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Monday, January 16, 2012

The difficulties in the acquisition of the process of concept of number and its application

With out any doubt, the difficulty that causes more grief and bitterness in children about their school work is the lack of expertise in handling numeric. No matter if the child is involved or not in this family game that tells them all the time: "No matter, I never understood beyond the multiplication, but it is better to take out a good grade in next evaluation" is always possible to tell the difference between a child who understands and even enjoys numbers and children who have no idea what persons talk about once they passed from counting objects to numbers.

That is why it is worth analyzing what happens between the first count that children enjoy, like when someone asks their age and they fast raise their fingers proud to know the answer, or while the family enjoy those games that require  dice and they must count to the number  to reach the finish of the game, and after some years,  the understanding and enjoyment of quantum physics.

Of course, many things happen between all those years, and it's not possible to forget the question that inspired the life of Jean Piaget: What is the origin of the ability to think the world in terms of numbers?.

Piaget believed that the acquisition of the concept of number happens around 5 years old, but it was clear that the process required prerequisite skills of logical reasoning as the transitive property, which allows understanding ideas like that if A is greater than B and B is greater than C, then A is greater than C.

Of course, it is also necessary the conservation of number, which is the ability to establish unique bi correspondences between two sets, which is not simply the ability to understand that the number of objects is the same regardless of the order or arrangement made of them (Piaget, 2001).

However, studies in the area of neuroscience have shown that children from before the first year are capable of developing rudimentary numeracy, independently of language (Jacubovich, 2006).

In fact quantities are so important that is not possible sometimes to describe images or sounds without numerical concepts. As an example, please try to describe the next image without using amounts, numbers, any quantity adjective, or words such as or no more than or less than...

And that's why numbers are a mystery if they are just looked only like a difficult area to solve in school, complex even for adults, as discussed in the article numbers and little numbers in this blog (Dzib Goodin, 2011). But how far is a difficulty of use and management to acquire numerical process due to non-teaching or brain damage?.

From the point of view neurocognitive, exists two ways to determine when is a problem in this sense, one is under the term acalculia, which is the loss of ability to perform mathematical tasks as a result of brain pathology (Ardila and Roselli, 2002) and on the other hand, it is possible to speak of developmental dyscalculy, which is a disorder that affects the acquisition of numeracy and calculation skills (Mogollon, 2010).

But the difficulty of acquisition is understandable since the process requires more than just recognizing 10 numbers, talking about Arabic representation, and combinations thereof, involves at least three processes:
The first process refers to the analog representation of the quantities, for example: **** can be represented by a four. This is analyzed in the brain in the inferior parietal region bilaterally.

The second process refers to the auditory verbal code. 4 sound like 4 and is the representation of 4. This is possible thanks to the relationship between the auditory and verbal processing, which is done at perisylvian areas of the left-hemisphere.

While the third process, which is what mostly repeated in school, refers to the visual code or Arabic, which gives meaning to 4. To this point 4 represents the number of objects linked to a word with a specific sound. The area is in charge fulsiforme gyrus of both hemispheres (Alonso and Fuentes, 2001).

But this is not achieved  only because children  reach certain age, or by school pressure, is a complex process of neural networks that has been knitting before it is aware of the number, as well as the language, since it is genetically predisposed (Radford and André, 2009), since apparently the counting is a necessary skill for survival. And you just need imagine this scene: it's not the same be surrounded by a wild bull, than  by 12 wild bulls, no doubt, the survival strategy is definitely different!.

However,  to complex mathematical  processes are needed many more brain areas involved, for example,  understanding of numerical magnitude is not restricted to the representation, because if you have 2, 4 and 6, even if they  are distinct entities, imply order of  quantities, and that is why mathematical problems require the parietal lobe, the cingulated cortex and subcortical regions, as well as the frontal lobe and the limbic system to remember  algorithms (Eger, Michel, Thirion, Amadon, Dehaene and Kleinschmidt, 2009; Serra Grabulosa, Adan Perez Pàmies, and Membrives Lachica, 2010).

It is then that any problem with the processing chain may involve an error from neuronal network, since the process is so fragile than any wrong process can produce another result, . At this sense, for example, forgetfulness is common when children are trying to solve any arithmetic problem or in the case of acalculia, the brain is not capable to recognize differences between numbers, or operations.

But challenges can be also attributed to errors of strategy or pressure on the student. For some reason students are less tolerant of frustration with math homework compared to any other type of performance, because it requires more attention, especially when it reaches the level of multi-digit operations (Hannula, Evans, Philippou and Zan, 2004).

That's why teaching mathematics requires support from a variety of other subjects, because although there are students predisposed to learning, others need more details, more concentration, and sometimes more than one way of doing the same operation.

That is why stress management is one key to motivate students with mathematical processing difficulties, because it is clear that a certain level of stress is good to keep the focus on the task (Anaya Anaya Durand and Huertas, 2010).

Learning mathematics then, depends on a chain of tasks, methodically arranged in the first place, and this helps to the brain to warn and manipulate them, searching results. But it has to start from the idea that mathematical concepts, unlike language,  depends on abstract content, so the big step is going from counting  objects to their representation with a number (Piazza and Dehaene, 2004).

But how is developed this chain of processes?, everything starts with a default network for  counting, babies are able to differences between one face, two faces, three faces, a crowd!. Then, socially adults begin to reinforce the language of mathematics. Everybody enjoys a baby when is asked: how old are you?, and then tiny  fingers get up to represent the word..

Then children start counting objects beyond their hands, and aligns toys, stones, marbles, images on the screen of a tablet ... they count dogs, they count flowers, they count ... first from 1 to 10, it is difficult to do it in a regressive way, but they try. Unfortunately they need to do it from 1 to 10 many times before eventually, make it gradually, but they are building neuronal networks.

And then, the big step, counting independently of the objects, it means they do not need 5 flowers to see a 5.

But there are some other needs to considerate, for example have clear differences between left and right, even if this is not a number, it can create a problem to the numerical manipulation. Just one example, it's not the same number 15 to 51, same components different position, different numbers.

The same applies to up and down. I have an example: 100 + 84,  simply ask to a child to arrange the numbers from top to bottom. The adult logic is:
+ 84

But a child can try something different
+100    There is a number under the other, right?

And children can go from there to the sum. And sometimes is able to sum well, but children simply change the sense left and right, or up and down:


Any teacher or parent will jump and shamelessly will tell to the child that the sum is poorly made, and will add another sentence:  YOU DO NOT know to sum. But look again, because children are doing it well in these cases:

100 → 1 +0 = 1
+84 →8 +4 = 12

The child is able to add, except that it’s not making it with the correct order.

And then they will face the subtraction, but also have a collision with the tenth number, the joker, the convenience, our never fully loved Zero.

Zero is worth nothing, we say. You are zero, at the left means nothing. But, what happens to the right? The relationship between laterality and mathematics it's more intense.

Let’s see an example:

 Here also can see different ways to get the result. One way is the classic 6 asks to borrow at 10, So 0 is not zero is 10 so we can take some of it ... or you can count up from 6 until it reaches 10.

Certainly is not the same 10 to 01. Clearly is not the same 1 +0 1 + 0.01 ... and mathematics become more complex and more and more until you reach a point where ordinary people can not understand what lawyers of the numbers are thinking (I love how that sounds).

Hence, the important thing is to create slow but steady steps in the formation of neural networks for acquisition of process of the concept and application of number, and who knows, perhaps to discover the mysteries of nature through mathematics. Never underestimate children.

Alma Dzib Goodin

If you would like to know more about my writing you can visit my web site:


Alonso, D. y Fuentes, LJ. (2001) Mecanismos cerebrales del pensamiento matemático. Rev Neurol. 33 (6) 568-576.

Anaya Durand, A. y Anaya Huertas, C. (2010) ¿Motivar para aprobar o para aprender? Estrategias de motivación del aprendizaje para los estudiantes. Tecnol. Ciencia Ed. 25 (1) 5-14.

Ardila, A. and Roselli, M. (2002) Acalculia and Dyscalculia. Neuropsychology Review. 12 (4) 179-189.

Eger, E., Michel, V., Thirion, B., Amadon, A., Dehaene, S. and Kleinschmidt, A. (2009) Deciphering cortical number coding from human brain activity patterns. Biology. 19. 1608-1615.

Hannula, M, Evans, J., Philippou, G. and Zan, R. (2004) Affect in mathematics education: exploring theoretical frameworks. Psychology of Mathematics Education. 1. 107-136.

Jacubovich, S. (2006) Modelos actuales de procesamiento del número y el cálculo. Revista Argentina de Neuropsicología. 7. 21-31.

Mogollón, E. (2010) Aportes de las neurociencias para el desarrollo de estrategias de enseñanza y aprendizaje de las matemáticas.  Educare. 14 (2) 113-124.

Piaget, J. (2001) La representación del mundo en el niño. Morata. España

Piazza, M. and Dehaene, S. (2004) From number neurons to mental arithmetic: the cognitive neuroscience of number sense. M. Gazzaniga (2004) Cognitive neuroscience. MIT Press. USA.

Radford, L. y André, M. (2009) Cerebro, cognición y matemáticas. Revista Latinoamericana de Investigación en Matemática Educativa. 12 (2) 215-250

Serra Grabulosa, JM., Adan, A., Pérez Pàmies, M., Lachica, J. y Membrives, S. (2010) Bases neurales del procesamiento numérico y del cálculo. Rev Neurol. 50 (1) 39-46.


  1. I do believe that your knowledge and experience can be used into the american education, actually I would think you are using basic/universal concepts to explain how children can understand easier. I notice a lot of experience, but at the same time you as many others, can't open a door to make it work. But I appreciate the level of your ideas.

  2. I agree Sandra, I think all these articles shows a new way to understand education and learning but the best I see it's so easy to read and understand. We keep forward to meet the author face to face!.

  3. Thanks Sandra and Dolores, I do believe we must change the education, the old model can't continue, simply because we are too many persons in the world and because new technologies are opening another doors. Children deserve much more than just read or sum, they deserve think, dream and build a better world.