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Carlo Marchini & Paola Vighi – Working document n. 7
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Carlo Marchini & Paola Vighi
Mathematics Department University of Parma
A preliminary remark.
In Italian School, Primary School occupies the first five years of compulsory school, starting with 6
year old pupils. The introduction of relative integers numbers is suggested in classes fourth or fifth.
The curriculum suggests that this kind of number must be presented in concrete contexts (The most
frequent: economics and temperature scale). Moreover the curriculum asks for representation of
relative integer numbers on number line.
The curriculum is divided in two parts: contents and procedures/applications. Relative integer
numbers are presented in the first part, only.
At the end of the fifth school year there is a written and oral examination of pupils. Teacher feels
that this examinations is an evaluation of her/his five years teaching, since the examination
questions are decide collectively. Customarily, mathematics test is a problem in which relative
integer number do not appear. For this reason few Italian teachers of primary school introduce
relative integer numbers, their activity is mainly concentred on fractions and decimal numbers,
since curriculum requires much more explicitly knowledge of the concepts and knowledge of the
calculus procedures in problems and in applications.
Relative integers numbers are one of the main concern of the Lower Secondary School (the next
three years of compulsory school, 11- 13 year pupils) and this subject is a ‘standard’ for the third
year. In Italian school book the arithmetic of relative integer numbers is (strangely enough) called
‘Algebra’ and presents also the first approach to equations and numerical – symbolic calculus with
brackets.
We have few protocols about integer relative numbers (two classes only) and pupils of this sample
reveal ‘classical’ misconceptions. Carlo had an interesting experience in a second class of primary
school, about this kind of numbers.
Many Italian teachers feel that they are not in position to treat negative numbers in their class. In
occasion in which the negative numbers can give a quick and correct answer, teachers prefer to
avoid the mention of negative number, paraphrasing them: not – 5 °C, but five degrees under zero,
not a count settlement of – 34 €, but a debit of 34 €, or, as in Italian standard book entry, 34 red.
The double entry bookkeeping used in balance, has the same origin.
The origins of a primitive double-entry system have been traced as far back as the 12th century.
Some sources suggest that the success of the most important European bank in the period 1397 –
1494, the Medici’s Bank was given by the genius of its founder, Giovanni di Bicci de’ Medici. Now
Luca Pacioli, is often called the "father of accounting" because he was the first to publish a detailed
description of the double-entry system, in a 1494 mathematics textbook Summa de arithmetica,
geometria, proportioni et proportionalita. (Pacioli, …).
All these ‘sleight of hand’ have a misconception as a consequence: a negative number is intuitively
associated with quantity (from one part of other of the double entry). This intuition conflicts with
order and (later) with operations.
Alessia wrote: “I must remember that negative numbers near to zero are bigger and they far from
zero are lesser”, but to the following tasks
“Put a digit in the square in order to obtain the smallest number possibile -3 < +35;
Put a digit in the square in order to obtain the greatest number possibile -2
she answers 0 (i.e. -30) in the former (i.e. -29) in the latter.
< +25;”
Mathematical Misconceptions international roject
Carlo Marchini & Paola Vighi – Working document n. 7
Domenico did not resolve the task
“Put a digit in the square in order to satisfy
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-3
< - 25;”
He comments “I cannot place any digit since 3 is always bigger than 2”.
This sort of mistakes could hides the previous discussed misconception.
Other important misconceptions take origin from the presence of the sign ‘ – ’ which is interpreted
not as ‘the opposite of’ but as the sign for subtraction, but we can highlight this phenomenon in
connection with operations. In Italian curriculum for primary school relative integer numbers are
presented only in connection with ordering, without operations.
Nevertheless Carlo was in December 2006 in a second class and he was present to this
conversation. The teacher gave as homework to fill the following table 1 (a similar one)
- 0 1 2 3 4 5 6 7 8 9 10 Many pupils presented the - 0 1 2 3 4 5 6 7 8 9 10
right answer (table 2). 0 0
0 0
Some
of them filled 1 1 0
1 1 0
completely
the table with 2 2 1 0
2 2 1
a
sort
of
reflection.
In this
3
3 3 2 1 0
way
they
‘change
4
4 4 3 2 1 0
operation’ in the sense that
5
5 5 4 3 2 1 0
the subtract the lesser to
6
6 6 5 4 3 2 1 0
the
bigger
term
7 7 6
3
(misconception that it 7 7 6 5 4 3 2 1 0
8
appears
in
column 8 8 7 6 5 4 3 2 1 0
9
subtraction). The left- 9 9 8 7 6 5 4 3 2 1 0
10
8
4
lower corner represents 10 10 9 8 7 6 5 4 3 2 1 0
the result of row entry minus column entry; the right-upper corner represents the result of column
entry minus row entry.
- 0 1 2 3 4 5 6 7 8 9 10 There are some shy - 0 1 2 3 4 5 6 7 8 9 10
0 0 1 2 3 4 5 0 7 8 9 10 attempt to fill the table, 0 0 -1 -2
negative 1 1 0
1 1 0 1 2 3 4 5 6 7 8 9 adding
2 2 1 0 1 2 3 4 5 6 7 8 numbers (table 4). 2 2 1 0
Teacher was tempted to
3 3 2 1 0 1 2 3 4 5 6 7
3 3 2 1 0
do not treat this case,
4 4 3 2 1 0 1 2 3 4 5 6 but gave a questioning 4 4 3 2 1 0
5 5 4 3 2 1 0 1 2 3 4 5 look to Carlo (he was 5 5 4 3 2 1 0
6 6 5 4 3 2 1 0 1 2 3 4 working to another 6 6 5 4 3 2 1 0
7 7 6 5 4 3 2 1 0 1 2 3 subject and he was 7 7 6 5 4 3 2 1 0
completely
silent).
8 8 7 6 5 4 3 2 1 0 1 2
8 8 7 6 5 4 3 2 1 0
Carlo’s replying look
9 9 8 7 6 5 4 3 2 1 0 1 convinced the teacher 9 9 8 7 6 5 4 3 2 1 0
10 10 9 8 7 6 5 4 3 2 1 0 to discuss this answer. 10 10 9 8 7 6 5 4 3 2 1 0
She ask the pupil where he saw this type of number. Pupils answer was: “On the car of my father”.
A girl said: “Me too, since there are infinite negative numbers!” . As a consequence teacher ask the
two children if they were able to fill the table (on the blackboard) and they wrote (Table 5). When
the pupils wrote the first row, teacher was favourably surprised, but at -13 she stopped the pupils,
saying that they have to learn a much mathematics more and they will be able to solve correctly this
problem, when they will be in Lower Secondary School. At the end of the lesson, Carlo discussed
with the teacher. She said that was a good surprise for her to ascertain the presence of negative
numbers in a second class, but she thought that it was to early to introduce in a correct way negative
numbers for the present moment, in order to avoid possible confusions. The pupils proposing the
Mathematical Misconceptions international roject
Carlo Marchini & Paola Vighi – Working document n. 7
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negative numbers were among the best pupils in mathematics, but the other classmates were not at
the same attainment level.
This example shows, in our opinion, what could happen when pupils’ and teacher’s attention are
focused only to number and not to number and operation among them.
0 1
2
3
4 5 6 7 8 9 10
0
0 -1
-2
-3
-4 -5 -6 -7 -8 -9 -10
1
1 0 -11 -12 -13
2
2 1
0
3
3 2
1
0
4
4 3
2
1
0
5
5 4
3
2
1 0
6
6 5
4
3
2 1 0
7
7 6
5
4
3 2 1 0
8
8 7
6
5
4 3 2 1 0
9
9 8
7
6
5 4 3 2 1 0
10 10 9
8
7
6 5 4 3 2 1
0
This anecdote has many interesting aspects.
- It suggests an arithmetical path for introducing negative integers numbers avoiding the
‘geometrical’ setting involved in the presentation of negative integers numbers by left
prolongation (or arranging symmetrically) of number line.
- In a certain sense we can justify the pupil’s mistake as a sort of reproduction of number line in
filling the free squares.
- Nevertheless it is relevant the implicit presence of the infinity.
- The episode reveals the presence of a huge children’s mathematical knowledge as result of the
outside school experience.
- This experience could be a motivating starting point for the learning of mathematical topics.
In this case the need of a correct knowledge was evident and the wrong answers would have
been transformed in a firm mathematical attainment.
- From teacher’s point of view, this evolution of the task was unexpected and she was afraid of
continuing to develop the topic (in that moment).
- The very common use of this sort of double-entry tables with heading of columns from left to
right and with top – bottom for the rows could conflict with the ordinary representation of a
Cartesian coordinate systems.
This year I received two third classes of primary school visiting Mathematics Department of
Parma University.
I ask to a student of mine to simulate a university lecture for these young visitors. We decide to
present the subtraction and we start justifying the results of subtraction as 4 – 6 with number line
‘jumps’. After few examples, children were able to observe and use the table symmetries along
diagonals, filling completely and correctly the eleven time eleven table.