Mathematical Misconceptions international roject
Carlo Marchini – Working document n. 4
British Academy Grant
Carlo Marchini
Mathematics Department University of Parma
Number misconceptions. Place value is a problem from the beginning of Primary School and it
remains for all the Primary School and longer (Moscucci & Piccione, 2003). For instance, some
protocols of ARMT show that the concept of place value could not be an active knowledge resource
for the solution of problems (Grandpa James’ tricks 1), in the first two years of Upper Secondary
School, too.
In the today Italian School, Primary School is compulsory and has an official curricula, with some
space for self-government. Before it, there are the Kindergartens, some of them public institutions,
some other private ones, and there is not a clear curriculum for them. Moreover the pupils’
attendance to Kindergarten is not compulsory. Most public Kindergarten, Primary School and, in
some case Lower Secondary School, are controlled by the same board of management, depending
of geographic constraints or the pupils’ number. And in this cases, the activity of the two last
Kindergarten classes anticipates some issues of Primary School: “pregrafismo” and “prenumeri”,
planned as a preparation for Primary School.
(We can ask to some Kindergarten teacher examples and documents to add here)
These activities induce pupil’s ability of writing her/his name (in capital or in cursive) and to know
digits (in same case excluding 0, in some case including it), but, more often, Kindergarten pupils do
not master the difference between number and digit.
In Kindergarten, the attention is mainly devoted to the right way of writing digits, in order to avoid
“reflections” such the following ones
(We can add here some example of pupils protocols)
1
14.II.17. GRANDPA JAMES’ TRICKS (Cat. 8, 9, 10) ARMT ©
Grandpa James is keen of riddles. Recently he proposed this riddle, to his grandchild:
“Cast the two dices and, without showing me the result of the throwing,
- multiply by 2 the numbers on a dice and add 5 to the result;
- multiply by 5 the new result and sum the number on the second dice.
Please, tell me the number obtained at the end and I will be able to give you the throwing result.”
How can Grandpa James succeed in hitting the numbers resulting by the throwing, which trick did he invent?
Justify your answer.
Remark that Rally Mathématique Transalpin is a mathematical contest for classes. On a sample of 173 classes of the
three grades 8 (71), 9 (76), and 10 (26), 79,19% of them give answers showing that place value does not operate as a
firm knowledge.
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Carlo Marchini – Working document n. 4
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We can say that in these years the semiotic graphic register (Duval) is involved. There are other
counting activities regarding number and digit; in that cases numbers (but we prefer digits) are only
sounds and the numbers from 1 to 9 are assumed as (nursery) rhyme, devoid of meaning, except the
orderings (increasing and decreasing ones) in which the name of digits are pronounced.
Nevertheless is not unusual to listen 5 year pupils aware of some Arithmetic, going beyond of ten
and using arithmetic operation (in Carlo’s experience, addition and subtraction, and the problem of
the infinity of natural numbers operating by “adding one more”). It is evident that there is a private
knowledge grown outside of the school walls, in family or somewhere else.
At the beginning of compulsory school we have pupils with very different levels of knowledge
about numbers, digit and Arithmetic, since in the same class of grade 1, you can find pupils coming
from different Kindergartens (in Italy 94%, by official national statistics), pupils which did not
attend a Kindergarten, and pupils coming from abroad, with understanding language difficulties.
The research of Baldazzi et al., 2004 proves that with a sample of 99 pupils, being 6 year old pupils
(from schools in Northern Italy), 88% of them are familiar with place value with three digit
numbers.
Therefore the primary school teacher must create a sort of “ground zero” of a common knowledge.
This necessary educational activity could be an important reasons for the birth of misconceptions.
Misconception could be an acquaintance, resulting form condensation (Sfard) of pre-conceptions,
intuitions and personal beliefs, but it is a knowledge that hampers the learning of school contents,
thence it can be ascribed to the category of obstacles, in Bachelard – Brousseau setting.
Number versus digit. In the first (two) year of Primary school, a mess between digit and number is
present. It is based on out-of-school language, on the teacher practice, on the difficulty to point out
the difference between a symbolic register and its semantic. Usually we speak about telephone
numbers, numbers on the license plate, numbers for the size of a dress, for the street number, and so
on. Children learn similar occurrences of digits (and they call them ‘numbers’) before their teacher
teaches them Arithmetic.
The difference between number and digit could be not pointed out by teacher, from the beginning.
Some of the interviewed teachers prefer to delay in that, waiting for the moment in which two digit
numbers will be introduced, or later on.
Some others teachers present digits as soon as possible, contemporaneously to the alphabet letters.
This attention to semiotic symbolic register, in our opinion, is useful.
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Carlo Marchini – Working document n. 4
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It could be not simple to distinguish between digit and one-digit number. A simple rule is that if we
consider digit as a letter, familiar sentences such that: “the sum (difference) of digits” or with other
operations, lose their meaning, immediately. More delicate is the problem of digit ordering since we
can consider a ‘natural’ ordering in the alphabet, and a similar ordering among digits, but this
ordering is deprived of meaning; ordering of one-digit numbers reflect semantics of numbers, and it
is a consequence of quantity.
Oral and written words. From the linguistic point of view, each word conveys a multiple
‘identity’: there is spoken (or oral) word, in which the acoustic part is more relevant (phonemes);
there is written word that can be analysed on the basis of its graphemes i.e. the blocks of letters
representing phonemes since the alphabet is by no means a perfect method of transcribing human
speech. Moreover the convention of writings tend to widen the gap between the living speech and
its alphabetic transcription. The analysis of each grapheme by letters is a necessary third phase. A
hidden rule is that writing (graphemes and after letters) must adopt, in general, the spatial
collocation (from left to right or conversely) on the paper, translating the oral timing of phonemes.
Until now we are in symbolic register, even if there is difference between the nature of symbols
used.
The well-know (Kindergarten o Primary) talk : “A as an apple, B
as a ball,…” in which there are drawings and letters, is a very
complicate interplay, at least, among three kinds of semiotic
registers:
oral,
written
and
semantics,
and
a
mental
representation. Oral register is firstly used by teacher, (s)he often
points (with finger or a stick) at the letter A (symbolic register)
and then at (the drawing of) an apple. The teacher pronounces well the word ‘apple’ (oral register)
and shows (the drawing of) it; in this case the (drawing of an) apple is the semantics of the word.
The sound ‘apple’ is now related with the actual object and this is strictly
tied with a mental object ‘apple’ that is useful for the child, when he wants
to remember the letter A in absence of the fruit.
This procedure looses its meaning when the child doesn’t know the name
‘apple’ for this kind of fruit, e.g. an Italian boy connects an apple with the
letter M (mela). In this case the interplay among registers, is more
complicate, since there are two different oral register to connect: the Italian
speaking and the English speaking ones.
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But the written register is far from to be acquired. Now from the apple (as an object or a drawing of
it) pupil must be able to pass from drawings of apple, pipe, lamb, elephant, to the letters A, P, L, E,
in order to write APLE. Moreover he must know the peculiarity of this word for the presence of two
letters P together, APPLE.
Oral and written numbers. I waste some time and space on linguistic reflections, since the first
steps in number learning are very similar to the learning of writing and reading: for numbers, also,
there is a writing, there is a reading, there are symbols, there are meanings.
The main difference is the risk of confusion: a letter in itself, e.g. ‘a’ can be also a word,
indeterminate article. This happens in English, but in Italian too and I think in several languages.
But very soon it becomes clear if the sign is used as a letter, or as a word.
The confusion number-digit is more difficult to extirpate. A possible explanation is offered by
history: the introduction of digits is the more recent (8th - 13th century) than introduction of alphabet
(10th century b.C.) by Phoenicians.
The word ‘digit’ in itself is misleading, it recall the finger (digitus, in Latin), but in English there is
also the word ‘cipher’ or ‘cypher’ more near to the Latin transcription of an Arabic word, used by
Leonardo Pisano or Fibonacci as zephirum, for the sign 0, and now used in connection with
cryptography.
In the living tradition there are instances of digits that are called ‘numbers’.
The learning of Arithmetic passes through the oral register, in which the numbers are red, to the
symbolic register, in which the numbers are written using digits, not letters, and a semantic of
number, in which number often became a measure of quantity, sometimes a rank place or a label.
One of our teacher observes with her children that we need letters to write all the words, and in a
similar way we need digits (the alphabet of numbers) in order to write numbers. There is a
difference between linguistic and arithmetic alphabets. In the (Italian) alphabet there are 21 letters,
in the English alphabet there are 26 letters, the alphabet of some language are different from the so
called Latin alphabet, and so on. The alphabet for Arithmetic is of 10 ‘letters’, the digits, and this is
the same all over the world, even for that cultures using ideography (Chinese and Japanese).
Moreover this convention can be modified reducing the symbols to two only (and this is the way
she introduces multi-base calculus).
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Another teacher presents an activity for reinforcement of reading and writing of words and
numbers. There is a box containing small card on which there is one letter, each 1. A pupil pulls out
some cards and the pupils try to compose a word with the drawn cards. These words are written to
the blackboard and on personal copybook. It is clear that on the cards there are letters and not
words.
For the numbers the tools are similar, a box with small cards on which there is one digit, each. The
number of cards pulled out from the box is according to class competence. The first use of this is a
writing activity, since. Pupil pulls out a card from the box, he must read the digit on the card, he
must write on the blackboard the name of digit in letters. The card reading it is also a good occasion
to remark that there is a right way and a wrong one to place the cards, excepting for the digit 6 or 9.
The ambiguity of the card with digit 6 gives the opportunity to remark that on the card it is written a
digit, not a number, a quantity invariant of its representation, since we cannot rotate the word ‘six’
and get ‘nine’! When the play goes on, the teacher stops the pulling out, returns all the cards in the
box and asks pupils to write on the copybook the words written on the blackboard, in ascending or
descending number order. It is important, in our opinion, to remark that ordering is a feature
peculiar of numbers.
The play is repeated afterwards, pulling out two, three… cards and writing by letters the name of
number composed with the digits presents on the cards. Teacher uses also the activity with two or
more digits asking pupils to identify which is the smallest number or the greatest than can be
written with the digits appearing on cards.
This activity is suitable also at Level I (numbers up to 20), since numbers are not used for
arithmetic, but only as names. If the number is greater than 20, the teacher tells it to students, but is
not rare that pupils know the name of the numbers greater than 20, at the first level, too, since they
use their own knowledge, independent from schooling. This activity can have contraindications, as
shown by Baldazzi et al. 2004: the number 365 is written in Italian ‘trecento sessanta cinque’ (three
hundred sixty five) and some pupils wrote it with digits as 300605, or 30065, translating separately
each Italian word denoting numbers, with an attempt of ‘place value writing’ (respecting the order
of pronouncing). Conversely the practice of writing numbers in letters, is suggested also by
Speranza, 1981, since, “…the pupils must realize that the numbers are wrote with words, not with
magic signs”.
1
Commonly we say that digits are ten, but in the box it is enough to place only nine different types of cards, since the
same card with the digit 6 can be used for the digit 9, too.
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Therefore with this approach there can be the same problem that language teacher meets passing
from graphemes to words.
In the play with box and cards, the possibility of exchanging digits give the occasion to compare the
words used in each case and also to single out the name of digits in that words, with their place:
twenty-six (in Italian ventisei) and sixty-two (sessantadue). This is an activity of ‘written place
value’ preparing the problem given by the use of digits only.
The activity of writing numbers by letters is suitable for focusing on the ‘change of place’ of
reading digits (in Italian), passing from sixteen (16, sedici) to seventeen (17 diciassette): in numbers
from 11 to 16 it is written (and red) before the digit of unities and after the ten (sedici), from 17 to
19 the order changes, first the ten, after the digit of unities (diciassette).
This card play allows other reflections about the linguistic difference between thirteen and thirtyone, pointing out that digits in itself has no meaning (symbolic register), but they acquired meaning
(semantics) in connection with writing order. We can say that is the rising of place value.
It is also relevant to remark that the same digit ‘changes’ name depending on the place value.
Sometimes is only a shortening or redoubling of consonants, due to euphony, sometimes is a true
change coming from old linguistic roots. We refer here only to Italian language
1: is uno, un or nothing 1: 71 = settantuno; 11 = undici; 134 = centotrentaquattro; 1,000 = mille;
21,000 = ventunmila; 1,000,000 = un milione; 1,000,000,000 = un miliardo.
2: is due, do or ve depending on its place value, we think, based on an ancient concept of number: if
2 is used for tens, ten thousands, ten millions, and so on, its name is not ‘due’, but is ve, coming
from Latin viginti. Therefore 22,222, in Italian, is red: ventiduemiladuecentoventidue. The form do
is active only in 12 = dodici; 12,000 = dodicimila.
3: is tre everywhere.
4: is quattro, quattor, close to the Latin name quattuor, or quar:
4,414,144 =
quattromilioniquattrocentoquattordicimilacentoquarantaquattro. The changes of name for 4
follows the same rules applied to 2.
5: is cinque, quin, cinqu, as happens for 2 and 4: 5,555 = cinquemilacinquecentocinquantacinque,
but 15 = quindici; 15,000 = quindicimila.
6: is sei, se, sess, has happens for 2: 66,616 = sessantaseimilaseicentosedici.
1
The fact that in some cases 1 is red and in other not, support the statements that 1 is not accepted, in Italian, as a
number, but is confused, with the indeterminate article.
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Carlo Marchini – Working document n. 4
7:
is
sette,
British Academy Grant
assette,
sett:
77,717,427
=
settantasettemilionisettecentodiciassettemilaquattrocentoventisette.
8: is otto, ott: 88,218,328 = ottantottomilioniduecentodiciottomilatrecentoventotto.
9: is nove, annove, nov: 90,919 = novantamilanovecentodiciannove
In this inventory of names of numbers we must pay attention to the powers of ten, used in reading
and writing numbers.
10: is dieci, dici, as number, nti, nta, anta, nt, ant 1, for place value: 10 = dieci 2; 10,000 =
diecimila; 10,000,000 = diecimilioni; 10,000,000 = diecimiliardi. The form dici is used in numbers
from 11 to 19: 11 = undici; 16 = sedici; 17 = diciassette; 18 = diciotto; 19 = diciannove. The
remaining form are used explicitly for place value nti for numbers from 20 to 29, excluding 21 and
28, since Italian names for 1 and 8 begin by vowels: 20 = venti 3; 24 = ventiquattro; 29 =
ventinove. The forms nta are used for the numbers from 30 to 39, excluding 31 and 38: 30 = trenta;
33 = trentatre; 39 = trentanove. We have the form nt in 21 = ventuno; 28 = ventotto; 31 =
trentuno; 38 = trentotto. For numbers from 40 to 99 the form used is anta, and the short form ant
when 1 or 8 are involved as unities (thousands, millions,…): 46 = quarantasei; 51 = cinquantuno;
68 = sessantotto, 97 = novantasette.
100: is cento
4
as a number and (the same) cento, when place value is involved, always: 100 =
cento; 453 = quattrocentocinquantatre; 280 = duecentoottanta, even if there is hiatus; 118,321 =
centodiciottomilatrecentoventuno
1,000: is mille, as number, mila, for place value: 1,000 = mille; 1,813 = milleottocentotredici; 2,813
= duemilaottocentotredici
1,000,000: is milione and milioni. In this case the use is always for place value, with the difference
of the conjugation with singular or plural. Therefore 1,000,000 is unmilione 5; 2,317,432 is
duemilionitrecentodiciassettemilaquattrocentotrentadue.
1,000,000,000: is miliardo and miliardi (the same as for milione).
1
Remark the presence of the same consonant of the word ‘ten’, as a possible common root of these words in Italian and
in English (German also)(Grimm’s laws).
2
Carlo is not sure that in 10 = dieci, this is the name of the number or it is the place value operating, with a silent
presence of 1.
3
The sound of Italian ‘venti’ and of English ‘twenty’ is very similar.
4
In this case is more clear the silent reading of 1: 200 is red duecento. The problem with dieci (ten) in Italian is more
complicate since 20 is not duedieci, 30 is not tredieci,
5
We adopt here the convention in use for the bank cheque: the numbers must be written in letters without blank space
among letters.
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In everyday language there aren’t other names for the power of ten. In scientific language there is a
code for other powers of ten with positive and negative exponents.
In all, the Italian names for digits and power of ten used in everyday language are 36 (37 with the
‘silent’ 1) different words that pupils must learn and use for writing correctly the names of natural
numbers in letters.
We think that in other languages it is the same.
The coincidence with symbolic representation register and written or spoken one begins from 30 on,
since the registers have the same rule of construction. These literary remarks show that (in Italian, at
least) place value problem rises for numbers greater than 30 (20), since for less numbers there is a
lot of different rules which witness of different origins of the name of numbers. The symbolic
representation register has few rules than the oral and written one.
Remark that there isn’t different names for 0, the only one is zero as number and digit.
Today in Italian, but I think in many other languages (excepting French), we read numbers as
Fibonacci stated in his (1202) Liber Abaci. He introduced the rule of writing numbers by grouping
digit by three, and reading them by three : 22,222 = (ventidue)mila(duecentoventidue);
421,734,432 = (quattrocentoventuno)milioni(settecentotrentaquattro)mila(seicentoquarantadue).
This multiplicity of names used to communicate orally or by writing a number requires a long
training. Only when there is a complete coordination of three semiotics registers, oral, written by
letters, written by digits, together with the semantics of numbers, the problem of place value can be
dealt and solved.
Zero, nought, null, or cipher. The main ingredient of place value is 0. Before Fibonacci’s
introduction of digits, there were the numbers and there were different ways of representing them,
with (partially) additive systems. We suggest that, for instance, in the ancient Egypt, the signs used
for write the numbers were not digits. The Egypt numerical system denotes number as a sum,
disregarding order in which signs are written. The Babylonians used a partially additive system,
therefore they wrote numbers in order to calculate a sum. Only the smallness of their clay small
boards and the poorness of writing tools, compelled them to introduce signs in a sort of positional
system, but the number as a sum remained. The introduction of digits (nouem figuras indorum) can
be considered as a transformation of numbers in signs, used in symbolic way. The history of 0 is
completly different and goes reverse side: from the sign to a symbol and then to a number.
Fibonacci says, “cum hoc signo 0, quod arabice zephirum appellatur”, with this sign 0, that Arabs
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call zephirum. Hence 0 is introduced as a sign. The status of number for 0 dates from 1898, when
Peano agree that 0 is a natural number.
The main prejudices about 0 are that 0 means nothing, 0 doesn’t matter, therefore 0 can be deleted
without problems. The English names for 0 are exemplar. Writing by letters, we pass the zeros over
in silence: 56,004 = fifty six thousand four (cinquantaseimilaquattro).
The use of abacus can be introduce 0 as a place card, similar to a punctuation sign. From this to the
consideration of 0 as a number, it is a big step.
A possible reason of all these difficulties remains in the fact that 0 doesn’t represent a quantity,
hence it isn’t a number (the same can happen, in Italian, for 1, when it is confused with the
indeterminate article).
Moreover 0 can be associated with the empty, and there is a long philosophic tradition, from
Parmenides on, refusing the existence of the empty. In this view 0 in an epistemological obstacle,
that is widely treated in the literature.
Numerical systems without zero didn’t have a place value problem. We can say that the
introduction of zero, gives the opportunity of representing the (potential) infinity of the natural
numbers with a ‘small’ alphabet of digits, and with solid rules. For instance, the alphabetic (and
additive) numerical system of ancient Greece was an obstruction to a development of Greek
Arithmetic in comparison with the contemporaneous flourishing of Geometry.
Therefore we can say that many misconceptions related with place value has their true origin with
the introduction of 0.
The introduction of 0 at school. We report here some answers of teacher interviewed. Some
teachers begin with 1, and postpone the introduction of 0, later, after 9. In this way it could be
difficult to understand which digit is the least between 0 and 9, since some pupils could remind the
(time) order of introduction of digits.
In a second moment they use the intuitive set theory (putting objects in bags) and associate natural
number to set cardinality. In this case 0 is associated to the empty set. Some other teachers associate
0 to the indefinite adjective (or pronoun) ‘no’, ‘any’, ‘nobody’ (in Italian ‘nessuno’)
We do not agree with this approach. It seems to us that 0 could be a teacher’s own difficulty. We
know that empty set (and singletons) is difficult to learn and to teach, at University too. There are
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well known set-theorists, as Kuratowski and Sierpinski, having trouble with the empty set (in the
twenty’s of 20th century).
Some other teachers, after 9, do not introduce 0 expressly, but use it as a sign necessary to write ten.
It is evident that these approaches could be unsuitable for pointing out the difference between
number and digit, since cardinality is understood as a quantity.
Some other teachers prefer a graphic register: the number line. Some of them take the small scale
contained in the school holder, and remark that the first (big) notch is labelled with 0, followed by a
notch labelled with 1, and so on. Some other draw at the blackboard the number line and starts with
1, or with other numbers, e.g. a number in the date (day or month). It seems that this approach is
more suitable since pupils ask for ‘numbers before’ and then 0 appears as a necessity of
completeness. Some pupils ask also for numbers before 0.
In our opinion these activities do not distinguish between digits and numbers.
Another approach to zero is mediated by addition: the necessity of a neutral element.
We observe that the role of zero as a label, or as set, and so on, can be useful for operation, but this
role can produce misconceptions when place value is involved.
An example: We can find in books that multiplication by ten is made simply by adding a 0 to the
end. Some pupils observed that adding means that must be performed an addition and concluded
that multiplying by ten leaved unchanged the number.
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