WO1990004243A1

WO1990004243A1 - Mathematical aid

Info

Publication number: WO1990004243A1
Application number: PCT/AU1989/000435
Authority: WO
Inventors: William Angus Davidson; Silvia Davidson; John Dumergue Charters
Original assignee: William Angus Davidson; Silvia Davidson; John Dumergue Charters
Priority date: 1988-10-06
Filing date: 1989-10-05
Publication date: 1990-04-19

Abstract

A mathematical aid which represents relationships between numbers in a physical form so as to provide assistance in teaching or learning mathematic concepts. The mathematical aid comprises a series of symbols (A) each representing a number in a mathematically ordered sequence. The relationship between the numbers is represented by physical properties and each symbol has the physical properties relevant to the number it represents. In this way, the relationships between the numbers in the sequence can be determined by examination of the physical properties of the symbols representing the numbers. The physical properties may include shape colouring and surface texture applied to the symbols. The symbols may be of a three-dimensional form such as beads (A) strung together on string B. Alternatively the symbols may be of a two-dimensional form such as pictorial symbols on a chart.

Description

LU__l_I_____i MATHEMATICAL AID

FIELD OF INVENTION

This invention relates to a mathematical aid which represents relationships between numbers in a physical form.

BACKGROUND OF THE INVENTION

When numeracy is being taught, it is beneficial to represent relationships between numbers in a physical form so as to facilitate an understanding of basic mathematical concepts such as addition, subtraction, multiplication, division, mathematical progression and factorisation.

To a limited extent these relationships can be represented using counting blocks, but the process can become unwieldy where large numbers are involved. Furthermore, counting blocks have little appeal to many children and so use of such blocks does not generate an interest in learning mathematical concepts in the children. As a result of these deficiencies counting blocks are not altogether useful in teaching relationships between numbers. Similar deficiencies arise with Cuisenaire rods, which are coloured wooden blocks graduated in length to represent various numbers.

SUMMARY QF THE INVENTION

The present invention seeks to provide a novel and useful mathematical aid which represents relationships between numbers in a physical form so as to provide assistance in teaching or learning mathematic concepts. -

In one form the invention resides in a mathematical aid comprising a series of symbols each representing a number in a mathematically ordered sequence, the relationship between the numbers being represented by physical properties and each symbol having the physical properties relevant to the number it represents.

In this way, the relationship between the numbers in said sequence can be determined by examination of the physical properties of the symbols representing the numbers.

For such purpose, the symbols may be of various shapes which represent certain relationships between the numbers. For example, numbers which have a particular relationship in common may be represented by symbols which have a common feature of shape, such as a common cross-sectional shape, common profile or common elongation. Other relationships between the numbers may be represented by particular surface structures on the symbols. Relationships between the numbers may also be represented by particular colouring on the symbols, although this of course only accommodates visual examination.

As an example, a particular number and subsequent numbers which are multiples of that number can be represented by symbols having a common feature of shape. For instance, odd numbers may be represented by symbols which have one particular colouring and/or cross-sectional shape (such as circular) and even numbers may be represented by symbols which have a colouring or cross-sectional shape (such as rectangular) discernibly different from the symbols representing odd numbers. Furthermore, the number "3" and numbers which are a multiple of "3" may be represented by a particular surface structure on the symbols. The number "5" and numbers which are a multiple of "5" may be represented by symbols which are elongated in one direction. The number "7" and numbers which are a multiple of "7" may be represented by symbols which are elongated in another direction.

In one arrangement, the symbols are three-dimensional to permit both visual and tactile examination.

Conveniently, the three-dimensional symbols are in the form of beads strung together. For the purpose of appealing to children, the beads may be brightly coloured.

In another arrangement, the symbols may be two dimensional such as pictorial symbols which permit visual examination. Numbers which have a particular relationship in common may be represented by pictorial symbols which have some visual feature in common. The pictorial symbols may be modified visually in particular ways to represent other relationships which exist between numbers represented by the pictorial symbols. For the purposes of appealing to children, the pictorial symbols may be in the form of animal characters.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be better understood by reference to the following description of several specific embodiments thereof as shown in the accompanying drawings, in which:

Figure 1 is a schematic view of a mathematical aid according to a first embodiment which uses symbols of three-dimensional form;

Figure 2 is a plan view of a mathematical aid according to a second embodiment which uses symbols of two-dimensional form while having three- dimensional computing elements associated with the symbols;

Figure 3 is a perspective view of the mathematical aid of Figure 2, without the two-dimensional symbols being shown;

Figure 4 is a schematic view of a mathematical aid according to a third embodiment which uses symbols of two-dimensional form; and

Figure 5 is a schematic view of a chart which forms part of a mathematical aid according to a fourth embodiment which is a variation of the third embodiment.

DESCRIPTION OF PREFERRED EMBODIMENTS

The embodiments are directed to mathematical aids which have been devised particularly to appeal to children while they are being taught numeracy. The mathematical aids according to the various embodiments represent the relationship between numbers in a physical form for the purpose of assisting children in learning and understanding such relationships. In this way, children can relate to mathematical concepts such as addition (particularly adding in lots), subtraction, multiplication, division, mathematical progression and factorisation. The embodiments have been structured at various levels as children proceed through their education in numeracy, as will be explained later.

The embodiment shown in Figure 1 of the drawings is directed to a mathematical aid in which the relationship between numbers is represented in a physical form by symbols A which are three-dimensional. The symbols A are in the form of beads which are strung together on string B which is elastic in character. Each symbol A on the string B represents a number in a mathematically ordered sequence. For identification purposes in the drawings, the number represented by each symbol has been placed on the respective symbol, although such numbers would not normally appear on the actual teaching aid.

In this embodiment the symbols represent numbers "1" to "30" inclusive and a spacer C separates the two symbols which represent numbers "1" and "30". For identification purposes, the symbol representing number "1" is larger in diameter than the remaining symbols. The elastic nature of the string B allows neighbouring symbols to be moved apart when the mathematical aid is being handled for tactile examination.

The mathematical relationships between the various numbers are represented by physical properties given to the beads, .as will now be explained. The beads which represent odd numbers are of one colour (such as green) and the beads which represent even numbers are of another colour (such as yellow) . In this way, odd and even numbers can be readily distinguished from each other upon visual examination of the mathematical aid. To demonstrate this difference in colour in the drawings, beads which represent odd numbers are shown with surface hatching and beads which represent even numbers are shown without hatching. In addition to the colour differences which allows odd and even numbers to be distinguished, the beads representing those numbers also have a feature of shape which provides for discrimination between odd and even numbers. This is achieved in this embodiment by providing the beads with two different cross sections. Specifically, the beads which represent odd numbers are of a generally circular cross-section in at least one direction, and beads which represent even numbers are of generally rectangular cross-section in at least one direction.

Other mathematical relationships are represented by particular surface structures on the beads. More particularly, the number "3" and subsequent numbers which are a multiple of "3" are represented by a discernible surface texture on the beads such as a combination of protrusions D and recesses E. This surface structure is arranged so that its presence can be determined upon both visual and tactile examination of the beads.

The number "5" and subsequent numbers which are a multiple of "5" are represented by a common feature of shape which can be 'discerned from features of shape existing in other beads. In this embodiment, the feature of shape is in the form of elongation extending across the length of the string B. In this regard, it can be seen from the drawings, that the numbers "5", "10", "15" and subsequent numbers of which "5" is a divisor are represented by beads which are elongated in the direction crosswise of the string B.

Beads which represent the number "7" and subsequent numbers which are multiples of "7" are also distinguished from other beads by a feature of shape which in this embodiment is elongation in a direction along the length of the string. In this regard it can be seen from the drawing that the numbers "7", "14", "21" and subsequent numbers of which "7" is a divisor are represented by beads which are elongated in the direction of the string. Where beads represent numbers which have several divisors, such beads incorporate the distinguishing features given to the divisors. For example, the bead representing the number "6" is of a colour which represents an even number and is also rectangular in cross-section for the same reason. Additionally, the bead has a particular surface structure which indicates that .it is a multiple of the number "3". As another example, the bead which represents the number "15" is of a colour which represents that it is an odd number and is also generally circular in cross- section for the same reason. Additionally, the bead is elongated in the direction across the string to represent that it is a multiple of the number "5" and has a particular surface structure to represent that it is also^" a multiple of the number "3".

The beads can be used to assist in teaching and understanding mathematical concepts such as addition, subtraction, multiplication, division, mathematical progression and factorisation. As an example of adding in lots, a child can be given the task of counting in groups of three from the number "1" to the number "15". In carrying out this task, the child would first identify the bead representing number "1". The child would then identify each successive bead having a discernible surface texture in the knowledge that it represents either the number "3" or a subsequent number which is a multiple of "3". In this way, the child could perform the task of counting to 15 in the manner prescribed. This teaching process can establish patterns which introduce children to the processes of multiplication and division and to the concept of mathematical progression. In a similar way, a child can perform the mathematical operation of subtraction.

As multiplication is an extension of counting in groups, the concept of multiplication can be introduced to a child by way of the teaching aid. For example, if the child is given the task of multiplying the number "3" by the number "5", the child can approach such task by locating the fifth bead representing the number "3" or a multiple thereof in a sequence commencing from the bead which represents the number "1".

Similarly, the child can be introduced to the mathematical operation of division. For example, if a child is given the task of dividing the number "15" by the number "5", the child can approach the task by counting the number of beads which represent the number "5" or a multiple thereof in the series of beads from number "1" to number "15". The child can also be introduced to the concept of a remainder in the process of division. For example the child can be presented with the task of dividing the number "11" by the number "3" . The child would approach the task in the way described in the immediately preceding example and would determine that there are three groups of number "3" in number "11" with two beads remaining. The remaining two beads represent a remainder of "2".

The child can also be introduced to the concept of factorisation which will involve identifying the particular divisors in a given number. This can be done by inspecting the particular bead representing the given number.

A marker (not shown) can be used in association with the string of beads so that any particular location in the sequence of numbers can be marked and used as a reference from which further computations can be carried out.

The mathematical aid according to the first embodiment is particularly suitable for introducing children to mathematical concepts as it involves both visual and tactile examination. Additionally the beads can be handled and used as counting elements in much the same way as fingers.

As a child develops, he or she can grasp mathematical concepts without as much reliance on tactile examination and so the child may advance to the mathematical aid according to the second embodiment which is shown in Figures 2 and 3.

The mathematical aid of the second embodiment is in the form of a ruler F, one edge G of which is straight and the other edge H of which is provided with protrusions I which constitute counting elements. Thus a child can count along the counting elements in much the same way as he or she would do on fingers.

Each counting element is associated with a number which is represented by a symbol having physical properties relevant to the particular number. The symbol is of two dimensional form such as pictorial symbols representing animal characters. Numbers which have a particular relationship in common are represented by pictorial symbols which have some visual feature in common.

The animal characters are intended to appeal to children and for such reason characters in the form of a Koala, Kangaroo and Platypus are chosen. In this embodiment, the symbols represent a numerical sequence from "1" to "20" in which all numbers are represented by a Koala character, with the exception of the number "5" and subsequent numbers which are a multiple of "5" which are represented by the Kangaroo character and the number "7" and numbers which are a multiple of "7" which are represented by the Platypus character.

As with the first embodiment, the symbols are coloured to distinguish odd numbers and even numbers. In the drawings for this embodiment, the animal characters representing odd numbers are shown with a dark image and animal characters representing even numbers are shown in a light image.

Characters which represent the number "3" and multiples of that number are shown with surface hair which corresponds somewhat to the discernible texture structure on symbols of the- first embodiment. The character of a Kangaroo has been chosen to represent the number "5" and multiples of that number, as its height corresponds to the concept of elongation in a direction transverse to the sequence of numbers and so corresponds somewhat to the feature of elongation crosswise of the string in the teaching aid of the first embodiment.

Similarly, the character of the Platypus has been chosen because its length represents elongation in the direction of the sequence of numbers and so corresponds somewhat to the feature of elongation in the direction of the string in the teaching aid of the first embodiment.

By maintaining the concept of elongation crosswise of, and along, the sequence of numbers, it is easier for a child when moving between the teaching aids of the first and second embodiments. As with the first embodiment, where animal characters in this embodiment represent numbers which have several divisors, such characters incorporate the distinguishing features given to the divisors. For example, the symbol representing the number "6" is a Koala character having a light image which represents that it is an even number. The Koala character has surface hair which indicates that it is a multiple of the number "3". As another example, the symbol which represents the number "15" is of a dark image which indicates that it is an odd number and is in the form of a Kangaroo character which indicates that it is a multiple of the number "5". Additionally, the Kangaroo character has surface hair which indicates that it is also a multiple of the number "3".

The teaching aid according to the second embodiment can be used for teaching purposes in much the same way as the teaching aid according to the first embodiment. There are, however, several additional features present in the teaching aid according to the second embodiment which are useful for teaching and instruction purposes. Specifically, each animal character which either represents the number "4" or a multiple of that number is further distinguished by the presence of a shoe on one of its feet. Additionally, each animal character which represents the number "8" or a multiple of that number not only has a shoe on one foot (because it is a multiple of the number "4") but also has a shoe on the other foot. Additionally, eacti animal character which represents the number "9" or a multiple of that number is further distinguished by the presence of a bow tie around its neck.

The additional features of shoes and bow ties demonstrate further relationships between the numbers and so facilitate a higher level of teaching. A still further benefit of the teaching aid according to the second embodiment is that the protrusions I which constitute counting elements are actually marked with the number to which the protrusion and associated animal character relate. This assists children in relating the particular characters to numerals.

As a child develops still further, he or she can grasp mathematical concepts at a still higher level without as much reliance on tactile examination and so the child may advance to the mathematical aid according to the third embodiment which is shown in Figure 4.

The mathematical aid of the third embodiment is in the form of a chart J which carries a legend K and a grid L comprising a plurality of rows and columns of squares.

Within each square of the grid there is an animal character of the same form as the animal characters used in the mathematical aid of the second embodiment. The mathematical aid according to the third embodiment does accommodate numbers "1" to "100" and provides additional information about relationships between the numbers. The relationships are set out in the legend K which appears on the chart beside the grid.

In this embodiment, prime numbers are identified by the presence of a hat on each animal character representing a prime number.

Numbers which are multiples of both "5" and "7" are represented by the symbol of a Kangaroo with a Platypus on its back.

The presence of an ice cream cone represents the number "2" or a multiple of that number. More particularly, an ice cream cone having two scoops represents a multiple of the number "2", an ice cream cone having four scoops represents a multiple of the number "4" and an ice cream cone having eight scoops represents a multiple of the number "8" .

The pictorial symbol of a balloon being held by an animal character represents either the number "3" or a multiple of that number. A balloon symbol having three sections represents a multiple of the number "9". Thus the number "63" is represented by the symbol of a Platypus with hair carrying a balloon having three sections. The Platypus is shaded to indicate that it is an odd number and has no hat thereby indicating that it is not a prime number. The Platypus character represents that the number is a multiple of "7" and the presence of hair on the Platypus represents that it is also a multiple of the number "3". The presence of the balloon having three sections indicates that the number is a multiple of the number "9".

The pictorial symbol of a star attached to the ear of a character represents a number which is either "6" or a multiple of "6". Thus the symbol representing the number "54" is in the form of a Koala with hair carrying an ice cream and a balloon, and having a star attached to one of its ears. The Koala is not shaded so indicating that it is an even number. The presence of hair on the Koala character indicates that it is a multiple of "3" and the presence of an ice cream cone with two scoops indicates that it is a multiple of "2" but not a multiple of "4" or "8". The presence of a balloon with three sections indicates that the number is a multiple of "9". Finally, the presence of a star attached to one ear of the Koala indicates that the number is a multiple of "6". The teaching aid according to the third embodiment is used in a somewhat similar way to the teaching aids according to the first and second embodiments.

The teaching aid according to the fourth embodiment is shown in Figure 5 and is a variation of the teaching aid according to the third embodiment. With a teaching aid according to the fourth embodiment, the grid L is marked with the numbers "1" to "100" and one hundred tablets (not shown) are provided one corresponding to each number. Each tablet bears a symbol representative of the particular number on the grid to which it relates and a child is given the task of placing each tablet in its correct position on the grid.

From the foregoing it is evident that the present invention provides a novel and useful way of teaching mathematical concepts to children and provides a way in which ^• children can instruct themselves in relation to mathematical relationships. It is believed that the teaching aid according to the various embodiments would be particularly appealing to children as they progress through their education in numeracy and would create an interest both in the relationship between numbers and in mathematical concepts.

While the invention has been described with reference to several specific embodiments, it should be appreciated that it is not limited thereto and various modifications and changes may be made without departing from the scope of the invention. In particular, it should be appreciated that a teaching aid according to the invention may take other forms such as an electronic device arranged to display various symbols representing relationships between characters.

Claims

C_ A1E£

1. A mathematical aid comprising a series of symbols each representing a number in a mathematically ordered sequence, the relationship between the numbers being represented by physical properties and each symbol having the physical properties relevant to the number it represents.

2. A mathematical aid according to claim 1 wherein numbers have a particular relationship in common are represented by symbols which have a common feature of shape.

3. A mathematical aid according to claim 2 wherein said common feature of shape comprises a cross-sectional shape.

4. A mathematical aid according to claim 2 wherein said common feature of shape comprises a common profile.

5. A mathematical aid according to claim 2 wherein said common feature of shape comprises elongation.

6. A mathematical aid according to any one of the preceding claims wherein a particular relationship between some of the numbers in said sequence is represented by a surface structure on the symbols representing said numbers.

7. A mathematical aid according to any one of the preceding claims wherein a particular relationship between some of the numbers in said sequence is represented by particular colouring on the symbols.

8. A mathematical aid according to any one of the preceding claims wherein said symbols are each of a three dimensional form.

9. A mathematical aid according to claim 8 wherein said three-dimensional symbols comprise beads strung together.

10. A mathematical aid according to claim 9 wherein said beads are strung together on a string which is elastic in character.

11. A mathematical aid according to claim 9 or 10 wherein said beads representing odd numbers are of one particular cross-sectional shape and said beads representing even numbers are of a cross-sectional shape discernably different to the cross-sectional shape of said beads representing odd numbers.

12. A mathematical aid according to any one of claims 9, 10 or 11 wherein a particular relationship between some of the numbers is represented by a discernible surface texture on the respective beads.

13. A mathematical aid according to any one of claims 9 to 12 wherein a particular relationship between some of the numbers is represented by the beads being elongated and disposed cross wise of the direction of the string.

14. A mathematical aid according to any one of claims 9 to 13 wherein a particular relationship between some of the numbers is represented by the beads being elongated and disposed in the direction of said string.

15. A mathematical aid according to any one of claims 1 to 7 wherein said symbols are of a two-dimensional form.

16. A mathematical aid according to claim 15 wherein numbers having a particular relationship in common are represented by symbols having a common profile.

17. A mathematical aid according to claim 15 or 16 wherein said two-dimensional symbols comprise animal characters.

18. A mathematical aid according to claim 17 wherein a particular relationship between some of said numbers in said sequence is represented by hair on said animal character.

19. A mathematical aid according to any one of claims 15 to 18 comprising a ruler to which said symbols are applied.

20. A mathematical aid according to claim 19 wherein said ruler includes protrusions one corresponding to each symbol- applied to the ruler, said protrusions comprising counting elements.

21. A mathematical aid according to any one of claims 15 to 18 comprising a chart to which said symbols are applied.

22. A mathematical aid substantially as herein described with reference to Figure 1 of the drawings.

23. A mathematical aid substantially as herein described with reference to Figures 2 and 3 of the drawings.

24. A mathematical aid substantially as herein described with reference to Figure 4 of the drawings.

25. A mathematical aid substantially as herein described with reference to Figure 5 of the drawings.