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Using visual and other aids for working with children with learning
difficulties
Irena Budínová
When working with children who have difficulties in learning mathematics (brought
about e.g. by various causes connected to deficiencies of partial functions of mathematical
abilities or specific learning difficulties) we look for such teaching purposed that would
appeal to them and mobilize the activity of their minds in such a way that the work of their
hands would transform into the work of their brains.
Within the individualized teaching, we are using a set of simple aids, from which we
can form a portfolio for each pupil. We have had good experience with the pupils‘
participation on the production of the aids. The financial costs for the aids are minimal. The
pupils then work with the aids in ways fitting for them – if they are not certain and need some
support. or when they need concrete models. It shows that respecting multisensorial approach
in constituting the individual notions, because when we make use of as many senses as
possible, at least one of them is usually the constituent one for each pupil.
Some sets of visual aids:
1. Aids suitable for constituting the notion of a natural number and constituting the
operations with natural numbers: concrete objects – smaller toys (cars, dolls),
pebbles, cubes, sticks, PET-bottle lids of different colours and sizes, bigger buttons,
etc. We can also make good use of cards with dots from a blank card to a card with
twenty dots, a set of cubes, and a set of squares.
2. Aids suitable for understanding the positional decimal system:
Sets of ten straws: Sets of ten straws or sticks are suitable for modelling and
understanding two-digit numbers. The child sees ten units tied into a one block of ten.
If the child for example takes 42 to mean 24 and vice versa (right-left orientation
disability), the aid is very suitable.
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Figue 1. Sets of straws
Cards with numberswith more dgitis,e.g.:
50 400 7 000 60 000 300 000 8 000 000 20 000 000
When putting the cards one on the other, the pupils have a better idea of a number,
because they work with all the orders of magnitude (which are on the bottom card).
The use of cards for calculations is described below.
Money model: Paper models of money are suitable when the children understand the
value of the individual coins and banknotes. The so-called fairy-tale money are less
suitable, since the children’s attention is often drawn to the picture, and not to the
value of the banknote.
3. Abacus (twenty-bead, hundred-bead, and order of magnitude abacus).
Figure 2. Twenty bead abacus
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Figure 3 Hundred-beads abacus
Figure 4. Counting frame
4. One-hundred table
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
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The one-hundred table is used by pupils to understand two-digit numbers (reading, writing,
order, transfer between the tens) and also to identifiy multiples of numbers.
5. Orderof magnitude tables for writing and reading numbers.
MILLIONS HT TT THOU-
SANDS
HUN-
DREDS
TENS UNITS , tenths hun-
dredths
thou-
sandths
6. Frames for transfer between units, models of clocks an various measuring devices.
Frames are made of thick paper and are supplied with squares witht the individual
digits that can be laid on the second line of the frame.
Frame for transfer between length units
km m dm cm mm
0 0 0 0 0 0 0 0 0
km m dm cm mm
0 0 0 0 0 6 0 0 0
From the picture, the transfer is obvious: 6 m = 60 dm, 6 m = 600 cm, 6 m = 6 000 mm, 6 m
= 0,006 km.
Frame for the transfer of area units:
km2
ha a m2
dm2
cm2
mm2
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
km2
ha a m2
dm2
cm2
mm2
0 0 0 0 0 0 0 0 0 0 2 5 0 0 0 0 0 0
25 m2
= 2 500 dm2
= 250 000 cm2
, 25 m2
= 0,25 a = 0,0025 ha
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Frame for the transfer of volume units
.
34 dm3
= 34 000 cm3
34 dm3
= 34 000 000 mm3
34 dm3
= 0,034 m3
34 l = 340 dl = 3 400 cl = 34 000 ml.
7. Paper board games – loto, domino, memory, bingo, etc. (as a way to practice and
consolidation of operations with numbers).
8. Models of the fraction as a part of the whole (the rectangle and circle model are
especially useful, but there are also other models).
9. Wooden sticks for crafts of various lengths and colours to model straight line
sections (comparing length of the sections, triangle inequality etc..).
10. Models of geometrical shapes from coloured paper of a foil (triangles, squares,
rectangles, circles).
11. Solid nets made from items of daily use (boxes for tea, toothpaste, toothpicks, etc.).
12. Set of dice for Ludo.
13. Sets of solids (cuboid, cube, prism, roller, pyramid, cone, sphere).
14. Boxes and bottles for drinks in various volumes.
15. Models for the units of length and area (1 dm2
, 1 cm2
, 1 mm2
).
Although these aids are already at schools for demonstration, it is useful for the pupils to
have their own set of these aids so that they can use it in a suitable moment.
From the point of view of didactics, Montessori material is very suitable. We will show their
possible utilization in the following text.
Pupils‘ success in mathematics can be limited by their inability to count with natural
and decimal numbers. As these calculations are needed in mathematics throughout, this
handicap is very serious for the concerned pupils. Although some teachers are benevolent to
numerical mistakes, the fact that they are unable to perform the calculation correctly is usually
very frustrating for the pupils
m3
dm3
cm3
mm3
hl l dl cl ml
0 0 0 0 0 0 0 0 0 0 0 0
m3
dm3
cm3
mm3
hl l dl cl ml
0 0 0 0 3 4 0 0 0 0 0 0
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When counting with natural numbers, we often encounter mistakes related to written
operations, which are usually grounded in some specific disability – this may be dyslexia,
dysgraphia, or discalculia. We will now introduce the most common mistakes encountered in
written addition and subtraction of natural numbers and the possibility to introduce
Montessori material when teaching these pupils. This material can be helpful in overcoming
their problem.
Written addition of natural numbers
1. Not respecting the order of magnitude within the natural number: Pupils do not
write the numbers down with the corresponding orders of magnitude below each other,
but they write the highest order of each number below each other:
2. Not understanding the written form of the number in the decimal positional
system: Pupils make various mistakes in additions connected with counting up to 20.
They do not hold the „one“ and do not add it to the higher order of magnitude. For
example, when calculating 8+9=17, they are unable to add ten units of the lower order
of magnitude to the higher one..
The pupil writes the numbers as they come up in the calculation.
We encounter a number of other mistakes connected with the base 10 positiona
notation (decimal positional notation).
To eliminate the above-mentioned mistakes and also other ones, Montessori material Bank
may successfully be used. This material contains small beads, representing units, then rows of
ten beads, representing tens, ten rows of ten beads representing one hundred, and ten onehundred
tables piled on top of each other – thousands. The material further contains two sets
of cards – a small one for calculations and the big one for results. Children can work with it
already in the kindergarten age, where they use the Bank to exchange different quantities,
they e.g. exchange 10 tens for 1 hundred. In primary school, they use the Bank for addition,
subtraction, simple multiplication and division of natural numbers.
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Figure 5. Montessori material: the Bank
Let us show the work with the Bank step by step, using the example .
Step 1: We model numbers and
from the material. We find the corresponding
numbers from the set of small cards. W leave
the cards laid out next to each other,
Step 2: We manipulate the cards in sucha way
that the notation for that number in decimal
positional notation be visible.
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Step 3: If the children are comfortable with this,
we can turn the material into chaos. That means
that we mingle the material regardless of the
orders of magnitude.
Step 4: We arrange the material according to the
orders of magnitude. We have 7 units, 11 tens, 5
hundreds, and 3 thousands
Step 5: We exchange ten tens for one hundred,
and we can count from the units and write down
the result:
.
An experience: Two students of social pedagogy of the branch teacher’s assistent, Jana and
Linda worked with the Bank. One of them, Jana, had troubles with writing numbers when she
was a child. The other one, Linda, loved order. They first started to discuss whether to go
through the step of chaos, or not. Jana liked the idea, but Linda did not like it. They agreed to
leave out the step of chaos at first, and Linda was happy. They worked properly with the
material as well as with the number cards. Next time, they thus decided to go through chaos,
but Linda was very much looking forward to having everything organized. When they first
reached the stage of counting over 10, Jana said, „Shall we go to the bank?“ Later, I could see
that they could not wait for the point of going to the bank and when that happened, they
loudly rejoiced.
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Both students were excited about the work with the Bank. Jana said that the Bank would
probably have been helpful for her as a child, because she did not care for the orders of
magnitude and claimed that „zero does not matter“. Her father was quite upset about that, but
could not change anything, because for her, the orders of magnitude meant nothing.
Written subtraction of natural numbers
We will most often encounter the following mistakes in performing written subtraction:
1. Misunderstanding substraction when going over the base of 10: The pupil always
subtracts lesser number from the greater one; e.g. when calculating , they
calculate instead:
2. When subtracting, they do not add „one“:
They calculate the task as follows: 9 and how any are 13? 4, they do not hold
the one. The next step is 1 and how many is 4? 3.
We also encounter a number of other mistakes.
The Bank material can help the pupils also in this case. On one hand, it facilitates the
understanding of the order of magnitude and respecting them, and on the other, it does not
allow the pupils to turn around the digits so that they always subtract the smaller number from
the grater one.
An example of subtraction without going over the base of ten: We may present the kuds with
the following situation: there were 7 254 crowns in the Bank. A robber came and stole 3 142
crowns. The kids will then take 3 142 beads from the original amount. They will place these
beads on a tray, so that they are on the side (which evokes the situation when the money is in
fact somewhere else), but keep them so that they are able to check the task later. The pupils
gradually learn to write the calculation in a column. In the algorithm for written subtraction,
we start with units, and therefore we also start with units when using the material.
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An example of calculation involving transfers from one order of magnitude to another. In the
next phase, we deal with substraction involving one or more transfers from one order of
magnitude to another. From the original amount of 7 254 crowns, the robber stole 3 461
crowns. This example involves two transfers between orders of magnitude.From 4 units, the
pupils take 1 unit and put it on the tray. We cannot take 6 tens out of 5 tens, and thus the
pupils first have to exchange 1 hundred for 10 tens. Now they take 6 tens from the 15 tens,
and 9 tens remain. They are left with 1 hundred, from which 4 hundred cannot be taken. They
therefore exchange 1 thousand for 10 hundred, and take 4 hundred from 11 hundred; thus 7
hundred remain. Finally, they take 3 thousand from the 6 thousand and 3 thousand are left.
The final result is: . The pupils realise this way what it means to
„hold one“ and to which number we need to add 1.
Addition and subtraction of decimal numbers
1. Children add or subtract digits of different orders, , e.g.:
2. They do not respect the transfer between the orders, e.g.:
3. They always subtract the lesser number from the greater one, e.g.
4. They do not respect the rules of the positional decimal system, e.g.:
5. When performing written subtraction, they do not write the digits in the
corresponding columns, e.g.:
And many other mistakes may be encountered in mental and written calculations.
Montessori materia Table for decimal numbers is an aid that helps children eliminate
mistakes and understand the principle of calculating with decimal numbers. We will show
how it works on the example (the first number contains non-zero number of
hundreths, while the second one does not).
Step 1: We ask the pupil to enter the example onto the board. The
pupil can immediately see that in the order of units, we now have 5
cubes, in the order of tenths also 5 cubes, and in the order of
hundreths 4 cubes. This can now be modelled with the help of
cards with numbers.
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Step 2: The pupil puts the cubes together and puts the cards with
numbers on each other. He can write down the result:
.
In the next phase, we can can try examples for addition involving transfer between orders of
magnitude. The children that already know the Bank or the table for division do not have
problems with exchanging the cubes between different orders. Let us show how to solve the
task 4,14+2,28.
Step 1: The pupil enters the numbers with the use of cubes.
Step 2: He will move the cubes towards each other. He now has
12 cubes in the order of hundredths. He thus takes 10 of the
hundredths cubes and exchanges it for one tenth-cube. He now
has the result: . He writes the result with the
help of number cards and also copies the result into his exercise
book.
Later, we can also choose more demanding and interesting tasks, as 8,001+1,999. After
the pupil performs all the exchanging between orders, he will arrive to the result: 10.
References
Feez, S.: Montessori and Early Childhood. SAGE, 2010