Joel
is an engineer who has aphasia following a stroke he underwent at the age of
72. Since the stroke, Joel expresses
himself mostly in single words or in single utterances; He can't produce whole
sentences. He also can't read single
words. Does he have difficulty reading
numbers as well?
Stanislas
Dehaene, a renowned scholar of numerical cognition, developed in 1992 the Triple
Code Model of numbers (Dehaene and Cohen,
1995). According to this model, numbers
are represented in the brain in three ways:
a) Verbal Code: numbers
are represented as a series of words with a specific syntactic order ("one
thousand and twenty four"). Basic
arithmetic facts are coded in verbal form as well ("nine times nine equals
eighty one"). Impaired verbal code
affects a person's ability to name digits and numbers and to retrieve basic
arithmetic facts. b) Visual Code: numbers are
represented as a series of digits (for instance, 4521). This code underlies an internal representation
of a number line. This is akin to an
orthographic representation. c) Analogical/Quantitative Code: here
the quantitative meaning of number is represented (this is actually a semantic
representation). This code enables one to
determine that 9 is greater than 3.
The
three codes are separate and there are dissociations between them (one of them
may be impaired while the others are intact).
They are supervised by different brain areas. However they are closely related and there
are interactions between them. Dehaene argues that there are direct
"translation" paths between each pair of representations. This means that when we process a number we
don't always represent it in all three codes.
For example, the direct path between the verbal and the visual
representation enables people to read or write numbers even without processing
their semantic, quantitative meaning; the
direct path between the quantitative and the visual representations does not
take syntactic information into account, etc.
We move between codes in order to perform tasks that can be performed
only in a specific code. For example,
magnitude comparisons ("which is larger, 9 or 3?") are done using the
quantitative code; in order to retrieve
the answer to the problem 2x3 one has to transform it to verbal code (Dehaene argues that arithmetic facts are stored
in a verbal code).
Dehaene's model reminds me of the single word reading model (Friedmann and Coltheart, 2017).
According to this model, when a typical reader reads a familiar word, he
activates three lexicons: the Orthographic Lexicon (which
stores the written form of words), the Semantic Lexicon (which stores word meanings) and the Phonological Lexicon (which
stores the oral form of words – the sequence of sounds that make them up). The Orthographic Lexicon may be analogous to
the Visual Code in Dehaene's model; the Semantic
Lexicon may be analogous to the Quantitative Code; and the Phonological Lexicon
can roughly be analogous to the Verbal Code.
To
what extent do these models overlap?
I'm
sure they don't fully
overlap. Not every person who has
difficulty reading single words also has difficulty reading numbers, and vice
versa. The linkage between the models
may be strained, but I still find it interesting to think about:
The Orthographic
Lexicon enables us to identify known words.
Is there a relation
between the Orthographic Lexicon and the Visual Code of numbers? Alexia is an acquired disorder which renders people unable to
read written words but able to write words (!). Their other language skills are intact as well. One of the features of Alexia is slow and
effortful but mostly accurate word reading.
The longer the word, the longer it takes for a person with Alexia to
read it. Even people with pure alexia
tend to be less impaired in reading digits than in reading letters and words. Under short exposure times, all people identify
digits better than letters. It is
possible that the difference pure Alexics have between known word recognition
and digit and number reading is an amplification of the normal difference
between letter and digit processing (Behrmann and Starrfelt, 2011). We said earlier that the Orthographic Lexicon
enables us to recognize familiar words.
I wonder if the ability to recognize "familiar numbers" like
1492, 1776, also resides in the Orthographical Lexicon (this means that a
familiar number is processed like a known word).
The Phonological
Lexicon stores the phonological code of words.
A person with disability in the Phonological Lexicon can read words
silently but has a difficulty reading words out-loud, or reads them incorrectly
(for example, switches between phonemes).
An impaired Phonological Lexicon also affects speech. Is there a connection between the Phonological Lexicon and the Verbal Code
of numbers? Since the verbal
representation of a multidigit number is made out of several words ("three
hundred and forty two") there are syntactic relations between the
words. In this aspect the single word
reading model cannot be an exact parallel of the Triple Code model. On the other hand, reading single words requires
a grammatical analysis (parsing the word into a prefix, a suffix and a stem; Work+ed).
Prof. Naama Friedman writes that the initial morphological analysis of a
word is made in the Orthographic Input Buffer - that is, at a very early stage
before the word reaches any of the lexicons.
The
verbal code represents numbers as series of words in a specific syntactical
order ("three hundred and twenty four"). Syntax is the meaning derived from the order
of words in a sentence. When the order
of words is changed, the meaning is also changed (a dog bit a man vs. a man bit
a dog). Likewise in arithmetic: the order of digits in a number has meaning,
and when it is altered, the meaning of the number is altered (1984-1948). People with Broca's Aphasia understand the meaning of words
and sentences (their semantics is intact) but their syntax is impaired. These people might make syntactical errors in
reading and writing numbers. For
instance, they may read 14 as 4. When
they read "three hundred twenty six thousand four hundred fifty one"
they have difficulty distinguishing the different meanings of each of the words
"hundred". They find it hard
to translate from verbal code to visual code and vice versa (Ardila and
Rosselli, 2002).
Going
back to "Joel", we've seen that he finds it hard to express himself
in sentences and to read. Prof. Naama Friedman,
Dror Dotan and prof. Dehaene tested his number processing abilities. Apparently, Joel has a difficult time reading
multidigit numbers out loud. He reads
them digit by digit (for example, he reads the number "47" as
"four, seven"). He has
difficulty producing the tens form (eg.
Forty) and the "teen"
form (thirteen, fourteen etc.). This
means that he has difficulty translating multidigit numbers into number
words. His arithmetic disability has a
syntactic character, but Joel did not lose all his syntactic abilities: in
number reading tasks Joel never said the units digit before the tens digit. This means than he successfully codes the
relative order of the digits, an information that can be conceptualized as
syntactic.
Joel's
visual code of numbers is intact.
Contrary to his word writing difficulty, Joel can write numbers correctly,
even by dictation. This means that
despite his difficulty translating visual code to verbal code, he can transform
verbal code to visual code. Joel
understands the quantitative meaning of number, and his quantitative
representation is intact (for example, he successfully places numbers on a
number line and solves double digit addition problems, as long as he does not
have to say the answer out loud).
The Semantic Lexicon stores word
meanings. An impaired semantic lexicon
badly affects a person's ability to understand the meaning of words he
reads. Such an impairment affects not
only reading but also the ability to understand spoken words. Is there a connection between the semantic lexicon and the
analogic/quantitative representation of numbers? The main feature of dyscalculia is impaired
number sense – an impaired grasp of quantity and the quantitative meaning of
number. Impaired number sense can be seen
in several ways: an adolescent computes the
answer to simple math facts, sometimes using his fingers; an adolescent solves
22+5 by drawing 22 lines, drawing 5 more lines and then counting all drawn
lines from 1 to 27; an adolescent doesn't use the Commutative law of addition (doesn't
know that 7+5=5+7); an adolescent does not feel that the result he got is
utterly unreasonable; an adolescent can't put a number on the number line (I used the word "adolescent" in
this paragraph to clarify that this does not apply to the performance of young children
whose basic arithmetic skills are still developing).
People
with Wernicke's Apasia have
difficulty understanding the meaning of words and sentences. Consequently, they make very significant
lexical and semantic mistakes. This is
apparent both in their use of language and in their reading and writing of
numbers. For example, when they write
numbers from dictation, they may write entirely different numbers (a person is
asked to write 257. He says "820" and writes "193"); people
with Wernicke's Aphasia may read 37 as 27 or 1527 as 15,27 ( Ardila
and Rosselli, 2002). The scant descriptions I found
in the literature of arithmetic mistakes committed by people with Wernicke's Aphasia
are not very similar to the descriptions of mistakes committed by people with
impaired number sense.
Prof.
Naama Friedman describes different kinds of dyslexia caused by impairments in
different components of the model for single word reading. Is it possible to describe different kinds of
arithmetic disabilities caused by impairments in each code/representation or by
impairments in the translation processes between representations? I did not find literature that presents this
subject in a systematic way. In
addition, the boundaries between the three representations are permeable
(naturally, since they work together).
For example, the visual representation of number (say 1948) is built
according to a specific syntax (8 is the units digit, 4 is the tens digit etc.)
but the syntactic aspect is related to the verbal code.
Ardila, A., & Rosselli, M. (2002). Acalculia and
dyscalculia. Neuropsychology review, 12(4), 179-231.
Dotan, D., Friedmann, N., & Dehaene, S. (2014). Breaking down number
syntax: Spared comprehension of multi-digit numbers in a patient with impaired
digit-to-word conversion. Cortex, 59, 62-73.
(I
actually read the Hebrew version of this paper):
פרידמן, נ., דותן, ד., ודהאן, ס. (2014) הבנה לא מילולית של מספרים רב ספרתיים. שפה ומוח, 11,
25-47.
Dehaene, S., & Cohen, L. (1995). Towards an anatomical
and functional model of number processing. Mathematical cognition, 1(1),
83-120.
Friedmann, N., & Coltheart, M. (2016). Types
of developmental dyslexia. Handbook of
communication disorders: Theoretical, empirical, and applied linguistics
perspectives.https://pdfs.semanticscholar.org/b100/09373f19aba3f68d3568fc25a5352e1e42e5.pdf
(I
actually read the Hebrew version of this paper):
פרידמן,
נ וקולטהארט, מ. (2017). דיסלקסיות התפתחותיות. שפה ומוח, 12, 1-34. http://www.tau.ac.il/~naamafr/hebmain.html
Starrfelt, R., & Behrmann, M. (2011). Number reading in
pure alexia—A review. Neuropsychologia, 49(9),
2283-2298.