Education and demonstration – Mathematics – Arithmetic
Reexamination Certificate
1999-07-13
2001-03-27
Ricci, John A. (Department: 3712)
Education and demonstration
Mathematics
Arithmetic
C434S208000, C434S209000
Reexamination Certificate
active
06206701
ABSTRACT:
BACKGROUND
“The ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value . . . appears so simple to us now that we ignore its true merits. But its very simplicity puts our arithmetic in the first rank of useful inventions . . . remember, it escaped the genius of Archimedes and Apollonius.”—Laplace
Fundamental mathematical concepts are very difficult to grasp. Although most adults use numbers and mathematics daily in performing activities, the underlying concepts are difficult to learn. Children often require several years to master the premise of recognizing that numbers represent a numerical quantity for a group of real world objects. Further, these abstract numerical quantities can be added and subtracted, which correspond to the number of real objects represented by the numerical quantity.
The grouping of numbers into fields such as hundreds, tens and ones is an abstraction within itself. In the decimal representation, numerical quantities are grouped into sets of singles, tens and hundreds units (and continues into thousands, etc.), allowing any size numerical quantity to be represented with a number. The idea that ten ones is the same as one ten is fairly straightforward., but changes resulting from addition and multiplication can be troublesome. Children must understand that adding single units can affect the tens or even hundreds units of a number.
Subtraction requires an even greater abstraction for children to master. When performing subtraction on two numbers, if the second number has a unit place larger than the first number, the child must “borrow” from the next higher unit of the first number to obtain enough units to perform the subtraction. This concept of borrowing from a higher unit is difficult to grasp. Schools often teach the borrowing process by rote, without allowing the students to truly understand what is going on.
Several teaching blocks and rod systems have been widely used. These sometimes are used for demonstration, but when children use the block systems on their own, there is no feature of the blocks or rods to show a correct solution to a problem. The child can group the blocks or rods in any fashion, with no indication that any particular grouping is better or useful.
Accordingly, what is needed is a system or apparatus allowing children and others to visualize the process of quantifying a set of objects, and once quantified, to manipulate that quantity with various mathematical techniques such as addition and subtraction. The apparatus should also be visually stimulating and exciting to help maintain attention of the users. The apparatus should be fool proof, and allow users to repeat any operation many times and always perform the same steps to get the result, in effect, be self-educating.
SUMMARY
The present invention solves the above problems using a plurality of block elements which are stacked together to produce a numeric quantity. The front and back of each block element can include dimples in the form of indentations allowing the block elements to stack together and remain cohesive as a unit.
Once stacked, the block elements are inserted into a block element container which will organize and hold a predetermined number of stacked block elements. The block element container has two identical portions which will close and lock only when the predetermined number of block elements are contained within. The block element container includes indentations which align and lock with the indentations and protrusions on the stack of block elements.
Further, at least one large container for block element containers can hold several block element containers for organizing and holding a predetermined number of stacked block element containers. The design of the container for block element containers is similar to the design of the block element containers.
In the preferred embodiment, a block element container will hold ten stacked block elements, and a large container will hold ten stacked block element containers. Therefore, this system represents the decimal (Base 10) counting system. Other counting systems such as Base 8 or 12 are possible, and only require different size containers.
The present invention also includes counting devices for providing a visual display of the number of block elements. Any of the block elements or containers can be inserted into the counting devices. In one embodiment, the counting devices has space for nine such elements (for example, block elements). When a tenth block is inserted, the counting device will produce a signal to the user that there are ten such elements. Therefore, the elements should be removed, stacked together inserted into a container, and moved up to the next magnitude unit. For example, if ten block elements are collected during an addition, the singles counting device will signal the user to remove the block elements, insert them into a block element container, and insert the block element container into the ten's counting device. This reinforces the concept of carrying “overflow” to the next magnitude counter. The counting devices can use several different means for determining the number of elements present. This can include an electronic sensing device, a weighing device, or a balancing device.
In another embodiment, there are a plurality of first block elements, each of the first block elements having a dimension of 1×R×R
2
; a plurality of second larger block elements, having a dimension of R×R
2
×R
3
; and at least one third larger block element, having a dimension of R
2
×R
3
×R
4
; wherein R is the third root of a predetermined number.
The present invention can be implemented using physical block elements and containers. Alternatively, the block elements can be simulated on a computer display system using images of objects which are manipulated using a pointing device or keyboard.
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Dike Bronstein, Roberts & Cushman LLP
Neuner George W.
Ricci John A.
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