More than likely, when you learned how to add, you started on the right and moved to the left. If you were adding whole numbers, you added the ones, “carried” if necessary, and repeated for the tens, hundreds and so on. This works well on paper, and it is the most efficient paper and pencil method; however, adding in the other direction has several desirable advantages: the left to right method promotes a better understanding of place value, it can be done mentally with much greater ease, and it does not require that numbers be lined up in a column. Students can learn left to right addition, so they have another method to choose from when presented with addition problems.
Left to right addition involves adding the largest place values first. As you move from left to right, you keep a cumulative total, so it is simply a number of smaller addition problems. To give you an idea of how it works and what it sounds like, consider the example, 677 + 938.
Begin by adding the left most place values. In the example this is 600 plus 900 equals 1500. Add the values in the next place, one at a time, to the previous sum, and keep track of the new sum each time. In the example, 1500 + 70 is 1570, 1570 + 30 is 1600. For students who are more proficient at this algorithm, they don’t necessarily think “plus 70″ or “add 30.” Their thought process, if said out loud might sound like, “600, 1500, 1570, 1600, . . .” Continue adding the values in each subsequent place until finished. The final steps in the example are 1600 + 7 is 1607, 1607 plus 8 is 1615. The sum is 1615.
As you can imagine, students need to be proficient at single digit addition and have an understanding of place value before attempting left to right addition. When they are first learning it, they might try repeating sums as they go along (e.g. 1500, 1570, 1570, 1570, 1600, . . .) to help them retain the newest sums. They might also cross out digits as they are adding. There is no rule about having to add in this way mentally. Students could write down the sums as they proceed.
Left to right addition promotes a better understanding of place value than right to left addition. In right to left addition, single digits are carried or regrouped with little emphasis placed on what the value of those carried digits are. In the example, 1246 + 586, students add 6 + 6 to get 12; they write down the 2 and carry the 1 when they should be carrying the ten. In the next step, they add 8 + 4 + 1 to get 13; they write down the 3 and carry the 1 when they should be adding 80 + 40 + 10, writing the 3 in the tens place (i.e. 30) and carrying the hundred. Essentially, right to left addition excludes vocabulary related to place value. Left to right addition, on the other hand, promotes an understanding of place value as each digit is given its correct value. In the example, the one in the thousands place is one thousand, the two in the hundreds place is two hundred, and so on.
Left to right addition is well-suited to mental addition since the sum is cumulative with no steps in between; in other words, there is nothing for the student to keep in mind except for the cumulative sum. In right to left addition, several numbers must be remembered as the student proceeds. To illustrate this, consider the simple example, 64 + 88. In left to right addition, the sum is simple to find: 60, 140, 144, 152. Only one number had to be remembered at any point. In right to left addition, 4 + 8 is 12, so there are already two numbers to remember: the two in the ones place and the regrouped ten. The next step is to add 60 + 80 + 10 to get 150. At this point, the two must be recalled and added to the 150 to get 152. Although this sounds simple, it becomes more complicated with more digits.
Right to left addition does not require numbers to be lined up in a column, but it is often taught that way because the method tends to ignore place value and relies on a student’s ability to line up the place values to compensate. Many errors that students make in right to left addition occur because they don’t have a strong knowledge of place value, and they forget or don’t realize that like place values need to be lined up. They might, for instance, add a digit in the tens place to a digit in the hundreds place. Another scenario is a sloppy recording of numbers where a digit is mistakenly added to the wrong column. In left to right addition, the emphasis is on finding a certain place value in each number rather than relying on the place values being aligned. Students, of course, need to be able to recognize place value before they can be successful at this method. For instance, they should be able to recognize that the ones in the numbers: 514, 1499, and 321 are in the tens, thousands, and ones places respectively. If they can’t, further teaching on place value is required before addition can be taught effectively.
Although left to right addition has several advantages, it isn’t suggested that you scrap everything else. Learning a wide variety of addition methods allows you latitude in problem solving situations. By teaching students this method, you give them another option when they are tackling addition questions.
Your Brain – General Features
The Human Cortex
The most striking feature of the human brain is seen in the cortex. This is the folded, hemispherical structure which constitutes the bulk of the visible brain.
It is not present in reptiles.
The cortex is relatively recent. It is perhaps one hundred thousand years old and is the part of the brain most closely associated with our ability to form complex representations of the external world, to reason logically and to use language.
It is much more dominant in humans than in any other species.
Regions of the cortex control vision, our auditory senses, and voluntary movement and touch sensations. It is also crucial for long term memory.
Neurons and Networks
The central nervous system is composed of something like one hundred billion nerve cells or neurons.
Each nerve cell or neuron possesses a single axon along which it can pass electrical signals to other neurons. Incoming signals are carried by a neuron’s dendrites which form a tree-like structure around the neuron.
Neurons are about one micron (1 millionth meter) in diameter. The dendrites are perhaps ten times this in length while the axon varies from a millimetre up to one metre in length.
The signal from one neuron reaches another at the junction of axon and dendrite — the synaptic gap.
The typical voltages associated to these signals are small (tens of millivolts) and travel at about two hundred miles an hour (100 metres per second)
Typically neurons can only fire once every millisecond (one thousandth of a second)
Different patterns of electrical firing activity are associated with different brain functions.
Learning and Connections
The brain is both robust (able to function in the event of severed connections and/or dead neurons) and plastic – able to adapt to new memories and functions.
This is due to ability of the brain to form new connections between neurons. These connections take place at synapses and are
mediated by the release of neurotransmitter chemicals.
These neurotransmitters alter the effective strength of the signal which can pass between
neurons.
During our early years and during any kind of learning process these connections form and change their strengths.
The power of the brain as a computational device derives from the complex network of neural pathways and the simultaneous processing capability of all the neurons.
One such immensely powerful device belongs to you.
You can personally programme this device (your brain) to deliver everything you have ever truly desired.
This Genie within you is simply waiting to be told what it is you want.
So set your Genie some exciting tasks to perform and pilot yourself to a future of positive expectation.
Educational theorists, from philosophers like Socrates and Rousseau to researchers like Howard Gardner today, have addressed theories of learning. Many of their ideas continue to influence homeschoolers as well as traditional educators. A little familiarity with some of the ideas most popular among homeschoolers will help you make sense of the wealth of available materials when you begin to make choices for your family.
Jean Piaget and Cognitive Development
He proposed that children go through several distinct stages of cognitive growth. First comes the sensorimotor stage (birth to two years), during which the child learns primarily through sensation and movement. At the pre-operational stage (ages two to seven), children begin to master symbols such as language and start to be able to form hypotheses based on past experiences. At the concrete operational stage (ages seven to eleven), children learn to generalize from one situation to similar ones, although such reasoning is usually limited to their own concrete experience.
Finally, at the formal operational stage (eleven years older), children can deal with abstractions, form hypothesis and engage freely in mental speculation. Although the rate at which children progress through the stages varies considerably, the sequence of stages is consistent for all children.
Therefore, to be appropriate and effective, learning activities should be tailored to the cognitive level of the child.
Rudolf Steiner and the Waldorf Schools
Steiner divided children’s development into three stages: to age seven, children learn primarily by imitation; from seven to fourteen, feelings and emotions predominate; and after age fourteen, the development of independent reasoning skills becomes important. Waldorf education tends to emphasize arts and crafts, music, and movement, especially at younger ages, and textbooks are eschewed in favor of books the students make for themselves. Waldorf theories also maintain that the emphasis should be on developing the individual’s self-awareness and judgment, sheltered from political and economic aspects of society until well into adolescence.
Montessori and the Prepared Environment
Italian physician Maria Montessori’s work emphasized the idea of the prepared environment: Provide the proper surroundings and tools, so that children can develop their full potential. Montessori materials are carefully selected, designed to help children learn to function in their cultures and to become independent and competent. Emphasis is on beauty and quality, and that which confuses or clutters is avoided: Manipulative are made of wood rather than plastic tools are simple and functional, and television and computers are discouraged.
Charlotte Mason: Guiding Natural Curiosity
Charlotte Mason was a nineteenth-century educator advocated informal learning during the child’s early year contrast with the Prussian system of regimented learning then in vogue. She recommended nature study to develop both observational skill and an appreciation for the beauty of creation and extended that approach to teaching history geography through travel and study of the environment rather than as collections of data to master. She felt children learn best when instruction takes into account their individual abilities and temperaments, but she emphasized the importance of developing good habits to govern one’s temperament and laying a solid foundation of good moral values.
Holt and Unschooling
Educator John Holt wrote extensively about school reform in the 1960s. Although he originally proposed the word “unschooling” simply as a more satisfactory alternative to “homeschooling.” Unschooling now generally refers to a style of homeschooling, in which learning is not seperated from living, and children learn mainly by following their interests. Children learn best, he argued, not by being taught, but by being a part of the world, free to most interests them, by having their questions answered as they ask them, and by being treated with respect rather than condescension.
Gardner and Multiple Intelligences
Psychologist Howard Gardner argues that intelligence is not a single unitary property and proposes the existence of “multiple intelligences.” He identifies seven types of intelligence: linguistic, musical, logical-mathematical, spatial, bodily kinesthetic, interpersonal, and intrapersonal. Because each person has a different mix of these intelligences, learning is best tailored to each individual’s strengths, rather than emphasizing the linguistic and logical-mathematical approaches traditionally used in schools. A bodily kinesthetic learner, for instance, might grasp geometric concepts presented with hands-on manipulative far more easily than she would if they were presented in a more traditionally logical, narrative fashion. A teaching approach that recognizes a variety of learning styles might encourage many individuals now lost by conventional methods.
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