1. * GB780032 (A)
Description: GB780032 (A) ? 1957-07-31
Improvements in or relating to teaching apparatus
Description of GB780032 (A)
PATENT SPECIFICATION
Inventor: WALTER ALFRED GORDON BASS Date of filing Complete
Soecification April 29, 1955.
Application Date April 13, 1954.
(Patent of Addition to No. 713,731 dated Dec 6, 1950).
Complete Specification Published July 31, 1957.
Index ur acceptance:-Class 146(2), G(5B: 10).
International Classification:-G09b.
COMPLETE SPECIFICATION
Improvements in or relating to Teaching Apparatus We, COUNTERPLAY
LIMITED, a British Company, of Provincial House, 98, Cannon Street,
London, E.C.4, do hereby declare the invention, for which we pray that
a patent may be granted to us, and the method by which it is to be
performed, to be particularly described in and by the following
statement: -
This invention relates to teaching apparatus for elementary
instruction in counting and the principles of arithmetic and like
operations involving number. It is an improvement in or modification
of the apparatus described and claimed in the Complete Specification
of
British Patent No. 713,731.
In the course of instructing backward adults in the principles of
counting as a basis for all mathematical operations involving number,
as set out in the said prior specification, certain modifications of
the simpler forms of equipment therein described have been found
desirable. Thus, for example, the prior specification describes a set
of articles comprising a rigid carrier having a rectilinear scale
thereon calibrated from zero, one or more separate rectilinear scale
elements each calibrated from zero and each adapted for alternative
attachment to the carrier so that the zero calibration thereof is

2. always located at the same position on the carrrier, and index means
adapted to be selectively located on the carrier in registration with
a calibration mark of at least one of the said rectilinear scales.
The set of articles referred to above has for the index means a
plurality of square counters of side length equal to the spacing
between calibration marks on at least one of the scales, the scales
themselves bearing numerical symbols in the cardinal series against
the appropriate calibration marks. Where, however, the pupils have no
fundamental conception of the meaning of number, it has now been found
more satisfactory to omit the numerical symbols, or mark them only in
outline (as by means of dots), and to instruct the pupils to insert or
fill in, on the "blank" scales the appropriate numerical symbol as
each counter is moved - [trice 3.. 6d.] into register with the scale
at the next vacant position, whereby each successive counter is seen
to be in one-to-one correspondence with a respective cardinal number.
In this way, the)u serial nature of the numerical symbols is directly
associated in the pupil's mind with a visual and tangible
representation, by means of the counters, of the quantitative value of
each successive symbol. 55 Furthermore, it has been found of value to
mark each counter with a directional or other representation familiar
to the pupil such as an arrow pointing at right angles towards one
edge of the counter, the significance of the arrow nO becoming
apparent in differentiating between the operations of say, addition
and subtraction, as will be described more fully below.
The present improvement in or modification of the teaching apparatus
described and 65 claimed in the said prior specification resides
primarily in providing for insertion on the scale by the pupil of the
appropriate numerical values between successive parallel calibration
marks which run perpendicular to the boundary zone against which the
counters are slid, and divide the working surface of the generally
rectangular flat rigid carrier. The calibration marks are spaced by
the width of a counter, for which they constitute guides for its
correct,t movement into register with the scale, whilst the counters
themselves carry a representation of a familiar article, such as an
arrow or a bottle, to aid in identifying directional motion of the
counter-i.e. in the direction of the so arrowhead, or in the direction
for standing the bottle on its base.
Preferably, the side of the carrier up to which the numerical scale
extends in the direction away from zero is irregularly shaped to 85
suggest fracture of the carrier from a larger piece and so afford the
impression that the numerical series of identification symbols is
intended to extend to infinity.
Whilst the scope of the invention is defined 90 by the appended
claims, three forms of apparatus made in accordance therewith will now

3. be 780,032 No. 10846/54.
particularly described, by way of illustration only, with reference to
the accompanying drawings in which: Fig. 1 is a perspective view
partly broken away, of a simple form of the invention, and Figs. 2 and
3 are similar views of modified forms thereof.
In the first form shown in Fig. 1, the carrier is constituted by a
flat substantially rectangular board 1, having at least two straight
edges 2, 3 along which are demarcated respective boundary zones 4, 5
having their inner edges straight and at right angles to each other.
These zones, into which counters 6 described below are not to be
moved, are constituted by low flat ledges formed or secured on the top
surface of the carrier. They are of equal height and width, which may,
for example, be the same as the thickness and width of each one of a
set of flat square counters 6 to be used in conjunction with the
carrier. The longer ledge 4 terminates at its further end 4a in an
edge of irregular contour which is continued across the board at this
end and is intended to c; nvey to the pupil the impression that the
ledge 4 and board 1 are broken off at this point and would normally
extend an indefinite distance beyond the break.
The flat top surface of the ledge is finished-as by the application of
a matt paint or in any convenient manner-to take chalk or pencil
markings, and on this surface the pupil writes the cardinal numbers
serially to form a scale the shape of the respective numerals being
marked by dots through which the pupil can draw the full outline. The
purpose of the broken end of the ledge 4a is to indicate that there is
no uprer limit to the number series.
The flat surface of the board 1 which is bounded by the ledges 4, 5
and the other two free edges constitutes the working surface 7 and is
ruled in straight lines 8 which may, if desired, be. continued for a
short distance at 8a on the flat top surface of the longer ledge 4.
These lines 8, 8a are perpendicular to the longer ledge 4 and equally
spaced by a distance equal to the width of a counter 6. Counters can
thus be placed on the working surface 7 of the board 1 so as to fit
exactly between any pair of thes lines 8. It may be desirable in some
cases to emphasise these lines by grooving the working surface 7 of
the board or by forming narrow ribs which run into the longer ledge 4
and act as positive guides for the placing and subsequent sliding of
counters in the manner described below.
The area 9 of the top surface of the longer ledge 4 which is also
common to that of the shorter ledge 5 (i.e. the area of intersection
of the two ledges) may bear permanently in full lines the
identification symbol " 0", the line of the inward edge of the shorter
ledge 5 being continued partly across the surface of the longer ledge
4, either as an indelible contrastingly coloured mark or as a groove

4. or like structural formation, to represent the zero calibration of a
scale. By so locating the zero symbol, the location of a counter in
exact registration therewith is positively prevented. No further
numerical symbols are permanently applied or 70 engraved on the longer
ledge, except by dots.
The length of the shorter ledge 5 is at least equal to three times the
length of side of a counter 6, and its free end may also have an
irregular contour (not shown in Fig. 1) suggesting the breaking off of
the ledge from a lkner criginal fermation extending for an indefinite
distance. The corresponding edge of the working surface 7 of the board
1 may then repeat this irregularity of contour similar to 80 the
irregular edge 4a, so that the pupil is always given the visual
impression that he is working in one small corner of an otherwise
limitless field.
This feature of the irregular edge 4a or 85 edges, of the board 1 has
an important psychological effect in countering any impressionwhich is
very readily acquired otherwise-in the pupil's mind that the
mathematical processes he is taught to carry out on the board 90 only
apply up to the magnitude of the maximum numbers of counters which the
board can accommodate. For a similar reason, it is preferred to
continue the longer ledge 4 at least as far as the space for the
numeral 11, which 9s is not normally thought of as an end of the
cardinal series of numbers.
In an alternative form of board 1 shown in Fig. 2, the longer ledge 4
is formed by a flap 41 which is hinged along the corresponding lob
edge of the working surface 7 of the board 1 and can be swung into a
generally upright position as shown, when the board is to be used.
Suitable releasable struts or other support members (not shown) may be
provided 10i to hold the flap 41 in its erected position.
In either form of board, the flat top surface of the longer ledge may
be provided with clips, or like retaining devices (not shown) to
receive and hold numbered cards or plates (indicated 11C at 42 in Fig.
2) which perform the same function as the hand written symbols
referred to above.
In a third form of board shown in Fig. 3 the longer boundary zone 4,
into which l1i counters 6 are not to be moved, is defined by a
straight line, groove, or narrow rib 43 along its inward edge and by
an upstanding parallel ridge 44 along its outer edge, the spacing
between the inner and outer edges being such 12( as to receive a
series of plates, blocks, cards or the like 45 each of an equal width
to the width of a counter 6, and preferably of other dimensions
suitably equal to the other dimensions of a counter, each such plate,
block or card 45 12' bearing a respective numerical identification
symbol such as a numeral in the cardinal series.

5. When all the said plates or the like 45 are properly positioned in the
longer boundary zone 4, they present a ledge similar to the 13(
780,032 ledge 4, whereupon the pupil can now see a scale bearing a
continuous series of cardinal numbers together with a visual
representation of the quantity which the total number signifies.
Furthermore, his successive adtions of 70 sliding counters on the
working surface 7 of the board in the direction of an arrow 11 up to a
datum line has inculcated into his mind the essential one-to-one
correspondence between a quantity and its numerical symbol. 75 This
concept of one-to-one correspondence has been found in practice to be
of vital importance in teaching the correct approach to the theory of
number, on which are based all other arithmetical processes.
Additionally, the pupil so acquires a sense of direction associated
with number which becomes of importance when the processes of addition
and subtraction are taught, and later still when the pupil begins to
study vectorial problems. 85 The counting process described above has
a further particular advantage in that it emphasises, by virtue of
requiring the pupil himself to enter the cardinal number symbols in
their appropriate places, the meaning of each 90 individual number as
part of the unbroken sequence and related directly and consecutively
to the preceding numbers. No number is learnt in this way until the
preceding number is learnt, and all are seen to be inescapably of 95
value only in relation to zero, without which concept number is robbed
of reality.
Against the sequence of numbers or scale on the longer ledge 4 the
pupil learns to count in the following ways: - 100 a) in ones from any
starting point (including zero) this process being related to
Arithmetic by being expressed in terms of the + sign:
e.g. starting point two, add two, result four 2+2=4 105 b) in ones
from any starting point towards zero, this process being related to
Arithmetic by being expressed in terms of the - sign e.g.
starting point four, substract two, result two.
4-2=2 110 c) in equal groups starting from zero, this process being
related to Arithmetic by being expressed in terms of the sign "X":
e.g. in two groups of two there are four; written.: (2)X2=4 (The
numeral within the brackets 115 indicates the number in each group);
and read as "two multiplied by two equals four".
d) in equal groups from any starting point to zero, this process being
related to Arithmetic by being expressed in terms of the. sign: 120
e.g. there are two groups of two in four, written:4 -=2 (2) In the
teaching of addition a group of counters 6 corresponding in number to
the 125 starting point of the sum are placed on the board 1 and moved
up to the scale 4 with the arrows 11 pointing towards the scale. A
longer ledge of Fig. 1 and perform an identical function to be

6. described below. The term "ledge" as used hereinafter is deemed to
include all the above alternative constructions S or their mechanical
equivalent.
In all forms of board described above, the ledge 5 which has been
called the shorter ledge and which defines the other boundary zone may
be constructed similarly to the longer ledge 4, or may simply be
defined along its inner edge by a straight narrow rib serving as a
stop or guide against which a counter 6 may be positively located and
moved towards or away from the longer ledge 4.
S5 Each counter 6 for use with the board may have its upper surface
marked with a large spot from which extends an arrow 11 whose head
lies close to one edge of the counter. The spot and arrow 11 lie on a
perpendicular bisector of one pair of opposite sides of the counter.
Instead of an arrow 11, however, the counters may carry
representations of other familiar articles, such as bottles 13 (Fig.
1), which preferably figure in simple rhymes or zS songs involving
counting. The counters 6 of a set may be coloured, and may be divided
into groups of equal numbers which are differentiated by contrasting
colours.
In using any of the above-described boards or carriers, the theory of
counting in.the cardinal number series is taught by instructing a
pupil to place a counter 6 on the working surface 7 of the board 1,
arrow side up, against the shorter ledge 5 and between it and the
adjacent guide mark 8 with the arrow 11 pointing to the longer ledge
4, and then slide it against the shorter ledge 5 until it comes into
contact with the longer ledge 4. The pupil is first shown that this
counter cannot register with the zero symbol, and is then told that
this counter represents the quantity defined as " one ", and is told
to write or insert the numerical symbol " 1 " on the flat top surface
of the longer ledge 4 opposite the arrowhead.
The symbol " 1 " thus identifies the counter lying between the first
and second guide marks 8 on the carrier surface and against the longer
ledge 4. This done, the pupil is instructed to take another counter 6,
place it on the working surface 7 of the board 1 with the arrow 11
pointing in the same direction, and slide it up to the longer ledge 4
beside the first counter 6, between the second and third guide marks
8. He is then told that all the counters now on the board represent
the quantity defined as " two " and is told to write or insert the
symbol " 2 " on the flat top surface of the longer ledge 4 opposite
the arrowhead 12 on the second counter. The symbol " 2 " thus
identifies all the counters lying between the shorter ledge 5 and the
third guide mark 8 on the carrier, and against the longer ledge 4.
The process is repeated until no more counters 6 can be placed against
the longer 780,032 number of counters corresponding to the number to

7. be added are counted separately in a row behind the first and then
moved up to the scale 4 alongside those already there with their
arrows pointing towards the scale. The result of the addition is given
by the number of the last pairing between a counter 6 and a numeral of
the scale 4.
In teaching subtraction, the counters representing the subtractions or
subtrahend are reversed so that their arrows 11 point away from the
scale 4 and are then moved in the direction of their arrows away from
the scale.
The concept of sign is thus given a vectorial significance, further
enhanced by the physical movement of the counters in the opposite
direction (i.e. away from the scale) from that employed in addition.
In the teaching of multiplication by the modified apparatus according
to the invention the concept of groups of equal numbers of counters is
introduced. Groups are counted as groups, the shorter ledge 5 being
used as a guide for presenting the counters in the first group i.e.
the group which is in one-to-one correspondence with the numeral 1 on
the scale ledge 4. The shorter ledge 5 can be permanently or manually
calibrated for the purpose of counting the number in a group, as
indicated by the bracketed numerals in Figure 3.
This action introduces the aspect of counting from zero in a second
dimension, which is met again in the "second aspect" of division, as
noted below.
The first group referred to above is next used as the guide for
presenting the counters in the second and equal group, this group
being in ine-to-one correspondence with the numeral 2 on the scale
ledge 4. The second group is then used as the guide for the third
equal group, i.e. the one which will be in one-to-one correspondence
with the numeral 3 on the scale ledge 4.
The number of counters in so many equal groups is then counted as in
addition. Thus the product (3)X3 is represented by placing the
counters of the first group of three in oneto-one correspondence with
the numbers of the scale along the longer ledge 4, the last pair5s ing
between a counter and a numeral then being at 3. This group is
followed in like manner by the second group of three along the longer
edge 4 so that the last pairing between a counter and a scale numeral
is now at 6. Lastly, the third group of three is placed along the
longer ledge 4, and it is noted that the last pairing between a
counter and a scale numeral is at 9. Thus, the last pairing between a
counter of a group and the numerals of the scale tells the pupil how
many there are in so many groups of one, two, three, or more, e.g. in
threes:In one group of three-three In two groups of three-six In three
groups of three-nine Such results are expressed arithmetically as:(3)x
1=3 (3)x2=6 (3)x3=9 70 By the use of a bracket notation, the

8. fundamental distinction is made between the number in each group (the
multiplicand) and the number of groups (the multiplier). Furthermore,
the relation between multiplication and addition is made clear to the
pupil; i.e. multiplication is the addition of as many groups as the
numerical value of the multiplier, each group containing as many as
the numerical value of the multiplicand. 80 In the teaching of
division (in the first aspect -" Given a particular total or starting
point, how many equal groups, each of a lesser number, does that total
contain? ") counters 6 to the number of the starting point (dividend)
are 85 moved up to the scale 4 and then turned so that the arrows 11
point away from the scale (as in subtraction). They are counted in
ones, or twos, or threes or more (equal to the numbers in the
division) from the starting point to 90 zero and moved away from the
scale in equal groups, the number of groups being counted.
If, for example, the starting point (dividend) be six and the number
in each group (divisor) be two, it will be found that three groups are
so 95 counted. The result, expressed arithmetically, is:6 =3 (2) The
moving away of the groups from the scale i.e. the subtraction of so
many groups, 10( can be recorded in the form of " long" division as
follows:3 2/ 6 -6 0 Thus the relation between division and subtraction
is made clear to the pupil; i.e. division 105 (the first aspect
enunciated above) is the subtraction from the dividend of groups each
containing a number equal to the divisor until no more groups can be
subtracted, then counting the groups. 110 If after the subtraction of
so many equal groups some counters remain against the scale, these can
be counted and related to the associated group by being expressed as a
fractional part of the group. This is then added to the 115 number of
groups. Thus 780,032 780,032 1+.3; (or L) 4/ 7 -4 3 -3 Reference has
already been made above to the " second aspect " of division. This can
be illustrated by posing the problem-" Given a total number of
articles, and the number of equal groups into which they are to be
divided, how many will there be in each group? " Apparatus according
to the present invention solves this problem in the following way.
Let the given total number of counters be six, and the number of equal
groups into which they are to be divided be two. Then, referring to
Figure 3:One counter 6 is placed facing the numeral 1 on the longer
ledge 4, and a second counter 6 is placed facing the numeral 2 on this
ledge.
At this point in the division, there are thus two groups, of one
counter each, on the board, the number in each group being indicated
by writing " (1) " on the shorter ledge S opposite the first counter.
Next a third counter 6 is placed immediately behind the one already
facing the numeral 1 on the longer ledge 4, and a fourth counter is
placed immediately behind the counter facing the numeral 2. At this

9. point there are thus two groups of two on the board, the number in
each group being indicated by writing " (2) " on the shorter ledge 4
against the third counter.
Finally, a fifth counter is placed immediately behind those already
facing the numeral 1 on the ledge 4, and the sixth and last counter is
placed immediately behind those already facing the numeral 2 on the
ledge 4. There are thus seen to be three groups of two on the board,
the number in each group being indicated by writing " (3) " on the
shorter ledge against the fifth counter. In symbols, this result is
expressed as: 6 -(3) 2 The number in each group is counted from zero
in the second dimension, so that the exercise illustrates the concept
of two-dimensional counting from zero and serves to inculcate into the
pupil's mind the basis of graphs which follow at a later stage and as
a natural development from these elementary exercises.
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