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First, we need to define our terms. Spiral learning is based on behaviorism, which says we are programmable machines and we need endless repetitions to master something. Spiral curriculums cover the same material year after year in ever-widening circles, with the anticipation that increased exposure will eventually lead to mastery of the basics. The number of topics covered is broad, but they never go deep. It is more of an exposure philosophy.
The mastery approach curriculum builds sequentially. This philosophy states that there is no need to move to the next step until the preceding one is mastered. Therefore, lessons may take many days or even weeks if necessary for students to master the facts. Fewer topics are covered. Pre-testing and post-testing are done to assure mastery. Both approaches have some validity as well as some drawbacks.
The way the brain works is that it attaches new information to something already known. The more ways information is attached in the brain, the better it is learned. Children need more than one exposure and more than one way to learn a topic, but repeated exposures to the same material are not enough for mastery.
Thomas E. Clark, the author of VideoText Algebra, defines the goal of arithmetic as finding an answer, but the goal of mathematics is solving a problem. We need to teach students to be problem solvers. Therefore, RightStart™ Mathematics introduces a large number of topics, but they are built sequentially for greater understanding. Students need to be challenged by many topics in order to see the interconnectedness in mathematics. For example, one of the goals of mathematics instruction is that students be fluid in their basic facts. So, students learn strategies for mastering the facts. They master them by playing games, which gives them a reason for learning the facts.
In RightStart™, students learn techniques for thinking mathematically. Dr. Cotter has systematically introduced principles of mathematics that lead students to self-discovery through well-designed lessons.
Understanding is stressed throughout. The primary manipulative is the Cotter Abacus. Necessary repetition is provided through math card games.
The RightStart™ program is complete in itself. It is not a supplement nor does it need other supplements. Math card games are interwoven into the RightStart lesson plans. The games are often used as a supplement for other math programs. RightStart™ also includes the other branches of mathematics required from early on, such as geometry, algebra, probability, and statistics.
The Cotter Abacus, grouped in 5s and 10s, is designed to teach adding, subtracting, multiplying, dividing, money, and other concepts. The second side emphasizes trading: that 10 ones are 1 ten, 10 tens are 100, and 10 hundreds are 1000. As with all good manipulatives, children use it less and less as they construct their mental models.
The Slavonic Abacus, used in the UK, Australia, and New Zealand, is the same as the front side of the Cotter Abacus, but it does not have a reverse side.
A more serious limitation of rods 6 through 10 is that they cannot be visualized, or seen in the mind because they are not grouped in fives. Try to imagine 8 blocks in a row without any grouping – virtually impossible. Now imagine 5 blocks as red and 3 as green – this you probably can do.
Also adding two rods does not immediately give the sum. If you have a blue 8-rod and a green 7-rod, the answer will not be obvious. The child will have to find the yellow 10-rod and the orange 5-rod to come up with the answer. Some children, rather than learning that 8 + 7 = 15, remember blue + green = yellow + orange. That’s weird math and very strange art!
This idea of grouping in 5s and 10s has been around for a long time. The Romans grouped in fives (8 as VIII), and composers used two groups of five lines for writing music. Money and clocks group in fives too.
The purpose of a manipulative is not only to see the concept but to help the learner construct a mental model, for example, to learn the facts.
Any concept that can be taught with colored rods can be taught with the Cotter Abacus without the bother of little pieces.
The first edition RightStart™ Mathematics, which is still available, is packaged in Starter Kits and Add-On Kits to ensure the needed materials are available. Not only will the lesson book and worksheets be included, but the Cotter Abacus, card decks, Place Value Cards, and all the other materials will all be there for you.
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If you are using the first edition, once RS1 Level E is completed, we recommend you proceed to RS2 Level F. Once those lessons are complete, you will continue on to RS2 Level G and RS2 Level H.
Understanding mathematics is of prime importance. Since the vast majority of middle school students are visual learners, approaching mathematics through geometry gives the student an excellent way to understand and remember concepts. The hands-on activities often create deeper learning. For example, to find the area of a triangle, the student must first construct the altitude and then measure it.
Much of the work is done with a drawing board, T-square, 30-60 triangle, 45 triangle, a template for circles, and goniometer (device for measuring angles). Constructions with these tools are simpler than the standard Euclid constructions. It is interesting to note that CAD (computer-aided design) software is based on the drawing board and tools.
Levels G and H incorporate other branches of mathematics, including arithmetic, algebra, and trigonometry. Some lessons have an art flavor, for example, constructing Gothic arches. Other lessons have a scientific background, sine waves, and angles of incidence and reflection; or a technological background, creating a design for car wheels. Still other lessons are purely mathematical, Napoleon’s theorem and Archimedes stomaching. The history of mathematics is woven throughout the lessons. Several recent discoveries are discussed to give the student the perspective that mathematics is a growing discipline.
Good study habits are encouraged through asking the student to read the lesson before, during, and following the worksheets. Learning to read a math textbook is a necessary skill for success in advanced math classes. Learning to follow directions is a necessary skill for studying and everyday life. Occasionally, an activity or lesson refers to previous work making it necessary for the student to keep all work organized. The student is asked to maintain a list of new terms.
RS2 Levels G and H are written with several goals for the student:
- to use mathematics previously learned,
- to learn to read math texts,
- to lay a good foundation for more advanced mathematics,
- to discover mathematics everywhere, and
- to enjoy mathematics.
For the child using the second edition and is new to the program, review lessons are included in the beginning of each level B through F, so transition lessons are not needed.