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Teaching of Fractions
How does the teaching of mathematics and, in particular, fractions tie in with children at this stage of development?

The study of fractions is taught by teaching from the whole to the parts. This process is a parallel activity to the growth pattern of development that is happening in children around nine/ten.

It follows, therefore, that while many educators ask the question "Is this child ready to handle this content at this particular time?" a teacher using the approach described would tend to ask such questions as "What will help us feed and enrich this child at this particular stage of development?"

In addition, as the elementary school child's world is also one of color and pictorial representation, approaches which stimulate children's imagination or which engage their creative and artistic abilities are recommended.

Our Approach to Teaching a New Concept in Fractions
First Stage
The first stage is to relate material to the experience of the student. It is accepted that this experience is different from the adults' experience. In essence children are not miniature adults. When teaching adults we would probably teach immediately to the "head" whereas for elementary school children we need to teach to the "heart" and "hands". Wherever possible content is introduced so that it relates to artistic and pictorial representation. One way of doing this is through storytelling where the student's imagination is stimulated although it should be added that a subject area such as fractions does not lend itself to the creation of stories as easily as say history does. Through stories, information is absorbed in a way that is in empathy with the students' experience. (storytelling was dealt with in the first newsletter - for sake of ease it is repeated in Note 2.)
Second Stage
The second stage is to encourage the students to express their experience through a variety of artistic forms. Some children will want to do this quickly; others will want to take their time. This stage should not be hurried and the children should be given enough time to work through a number of forms according to ability, aptitude and temperament.
Third Stage
The third stage is to work through concrete examples.
Fourth Stage
The final stage is to introduce the abstract concepts and to work symbolically with numbers.

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Our Lesson Plan
Students need to be able to convert a fraction to its equivalent form in order to carry out all but the simplest computation involving fractions. It is important, therefore, to introduce this aspect quite early in the study of fractions. We begin with the story "Distributing the Hay." We recommend the story is first read to the students. Next, students learn the words in the glossary (see Teacher Notes.) Students can then read the story independently (readability level of Student Version is Grade 3.7). Thus, students are introduced to the concept of equivalent fractions. As stated earlier this is a math/language arts program and Assignment 1 involves students in identifying (through a pictorial approach) nouns, verbs and adjectives.

The activity sheets (Activities 1, 2 and 3, Assignment 2) are designed to provide students with a visual experience of equivalent fractions. They are also designed to encourage students to work first through the imaginative realm, then through the visual, followed by the concrete, and finally the abstract. Instructions are given to guide the student through the steps of creating a Fraction Chart (Assignment 3). In this assignment they gain valuable experience in following written instructions. There is sufficient repetition of vocabulary and sentence structure to assist the less able reader.

2. Methodology of Teaching
Introduction
The best results of any teaching process will occur when a good relationship has been established between teacher and child. Yet the qualities that determine the way in which we react and respond to each other are virtually ignored in the educational sector. Teachers have no option but to depend entirely on their own perception and awareness to meet the needs of each individual child. Contained in the Waldorf approach is the belief that each of us possess four temperaments or personality types, one of which will predominate. Once teachers appreciate different temperaments or personality types they can perceive "needs and interests" according to the temperament or personality type of each particular student.

Background
The Greeks were the first to write about the four temperaments found in human beings and how one predominates. Hippocrates described how nature as a whole and the elements that comprise it (the macrocosm) were reflected in our own nature (the microcosm). He was also of the opinion, and this, of course, was long before the days of endocrinology, that the four elements were represented in the human body in the form of four "humors". The names the Greeks gave to the temperaments (choleric, melancholic, phlegmatic and sanguine) were related to the humors and man's physical make up. Although these names are still used in the Waldorf approach they do not relate to modern terminology. For example, the description of a melancholic refers to a particular temperament and not to a depressed or gloomy state of mind.

Basis for interaction between teacher and student
Doesn't any teacher want to know how to motivate his/her students? Doesn't any teacher want to see that their students' learning is somewhere near the optimum? This area is therefore crucial for it is part of the craft of the teacher to formulate lessons so an appeal is made to each of the temperaments.

What is the basis for interaction between teacher and student? It is that the teacher should be working with the main temperament of each individual child and not against it. If the inner needs according to temperament are not met a barrier to learning will almost certainly occur.

What is the reality in any interaction? Each one of us at any particular time possesses a certain disposition and attitude towards whatever is happening in that interaction. It is assumed that a teacher would find it helpful if they knew why a student was thinking or acting in that particular way.

Material in "Equivalent Fractions"
All examples in Student Version of the story.
When there is something in a story that appeals to that child's temperament he/she is more likely to be attracted to, and show greater interest in, the content. Usually it is enough to include a brief description in order to gain the interest of that particular temperament.

For example, many children with a predominantly melancholic temperament are concerned or worried about things or circumstances. It is reassuring to them to know that others also worry about things. We see that Mr. Austin worries a great deal "Each winter he worried whether there would be enough hay for him to buy ...." Melancholic children, when things go wrong, tend to complain or grumble as a natural part of their social interaction. Hence we have Mr. Austin grumbling to his wife "I'm not sure how long I can do all this work ...."

Cholerics are usually very practical. They tend to view the world in terms of black and white, their energy is outgoing and they love challenges and solving problems. In the story Mrs. Austin has a choleric nature and she accepts the challenge presented to her and immediately comes up with an answer.

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Note 1
I believe that the majority of child psychologists and educators would accept the concept of stage in relation to child development. The following may be helpful in clarifying what we mean by stage.
"The enlargement of the life space proceeds by stages, each of which may involve a varying period of time for acquisition followed by a period varying in length during which the growing person adapts to his/her new found functions and properties. Thus, there are sudden as well as gradual transformations of behavior with each change followed by a period of gradual adaptation." J.E.Anderson
"1. There is an invariable order of the stages of development;
2. no stage can be skipped;
3. each stage is more complex than the preceding one: it represents a transformation of what existed before in a new form; and 4. each stage is based on the preceding one and prepares for the succeeding one. The development model assumes an inner logic, a built in plan that gives direction to the sequence of development." L.Breger

Note 2: Storytelling
Why does a lesson on mathematics begin with a story? Stories, more than anything else, engage the child's imagination. They transport children into a world where they can easily create pictures, and visualize a situation. Part of the task of teachers is to transmit knowledge to students. However, the act of learning is not complete until students have internalized the knowledge and made it their own. This process is far more effective when it is done through storytelling. We need not be concerned, at present, with the extent to which this occurs or with the influence on hemispheric development (this will occur in a later newsletter). The main point is to establish that storytelling is a way in which we can pass on information to students while at the same time enhancing and optimizing their learning.

Storytelling and Mathematics
We believe that this enhancement and optimization can occur even in subjects like mathematics. For example, in our material we try and introduce many concepts with a story (an overview of our material is available at: http://members.home.net/waldorfedu/Math.html).

When teaching our lessons we recommend that, initially, the teacher reads (even better, if the teacher can tell) the story to the class. At this point there is no pressure on children to do anything but listen to the content and create pictorial representations according to their ability to do this.

They will, instinctively, create pictures from the story content, and in so doing they create their own context within which to work. Any information they now receive will be absorbed into this context. Thus the base has been laid for the teacher to move on to the next stage.

In other words, the teacher by reading or telling the story has created a context with which the children can identify, and through which they can absorb information.

As far as our approach is concerned this point is crucial. Students start the process of understanding by creating pictures in their imagination. It is only when this has occurred should they move on to work through hands-on experiences with manipulatives. Finally, and only when they have worked through the first two stages, are they ready to work with the abstract and theoretical.

Contents of this lesson plan:
Teacher Notes
Story - Distributing the Hay (Teacher Version)
Story - Distributing the Hay (Student Version)
Assignment 1: Language Arts
Assignment 2: Distributing the Hay
Assignment 3: Making a Fraction Chart
Assignment 4: Equivalent Fractions
Assignment 5: More Equivalent Fractions

Distributing the Hay (Teacher Version)
A farmer, Mr. Austin, had 18 horses of which he was very proud. During the summer months the animals fed on the rich grass of the pastures, and were strong and healthy. But when winter came, the grass in the fields grew slowly and there was not enough for the horses to eat.Each year Mr. Austin bought hay so that his horses would not go hungry and would stay healthy during the cold months. Mr. Austin stacked the hay in a large barn. The hay was just the right food for the horses and it was quite easy to store.Once again winter arrived and the hay was ready, but Mr. Austin wanted to make sure that he treated all the animals fairly. He wanted to make sure that each horse had the same amount to eat; that each of the 6 horses in his first field did not get more or less hay than each of the 12 horses he kept in the second field. He therefore had to work it out very carefully and he wrote down the following to help him make his calculations.

He also discovered:

If you start with 12/18 and divide the numerator and denominator by 2	it becomes 6/9
If you then take 6/9 and divide the numerator and denominator by 3	it becomes 2/3

So Mr. Austin found that instead of dividing the hay into 18 portions he now only needed to divide each day's hay into 3.He gave 1/3 of the hay to the horses in the first pasture and 2/3 to the horses in the second pasture.He soon found that caring for his horses took far less time. He also knew that the horses in each of the pastures would now be able to have a fair share of the hay.

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Distributing the Hay (Student Version)
A farmer, Mr. Austin, had 18 horses. He was very proud of them. During the summer months the animals fed on the rich grass in the fields. The horses were strong and healthy. In the summer Mr. Austin was happy because he knew his horses had enough to eat.

When winter came, the grass in the fields grew slowly. At this time the horses did not have enough to eat.

In the winter Mr. Austin bought hay. Each winter he worried whether there would be enough hay for him to buy. Each winter he worried whether his horses would have enough to eat. Each winter he worried whether they would stay healthy during the cold months.

Mr. Austin stored the hay in a large barn. The hay was just the right food for the horses. Also, it was quite easy to store.

Once again winter arrived and the hay was ready. As usual Mr. Austin wanted to make sure that he treated all his horses fairly.

He wanted to make sure that each horse had the same amount to eat. That meant that each of the 6 horses in his first field got the same amount as each of the 12 horses in the second field.

He knew that he had to work it out with care. He wrote down the following to help him work it out.

This is the drawing of the hay that Mrs. Austin made. The hay has been divided into 18 portions, one for each horse.
Shade the hay for the larger field in green. Shade the hay for the smaller field in brown. Look carefully at the hay you have colored in. Can you see that 6/18 is the same as 1/3? Can you see that 12/18 is the same as 2/3?	This is how Mr. Austin checked the solution: 6/18 = 3/9 = 1/3 12/18 = 6/9 = 2/3

• Last year the Austins had fewer horses.
• There were 9 horses in the large field and 6 horses in the small field.
• Draw the horses in their fields.

• Make a drawing of the hay they will need as Mrs. Austin did.
• Shade the hay for the larger field in green and shade the hay for the smaller field in yellow.

Assignment 3: Making a Fraction Chart
You will need some squared paper. Choose one with large squares to make a large chart. Draw a rectangle that is sixteen squares along one side and five squares along the other.
	Shade the squares in the first line a color of your choice. Label it "1".
Shade the squares in the second line a color of your choice. Divide the strip into two equal parts and label the two halves.
	Shade the squares in the third line a color of your choice. Divide the strip into four equal parts and label the four quarters.
Shade the squares in the fourth line a color of your choice. Divide the strip into eight equal parts and label the eighths.
	Shade the squares in the fifth and last line a color of your choice. Label the sixteenths.
In the next assignment you can use your Fraction Chart.

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