all make mistakes. Remember, some of the most powerful learning stems from making a mistake.

(http://math.about.com/od/reference/a/anxiety.htm)

Six key principles for effective teaching of Mathematics

Principle 1: Articulating goals

Identify key ideas that underpin the concepts you are seeking to teach ,communicate to students that these are the goals of the teaching, and explain to them how you hope they will learn.

Principle 2: Making connections

Build on what students know, mathematically and experientially, including creating and connecting students with stories that both contextualise and establish a rationale for the learning.

Principle 3: Fostering engagement

Engage students by utilising a variety of rich and challenging tasks that allow students time and opportunities to make decisions, and which use a variety of forms of representation.

… students would benefit from more exposure to less repetitive, higher-level problems, more discussion of alternative solutions, and more opportunity to explain their thinking.(Hollingsworth et al., 2003, p. xxi)

… opportunities to appreciate connections between mathematical ideas and to understand the mathematics behind the problems they are working on.

Principle 4: Differentiating challenges

Interact with students while they engage in the experiences, encourage students the interact with each other, including asking and answering questions, and specifically plan to support students who need it and challenge those who are ready.

Principle 5: Structuring lessons

Adopt pedagogies that foster communication and both individual and group responsibilities, use students’ reports to the class as learning opportunities, with teacher summaries of key mathematical ideas.

lesson review:… involves much more than simply restating the mathematics. It encourages children to reflect on their learning and to explain or describe their strategic thinking. The end of the session gives the opportunity for teaching after children have had some experience with mathematical concept. (Cheeseman, 2003,)

Principle 6: Promoting fluency and transfer

Fluency is important, and it can be developed in two ways: by short everyday practice of mental processes; and by practice, reinforcement and prompting transfer of learnt skills.

How to Understand the Different Areas of Mathematics

1.      Understand that mathematics consists of a broad range of topics and is not a single subject. The following steps detail the different areas with which you will need to become familiar as you are studying mathematics.

2. Begin with arithmetic. Arithmetic is the first branch of mathematics that you will have studied in elementary and middle school. It deals with the study of numbers and the use of the four fundamental processes:

3. Be aware that arithmetic is everyday math. It is important to get a solid grounding in this aspect of mathematics because you use it in your personal affairs, and arithmetic is the basis for most other mathematics.

4. Learn about algebra. Algebra is used widely to solve problems in business, industry, and science by using symbols, such as x and y, to represent unknown values. The power of algebra is that it enables us to create, write, and rewrite problem–solving formulas. Without algebra, we would not have many of the items we use on a daily basis, for example, television, radio, telephone, microwave oven, etc.

5.Proceed to geometry. Geometry is the branch of mathematics that deals with shapes. More specifically, geometry is the study of relations, properties, and measurements of solids, surfaces, lines, and angles. It is most useful in building or measuring things. Architects, astronomers, construction engineers, navigators, and surveyors are just a few professionals who rely on geometry.

6. Become familiar with trigonometry. Trigonometry is mathematics that deals with triangular measurements. Plane trigonometry computes the relationships between the sides of triangles on level surfaces called planes. Spherical trigonometry studies the triangles on the surface of a sphere.

7.Learn calculus. Calculus is high-level mathematics dealing with rates of change. It  many practical applications in engineering, physics, and other branches of science. Using calculus, we understand and explain how water flows, the sun shines, the wind blows, and the planets cycle through the heavens. Differential calculus deals with the rate of change of one quantity with respect to another, for example the rate at which an object’s speed changes with respect to time. Integral calculus deals with adding up the effects of continuously changing quantities, for example, computing the distance covered by an object when its speeds over a time interval are known.

8.Understand the field of probability. Probability is the study of the likelihood of an event’s occurrence. It is useful in predicting the outcomes of future events. Probability originated from the study of games of chance. It is now used for other purposes, including (1) controlling of the flow of traffic through a highway system; (2) predicting the number of accidents people of various ages will have; (3) estimating the spread of rumors; (4) predicting the outcome of electronics; and (5) predicting the rate of return in risky investments.

9.Learn statistics. Statistics is the branch of mathematics that helps mathematicians organize and find meaning in data. Anyone who listens to the radio, watches television, and reads books, newspapers, and magazines cannot help but be aware of statistics, which is the science of collecting, analyzing, presenting and interpreting data. Statistics appear in the claims of advertisers, in cost-of-living indexes, and in reports of business trends and cycles.(http://www.wikihow.com/Understand-the-Different-Areas-of-Mathematics)

Several Ways to Achieve Encouragement and Rapport in Teaching

1.      be approachable.

                       A. Students feel a connection with you

                       B. Always remain professional

2.     Reward hard work

3.     Distinguish mathematical achievement from intelligence.

4.     Relate an experience to them

YOU CAN DO MATH!

Attitudes and Misconceptions

Do your experiences in math cause you anxiety? Have you been left with the impression that math is difficult and only some people are 'good' at math? Are you one of those people who believe that you 'can't do math', that you're missing that 'math gene'? Do you have the dreaded disease called Math Anxiety? Read on, sometimes our school experiences leave us with the wrong impression about math. There are many misconceptions that lead one to believe that only some individuals can do math. It's time to dispel those common myths. Everyone can be successful in math when presented with opportunities to succeed, an open mind and a belief that one can do math.

True or False: There is one way to solve a problem.

False: There are a variety of ways to solve math problems and a variety of tools to assist with the process. Think of the process you use when you try to determine how many pieces of pizza will 5 people will get with 2 and a half 6 slice pizzas. Some of you will visualize the pizzas, some will add the total number of slizes and divide by 5. Does anyone actually write the algorithm? Not likely! There are a variety of ways to arrive at the solution, and everyone uses their own learning style when solving the problem.

True or False: You need a 'math gene' or dominance of your left brain to be successful at math.

False: Like reading, the majority of people are born with the ability to do math. Children and adults need to maintain a positive attitude and the belief that they can do math. Math must be nurtured with a supportive learning environment that promotes risk taking and creativity, one that focuses on problem solving.

True or False: Children don't learn the basics anymore because of a reliance on calculators and computers.

False: Research at this time indicates that calculators do not have a negative impact on achievement. The calculator is a powerful teaching tool when used appropriately. Most teachers focus on the effective use of a calculator. Students are still required to know what they need to key into the calculator to solve the problem.

True or False: You need to memorize a lot of facts, rules and formulas to be good at math.

False False! As stated earlier, there's more than one way to solve a problem. Memorizing procedures is not as effective as conceptually understanding concepts. For instance, memorizing the fact 9x9 is not as important as understanding that 9x9 is 9 groups of 9. Applying thinking skills and creative thought lead to a better understanding of math. Signs of understanding include those "Aha" moments! The most important aspect to learning math is understanding. Ask yourself after solving a math problem: are you applying a series of memorized steps/procedures, or do you really 'understand' how and why the procedure works

Answer the questions: How do you know it's right? Is there more than one way to solve this problem? When questions like this are answered, you're on your way to becoming a better math problem solver.

True or False: Keep giving more drill and repetition questions until children get it!

False False, find another way to teach or explain the concept. All too often, children receive worksheets with drill and repetition, this only leads to overkill and negative math attitudes! When a concept isn't understood, it's time to find another method of teaching it. No new learning has ever occurred as a result of repetition and drill. Negative attitudes toward math are usually the result of overuse of worksheets.

In summary:

Positive attitudes towards math are the first step to success. When does the most powerful learning usually occur? When one makes a mistake! If you take the time to analyze where you go wrong, you can't help but learn. Never feel badly about making mistakes in mathematics.

Societal needs have changed, thus math has changed. We are now in an information age with technology paving the way. It is no longer enough to do computations; that's what calculators and computers are for. Math today requires decisions about which keys to punch in and which graph to use, not how to construct them! Math requires creative problem solving techniques. Today's math requires real-life problems to solve, a skill highly prized by employers today. Math requires knowing when and how to use the tools to assist in the problem solving process. This happens as early as pre-kindergarten when children seek counters, an abacus, blocks and a variety of other manipulatives. Family involvement is also critical in nurting a positive and risk-taking attitudes in math. The sooner this begins, the sooner one will become more successful in math.

There are many reasons to learn math and it's never too late to start!