On some mathematical concepts / themes to be faced in the 2nd part of primary school
("interwoven" within teaching units)

[primary - A   primary - B   lower secondary   upper secondary - A   upper secondary - B  school]

The concept of model
The concept of number
Elementary arithmetic
The concepts of ratio and proportionality
The diagrams
Approximations. Calculators. Computer
Descriptive statistics
Formulas, terms, graphs
The concepts of function and resolution of an equation
Theory of probability
Relations with the other disciplinary areas

The same contents present in the document relating to the next school level are more or less faced, with a different slant; however, many topics are also dealt with in the one relating to the previous level.
The indications refer to how much should be faced on average in the 2nd part of primary school. They are presented articulated by thematic areas but, as it is clarified, in the didactic paths the different mathematical concepts must intertwine with each other and with the other disciplines.

The concept of model

    Mathematics can be called the science of models. The concept of model must therefore play a central role in its teaching from the first levels.

    The theme of modeling unites all forms of knowledge; this offers numerous and fruitful opportunities for interactions between the teaching of mathematics and that of the other disciplines.

    The following figure illustrates the relationship between "model" and "reality".  Given a real situation, a model is a simplified representation that illustrates some aspects, for certain purposes.  Depending on the purpose there may be different models of the same situation (an airplane model may have the same shape and color as a real airplane but not fly, or it may not resemble a real airplane but fly).  The "goodness" of a model depends on its adequacy to the objectives for which it was built (to better highlight some aspects, to generalize some properties, to facilitate comparison with other situations modeled in a similar way, ...).  The preliminary phase of the modeling circumscribes the aspects of reality involved in the problem to be studied.  In the following diagram, the environment is the part of reality that is isolated, not in detail, in this way, and the situation is the complex of aspects of the phenomenon to be modeled (including assumptions, intuitions, perceptions, intentions, ... of who builds the model) which will be taken into account in the representation:

model and 

    The same model can be used to represent different situations. A very simple example is the concept of arithmetic mean, which can be used to indicate, in a given country, per capita consumption of meat, the average family income, the average height of the twenties, …  Another common example is the direct proportionality, which can be used to represent the relationship between the weight of a food product and its cost, between the lengthening of a spring and the weight of the object hanging on it, between the distances among the parts of a car and those among the corresponding parts in a its model, …

    And the same situation,depending on the needs, can be represented with different models.  The figure below on the right reproduces part of the graphic index printed on the first pages of a railway timetable:  it is a map in which the railway lines are reproduced and the relevant time frames are shown; it is a different model from a usual map in that it does not correctly represent distances and directions.


    Let's see another, simple, example in which we face a problem with two different mathematical models.


    I have a die made of cardboard. I cast it repeatedly. I want to evaluate the probability that a number less than 3 will come out.

    I can study the problem empirically:  do 100 tests, obtain a histogram like the one on the side and conclude that the probability sought is about 29 + 23 out of 100, or about 52%.

    Or I can deal with the problem in a conceptual way:  I can hypothesize that the die is perfectly balanced and therefore believe that "1 and 2" have 2/6, or 1/3 as the probability of exit.



    The "conceptual" model allows me to find an answer quickly, but it is based on the hypothesis that the dice is perfectly balanced (as do not be the made with cardboard dice and not even the usual game dice, in which, for example, the face with "6" weighs less than face with "1").  The "empirical" model, based on experimentation with "my" dice, would allow me, if I had a lot of time, to find a more correct answer (see the figure on the left); but I would need probabilistic tools to evaluate its accuracy.  Then there is a relationship between the two models: I use the empirical model to evaluate the adequacy of the conceptual model. In short, the use of mathematics does not consist in brutally applying formulas, but involves "non-mechanical" cultural attitudes; beyond this example, it is also what school must train the pupils to.

    In many cases, as in those now exemplified, it is necessary to reinterpret the model by verifying its adequacy to represent the studied phenomenon and possibly to clarify the situation to be modeled. In the diagram we put a double arrow model ↔ situation to highlight this refit.

    The figure examines not only the model-reality relationships but also the "tools" used to build the models and how their development interacts with the educational process, i.e. with pupils, teachers and the whole of knowledge (disciplines, disciplines, techniques, .…).

    Models are abstract representations of "real" objects or phenomena (of a material type: a topographic representation is the model of a territory; of a social type: the concepts of "verb", "noun", ... are models for representing certain elements of verbal communication; ... or abstract type: the "commutative property" is a model to describe an aspect of some mathematical operations).

    Models are built using cognitive artifacts, i.e. "material objects"  (paper, signs, sounds, colors, ...)  or ad hoc "artificial constructions"  (language, concepts, ...)  that man uses as a prosthesis of his mind. The term "cognitive artifact" was introduced by Donald Norman in 1993 (Norman D.A., Things that Make us Smart, Wesley Publishing Company, Addison, 1993), , while that of "prosthesis tools" was proposed by Jerome Bruner in 1986 (Bruner J.S., Actual Minds, Possible Worlds, Cambridge, Mass., Harvard University Press, 1986).

    The development of cognitive artifacts to model situations does not happen episodicly, but in a social context of cultural growth that is gradually being organized into structured forms of knowledge, which are transmitted from one generation to another through educational processes, in which pupils and teachers interact with each other and with knowledge, the one under construction and the consolidated one.  These reciprocal interactions are described by the triangle shown at the bottom right in the initial figure.

    What has been observed so far applies to all disciplines and organized forms of knowledge. But mathematics has the specificity of not being characterized by a particular area of problems or phenomena that seeks to model (such as physics, history, linguistics, ...), but by the type of artifacts it employs for the construction of models, and which are used in all other disciplines.  So the graphic diagram seen above (which in itself is a simplification) in order to represent the situation of the teaching of mathematics must at least be enriched with the vertical hatch depicted on the side:  the artifacts, for mathematics, soon, from cognitive tools become objects of knowledge, from models that "abstract" from situations become gradually "concrete" tools to develop new abstractions, in a never-ending spiral.  Mathematical knowledge, which arises from modeled contexts, is organized internally not on the basis of these, but on the relationships and structural analogies between its artifacts.

    The situation, then, becomes more complex as both the "triangle", and the type of "artifacts", with the advent of computer science, have become more articulate: the software has become an "animated" interlocutor that interacts between the different subjects, in ways depending on the use that is made of it and the awareness with which it is used. Without further complicating the previous schematization, it should be borne in mind that the software now has, and will increasingly have, a decisive impact in the way different aspects interact with each other.

    In summary, however, we can say that the study of mathematics is articulated in the (non-linear) relationship between  posing problemsmodeling  situations to deal with the solution of problems,  building and resorting to  theories  that internally organize the relationships between the artifacts used for the construction of models and develop new artifacts.

    All this makes the role of the teacher crucial in mathematics education. He must:
  design and curate educational pathways that give concreteness to artifacts as they go further away from elementary forms of perception,
  to bring out "reality"-"abstract concept" conflicts (from those between "physical" and "mathematical" mirrors to those between "common" language and "mathematical" language, for example when talking about angles and sides or about rectangles, rhombuses, …, and to the many other conflicts present since the first teaching experiences)  They are conflicts that, if not explicit, risk being sources of misconceptions, while, if faced, they are an opportunity to transform a "destructive opposition" into a "productive dialectic", which contributes to building an adequate image of mathematics as discipline.
  educate to the choice of models (there is no "best" model) depending on the needs and "resources" (physical and conceptual artifacts) available,
  organize teaching so that references to real objects or situations are not only pretexts  but also establish virtuous relationships with the extra-curricular knowledge (and motivations) of the pupils,  decentralizing,  trying to have as a reference not only their own knowledge and motivations but, in a dialectical relationship, also those of the pupils,
  and give organicity to the knowledge they have acquired in the mathematical field  (even integrating the study of problematic situations with the development - starting from them - of new concepts and the consolidation of some operational skills through appropriate exercises, which depend on the level of knowledge and technologies historically available) so that they become a solid starting ground for new abstractions,
  all this taking into account that, especially in the first levels of education, the building of virtuous relationships with the extra-school also depends on the involvement of "families": it is necessary to make them participate "culturally" in the educational project that is being carried out (participating "culturally" does not mean "to make the repeaters", but to collaborate with teachers in the construction of relationships between school activities and extracurricular life); this is one of the most difficult tasks; this aspect should also be properly included in the "educational triangle" considered above ...

    HERE you will find some examples that illustrate the "the limits of the models".  This is an aspect, also mentioned earlier (when we have observed that the same situation can be represented with different models) and that is also taken up in later places, to which it is very important to emphasize in teaching.

The concept of number

    The first three of the following images illustrate some (conceptual and practical) aspects of numerical mastery that should have been started in previous years:  using numbers to count,  to represent and operate with monetary values,  to read and represent quantities by using measuring instruments of daily use,  up to the first representations on the timeline (for example to mark events that occurred during the month or during the year).  In the second part of primary school, events can be read and represented on a "longer" time line (first up to the age of great-grandparents, then moving back through the years).


How many minutes are there in a quarter of an hour?  And in three quarters of an hour?

    By referring to the daily uses of the number,  • some writing and numerical processing skills must be developed,  • the habit of making (implicitly) transference from one context to another to interpret some simple operations (translate quantitative problems into problems on the number line or vice versa) must be consolidated; and  • the habit of operating on divisions by reasoning either by quotition (200/50 is 4 since 50·4 = 200) or by partition (200/4 is 50 since dividing 200 into 2 parts I get 100 and dividing it further in 2 parts I get 50) must be consolidated, even if the concrete problems faced with them were, respectively, of partition or quotition.
    Starting from the most varied contexts, the interpretation of "abstract" numbers as positions on the graduated line must be started, which must then be consolidated in the lower secondary school:  it is one of the few automatisms (processing, association, passage from one representation to another, ...) that everyone must be able to exercise without mnemonic or reflective efforts (to free mental resources to devote to more conceptual aspects);  they are skills / attitudes that should be continuously consolidated / maintained.

    It is also appropriate to make students reflect on the ambiguities with which, in verbal language, numbers with "decimal point" (or "decimal comma") are expressed as well as to point out (eg by observing the menu-options of a computer or mobile phone) the various conventions ("," and ".") which, in the commercial sphere and in daily life, are used on the one hand in Italy, on the other in Anglo-Saxon countries and in the scientific world, to separate the integer part from the fractional part.

The decimal mark in oral language is often read "and"; for example "1.23" is read "one, point, twenty-three" or "one and twenty-three", as well as "one and twenty-three cents"; in front of "5.03", if you don't add "cents", you have to say "five and zero three", to avoid confusing it with "5.3".  In everyday dialogues, different conventions are often used depending on the context in which we find ourselves. For example, when we talk about cars, saying "displacement thousand and three" we do not mean 1003, but 1300, i.e "thousand and three hundred"; in other words we mean "one hundred". Speaking of money, "one euro and three" generally means 1.03 , i.e. "one euro and three cents", not 1.3 (that is, "one euro and thirty cents").

    Intertwined with the consolidation of the concept of "number" are the first reflections on the powers, on the approximate calculation, on the line of numbers (and on negative numbers), ... which are mentioned in the subsequent comments.

    The mastery of limited decimal numbers, the ability to use graduated measuring instruments, ... are fundamental, in particular, to informally start the concept of unlimited numbers (concept which will be taken up in the lower secondary school and which in the upper secondary will also be called real number).

    Without a first development of the concept of unlimited number, the use of square roots would make no sense (there would be no number that squares 2), nor, for example, the use of π.  Nor would "abstract" geometry exercises, in which the sides or corners have "exact" measurements, make any sense. It is then necessary, in a simple way, without formalisms, to focus on the fact that exercises are performed with exact measures to simplify life without taking account of approximations, but that in reality there are no objects whose length can be measured exactly (see the animation):

    And it is necessary to focus, always without formalisms, that while the lengths of objects do not have the contours precise, in the case of the times we do not have these limitations: the time varies continuously, not in jerks. It is we who cannot measure it exactly.

    There is no need to dwell on how sums and products of unlimited decimal numbers can be made or how we can calculate the square root of a number, which we will focus on in subsequent classes.

    Obviously, it makes no sense (neither "didactic" nor "cultural") to address the calculation in every numbering bases in general terms.  Another thing is to start using the numerical notations used in time measurement (bases 12, 24, 60), on which pupils already have operational skills acquired gradually over the course of their lives (see the figure at the beginning of this paragraph) , and from which it is necessary to start by referring, operationally, to the concepts of "ratio" and of "change" of units of measurement.


    In primary school the use of powers must be started in relation to the measurement of areas and volumes:  it is learned, operationally, that in 1 decimeter there are exactly 10 centimeters and that in 1 cubic decimeter there are exactly 1000 cubic centimeters, and that not randomly 1 cubic centimeter is 1 milliliter, being 1 liter equal to 1 cubic decimeter.


    It is necessary to focus the utility of the powers not only in relation to the use of different units of measurement, but also to the gradual education in the use of pocket calculators. The use of exponential notation to describe large and small numbers will be further explored in secondary school.  


Elementary arithmetic

    We have already discussed how the first forms of calculation should be started by discussing numbers, which inevitably can only be introduced contextually with the exercise of some calculation activity (we highlighted how in order to talk about "numbers" it is inevitable to talk about "operations ", we discussed the relationship between the divisions by quotition and those by partition, …).

    The formation of the ability to work with numbers must be based on concrete models and closely linked to problematic situations; this is not at the expense of acquiring a mastery of the algorithms, but is for their less superficial and longer lasting acquisition.

    It is worth remembering that concrete models do not mean at all the so-called "structured materials", which are a typical example of abstract models.  There are many problematic situations that refer to reality, and which, moreover, offer opportunities for manipulating materials that have concreteness and richness of mathematical content (and in this sense are "structured"):  measurement of physical quantities , analysis of economic and demographic data, …, geometric and statistical activities, …, in addition to the many other opportunities offered to the teacher by the themes and activities he faces with his class.

    Another suggestion is to limit the number of digits used in the calculations by favoring the acquisition and consolidation of the positional notation and of the "changes".  The concrete models, both through practical activities (e.g. working with coins and banknotes, with measurements of capacity, lengths, …), and then, through verbal and graphic references (graphically and verbally illustrating situations of economic exchange, measurement activities, …), can help in limiting and isolating the figures that actually come into play in the calculations and in focusing on the idea of the "changes":   to add 15 cents and 20 cents or to add 200 euros and 300 euros I do not have to add up numbers expressed in cents or hundreds, but, using the unit "cent" or "one hundred euro", I can make 15 cents and 20 cents = 35 cents or 2 hundred euros + 3 hundred euros = 5 hundred euros;   dividing a liter of milk into three parts does not involve a division in which decimal figures intervene if we consider that 1 l = 1000 ml and we make 1000:3, eventually transforming the result into liters;   or to calculate thirty cents by 5 you can make 30 cents · 5 = 150 cents = 1 euro and 50 cents before having developed the technique for multiplying decimals; …

    Activities of this kind, in addition to allowing to perform calculations conceptually within the reach of the pupils (without waiting for the intervention of new "magic" rules and techniques that allow to face the more "difficult" numbers), develop numerical competence and "operational" mastery of arithmetic properties (possibility to reorder a sum or a product, distributivity, multiplications and divisions by 10, 100, …).  Once these skills are consolidated in "contexts", it is easier (thanks to the presence of these reference models that act as conceptual "prototypes") to transfer them to activities with pure numbers.  Similar considerations apply to the introduction of relative numbers based on significant situations (temperatures, floors by an elevator, level of rivers, time difference compared to a day or a fixed time, …).

The temperature goes      
from −4° to 1°:  rises by°.  

    At the same time as the suggestion not to carry out early calculations in writing involving too many figures, the opportunity to gradually develop the calculation techniques is underlined, of which the pupils must be able to understand the meaning.  The following figures exemplify some possible developments (subsequent to those exemplified by the figures of the entry "the concept of number") through which some techniques can be introduced.


    Some animations

    Calculation techniques that employ schemes similar to those illustrated above can constitute an intermediate step for a motivated introduction of more "automatic" procedures, but they can also be used later:  in addition to being more "controllable" than the latter ones, in many cases they are also more effective.  In fact, they do not provide for a univocal development, but allow to proceed in different ways according to the characteristics of the numbers involved (and the tastes of the person performing the calculation).  Activities of this kind (and, more generally, the use of different algorithms to obtain the same result) are very important.
    Obviously it is necessary to consolidate the multiplication table "mnemonically".  There are those who neglect this aspect, there are those who resort to funny memorization strategies (there is a large repertoire of odd techniques and tricks), but there is no other way than to fix in the mind (even through repetitive activities, which are easy to motivate and can easily be integrated with the use of calculators and with the rapid mental calculation, which we mention later) the results of these (64) multiplications.
12345 6789
246810 12141618
3691215 18212427
48121620 24283236
510152025 30354045
612182430 36424854
714212835 42495663
816243240 48566472
 918273645 54637281

    We quickly mention other related activities:  different descriptions of the same algorithm (different formulas, flow graphs, …),  approximate calculationstatistical activities,  activities with different bases (used in "life") that can be offered by measurement activities and use of physical quantities (time, rotations).

    We refer to these exercises, and to those present in the following pages, for further ideas.

    It is also necessary to keep in mind the possible presence of pupils with dyscalcium or dyslexic disorders.  For this problem, we refer to the considerations made HERE in the section dedicated to secondary school.

    Arithmetic operations are the first numerical functions with which one has to do in school life. We will come back to this later.

The concepts of ratio and proportionality

    The concepts of relationship and proportionality are gradually built up within each mathematization activity; we have already seen it in the items considered in the previous points (when using different units of measurement, when referring to the use of the clock, …).  Concepts are built in contexts in which they intertwine with each other; then gradually the school should shed light on the differences and analogy relationships between them.

    Among the contexts in which these concepts can be developed, there are many related to historical, technological or geographical issues, also developed during visits to companies. Think, for example, of the observations on the forms and functioning of machinery (possibly also usable for production activities that can be tackled at school), …; for example a machine that rolls out the dough, which divides the thickness by 3 and multiplies the length of the sheet by 3, a whisk with the driving wheel which has four times the teeth of those of the whips, a mill in which the ratio between the pegs of the two wheels that mesh is equal to the ratio between the speed of rotation of the paddle wheel driven by the water and that of the grinding wheel:

    Other contexts allow us to focus on the idea of proportional transformation: if I cut the dimensions of a rectangle in half, the ratio between them does not change (b), if I change the units of measure the ratio between the measures does not change (a); and the relationship between two numbers doesn't change if I multiply them or divide them both by the same number. This is the conclusion at an adult level: the pupils must arrive (through activities in various contexts) to operationally master (not recite) this property.


    In secondary school mastery of these concepts will later have to find more formal ways of consolidation.  But already in primary school pupils must gradually acquire the ability to switch "spontaneously" from one form of representation to another of the most common use ratios (0.75, 3/4, 75%, 45/60;  15/60, 1/4, 25%;  1/3, 20/60, 0.333…, 33%; …), also through exercises that consolidate the various graphic representations (strip diagrams, circular sectors, suitably graduated).  They must master the equivalence between division by quotition and by partition to immediately realize, for example, that 1/0.1 is 10 because 0.1 is 10 times in 1.  They must acquire the idea that "%" stands for "/100".

    The intertwining with the graphic representations is also important for expressing the ratios in percentage form: for the ratio between 264 and 763 (data referring to some context) we can resort to graphic representations first on graph paper (see) then, possibly, also using software (see the figure below on the left), without necessarily moving to numerical calculation, which, towards the end of primary school, can be performed with a calculator (divide and express the result in hundredths: 264 763 100 ).  In particular, it should be noted that the symbol ":" (which appears on the maps) is simply the division symbol, and that this can also be expressed with "/", and that, for example, to find the angle with which to express 264 if 763 is represented with the whole circle, I can, as seen in the previous figures, use a circle divided into 100 parts.


We can also use these simple scripts (see)

    The concept of fraction should be presented as a particular case of ratio: this allows the different aspects with which fractions occur in applications to be well intertwined.  The contexts in which the concept of ratio is used are innumerable; in addition to those considered previously, for example, we refer to the transmission ratios of a bicycle, to the small map of an apartment (in which I can initially represent 1 meter with 5 or 10 squares) …  But we also remember the interpretation of the road signs of steep ascent or the first reflections on the slopes, linked for example to the experiences of scouting.
    It is good to pay attention, especially in primary school, not to confuse the meaning of number with that of the term, which will be focused on in subsequent school levels, and not to call the fractions "numbers". For example, while 6 or 1.333… can be called "numbers", 2/3 (like 2+3 or 2·3) has as its result, or value, a number, but is not a number.  And it is appropriate to use fractions only to express ratios between quantities or quantities referred to concrete contexts, while activities with what we call "pure numbers" should be faced in relation to early activities on the number line (such as those seen above), to be consolidated in subsequent school levels, or to the execution of calculations, by hand or with a calculator. These considerations are not "puristic", but must be reported to the misconceptions that, indirectly, are likely to be created in pupils.

The diagrams

        The use of the various graphic representations greatly facilitates the reasoning. It is also the way in which mathematical elaborations (relating to sociological investigations, economic situations, technical-scientific phenomena, …) are generally communicated to us by the mass media.  It is therefore essential that the teaching educates to their use (both to model situations and to carry out theoretical considerations), to the transfert between them and other forms of representation, to the intertwining of graphic methods and symbolic and numerical methods, …  Specific insights on the Cartesian plane and on the curves they can be started later, at the end of the lower secondary school.

    The following images give an idea of the activities to which the use of graphic representations can be intertwined.  The following graph and the first of the following graphs are related to activities that can be carried out in the first years of primary school.

hours of light and dark recorded weekly from December to July

    The other representations are some examples of the various activities carried out in the last years of primary school  (we observe the graph of a lap of the Giro d'Italia, of the type of those that appear on television - "is the slope of the roads the same as that of their representations in the picture?";  we read the chart of the body temperature of a sick person;  we zoom in or out on a figure);  activities that must be included in specific activities  (the Giro d'Italia passes near the school;  in this month many were sick with a high fever;  let's make plans of the classroomor;  let's make small-scale photocopies of …).




    These brief considerations must be intertwined with the numerous others on the graphic representations developed in the other paragraphs of the document.

Approximations. Calculators. Computer

    We have already pointed to the approximations towards the end of the section dedicated to the "number concept". The important aspect to focus on is that exact measurements of lengths, times, areas, … cannot be obtained, but only approximate values: it does not make sense to write many figures like result of a concrete problem, thinking of representing exactly the solution.  Obviously, in this scholastic level, it is necessary to limit oneself to approximate the final result to a significant order of magnitude based on the context, without too sophisticated mathematical reflections, thinking about the concrete meaning of the digits that are considered.  If I have monetary values I will have to approximate them to euros or cents, up or down, depending on the context.  If I have a rod 48.5 centimeters long and I have to divide it into three equal parts, I will round the result to 16.2 centimeters.  If I want to express the height of a building I will say that it is, for example, about 12 meters high, without considering the centimeters.  And if I perform operations with a calculator, at the end (and only at the end) I will have to round off the result taking into account the nature of the problem. And I have to learn to read the measurements on a measuring tape, on a spring balance, on a watch, …

    We must develop these "attentions" from the first years of school, if we want to make sense of mathematics teaching. A more rigorous treatment will be carried out in the various successive school levels.

    We must also begin to make estimates (without calculator, by hand or in mind) of the result of calculations of which we do not care to have the exact result or to check the calculations made in detail. Here are a couple of examples of approximate calculation carried out by rounding the numbers to 1 or 2 significant digits and performing the calculations on the rounded values, which can be faced towards the end of primary school:

2850  ≈  3000  ≈  30  ≈ 7
—— ——
370 400 4
      1530·18 ≈ 1500·20 = 30000


    It is advisable, already from the basic school, to make extensive use of pocket calculators.  It may be helpful for the teacher to review the calculators available to the pupils.  Remember that the use of calculators is indispensable if pupils with dyscalculic difficulties are present, as mentioned in the link in the elementary arithmetic section.

    The objectives should be eiher to acquire a greater mastery of this means of calculation, to understand its limits in order to interpret the results it provides, …, or to introduce and/or consolidate (at a first level) some mathematical knowledge: numbers (approximations, …), on operations (properties, priorities, …), on functions (1 input and 1 output functions, such as , , …, 2 inputs and 1 output, such as , , , , , …, 2 inputs and 2 outputs, such as , …, and  composition of functions, …), …, or to prepare the ground for understanding the functioning of computers (which, as mathematical knowledges and arithmetic calculation possibilities. do not differ "essentially" from a pocket calculator, where we have the possibility to operate with the keys , , , …).

    It is evident that the use of calculators and computers is to be introduced at the same time as the treatment of other mathematical themes, albeit with specific insights.  Examples of activities in this direction that can be addressed at the end of primary school can be suggested by the exercises present HERE.
    Some examples of various types of software can be found HERE.

    Two examples on the right, "division" and "pocket calculator" (usable from the very first years of school).

Descriptive statistics

    Descriptive statistics (to which we have mentioned several times discussing the previous topics, and in particular the concepts of ratio and proportionality) lend itself to the introduction in significant contexts of many basic mathematical concepts, from numbers to approximations, from the concept of function to graphical representations of numerical relationships.  This, at least, if the statistics - which more than 20 thousand years ago constituted the first mathematization activity of man (the use of marks cut into bones and other artifacts to compare quantity of days, objects, people, ...) and can be addressed even in nursery school - is not reduced to being an additional theme to be taught separately from the other themes.  The descriptive statistics tools then serve as points of reference for the subsequent introduction to probabilityHERE are links to programs that can be used for statistical processing.   

    The previous figure referred to a typical activity that can be faced in kindergarten (in the last year of kindergarten, the weather conditions can be recorded on a board and then the relative histogram can be built month by month), and which can also be addressed at the beginning of elementary school, adding other information in addition to meteorological information (absences, written notes on significant facts, …). The figure alongside represents the ways in which the pupils of a class arrive at school (all, and the boys and girls), with a histogram which is flanked by the numbers that express the height of the columns, histogram which can be used at the beginning of primary school.   
    Then it is possible to face many other activities, integrated into didactic paths that involve other "know-how".  An example: a research on previous generations to that of pupils.


    It is an analysis that can be developed through the administration and analysis of a questionnaire and that can be extended to various previous generations (here up to great-grandparents), within a path to build the sense of "history" and the "documentation" that can be addressed for example in the third form, and which can be worked on in subsequent years.  The collection of information on previous generations can also involve several classes from the same school, who can then compare and put together the data and other collected material (photos, notebooks, videos, …).        
    The quantitative aspects can be analyzed through graphs, block diagrams (histograms with crosses) and then also using the percentages (as we have seen, discussing the concept of ratio).  On the side, for example, a histogram on the number of great-grandparents' children and, on the right, that of the percentage distribution of the great-grandparents' age at the time of the grandparents' birth.
    After the pupils have learned to make histograms with crosses, at the end of primary school they can be educated to resort to simple programs like this or this, which can be used online.


    Below on the left the graph of the median heights of pupils of a fifth grade from when they were two years old up to they reached ten (see here).  On the right the graph of the trend of the Italian population detected at the censuses.

height in
various ages
at censuses

    The following figures recall the use of bar charts and areograms to represent classifications in generic classes, not in numerical ranges.  They are different mathematical models of the same phenomenon  (achievable on graph paper and circles graduated in hundredths,  quantifiable with percentages) which visually communicate different information  (the first ones allow you to quickly compare, in both a visual and numerical way, the different per cent frequencies;  the second ones allow to understand what relationship exists between the various parts and the total).

Liguria -  Food  Healthcare  Transport  Communication  Clothing  Housing  Other

    The examples we are seeing highlight another aspect: statistics must inevitably refer to non-mathematical contexts, and is therefore a stimulus  (for pupils and teachers)  to interact with other disciplines, as well as with the different areas of mathematics.

    Statistics also allow us to focus well on how, depending on what interests us, different mathematical models can be considered for the same phenomenon, highlighting or neglecting different aspects, as we have also observed above.  There is no the best model.

    In this regard, we make some observations on the concepts of average and median.  These are non-trivial concepts whose acquisition (which, together with that of the percentages, will continue in secondary school) should be done gradually, without early formalization.  Consider, for example, the differences between situations such as the calculation of the average consumption of milk, sugar, … in which the introduction of the arithmetic average can be controlled as a way of expressing how all consumption would be shared if there was a division into equal parts and situations such as the calculation of an average age or an average height:  while we can speak of "total consumption" it does not make sense to speak of "total age" or "total height".
    This suggests that the concept of arithmetic mean should be introduced for example for the understanding of data relating to the change in consumption of some foodstuffs in the last century rather than to evaluate the anthropometric changes of pupils.  For instance (as we have already observed) the heights of pupils at different ages can initially be compared by building and examining their histograms related to the various ages or, instead of calculating and comparing the average heights, comparing the "median values", ie the values that they are at the center in the ordered lists of the heights relating to the various ages  (if, to give an example relating to a few data, the values in centimeters are neatly:  Mario 108, Lucia 112, Enrico 112, Aida 118, Giorgio 120, the median value is that relating to Henry, that is 112).
    This is not a didactic trick:  for example in the official statistics on changes in the Italian population there are data relating to the median age of death (if the median age of death in a certain year was 42 years, this means that 50% of the dead were aged not more than 42 years and the other 50% age not less than 42 years).

Formulas, terms, graphs

    In primary school it is necessary to introduce the use of "trees", important for gradually learning the meaning of formulas and for moving from one language to another, to focus on the first conventions on the writing of terms, … and to describe procedures that are not easily described as formulas, to solve problems, …

    Moreover, they are used in various other disciplines:  the image below on the left recalls a typical use.  The image on the right recalls a particular use in mathematics, important to focus on the structure of terms.

    The following figures illustrate other typical examples made in various areas, which have strong intertwining with mathematical education:  the division into percentage parts, the division of a class into subclasses (to be dealt with operationally, not with abstract reflections on the "sets").

    It is then necessary to consolidate, through images and practices, the possibility of reordering the terms of a sum or product:

    Finally, by using graphic methods it is possible to face "algebraic problems" which are conceptually within the reach of pupils, without resorting to "algebraic techniques" that can be used in subsequent school levels. An example:


Which "percentage" of 434 is 116?
Which "datum" is 65% of 434?


The concepts of function and resolution of an equation

    The concepts of function and equation are perhaps the most important concepts of mathematics; they intertwine with almost all the other voices discussed here. They are present in programs of all school levels.  The first functions that the children explicitly meet are, at the beginning of elementary school, the four operations (two inputs), the unitary increments and decrements and the change of sign (one input).  But in elementary school, children also encounter functions with any amount of input, such as the maximum and minimum of a set of data, and functions to which a calculation procedure does not correspond (for example the height or weight of a person, or the population of a city, as a function of time, or tariffs of various kinds, in which the monetary value is expressed as a function of various quantities). They also come across multi-output functions (division with remainder, for example, is a two-input and two-output function).

    After all, histograms with crosses are also functions. They can be tackled even before primary school, in kindergarten, as in the cases present HERE:  the number (represented by a column of crosses) of the pupils who use it is associated with each way of getting to school,  each type of holiday resort is associated with the number of pupils who have spent the holidays in that way,  with each weather condition the number of days of the month in which the weather was such is associated, …

    Recall that the name "operation" is an appellation used to indicate some functions, generally with 1 or 2 inputs, but not only.  Also on calculators all the keys to perform calculations with 1, 2 or more inputs are called "function keys".

    The simplest and "cleanest" way to introduce a first description of the concept of function is precisely to refer to images as those above and to the idea of "means of calculation":  functions are ways to associate inputs to outputs.  Then, in later school levels, this concept can be better focused.

    For 1-input and 1-output numeric functions, the related graphs can be considered. Indeed, often, as we have also seen in the previous entries, functions can be presented directly in graphic form.

    Which of the graphs on
the side could represent
how (in the course of
hours) on a sunny spring
day the temperature in
the air changes?

    It is good to indicate the variables in various ways, as happens in everyday life.  Only at the end of lower secondary school will it be possible to gradually deal with abstract exercise and consolidation activities in which variables with names that are independent of the various application contexts are used.  And it is good to ensure that pupils immediately realize that a formula can be transformed by expressing a variable as a function of others in different ways, depending on the needs, informally resorting to the idea of inverse function, as a tool for transform formulas.
    On the other hand, the pupils must have already seen this, operationally, when they had to trace back a problem of subtraction to an addition problem (to find what 100-80 makes I can find what I have to add to 80 to get to 100), or a problem of division to one of multiplication (to find what 100/20 makes I can find how much I have to multiply 20 in order to get 100).  Some other examples:

 Invent realistic problem situations that can be modeled as follows:
 (a) 13 + ? = 29     (b) ? + 13 = 29     (c) 6 – ? = 4     (d) ? – 1.50 = 13.50     (e) ? · 6 = 15  

[Examples.  (a) today is 13th; in how many days is 29th?  (b) today is the 29th, 13 days have passed since she left; when did she leave?  (c) I had 6 euros, I have 4 left; how much did i spend?  (d) I spent 1.50 euros and I still have 13.50 left; how much did i have?  (e) we are 6, we must form 15 euros; how much should each person put if we contribute in the same way? ]

    We have seen other examples under the items "number", "ratio and proportionality", "formulas, terms, graphs".


    A reflection on the approach of geometric teaching in primary school should be addressed starting from some general consideration on geometric teaching, which we can entrust to this "digression", fantastic but very concrete, on the concept of angle.  In it we pose the problem of how to differentiate, by formalization and "technical" development, the teaching of the various themes in the different school levels.  A geometric theme has been taken as a "symbolic" reference because this area of mathematics is the one that in teaching is dealt with in a more "dissociated" way compared to the current state of the discipline.

    The many images in the comments relating to all the other topics highlight how this area of mathematics is also intertwined with the rest.  The aspects of this interconnection are essentially two:  on the one hand, to represent and study geometric objects, measures of distances, angles, extensions, displacements come into play … and realization or study of geometric transformations that involve numbers, arithmetic operations, relationships and more generally functions,  on the other to represent other mathematical objects (from numbers to functions, up to statistics and probability), are used,  from the first cultural (not only didactic) experiences,  geometric quantities and concepts, as we already have highlighted with many examples.
    Referring to the reference systems, on the one hand we will have representations in which the proportions between the two or three dimensions are kept, on the other hand we will have representations in which the different scales are chosen based on the order of magnitude and on the location on the line of numbers of the different entities represented.
    Let us dwell on the first aspect.

    During primary school pupils must acquire (through experiences, observations, drawings, visualization of images obtained with a camera, ...) the ability to associate visions with points of view:

    They must acquire some lexicon, correctly (in the figure below on the left (A) and (B) they are two hexagons because they have 6 sides or 6 corners - not 6 "internal corners": it makes no sense to talk in school about internal angles; what are the "external" angles?), and they must learn how to handle figures and calculate areas of simple figures without learning formulas or recipes by heart (how wide is the blue fabric part of the flag below?  how wide is the tiling under left? - see here):

    They must understand what an angle is (intended as the rotation with which to move from one side to the other, not as the "part of the plane" between the two sides, as books often present it, foreboding of the misconception for which, in figure 1, the angle A is greater than the angle B),  that the quadrangles are not as rigid as triangles (see figure 2),  that by cutting the "corners" of a triangle and joining them, an angle of 180 is obtained (figure 3)  and that it can be "demonstrated" that this fact applies in general since (figure 4) by moving a stick along the sides until it returns to its initial position it rotates by half a turn (see here):

        It can be observed that what has been said above for the proportional transformations of the rectangles also applies to the circles.  If I enlarge or reduce a circle, the ratio between its width, i.e. its diameter, and its length, i.e. its perimeter (which in the case of circles is called circumference), it does not change, and it is seen (as can be observed by measuring the circumference of a bicycle wheel with a tape measure) which is 3 and change.  At this point it can be said that it is more precisely 3.141592653… and that it is indicated by the Greek letter π, which is the initial of the Greek word perimetros:  the perimeter of the circle of diameter D is π·D.
    And it can be observed that a circle of radius R can be unrolled (as seen in this animation) in a right triangle with a cathetus as long as the circumference and the other as long as the radius, and conclude that its area is π·R²:

    The free-hand drawing activities (and those of drawing houses, streets, gardens, ... whose outlines are traced on transparent sheets hanging from the class windows, those of photographic reproduction, ...) are important occasions for space education , which must be significantly intertwined with other educational aspects. An important aspect is also the depiction of the shadows, which must first be freely addressed in the context of drawings invented by children and then with reflections linked to direct observations of shadows generated by the sun or a lamp, which must be followed by verification activities (or starting points for new experiences) such as explaining in words which or which of the three figures A , B and C below are wrong and why:

    They must  face calculations of "areas" of small territorial extensions using cartographic representations (the area of the park below on the left, visited by the pupils, is about 12+5+4 squares with side of 50 m, that is …),  calculate "lengths "of paths counting the steps and then reading the maps (what is the shortest way to go from one to the other of the two points marked on the map),  rotate 90 or 180 or overturn figures on the squared paper (like the half leaf in the figure below),  record (after some drawing activities such as those described above) the shadow of a vertical stick as the time goes to understand how the position of the sun changes,  or describe (in a way understandable by others) the functioning of a treadmill observed in a company visited:


    They must learn that if I have a pile of papers, the volume occupied by it does not change if I tilt or rotate it, but that if I have a box like the one below on the right (made up of hinged metal bars and covered with stretch fabric), by tilting it the volume changes (up to zero if I completely lower the side faces):

    These are important conceptual and operational acquisitions, to which we will come back in secondary school, with further insights and formalizations, and starting to detach ourselves from the "contexts".
    We refer to the considerations on the concept of "space" in the part dedicated to the lower secondary school for further information in this regard; in them the objectives of this scholastic level are focused and those of primary school (to which they should join) are further deepened.
    We only remember that the conflicts between mathematical terminology and common language (the different meanings of angle, direction, distance, curve, ...) should be highlighted.  And the fact that in practice we can only obtain approximate values of areas (and volumes) should be focused.

    As already mentioned, the concepts will have to be taken up and deepened later, with a view to a "spiral" recovery of the topics: as the school curricula for many years have underlined, it is not a question of making "anticipations", but of using, in appropriate ways, terminologies and concepts that find natural intertwining with other concepts and lay the foundations for the construction of subsequent levels of formalization.  This aspect seems fundamental to us, both to give a correct and "lively" image of mathematics, and not to favor mental stiffenings that identify concepts with particular definitions, particular calculation procedures, … and classify them into areas (geometry, arithmetic, ...) without communication between them.  HERE you can find various exercises that exemplify the considerations made in this paragraph.

Theory of probability

    Addressing the descriptive statistics, the basic concepts and techniques for the representation and study of random variables should already have been introduced. By addressing the  calculation of probabilities  the transition is made to the case in which "forecasts" are made, based on statistical considerations, beliefs or information of various kinds.
    At the level of primary school, complex problems must not be tackled, but the concept of "probable" (which is present in communications and verbal messages with which children have to do from the first years of life) must be focused on.
    We make only four examples, referring to abstract contexts with respect to the many situations that can occur in a school class:

• We have the histogram of the weights of a group of 27 children of 7 years (those who weigh 21 kg odd, those who weigh 22 kg odd, ..., those who weigh 31 kg odd). If I randomly pull a child out of this group, is he more likely to weigh less than 26 kg or weigh at least 26 kg? (to answer just count the squares that correspond to the two cases).

• We cast a die built with cardboard and we ask ourselves the problem "which face is more likely to come out?". By making a few throws, it is not possible to understand, but by doing many others it is seen that the most frequent exit tends to be that of the face to which no flaps are glued.

• The Milan-Inter match is about to take place. For Luigi, 60 out of 100 will win Milan. What is the probability for Luigi that Milan won't win?

• 3 boys, A, B and C, participate in a swimming competition. Based on the results obtained in alignment, it is believed that A and C have a double chance of victory than B. What are their respective chances of winning?  (it's easy to answer - 40%, 20%, 40% - if the situation is represented on squared paper: |−−|−−|−−|−−|−−|).

    The theme of probability is often developed in "harmful" ways:  think of the many books of subsequent school levels in which probability is "defined" as a ratio of favorable cases/possible cases, demonstrating ignorance of the most basic mathematical concepts, and favoring in boys, the development of misconceptions which will then be difficult to take apart.  The issue of probability, although involving, at first, very basic mathematical concepts (the same as in statistics), is difficult to deal with because it requires a deep understanding of the situations to be mathematized, not a banal application of some formulas.  For this reason it is good to focus the concept of "more probable" by preceding it by statistical reflections and considering various situations, also not confrontable with the "cases/cases" scheme mentioned above, such as those seen in the previous examples.

    The fact that probabilistic tools (and statistical ones) are to be used (and whose outcomes are to be interpreted) in contexts that are not purely mathematical is perhaps the reason why they are often overlooked by teachers or developed in completely incorrect ways. On the other hand, it is essential, from an educational point of view, to tackle this issue, also trying to highlight the difficulties and errors that pupils can commit by tackling probabilistic issues. This is a decisive aspect for this area of mathematics, also to encourage the development of attentions towards the nonsense that various mass media often propose by referring to probabilistic evaluations.

    In the last years of secondary school, the areas of statistics and probability will find a stronger intertwining when the intertwining between the two areas is theoretically focused, i.e. the reasons why, in a statistical experiment, as the number of tests increases the ratio between the number of favorable exits and total exits tends to stabilize on probability.

Relations with the other disciplinary areas

    In the initial paragraph, on the concept of model, we discussed at length the relationships between reality and the various models, disciplinary and non-disciplinary, with which it (or its particular aspects) can be represented and studied.  We also highlighted the differences between the mathematical models and the other disciplinary models, and the didactic attentions that all these aspects must urge in those who have the task of educating the new generations, especially in the early schoolastic bracket.
    While the other disciplines refer to different types of phenomena, mathematics is characterized by the type of artifacts it employs and is organized on the basis of their properties and the formal relationships between them, even if some border sectors (computer science, econometrics, mathematical physics, ...) are characterized by the relationship with certain disciplines and technological areas.
    But, as we have highlighted above, the birth and historical development of the various mathematical concepts has become intertwined with the other branches of knowledge, and the ability to use mathematical artifacts depends on the knowledge of the contexts in which they are used.  In the long journey of scholastic and extracurricular learnings through which we become adults, things initially acquired as abstractions then become concrete "objects" to describe "reality" on which to create new abstractions.  But, in order for this process to develop effectively, this concrete-abstract passage must be gradual, must be "perceived" by the pupils, must refer to the various cognitive aspects with which they have to deal in everyday life, to progressively arrive at forms of knowledge organized in disciplines.

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