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If in When and how do children enter the world of numbers?, we talked about children’s relationship with the concept of number from birth to around age three, today we will focus on the following year. What will children of these ages encounter in relation to the concept of number? What will they learn? What abilities will they develop? In general, we can say that they will have more experiences related to this abstract notion, and this will make them learn things, for example, counting beyond basics, identifying, comparing and relating small quantities, as well as reading and ordering new symbols…
But before going into detail about What they will learn about numbers around age of four, I will make a brief note on How this content is learned at those ages.
How are number concepts learned in children?
Based on the idea of Bruner’s spiral mathematical curriculum by Bruner, we know that there is an appropriate version of each new knowledge for each age. That is, it is much better to begin with a simple, initial and preparatory version of the new knowledge, than to try to achieve total assimilation of the content at the moment of learning it. This is a key idea for understanding the learning process of the concept of number and everything that surrounds it.
Still following the idea of spiral learning, Bruner proposes that, at the beginning, when new content is introduced, the goal is more to awaken curiosity, interest and connection of new ideas with reality, than the student’s total mastery of it. And we can ask ourselves, then, what happens with this knowledge? Does it remain incomplete? Don’t we expect children to “assimilate” the concept of 4, of 5, of 6… when we present it to them? Well, the answer is no: that is not the initial goal, they are not yet ready. Our initial goal is for them to feel curious about numbers, to want to know more about them, to try using them in appropriate contexts and, therefore, to gradually build these notions as they apply them.
Research tells us that the deepening and enrichment of this first approach will be carred out gradually, for example, in activities such as workshops, games, stories, spaces, in real classroom situations and even in following grades. We call this spiral learning, in which fundamental contents appear and reappear repeatedly throughout the school year and in following grades, gaining in each step in understanding, connections and depth. In this way, the network of basic, abstract and solid notions that will be the foundation of all subsequent mathematical construction is built.
What will they learn about numbers around age four?
Now yes: what do children of these ages learn about this topic? In general, we can say that in this grade they begin to build the concept of number, they are creating initial ideas about what numbers are, how they are organized and how, when and why adults use them. If we observe in detail what this What do they learn? includes, we see that we can distinguish, broadly speaking, two types of knowledge: social knowledge of numbers and logical-mathematical knowledge (Piaget).
social knowledge of numbers
Association of each written form with a quantity and the word that designates it.
Increasing competence in rote counting while associating each written form with the corresponding word.
In general, all this social knowledge leads to the construction of some inicial ideas:
- The same quantity can be recognized, identified, and named in different ways: as a set of objects, with a word, and with a written form.
- These number-words and these written forms can be sorted.
This social content of numbers, for years, has been the main —and often the only— content taught in early childhood education. Now we know, however, that this is neither the only content they can learn, nor even the most important. Let’s see what else they will learn during this school year.
The logical mathematical knowledge of numbers
Logical-mathematical knowledge cannot be transmitted from outside; each person must construct it for themselves. This knowledge is not directly perceptible from reality; to construct it, it is necessary to establish relationships between various elements, it is necessary to reason. So, what ensures that this happens? That children construct this type of knowledge? Well, the role of teachers and the school and family environment is fundamental. We must create numerical and quantitative situations with challenges, questions and provocations that stimulate children to reason and seek answers to solve these issues.
Focusing again on logical-mathematical knowledge, in relation to the concept of number, we expect that the child, around four years old, begins to discover the two key relationships that underpin this notion:
- The equivalence relationship: “the same as,” “the same quantity,” “there are equal amounts.” That is, beginning to recognize that “the same quantity can take very different physical forms.” Specifically, with small quantities, being able to recognize that in two different groups of objects there are “as many in one as in the other”. Beyond perception, size or volume of each collection, there can be the same number, that is, the same quantity.
Math Materials Table Space. Quantity: “as many as.” Here we establish equivalence relationships, placing as many elements as dots the reference quantity has.
- The order relationship: “more than and less than,” “being bigger and smaller than.” That is, beginning to identify where there are more or fewer elements when comparing two or more groups. Specifically, to do this you must not be influenced by perception; it is necessary to ignore the space and volume that each collection occupies to base it on reason, solving the question by establishing relationships.
Counting Workshop 0-5. Board game: The highest card wins in I4. Here we compare quantities and establish order relationships to determine which is the largest quantity.
This knowledge in the classroom will emerge step by step; at the beginning, they will start to solve it correctly with small quantities: 2, 3, 4 or 5. Little by little, children will become capable of comparing more than one group and being able to answer: Where are there more? Or, where are there two groups with the same quantity? And to do it correctly, they will gradually set aside “perceptual arguments” to gradually choose arguments based on “reasoning.” Even so, at these ages these arguments are still very weak. It is normal that, often, perception dominates over reason, especially with groups of 6, 7 or more elements.
Do you remember Piaget’s “conservation of discrete quantity” test? It showed us that children of these ages, after having matched two groups of 7 or 8 elements and having recognized that there were “as many objects in one group as in the other,” if an adult moves the pieces of one of the groups leaving more space between the objects and asks them again if there is still the same quantity in both groups, most children say that “there are more” in the group where the pieces occupy more space.
Therefore, at this moment perception often still dominates reason, especially with large groups. This is completely normal because perception has been the main channel for new knowledge since birth, but now, at school, we accompany them to discover these new tools that are beginning to develop. Without rushing, without expecting them to master everything immediately, without expecting all children to get it right at the same time, we present them with new challenges and accompany them to discover and apply these emerging reasoning abilities, avoiding focusing on errors and reinforcing each success on the path toward building the concept of number.
What does what we have just explained tell us pedagogically?
To summarize, we, the teachers, the adults, cognitively challenge children by creating situations in which numbers and quantities are present and are the necessary content to understand or solve what is happening. As Bruner proposes, we aim to awaken curiosity, interest and connection of new ideas with reality, and we give time for children to apply them and make them their own. We do not expect from the beginning that they solve everything correctly. We accompany them to read the written forms, to recite the series of number-words in order, to make associations between quantities, words and written forms, but we also present them with situations in which it is necessary to compare quantities, order them or make them equal, and we give time for them to find their own resources to solve it. We help them master and feel confident making these comparisons with small numbers before moving on to ordering and matching with larger quantities.
We establish equivalence relationships: “As many as,” the same quantity.
We establish order relationships : ‘ from smallest to largest ‘
Is that all? What else will they learn by combining this knowledge about numbers?
What we have talked about today is just the beginning; there is much more, but everything will be based on what we have just explained. The list of what they will learn this school year about this topic is long. Children in I4, in addition to what we have presented in this blog, will increase their knowledge about:
- The subitization,
- Cardinal counting,
- Transformation of quantities,
- Addition and substraction,
- Decomposition of each number,
- Inclusion of smaller quantities,
- Problem solving,
- Introduction of zero,
- Appearance of new mathematical symbols: +, – and =,
- Counting backward…
Since the list is long, and these topics deserve a calm look accompanied by theoretical insights, we believe they require a new blog entry.
See you soon!
Mequè Edo
References
Bruner, GS (2018). Cognitive development and education . Selection of texts by Jesús Palacios. purple
Kamii, C. (1984). The number in preschool education . viewer
Piaget, J., Szeminska, A. (1975). Génesis del número en el niño, (4a. ed.). Guadalupe.
