How To Teach Mathematics II

When looking to teach maths, sometimes you are faced with a complex curriculum which outlines various ways to manipulate numbers – these are not as easy to understand as the writers think – thus the following list outlines the ways to find your way through the maze!

Number Skills

Sorting- objects sharing a common attribute are grouped together.
One-to-one correspondence – this is necessary before counting can begin.
Counting- matching number names in one-to-one correspondence to a set of object.
Ordinality- ordering according to position
Cardinality- identifying sets that is the same number of members.
Conservation- recognizing that number is conserved whatever the arrangement of the members of the set.
Symbol recognition- recognising the abstract number symbol and relating them to the sound of the number words.
Place value- the position of a digit indicates its value, for example, 7, 73 and 741 each assign a different value to the digit 7.
Number operations
Addition- counting on from an identified point
Subtraction- taking away object from a set
- Working out the difference between 2 sets
Multiplication- adding equal sets (2+2+2=)
- enlargement (2,4,6,8,10)
- arrays e.g. 2 types of bread = 6 types of
3 types of fixing sandwich

Division- sharing amounts equally between a given number of sets.

Division- establishing how many sets of a given number can be made
There are 6 fish
(e.g. how many cats can have 3 fish each? )
Fractional parts-
1/3 can represent:
- one whole divided into there parts
- a share of a number that is divisible by three
- a share of a number less than three
- a share of a number not divisible by three
- a share of three
- a share of a number less than one

ALGEBRA- a generalisation of numerical relationships

Number patterns- portrayed numerically, algebraically and geometrically
e.g. patterns of number bonds
Patterns in place value
Patterns of multiplication and division
Patterns of equivalence and specific number patterns
Manipulation of unknown quantities.
- use of letters and symbols
- use of graphs and co-ordinates

Bibliography

How to teach maths by Stephanie C Mills 2008

Myers - Briggs Type Indicator INTRODUCTION!

Understanding one's temperament is a central part of any personal development. The purpose of this site is to give a detailed explanation about one of the most popular and influental psychological methodology known nowadays.

When Katharine Briggs and her daughter Isabel Myers designed Myers-Briggs Type Indicator(MBTI), they took Jungian typology as the basis. Here you will find some facts about Carl Jung, as well as the most important chapter of his book "Psychological types". Jung used 3 scales to measure people.

• The first one was (E)xtroversion vs. (I)ntroversion. This stands how people prefer to focus their attention(interest) and get/spend energy. In the extraverted attitude the energy flow is outward, and the preferred focus is on people and things, whereas in the introverted attitude the energy flow is inward, and the preferred focus is on inner thoughts and ideas.
  
  
• (S)ensing vs. i(N)tuition. Sensing and Intuition are the perceiving functions. They indicate how a person prefers to receive data from the environment around him. These are the nonrational functions, as a person does not necessarily have control over receiving data, but only how to process it once they have it. Sensing prefers to receive data primarily from the five senses(sight, hearing, taste, smell, touch), and intuition prefers to receive data from the unconscious, or seeing relationships via insights. Often it is called "sixth sense" or "gut feel".
  
  
• (T)hinking vs. (F)eeling. Thinking and Feeling are the rational functions. They are used to make rational decisions concerning the data they received from their perceiving functions, above. Thinking is characterized as preferring to being logical, analytical and thinking in terms of "true or false". Thinking decisions tend to be based on more objective criteria and facts. Feeling, which refers to subjective criteria and values, strives for harmonious relationships and considers the implications for people. Feeling decisions tend to be based on what seems "more good or less bad" according to values.
  
  
• And the 4th scale even it could be met in the Jungian works was truly added by Myers-Briggs team. (J)udging vs. (P)erceiving. It shows how people relate to the world around them. The J person tends to prefer to create and live in an ordered environment. Words like "structured" and "controlled" come to mind. Js tend to "plan their work and work their plan." They often come across as "decisive". The P individual tends to prefer a flexible, wait-and-see environment. Words like "spontaneous" and "adaptable" and "open-minded" best describe him or her.

All these scales explained above, describe contrasting preferences. Practically it is hard to find someone that is extrovert or thinking to the absolute extreme. He/she could be 80% thinking and 20% feeling type and usually one of the preferences is dominant. So using MBTI short-hand, we can describe people by their expressed preferences: ISFJ, ENTP, and so on. There are 16 different combinations of letters - giving us the 16 different "psychological types".

From : E-MBTI.com

Proof by Intimidation

Proof by intimidation is a jocular term used mainly in mathematics to refer to mathematical proofs which are so complex, so long-winded and so poorly presented by the authors that others are simply forced to accept it, lest they be forced to sift through its minute details. The term is also used when the author is an authority in his field presenting his proof to people who respect a priori his insistence that the proof is valid or when the author claims that his statement is true because it is trivial or because he simply says so. Usage of this term is for the most part in good humour, though it also appears in serious criticism. More generally, "proof by intimidation" has also been used by critics of junk science to describe cases in which scientific evidence is thrown aside in favour of a litany of tragic individual cases presented to the public by articulate advocates who pose as experts in their field.

Gian-Carlo Rota claimed in a memoir that the term 'proof by intimidation' was coined by Mark Kac to describe a technique used by William Feller in his lectures

taken from Wikipedia.com

Mock Mathematics

A form of mathematical humor comes from using mathematical tools (both abstract symbols and physical objects such as calculators) in various ways which transgress their intended ambit. These constructions are generally devoid of any "real" mathematics, besides some basic arithmetic.

Mock mathematical reasoning

A set of equivocal jokes applies mathematical reasoning to situations where it is not entirely valid. Many of these are based on a combination of well-known quotes and basic logical constructs such as syllogisms:

Example:

Premise I:	Knowledge is power.
Premise II:	Power corrupts.
Conclusion:	Therefore, knowledge corrupts.

This is used to demonstrate that studying causes one to fail.

Study = No fail	No Study=Fail
Study + No Study = Fail + No Fail
Study (1 +No) = Fail (1+No)
Study (1 +No) = Fail (1+No)
Study = Fail

There are also a number of joke proofs, such as the proof that women are evil:

Women are the product of time and money:	women = time × money
Time is money:	time = money
So women are money squared:	women = money2
Money is the root of all evil:	money = √evil
So women are absolutely evil:	women = (√evil)2 = abs(evil)

Another set of jokes relate to the absence of mathematical reasoning, or misinterpretation of conventional notation:

Examples:

That is, the limit as x goes to 8 from above is a sideways 8 or the infinity sign, in the same way that the limit as x goes to three from above is a sideways 3 or the Greek letter omega.

(That is, the "n" in "sin" cancels with the "n" in the denominator, giving "six" and 1 respectively.) See also Anomalous cancellation.

taken from Wikipedia.com

Stereotypes of Mathematicians

Some jokes are based on stereotypes of mathematicians tending to think in complicated, abstract terms, causing them to lose touch with the "real world".

Many of these jokes compare mathematicians to other professions, typically physicists, engineers, or the "soft" sciences in a form similar to those which begin "An Englishman, an Irishman and a Scotsman…" or the like. The joke generally shows the other scientist doing something practical, while the mathematician does something less useful such as making the necessary calculation but not performing the implied action.

Examples:

A mathematician and his best friend, an engineer, attend a public lecture on geometry in thirteen-dimensional space. "How did you like it?" the mathematician wants to know after the talk. "My head's spinning," the engineer confesses. "How can you develop any intuition for thirteen-dimensional space?" "Well, it's not even difficult. All I do is visualize the situation in n-dimensional space and then set n = 13."

A mathematician, a biologist and a physicist are sitting in a street café watching people entering and leaving the house on the other side of the street. First they see two people entering the house. Time passes. After a while they notice three people leaving the house. The physicist says, "The measurement wasn't accurate." The biologist says, "They must have reproduced." The mathematician says, "If one more person enters the house then it will be empty."

An example of a joke relying on mathematicians' propensity for not taking the implied action is as follows:

A mathematician, an engineer and a chemist are at a conference. They are staying in adjoining rooms. One evening they are downstairs in the bar. The mathematician goes to bed first. The chemist goes next, followed a minute or two later by the engineer. The chemist notices that in the corridor outside their rooms a rubbish bin is ablaze. There is a bucket of water nearby. The chemist starts concocting a means of generating carbon dioxide in order to create a makeshift extinguisher but before he can do so the engineer arrives, dumps the water on the fire and puts it out. The next morning the chemist and engineer tell the mathematician about the fire. He admits he saw it. They ask him why he didn't put it out. He replies contemptuously "there was a fire and a bucket of water: a solution obviously existed."

Mathematicians are also shown as averse to making sweeping generalizations from a small amount of data, preferring instead to state only that which can be logically deduced from the given information – even if some form of generalization seems plausible:

An astronomer, a physicist and a mathematician are on a train in Scotland. The astronomer looks out of the window, sees a black sheep standing in a field, and remarks, "How odd. Scottish sheep are black." "No, no, no!" says the physicist. "Only some Scottish sheep are black." The mathematician rolls his eyes at his companions' muddled thinking and says, "In Scotland, there is at least one field, containing at least one sheep, at least one side of which appears black from here."

A variant has the punchline "No," says the mathematician, "all we can say is that there is at least half of a black sheep in Scotland." Another variant with cows was featured in the book The Curious Incident of the Dog in the Night-time by Mark Haddon.

Pure mathematicians are mainly concerned with the properties of the abstract systems under study, not their actual applications. However, such applications are sometimes found in mathematics itself, resulting in new insights as old problems are cast in new light. In striving not to miss such connections, mathematicians often see problems in novel (but theoretically valid) ways, which unfortunately are not always as illuminating as one could wish for:

A sociologist, a physicist and a mathematician are all given equal amounts of fencing, and are asked to enclose the greatest area. The sociologist pauses for a moment and decides to enclose a square area with his fence. The physicist, realizing he can fence off a greater amount of land with the same amount of fencing, promptly sets his fence in the form of a circle, and smiles. "I'd like to see you beat that!" he says to the mathematician. The mathematician, in response, takes a very small piece of his own fencing, and wraps it around himself, proclaiming, "I define my location to be outside of the fence!"

A small set of jokes involves only mathematicians, such as the following involving statisticians:

Three statisticians go duck hunting. Their dog chases out a duck and it starts to fly. The first statistician aims and takes his shot, and it misses a foot too high. The second statistician aims and takes his shot, and it misses a foot too low. The third statistician says, "We got him!"

The humor there is derived from the fact that the average of the shots hits the duck, and so it is dead.

An engineer and a physicist are in a hot-air balloon. After a few hours they lose track of where they are and descend to get directions. They yell to a jogger, "Hey, can you tell us where we're at?" After a few moments the jogger responds, "You're in a hot-air balloon." The engineer says, "You must be a mathematician." The jogger, shocked, responds, "yeah, how did you know I was a mathematician?" "Because, it took you far too long to come up with your answer, it was 100% correct, and it was completely useless."

taken from Wikipedia.com

Three is equal to four

Theorem: 3=4
Proof:

Suppose:
a + b = c

This can also be written as:
4a - 3a + 4b - 3b = 4c - 3c

After reorganizing:
4a + 4b - 4c = 3a + 3b - 3c

Take the constants out of the brackets:
4 * (a+b-c) = 3 * (a+b-c)

Remove the same term left and right:
4 = 3

One equal to one half

Theorem: 1 = 1/2:
Proof:

We can re-write the infinite series 1/(1*3) + 1/(3*5) + 1/(5*7) + 1/(7*9)
+...

as 1/2((1/1 - 1/3) + (1/3 - 1/5) + (1/5 - 1/7) + (1/7 - 1/9) + ... ).
All terms after 1/1 cancel, so that the sum is 1/2.

We can also re-write the series as (1/1 - 2/3) + (2/3 - 3/5) + (3/5 - 4/7)
+ (4/7 - 5/9) + ...

All terms after 1/1 cancel, so that the sum is 1.

Thus 1/2 = 1.

Teachers Need More Knowledge of How Children Learn Mathematics

by Constance Kamii

Teachers need as much scientific knowledge about how children learn mathematics as physicians have about the causes of illnesses. Because of this need, teacher-preparation programs must change. Specific examples from classrooms illustrate this need.

I once wondered why some first graders were getting such answers as 3 + 4 = 4 (*1+1=11). By watching them, I found out that they were putting three counters out for the first addend and then four for the second addend, including the three that were already out.

Errors of this kind result from prematurely teaching a rule to follow. According to this rule, one must put counters out for the first addend, more counters for the second addend, and count all of them to get the answer. This rule works for children who already know that addition is the joining of two sets that are disjoint. However, the rule is superfluous for those who have constructed this logic, and it causes errors for those who have not constructed it.

Another example of imposing a rule that is either superfluous or premature is teaching counting-on to children who are counting-all. Counting-all refers to solving 3 + 4 by counting out three counters, then four other counters, and counting all of them again ("one-two-three-four-five-six-seven"). In counting-on, by contrast, children say "four-five-six-seven."

With scientific research replicated worldwide, Piaget showed that all children construct, or create, logic and number concepts from within rather than learn them by internalization from the environment (Piaget 1971; Piaget and Szeminska 1965; Inhelder and Piaget 1964; and Kamii 2000). Studying the research leads teachers to understand that addition involves part-whole relationships, which are very hard for children to make and which cannot be taught through practice and memorization. To add two numbers, children must put two wholes together ("three" and "four," for example) to make a higher-order whole ("seven") in which the previous wholes become two parts. When young children cannot think simultaneously about a whole and two parts, they count-all by changing both the "three" and the "four" into ones. Making them count-on is harmful when they cannot mentally make the part-whole relationship necessary to count-on.

When teachers study Piaget's theory and replicate the aforementioned research, they can understand why some first graders cannot count-on. When children have constructed their logic sufficiently to make the part-whole relationship of counting-on, they give up counting-all, just as babies give up crawling when they can walk. I hope that the day will come when teachers entering the classroom and those already in the classroom have as much scientific knowledge about how children learn mathematics as physicians have about the causes of illnesses. To reach this vision, the teacher-preparation programs must change.

References

Inhelder, Barbel, and Jean Piaget. The Early Growth of Logic in the Child. New York: Harper & Row, 1964.

Kamii, Constance. Young Children Reinvent Arithmetic. 2nd ed. New York: Teachers College Press, 2000.

Piaget, Jean. Biology and Knowledge. Chicago: University of Chicago Press, 1971.

Piaget, Jean, and Alina Szeminska. The Child's Conception of Number. New York: W. W. Norton & Co., 1965.

Constance Kamii, ckamii@uab.edu, is a professor of early childhood education at the University of Alabama at Birmingham. She studied under Piaget for parts of fifteen years to become able to use his theory in early childhood education.

*in indonesia often happened

How to calculate square roots without a calculator

Many mathematical operations have an inverse, or opposite, operation. Subtraction is the opposite of addition, division is the inverse of multiplication, and so on. Squaring, which we learned about in a previous lesson (exponents), has an inverse too, called "finding the square root." Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …

The square root of a number, n, written

is the number that gives n when multiplied by itself. For example,

because 10 x 10 = 100

Examples

Here are the square roots of all the perfect squares from 1 to 100.

Finding square roots of of numbers that aren't perfect squares without a calculator

1. Estimate - first, get as close as you can by finding two perfect square roots your number is between.

2. Divide - divide your number by one of those square roots.

3. Average - take the average of the result of step 2 and the root.

4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.

Example: Calculate the square root of 10 () to 2 decimal places.

1. Find the two perfect square numbers it lies between.

Solution:
3² = 9 and 4² = 16, so lies between 3 and 4.

2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer)

3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667

Repeat step 2: 10/3.1667 = 3.1579
Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623

Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001

If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.

Note: There are a number of ways to calculate square roots without a calculator. This is only one of them.

http://www.math.com

The Advantages of Organic Food

You Are What You Eat

Do you really know what goes into your food? Discover the advantages of organic food on this site and see exactly what producers have been adding to your fruit and vegetables to make it less healthy than a few years ago.

In the rush to produce more and more crops to satisfy growing demand producers have had to resort to using a lethal cocktail of pesticides to control disease and insect attack.

Good news for their bank balances perhaps but not good news for your health, this is why you need to be informed of the advantages of organic food.

Did you know that if you consumed an average apple you would be eating over 30 pesticides, even after you have washed it?

The quality of food has definitely gone down since the second world war. For instance, the levels of vitamin C in today's fruit bear no resemblance to the levels found in wartime fruit.

Organic food is known to contain 50% more nutrients, minerals and vitamins than produce that has been intensively farmed. Read more about this here.

You will have to eat more fruit nowadays to make up the deficiency, but unfortunately that means eating more chemicals, more detrimental affects on your health eating something that should be good for you!

Also don't forget about the cocktail of anti-biotics and hormones that cattle and poultry are force fed.

What happens to those chemicals when the animal dies?

Digested and stored in human bodies is the answer, have you seen pictures of animals in severly cramped conditions in battery farms?

It just does not make sense to state that any animal kept in these conditions is healthy and produces high quality food.

If you are as worried as I am about the health of your family then you need to read the articles on this and seriously consider converting your family to the organic lifestyle with the organic food information you are going to learn on this site.

Trust me, once you try some organic produce and taste an apple the way it should be, and perhaps how you recall it tasting in your youth, you will never go back to mass produced fruit again.

Sure there are issues with availability and cost but with a bit of research you should be able to find local stores who stock organic produce.

Also, don't forget about your local farmer, I'm sure you will be able to find one that has seen the light and opened up a farm shop to supply local residents.

You should be able to get some very keen prices from these shops, why not take a look around and see who is offering produce in your area?

Some more startling facts now. Pesticides in food have been linked to many diseases including:

Cancer
Obesity
Altzheimer's
Some birth defects

Not a nice list is it? There are probably others but if you think about it, how can it be okay for you to eat chemicals and not expect some form of reaction in your body. Our bodies are delicately balanced wonderful machines. Any form of foreign chemical is bound to cause irritation at the least.

Please take advantage of the organic food articles and information on this site and do consider taking a closer look at what you are eating. It's for your health after all!

Virginia Louise

www.organicfoodinfo.net