2.1 About mathematical knowledge

By having access to historically elaborated knowledge, new generations ensure that it remains alive and accessible to future generations. This process allows people to build on what has already been consolidated, so they don't have to constantly reinvent existing knowledge, but simply move forward and create new qualities of survival.

Part of this knowledge is mathematical knowledge, which can be understood as an instrument that allows, among other things, the solution of social problems.

In this course, mathematics surely contributed to human development reaches its current dimension and everything indicates that it will continue to be a substantial part of the production of new goods, whether material or not, created to contribute to the solution of a problem that may have social relevance, or that simply mobilizes someone. (^{Moura, 2011}, p. 49).

Mathematics contributes to human development by enabling new interactions with the social environment and direct problem-solving, which has the potential to promote human development. ^{Rosa, Moraes and Cedro (2016}, p. 158) explain that "one of the essential elements for the development of the full potential of the individual lies in the possibility of appropriating knowledge and, in particular, mathematical knowledge".

In other words, mathematical knowledge is a human product resulting from the search for solutions to situations encountered in the objective world, and it enables the development of those who appropriate it. However, for this knowledge to promote human development, one of the challenges is to provide teaching in schools that allows students to get closer to the essence of knowledge.

To reflect on this means to understand the social knowledge that is present in the culture of each social environment and to understand the movement that human beings have gone through to universalize it, to bring it into the classroom, with a view to the learning of the students.

When learning mathematics as a language, we need to master its signs, the connections between them, and its syntax. Treating the learning of mathematics as an activity implies making sure that the child has a reason to learn it, that he or she defines the actions necessary for learning it, and that he or she uses tools that allow him or her to have access to the mathematical language and to have access to new knowledge in which it is present. (^{Moura, 2007}, p. 62)

To think mathematically is to articulate knowledge with the social environment in which we live, linking this knowledge to the requirements of human development. The acquisition of mathematical knowledge contributes to the process of humanization, understood as the possibility of appropriating the most elaborate human culture and promoting the cognitive development of the subjects who appropriate it. Meanwhile, we understand how important it is for teachers to participate in training processes that enable them to understand the concept in such a way that they can consider the essence of the concept they are going to teach to promote psychological development, as Davídov puts it:

in such a structuring of education that it is possible to regularly direct the rhythms and content of development through actions that exert an influence on it. Such teaching must actually lead to development and create the conditions and premises for psychological development in children. (^{Davídov, 2017}, p.219)

In order to achieve the development of the maximum potential of the subject, teaching can be organized in such a way as to enable the understanding of the relationships that constitute the concepts covered. Like ^{Sousa (2018}), we advocate the logical-historical movement of concepts as a didactic perspective for teaching mathematics that promotes the development of theoretical thinking in students (^{Davídov, 2017}).Based on this premise, we turn our attention to the possibility for future teachers to experience training from this perspective.

Aware of the fact that the subjects in the curricular matrix of undergraduate programs do not always offer this alternative, the training space to which this article refers was organized, which dealt with the thematic unit "Quantities and Measures", guided by the study presented below.

2.2 Brief essay on the logical-historical movement of quantities and measures

Different human actions and problems have led to the creation of knowledge to solve situations in the social environment. However, each people developed their own ways of satisfying them, since sharing information with other groups didn't always happen due to the objective conditions of the time.

Every piece of knowledge went through different stages until it became universal. This is because human beings had to understand the space around them and the relationships that were part of it, thus systematizing a great deal of knowledge, one of which was mathematical.

From the earliest times, quantities, and measurements were in demand because, when man began to relate to space, he was already using this knowledge.

Technology, productivity, transportation, consumer needs, and many other factors determined the type of measurement in each region. For example, in societies with vast fertile regions, the system of agricultural measurement was poorly developed, while in societies whose economy was based on mineral exploitation, units of weight or volume were more perfected than units of linear measurement. (^{Silva, 2010}, p.14)

The different social demands made people create and improve their ways of measuring, inserting them into the context that required their use. In this sense, when studying quantities and measures, the focus should not be on their numerical results, but rather, according to ^{Silva (2010}), on knowing the path taken by measures over time, as this is how we will be able to get closer to their logical-historical movement of constitution. ^{Kopnin (1978}, p. 183-184) explains that:

Historical means the process of change of the object, the stages of its emergence and development. History acts as an object of thought, as a reflection of history, and as content. Thought aims to reproduce the real historical process in all its objectivity, complexity, and contradiction. Logic is the means by which thought accomplishes this task, but it is the reflection of history in theoretical form; in other words, it is the reproduction of the essence of the object and the history of its development in the system of abstractions. History is primary in relation to logic; logic reflects the main periods of history.

Although it is not known exactly when measurement began to be part of human action, ^{Eves (2004}) points out that the first agricultural or Neolithic revolution resulted in the organization of a community that was no longer primitive, moving from gathering food to producing it. This led man to seek new practices, to organize himself into groups and communities, and consequently to establish rules for social coexistence.

At that time, very elaborate measures were not required, since a few indications of "more than" and "less than" were enough, as ^{Silva (2010}, p. 38, emphasis added) points out.

primitive man didn't need a very elaborate system of measurements. Their metrological needs were certainly only for some rough indications of positions, approximate distances and relationships of magnitudes such as "greater than" and "heavier than" or "less than".

These indications satisfied the context of that time, based on primitive actions, since few elements were available for their realization. However, the observation of celestial bodies and regular climatic changes resulted in progress when it came to measuring.

The records of the passage of time were crucial in the appropriation of the movement of measuring things, through the observation of daily events (dawn, dusk, shadow projections, star recognition, solar and star clock, etc.), monthly (lunar phenomena, days, months, and weeks), or even annual (seasons, behavior of the stars, etc.). (^{Pozebon, 2017}, p. 114)

With these records, humans began to perceive the regularities of the passage of time, which led to the organization of one of the first elaborate human acts, according to ^{Hogben (1952}), the calendar. From this process, humans began to measure time, which allowed them to advance in other mathematical knowledge, such as "biunivocal correspondence and organization by groupings" (^{Pozebon, 2017}, p. 114).

With the advances in observing the passage of time, the techniques of cultivating land and moving animals to more fertile land also led to an improvement in communication related to metrology⁵.

However, from the moment it became necessary to cultivate the land or move animals to more fertile pastures, there was also a need to communicate more conveniently in metrological terms, and it may have been at this time that the first units of measurement appeared. They were simply based on the dimensions of the human body (^{Silva, 2010}, p. 38).

With new practices in terms of cultivation and caring for livestock, new contexts emerged and, due to its practical and symmetrical nature, the human body was used as a unit of measurement. According to ^{Crease (2013}, p. 12), "the human body was the first and oldest measuring instrument. Feet are accessible, everyone has them. Almost every civilization, at one time or another, had a 'foot' unit, often divided into 'toes'".

Various parts of the human body became units of measurement: finger, palm, hand, forearm, foot, step, to measure length; handfuls and pinches, to check volume and mass; and heartbeats and generations, to check the passage of time.

Man took himself as the standard of measure. This was the oldest and most universal system of measurement because a system of measurements defined in this way was highly convenient; everyone understood it easily and carried it with them at all times. Individual differences mattered little (^{Silva, 2010}, p. 38).

As anthropometric measurements could be used at all times because they were practical and shared by everyone, they became important allies in solving many problems arising from social actions. However, as new interactions were established, individual measurements were replaced by collective ones.

As man evolved and formed the first civilizations, it was naturally noticed that individual dimensions were unfortunately not regular enough to have a uniform and precise system of measurements. There were tall men and short men, fat men and thin men. It was therefore impossible to have a single standard. It was then that, although less practical, the first material standards were created (^{Silva, 2010}, p.39).

Therefore, although the different measurements based on the body's limbs were a more practical way, they were no longer enough in the face of the new interactions that were being established, and these subsidized a new social practice.

Even in prehistoric times, however, humans discovered the need to select a particular object for many purposes to define a unit of measure - a foot in length, not yours or mine; a carob bean, not this or that. This is called a standard, which is a sample of a particular quantity that we choose as the value 1 of that quantity. When a standard is created, it embodies the unit, giving it a specific, concrete identity as an artifact. (^{Crease, 2013}, p. 22, emphasis added)

Thus, standardization, as a social issue, became part of new interactions, and anthropometric measurements had to be materialized to be used. Because of their practicality, anthropometric measurements never ceased to be used, but to standardize them, the measurements of someone in a position of power began to be used, especially in the Middle Ages with the feudal system.

In this way, quantities and measurements became a tool of power because "the possession of this standard became associated with political and social power, with the authority of kings and the grandeur of God" (^{Crease, 2013}, p.22). In short, standard units became the measure of whoever was in power.

In Ancient Greece, measurement was considered an attribute of sovereign power. In Athens, the standards of weights and measures were dedicated to the gods and kept on the Acropolis. The Athenians even maintained a company of 15 officers, called "conservators of measures," who were responsible for guarding the original standards and inspecting or calibrating the copies. In Rome, they were kept in the Temple of Juno on the Capitoline Hill; and likewise, other important cities had their own means of preserving standards of measurement (^{Silva, 2010}, p. 30, emphasis added⁶).

With different standards, often varying between communities in the same city, quantities, and measures were widely used, with each community adopting its definitions. This meant that the most commonly used units were improved to meet the new requirements of those people, which were not necessarily those of others.

The local measurement systems of different societies were as original and varied as their works of art, political systems, and other forms of cultural life, and their conceptions of the meaning and purpose of measurement were equally diverse. The greater the importance a society attaches to a particular aspect of the environment - gold in West African cultures, salt in Mesoamerican communities, court ritual in China - the finer and more elaborate the measures of that aspect tend to be, and the more specified and regulated those measures. (^{Crease, 2013}, p.27)

In this way, the different actions reflected directly on the development of the magnitudes and measures of each people. With the emergence of new practices, such as the activity of commerce, which resulted in the control of measures, together with the character of justice, new relationships with measures were being established.

In the same direction of control and power, since the first organizations in civilizations, weights, and measures have been crucial for trade, directly influencing exchange relations through the level of local commercial development, since a more elaborate system of measures could always be found where trade was also more developed. (^{Pozebon, 2017}, p. 117)

With the growth of trade, measures began to have a direct influence on the exchanges that took place, since they had to be fair, and since this movement involved other people and peoples, it required greater care. In this way, the practices of exchange triggered various criteria that were established between social spheres so that they could take place in the best possible way. In addition, since there were more interactions between peoples, the action of transporting materials began to consolidate new practices involving measurements.

The transportation of goods generates units. Those that had to be carried on animals' backs were measured in sacks, bags, or parcels, the size of which depended on the animal, the goods being carried, and the distance. Goods transported by other means could be measured in cartloads, wagon loads, and boatloads, or in barrels or casks specially made to fit in such vehicles. The evolution of needs and technology - new markets, better transportation - reshaped the old units and created new ones. (^{Crease, 2013}, p.68)>

As the transportation of materials required new units of measurement, new forms were developed, such as the creation of the bag or package unit. Since these units depended on external conditions such as distance, how they were transported, and what was being transported, measurements changed according to social conditions, since they are the product of interactions with the social environment.

In this way, external conditions led people to use different units of measurement, as each person used what was closest to them or transformed materials to fit their reality. Although the different units and measures used initially responded to the specific actions of each people, as trade and population intensified, this became unsustainable. A common language was needed to facilitate and harmonize this process. It was then that standardization became a social necessity for peoples.

The creation of a universal system of measurement was no accident. At the end of the 18th century, with the advance of trade and industry in Europe, it became necessary to establish a universal language of measurement designed to harmonize production and trade. (Lanner de ^{Moura, 1995}, p. 45)

Trade and industry brought new actions that led to the standardization of quantities and measurements, but at the same time, many movements were taking place. In addition to conflicts, politics, and power relations, science was advancing and searching for increasingly effective measurements.

Scientific thought also changed dramatically. Until almost the end of the Middle Ages, Aristotle's view of the universe was that it was a cosmic ecosystem, with very different regions - primarily the heavens and the earth - containing different kinds of things for which different measurements were appropriate. Localities required local measurements. Science was qualitative; rules were generalizations of how nature usually works, as we humans usually experience it (^{Crease, 2013}, p. 72).

This view of science ceased to be represented by rules and began to be regulated by laws produced not by general aspects but by measurements, making it a quantitative science. As a result, a new idea was introduced into studies because "this new world is measurable, calculable, and universal. Nothing is final, the world is open-ended, and everything can be measured and remedied with infinitely greater precision" (^{Crease, 2013}, p. 73).

As new contexts emerged, measurements and units changed and influenced this whole process. Quantities and measures, therefore, can be understood as a product of the movement of humanity, re-signifying themselves according to the transformations people made in the social environment. In short, various actions have led to the use of new units of measurement.

In the logical-historical movement of quantities and measures, while some systems have not been perpetuated, many others have made it possible to satisfy social needs and have transformed the quality of human life. Regarding the appropriation of this knowledge through schooling, it is worth highlighting the importance of the teacher's knowledge as a condition for its adoption as a didactic perspective for teaching. That's what we'll talk about next.