Educación, Psicología & Ciencia

From theories to practice: Alex, what does “1”mean ?

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Many authors have paid attention to the etiology of dyscalculia (here I use dyscalculia or math disability as equivalent terms, even is not very purist). In this post, we  address some of the most famous theories in this field and how we can link them to the real practice.
Almost everybody agrees with the two most famous theories in this area, which are actually an amazing work in numerical cognition. These theories propose that number troubles come from a weakness in number sense or numerosity coding.


Mental Number Line


  1. The Theory of the Approximate Number System (ANS), and how we use it for representing large and approximate numbers via a mental number line (Dehaene, 2011).
    Is “8” big or small? – If we consider that a number lower than 5 is small.
    Is “6” big or small?
    Probably it´s much easier for a kid to say that 10 is a big number, due to the “mental line distribution”
  2. The Numerosity- Coding Hypothesis proposed by Butterworth (2010), which states that Math Disability is caused by a deficit in the processing of smaller and exact sets of numbers.
    These theories state that we all have a preverbal ability which contributes to the foundation for the Symbolic Number System that we use to learn mathematics.



Blue or Yellow? – In the easy task, the blue stars are much more salient but in the difficult task, a number of stars are quite similar, thus the RT increases.


Through the development of the language, we develop also a language based symbolic system, linking the word with the number. Thus, when we acquire the exact symbolic number system, we start to do better approximations and estimations. (Piaza et al., 2013).

Alex, what does 1 mean … and 20?

Alex (not his real name) is a 7 years old kid with who I have been working lately. He has problems with language skills and also mathematics (and other disruptive behaviors at school). Alex has a very evident problem while connecting Arabic numbers or number words with the numerical magnitude; what I call the induction process while representing the natural numbers. He counts, or better said “he knows the counting sequence” up to 20.

But, does it mean that Alex knows what numbers mean? The answer is no

Saying the number´s name it´s a hard work (35= …), so when I ask, he hides his hands under the table, counting from 1 to 20, repeating the name of the numbers with bated breath, so he can provide an answer to the question. If I ask him to do a sum, it´s fine till he sees before the same operation (which means that he is only repeating the action). However if I change the operation he doesn´t know what to do with numbers, and if the sum exceeds 10 (number of fingers), he doesn´t know what to do with his fingers – to copy the action doesn´t mean learning. Kids copy all the time just because they don´t want to be different and feel bad because they are unable to do it. So, they use own strategies to survive in the classroom.
Of course, to “countdown” it´s even harder, and I don´t ask him to do it right now because it increases a lot his anxiety. When Alex goes to math classes with his colleagues, he feels bad because he doesn´t understand math problems, nor subtraction. He behaves badly in the classroom to be punished by the teacher and run away from mathematics (so he won the label of lazy, not silly). Another day we will talk about how poor mathematics performance kills self-esteem in children.

Alex also shows…

  • Certain confusion with sense of direction sense 
  • Problems understanding time
    … So now I go home? So, when I go home? 2 classes so then break and then home? Tell me when I am going home. – He repeats this aloud in order to understand better “time”.
    … How long will we work with this? 10 minutes, already? – Thus is very challenging for him to manage own activities.
  • Physical coordination
    When I say, – Hands up!. Right, hand up! left! He loses the notion of the body– However I am still not sure if it can be due to the phonological de codification when I come with long sentences (action1 + action 2).
  • Social Skills
    Anger management, social skills and lack of motivation (extrapolated to other academic activities).

Alex´s case is very difficult because he has also reading disability (and very poor vocabulary); I can´t use here language to glue the real world with abstract numbers.

What happens with the Underlying Semantic Representation of Numerical Symbols?

Noel and Rousselle (2011) didn´t agree with this connection regarding the core deficit in math disability. These authors defended that the non-symbolic impairment is preceded by impairment in symbolic and exact number representations. Here´s the third theory:

  • 3. The Access Deficit Hypothesis which states that children with math disability have a core difficulty relating numerical symbols to their underlying semantic representation (a reversed story than previous authors). (Noël & Rousselle, 2011).

This theory defends that the profile of Dyscalculia changes across the time; a developmental perspective. The problem is accessing the numerical magnitude information covered by symbols such Arabic numbers or number words.

How do children learn that verbal numerals represent the natural number?


There are two foundational concepts for learning math; magnitude and value. – Is the concept of more and less clearly developed? When It´s not yet ready, longer is more and bigger is heavier (Piaget´s conservation tests: size, value, length, and numerosity).

(See Carey´s Development Model, 2001, 2004, 2009)

a. The kid must map number symbols into de ANS (not the reverse)
b. It takes around 1 year to understand the cardinal value of the counting sequence. Thus, when kids count it doesn´t mean that they know what numbers represent.
c. Research showed that kids who don´t yet understand the precise meaning of the number words in their counting sequence do not show ANS.

All this process requires a mental representation (abstract). Kids must understand that adding 1 (+1) or more leads to a cardinal labeled by a further word (the successor function).

Is a Core Deficit or an Access Deficit?

How education enhances the acuity of the non verbal approximate system?

Which is the role of the language?

Are conservation and other logical operation simultaneous or successive processing?

How to help the development of math abilities?

And so… which differences do exist between dyscalculia, math disability and math learning difficulties? May we review our vocabulary?

Thanks for reading 🙂

Have a good day

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Autor: Lorena Álvarez

Psicóloga, activista, trotamundos y una apasionada por la ciencia y las letras.

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