Abstract In this paper we present some classroom activities for primary school to show how we can best develop children’s mathematical power via establishing a good classroom culture [in particular new socio-math-norms, see Yackel & Cobb, 1996] and by using appropriate tools [in particular adequate cultural artifacts]. These activities are more relatable to the experiential worlds of the pupils, and consistent with a sense-making disposition. We deem that is important to immerse children into a classroom culture that focuses on the importance of activities of realistic mathematical modeling, i.e., both real-world based and quantitatively constrained sense-making. These activities require also a directed effort to change conceptions, beliefs and attitude of the teachers towards school mathematics.

1. Introduction

The Research Centre in Mathematics Education at the University of Padua (IT) has been working on the ideas expressed by the Italian Programs for the primary schools and has been developing new approaches to the mathematical concepts that were the objectives of the curriculum.

In this paper, we propose an experience the Research Group has worked out for primary school to show how we can best develop children’s mathematical power via establishing a good classroom culture [in particular new socio-math-norms, see Yackel & Cobb, 1996] and by an extensive use of concrete materials with which children typically occur in real-life situations (supermarket bills, bottle and can labels, railway schedules, the weather forecast presented in a newspaper, and so on, see Bonotto, 1999).

The activities we introduced in the classroom can be considered ‘contexts’, ‘rich materials’ in Freudenthal’s sense: "Logical blocks are a striking example of the implementation successes which can be reaped with sharply structured material - cheap successes obtained thanks to love of ease. Rich material, open to structuring, which provides for more didactical opportunities, is more demanding and therefore less easy to implement ... A large number of rich contexts for mathematics instruction is now available, more than anybody can image".

This activitiy is more relatable to the experiential worlds of the pupils, and consistent with a sense-making disposition. We deem that is important to immerse children into a classroom culture that focuses on the importance of activities of realistic mathematical modeling, i.e., both real-world based and quantitatively constrained sense-making Ruesser & Stebler, 1997. Furthermore we will ask for a change in teacher conceptions, beliefs and attitude towards mathematics.

2. Connections between mathematics and reality

The current Italian Programs for the primary school emphasize that "the introduction to the mathematical thinking and practice should been firstly addressed to construct ... a large experencial base of facts, phenomena, situations, on which intuitive knowledge, procedures and computation algorithms and the first formalizations of the mathematical thinking can develop".

We stress that bringing real world situations into school mathematics is a necessary, although not sufficient, condition to foster "a positive attitude towards mathematics, intended both as an effective device to know and critically interpret reality, and as a fascinating thinking activity", as is specified in the Italian Programs. We contend that this educational objective can be completely fullfilled only if we can get the students to bring mathematics into reality. In other words, we should not only ‘mathematize everyday experience’ but also ‘everydaying mathematics’; we offer some ideas to achieve this goal.

There is a great deal of mathematics embedded in every day life. Effective learning situations in classroom can be carried out by encouraging the children to analyze some ‘mathematical facts’ that are embedded in some cultural artifacts. Children can be supported in grafting the mathematics they are learning on their everyday experience, and -conversely- to bring their previous knowledge up to higher levels of understanding.

The cultural artifacts we have introduced in classroom activities are: supermarket bills, bottle and can labels, railway schedules, a ski-race time table, the weather forecast presented in a newspaper, and others. They offered the children with the opportunity of making connections between the mathematics utilized in the real world situations and the mathematics which is the target of classroom education.

Having recognized a gap of competence between the mathematics individuals develop in out-of-school contexts and the mathematical reasoning which is usually performed in classroom, we propose to introduce in the school contexts, at least partially, the conditions that often make the out of school learning more effective. Many studies have pointed out that the local strategies developped in practice are more effective than the arithmetics algorithms, which are usually taught in school to give the students powerful general procedures that, in fact, are frequently useless in out-of-school contexts, see Schliemann [1995]. Related researches are Nunes, Schliemann, & Carraher 1993; Saxe, 1991, on street vendors’ mathematics; Lave, 1991 and 1995; Saxe et al., 1996; Bishop & Abreu, 1991.

Mathematics was introduced in the context of activities which could promoted the movement from the situations in which it is usually utilized to the underlying mathematical structure, and back, from the mathematical concepts to the real world situations, according to the ‘horizontal mathematization’, see Treffers, 1987.

The double nature of these artifacts, that of belonging to the world of everyday life and to the world of symbols, to use Freudenthal’s apt expression, makes it possible. A different use of the artifacts supported the opportunity to do ‘vertical mathematization’, from concepts to concepts. This manifested itself when symbols, embedded mathematical facts, become objects to be put in relationship, modified, manipulated, reflected upon by the children through property noticing, conjecturing, problem solving.

Children are asked to select other cultural artifacts in their day reality, to point out the embedded mathematical facts, to look for analogies and differences (for example, different number representations), and to generate problems (for example, discovering relationships between quantities). In contrast with the traditional classroom curriculum, children are offered endless opportunities to become acquainted with mathematics. This is a motivating starting point for the development of the ability of reality mathematization, and for a change of attitude towards mathematics. In order to reach the level of mathematical abstraction "children are expected to cover the long road that connects the observation of reality, the mathematizing activity, problem solving, the achievement of the simplest levels of formalization", as it is provided in the Italian National Programs. By engaging children in meaningful classroom activities, mathematics turns out to be a fascinating human activity, less harsh and torn off from reality than usually believed, given the traditional practice in school mathematics.

3. Cultural artifacts and the usual school practice

We contrast the traditional problems presented in school-books and still largely utilized in the mathematics practice, and the use of cultural artifacts in the innovative mathematics practice in classroom.

The concreteness of traditional word problems is only seeming. They acquire their relevance just because children repeatedly encounter them in their classroom life and learn that these problems are the crucial means to assess their mathematical knowledge. Time is spent in teaching and learning how to solve specific word problems, that correspond to specific categories by applying general purpose algorithms. The traditional word problems are stylized representations of hypothetical experiences, not slices of everyday existence. They are narratives "about assumed general cultural knowledge that (even) children can be espected to have. They are not about particular children’s experiences with the world ... Children’s intuitions about the everyday world are in fact constantly violated in situations in which they are asked to solve word problems. This discontinuity by itself may help create the division between ‘real’ and ‘other’ math by conveying the message that what children know about the real world is not valid ... ", Lave, 1988. Therefore "the supposedly everyday experiences of this genre are far from mundane, and problems are designed to provide occasions for practice at separating math from experience, rather than mathematizing it".

Furthermore recent researches [Schoenfeld, Gravemeijer, Greer, Verschaffel et al., Reusser, …] has documented that the practice of word problem solving in school mathematics promote in the students a "suspension of sense-making", Schoenfeld, 1991; primary and secondary school students ignore relevant and plausibility familiar aspects of reality and exclude real-world knowledge from their mathematical problem solving.

If we wish

we have to make changes.

The cultural artifacts we have introduced in classroom activities (supermarket bills, bottle and can labels, railway schedules, a ski-race time table, the weather forecast presented in a newspaper, and others, see Bonotto, 1999 and 2000), or cultural artifacts to be constructed by the children (e.g. calendars), are concrete materials with which children typically occur in real-life situations. So we have offered the children with the opportunity of making connections between the mathematics utilized in the real world situations and the school mathematics, that enter deeply into each other but they are governed by different laws and principles. These artifacts have relevance for the children; they are meaningful because they are part of the children’s real life experience, offering significant references to concrete situations, or at least more concrete ones.

"Contrast the supermarket bill, having few words, but a lot of implicit mathematical meanings, and the traditional sell-and-buy problems, full of words, but lacking significant mathematical implications", Basso & Bonotto, 1996.

They stir up curosity, attention, a positive attitude, even an emotional involvement towards mathematics. They bring forth the construction of new procedures, strategies, computational algorithms, would be they standard or not, that sometimes are close to the learning processes emerging in the out-of-school mathematics practice. This enables children to keep their reasoning processes meaningful, to monitor their inferences. As consequence, they can off-load their cognitive space and free cognitive resources to develop more knowledge (Arcavi, 1994).

In summary, the dualism between the ‘concrete’ (or ‘everyday’) and the ‘abstract’ must be overcome, and replaced by the dualism between meaningful experiences (both real and fictitious), and meaningless activities, whereas only the formers involve learners’intention and attention, give meaning to the activities children are involved in, and give raise to dilemmas and alternative solution strategies.

Furthermore, the cultural artifacts can be utilized to build up interdisciplinary classroom activities.

4. The role of children’s oral and written verbalization

In the experiences we implemented, the children were asked to write down their reasoning processes, in order to make explicit the mathematical facts embedded in the cultural artifacts, and make them objects of reflection. Afterwards, the children are encouraged to orally confront and reflect upon their own inferences. Children’s written texts, as well as the other texts, have, as Yuri Lotman said, a ‘functional dualism’ represented by a ‘univocal’ function and a ‘dialogical’ function. When a child writes a text, s/he transmits meanings to the others, teachers as well as peers (univocal function); in the same time, s/he offers a new symbolic space to be reflected upon and a source for new meanings (dialogical function).

In this stage that the child constructs a first level of understanding.

The teacher has the opportunity to single out the child’s level of competence and her/his difficulties, on the other hand it is possible to start a classroom discussion, focused on the children’s solution procedures. By asking the children to discuss and compare their solving procedures, would be they correct or not, a further reflection on one’s own and the others’ reasoning processes is fostered. In this second stage, children recognize similarities and differences among strategies. This recognition can denote steps towards a higher level of understanding and the intramental appropriation of the knowledge available on the intermental plane.

Finally, the children were asked to collectively work out and realize a collaborative text in which comparison among strategies and results, their generalization trials, and notes about their discussions are referred. In working out the collaborative text, each child is encouraged to overcome her/his own point of view, and share new mathematical knowledge.

This methodology can be useful for the teachers because the text makes explicit the stages of understanding of the mathematical principles underlying the classroom activity.

Furthermore, by letting each learner make explicit her/his own reasoning strategies the teacher can analyze individual reasoning processes, in the individual written expression stage, and can monitor the whole classroom activity, in the collective discussion stage; and can take as socially constructed, at least up to a certain level, the new mathematical knowledge in the stage of realization of the collective written text, as well as other systematizations.

The teacher constantly mediates among the different children’s reasoning processes in order to enable each child to move from her/his personal sense to the culturally shared sense, that is the necessary condition for taken-as-shared mathematical concepts.

5. A didactical proposal

We briefly present a classroom activity requiring the use of cultural artifacts we introduced in collaboration with the elementary teachers in our Research Group, and other connected teachers, working in schools located in the Northen-east Italy. The activities were conducted in the normal classroom environment, but offered the children new approaches to the mathematical concepts that were the objectives of the curriculum. For other activities see Bonotto, 1999, and Bonottto, 2000.

We present the tasks that have turned out to be relevant for making both the horizontal mathematization and the vertical mathematization emerge, and particular forms of what Freudenthal has called ‘anticipatory learning’ or ‘learning by advance organizers’ were apparent. Anticipatory learning takes advantage from rich contexts and it is hindered by poor contexts.

We analyze some protocols, in which some of the strategies the children utilized are displayed. The children needed to describe and comment each step of their reasoning processes, to clarify to themselves and to the others their solving processes.

The supermarket bills

This is a more articulate experience dealing requiring a mathematical reflection about five supermarket bills.

The teacher asked the children to collect a number of bills on which the total amount, the cost per unit and the total cost payed were marked.

Bills expressing interesting ratios were selected; in three bills either the information about the net weight (in the third and the fourth bills) or the information about the cost per one kilo (in the fifth one) was erased.

By the introduced modifications, the bills explicitly became a tool for the teacher to:

-make mathematical problems emerge;

-promote a ‘vertical mathematization’ activity in the classroom;

-enable the children to develop new mathematical knowledge.

III grade Make your observations on the numerical Netto Tara Prezzo Importo

 

operation did the machine carry on to find 0,268 4 12 000 3 215

out the total price to be payed?

IV grade Without performing the written computation, Netto Tara Prezzo Importo

 

make your considerations about the total kg g L/kg Lire

 

amount to be payed. 0,390 4 25 900 10 100

 

IV grade In this bill, an information is missing: Netto Tara Prezzo Importo

find out its value, without performing kg g L/kg Lire

he written calculation. ........ 0 2 290 4 650

IV grade In this bill, an information is missing Netto Tara Prezzo Importo

perform the written calculation to find kg g L/kg Lire

it out. Make your observations. ........ 4 15 400 7 270

IV grade In this bill, an information is missing: find it Netto Tara Prezzo Importo

out. Then, carry on the operation you perfor- kg g L/kg Lire

med and explain your reasoning process. 1.515 0 ........ 3 755

Our results are interesting, because children’s thinking strategies have emerged in:

-reading the supermarket bills;

-estimating the missed information in the bills;

-performing procedures to find out the exact information previously erased;

-making relations among the numerical values implied, which in turn lead them to a first grasp of the concept of proportionality;

-developping multiplication and division algorithms to deal with decimal numbers.

We briefly analyze some of the children’s cognitive processes (see Basso & Bonotto, 1996, for an extended version).

In the second bill task, the children were asked to make evaluations about the final price without performing the traditional written calculations. In order to achieve the task, the children mobilize cognitive resources that are usually left inert in classroom activities, such as:

-comparing numbers;

-searching for the price per unit of product;

-rounding numbers to make the computations easier;

-multiplying by exploitation of the distributive property of multiplication.

In explaining their procedures, the children express both their school knowledge (such as the distributive property of multiplication) and common-sense knowledge (for example, the rounding strategy to semplify the calculations).

Here we present some protocols:

Isabella: "I thought to perform 259x39. Mum spent less because she bought no Kilos. I thought to perform 259x39 since the zeros would have made it more difficult, and without it would have been easier.

259x39= (259x30) + (259x9)= 7,770 + 2 331,= 10,101.

The weighing machine rounded by L.1. I did 0.39x100 = 39 dag, then 25 900: 100= 259, which is the cost per dag; I round 259 to 260 and 39 to 40; therefore, 260x40 = 10,400."

Daniele: "The final cost is 10,100 because I multiplied 0.390x25 900, that is, the kilos I bought by the cost per kilo. To be sure that 10,100 is the right price, I have to carry on the calculation: 0.390x25 900, then I look for the difference between 10,100 and the result, and vedo how much more I paied. By rounding 0.390 to 0.4 and 25 900 to 26 000 the calculation becomes easier, and I can perform it mentally. I can transform 0.4 kg to hg and 26 000 L/kg to L/hg. Therefore, 4x2 600= 10,400. I should have paied 10,400, actually I paied 10,100."

In the third bill task, the children are asked to find out the net weight as the missing information.

In achieving the task, the children (both high achievers and low achievers) utilize estimation strategies to make a relation between the total price and the price per unit.

Many expressed their solving procedure as "the amount is more the double of the unitary price, therefore more than two kilos of goods were bought".

In the fourth bill task, the initial estimation favours a monitoring process on children’s following results.

The children who set up the division: 15,400: 7,270= 2.11,... get a very different result from their estimation, which was around half a kilo. Their previuos estimation enabled them to correct and set up a more correct division with inverted terms, as Thomas did.

He checked the result by relying on a ratio of the data and leaving the unitary price fixed.

Thomas: "L 7,270 is about the half of L 15,400, therefore the lady bought about half a kilo and a little bit more. Actually, you can’t say just an half because there are L 1,000 as difference, if I rounded L 7,270 to L 7,200 and I doublied it, I would get L. 14,400, and if I added L 1,000 the result is L 15,400. Anyway, to find out the exact weight you should perform: 15,400: 7,270=2.118. I think my first reasoning is more correct because it is impossible that the lady bought more than two kilos and spent less than L/kg. If she had bought 2 kg she would have spent L. 30,800, actually the final cost is L. 7,270, and I am right in saying that she bought about 5 hg, therefore i have to divide: 7,270:15,400=0.4". (He applied the written algorithm to carry on the division).

In the fourth bill task, many children uncorrectly choose the bigger number as the dividend, and smaller as the divisor, because they extend the operations that imply the natural numbers to decimal numbers, and they invert the role of the total amount and of the unitary price to find out the net weight. However, their previous estimation about the number magnitudes of the results enable them to check and revise their written procedure.

As noted above, in the #3, #4, #5 bill tasks, we had selected particular ratios between numbers, in order to lead the children toward the construction of situation sensible strategies. At the outset, many children did not immediately set up the written operation, but reflected and selected non-standard strategies in order to find out the appropriate solution for each type of problem.

The particular situations entangled with the supermarket bills lead the children to appropriate the local, heuristic, context-sensible strategies that are developed in out-of-school contexts and that have their own consistency and validity.

Another example regarding the fifth bill task:

Mauro: "To calculate the total amount to be paid, you need a net weight and a unitary price. I do not get the unitary price, and I have to find it out.

The net weight is 1.515 kg and the total amount is 3,755. If it were L.1,000/kg it would be 1.515x1,000, that is 1,515, but the result is too small. If it were L2,000/kg Iwould multiplicate by 2 and then by 1,000, that is 1.515x2=3.030 and then 3.030x1,000 and I would get 3,030.Now, 2,000/kg is a too small unitary price, and a bigger unitary price is required. Let me try with 2,500/kg:

1.515x1,000=1,515 and 1,515:2=L 750, more or less; therefore 3,000+750=3,750 Lire (more or less). The unitary price is about 2,500." 

Mauro proceeds by connected trials, he keeps the net weight and the total amount fixed, and varies the unitary price until the most satisfying result is founded out. He does not perform the division algorithm, but he applies strategies that ‘shiver’ the procedures in sequential actions, toward a good solution.

To find out the unitary price in the fifth bill task, some children performed scalar processes, and therefore seemed to get a first idea of proportionality. We suggest this can be an example of what Freudenthal has called ‘anticipatory learning’. Some children integrated integers and rationals, and selected the half a kilo as the unit of measure, because it is easily mastered, since it is contained a finite number of times in the net weight.

We offer a couple of examples:

Nicola: "To find out the unitary price I need to divide 3,755 (the money I pay) by 1.515 (the kilos I buy). The cost per kilo will be smaller than the total amount because I bought 1 kilo and half. I select 1,200 which is about 1/3 of the total amount (3,755). Therefore it corresponds to half a kilo. Therefore I have to multiply 1,200 which is (the price of) half a kilo x2. The price per kilo is around L2,400."

Daniele: "I spend around L 4,000 any kilo and half I buy. If I buy 3kg I spend around L 8,000. L8,000 is the approximate price for 3 kg, therefore 8,000:3= 2,600 more or less. I utilized the invariantive property, indeed I doubled both the the values. Therefore, if I performed 4000:1.5, I would get around 2,600."

As the operations and their executions are concerned, we developed problematic situations to avoid the traditional practice in italian elementary classrooms, that is the mastering of rigid procedure skills in simplified contexts. We engaged the children in mathematical reasoning activities in significant and complex problems, during which flexible procedural abilities can be developed to serve the solving processes. We think the classroom context should not divorce procedures from complex mathematical reasoning, but should support the children in mastering context-sensitive procedural skills in rich mathematical contexts.

In the #1, #4, #5 bill tasks, multiplication and division acquired their meaning before their introduction would be ‘mathematically’ justified.

Let us see Mauro’s protocol.

Mauro, for example,

-makes transformations: he finds out the net weight in grams and then the the price per grams, in order to get two integers to facilitate the multiplication;

-he is able also to perform the multiplication with the decimal number, then he compares its result with the former result and assigns the decimal point;

-justifies his decimal point assignement:

a) "I paid about of 12,000";

b) "in 0.268 there are tenths, hundredths, thousandths";

-analyzes the difference between the multiplication with naturals and the multiplication with decimals, giving an example: 5x5=25; 5x0.25=2.5 because, as he states "the decimals without 1 are smaller than 1, and the result will be smaller than the multiplicand".

Let us see Matteo’s reasoning about the fourth bill task:

Matteo: "The missing information is the net weight, expressed in kilos. To find out the net weight I need to utilize the following data: 15,400 (price L/kg) and 7,270 (amount in lire). I will perform a division: 7,270:15,400 because it is by dividing the total amount by the unitary price that I will find out the net weight:

Total amount : Price L/kg= Net weight

Total amount : Net weight= Price L/kg

Net weight x Price L/kg= Total amount

by knowing that 7,270 < 15,400 I can know in advance that the result will be about 0...., smaller than 1. (He performs the division) 7,270 : 15,400= 0.472. Yes, I was right because the result is smaller than 1: 1kg>0.472."

Matteo

-sets up reasoning procedures not only to solve the given problem, but also to develop more general rules to deal with related problems;

-makes an estimation about the division result (which should be smaller than 1);

-verifies his estimation by comparing it with the result of the written division.

Therefore, we can conclude that he:

-can categorize, that is, to put in evidence the solution pattern underlying the given problem, and its relationships with the patterns for related problems;

-can monitor his own cognitive processes.

Matteo, as well as other children, has developed a reasoning strategy which is similar to the IDEAL model expressed by Bransford and Stein, 1993. This model can be useful in enabling children to develop problem solving abilities and increasing their awareness about their reasoning processes.

IDEAL is an acronym which meaning can be expressed as:

I=identify the problems and the opportunities;

D=define the objectives;

E=explore the possible strategies;

A=anticipate the results and act;

L=look back and learn.

The model summarizes the five steps to be performed in any problem solving activity.

The child’s searching for inconsistencies in her/his problem solving process can take advantage by the ‘look back and learn’ strategy.

Matteo seems to be able to perfom this strategy.

6. The relationships between cultural artifacts and mathematics

The use of the cultural artifacts in mathematics education can be articulated in different stages in classroom activity; as not as simple triggers of mathematics understanding; instead they can be utilized for establishing mathematics education goals.

In a first stage, children can be asked to simply recognize the mathematical facts that are embedded and codified in the artifacts (horizontal mathematics); in a second stage thay can be asked to interpret and reflect on the mathematical facts, both in themselves and in connection with real world situations (horizontal and vertical mathematizations); in a third stage children can be asked to put mathematical facts in relation, make conjectures about procedures, notice properties (vertical mathematization, as we proposed in the supermarket bill tasks by erasing an information).

In the first stage, the cultural artifacts represent the real world situations; the mathematical decodification of the message contained in the artifact is the educational goal (often in this stage, mathematization is of the horizontal type).

In the following stages, the artifacts loose their fixed structures because the embedded numerical information is reduced and does not represent the out-of-school reality, although they are strongly tied with real world situations. Typically in the supermarket situations, bills do not lack information. Therefore in mathematics education, the artifacts can become mediational tools, and bridge the out-of-school experience with school activities leading towards new mathematical goals. Now artifacts become more explicitly tools of mediation and integration between in and out-of-school knowledge and experiences, between in and out-of-school mathematics; so they can constitute a didactic interface between the two different contexts. This different use of the artifacts supported the opportunity to do also ‘vertical mathematization’, from concepts to concepts. Vertical mathematization is the process of reorganization within the mathematical system itself, like, for instance, discovering connections between concepts and strategies and then applying these discoveries. In short "horizontal mathematization involves going from the world of life into the world of symbols, while vertical mathematization means moving within the world of symbols".

In this new role the artifact may become also real "mathematizing tools", able on the one hand to create new mathematical goals, on the other to provide pupils and students with a basic sense experience in mathematization. In the supermarket bill experience, the cultural artifact served the achievement of a number of mathematical goals: the development of estimation strategies; the extension of the multiplicative structure from the naturals to the decimals, the comparison of the two structures; the formalization of the decimal number multiplication algorithm; finally it seemed to foster a first understanding of proportionality.

The three stages in which the whole experience (as well as the methodological procedure) is articulated scan also the different cognitive (metacognitive respectively) levels the children have reached.

The schema Saxe et al, 1996, have proposed:

fits very well to the classroom reality, whereas ‘activity structures’ refers to the classroom practices that incorporate some portions of out-of-school experience and artifacts (journals, bills, labels, coins).

As highlighted in our experience, through the use of cultural artifacts that remind children of everyday practices, humans do not develop rigid and general algorithms, rather they construct specific heuristics, that are flexible, sensible to the contexts and the number magnitudes, quite meaningful, consistent, and cognitively manageable.

Given their properties, the cultural artifacts can be introduced and utilized in classroom as a starting point to facilitate children’s construction of new mathematical knowledge. According to our experience, children exhibit flexibility in their reasoning processes, by exploring, comparing and selecting among different strategies.

The strategies children develop in their out-of-school activities are cognitive resources and, if that is the case, easily re-constructed by children because they are assigned a precise meaning, and therefore they can be owned for a long time.

The use of appropriate cultural artifacts enables children to "creating, strengthening, and maintaining bonds with reality", as Freudenthal has foreseen; through their use, curricola starting from children’s out-of-school experience and leading towards the understanding of the mathematical facts embedded in real world situations can be developed. Furthermore, the children’s understanding can be generalized, istitutionalized in a shared mathematical knowledge, and applied in real world situation modelling.

Children develop more awareness about the mathematical facts embedded in the cultural artifacts, and personal understanding, which are useful in dealing with new problems in school as well as out of school.

7. Discussion and open questions

From an educational point of view, the classroom practices we implemented have increased our understanding about the mechanisms to make the connections between out-of-school and classroom activities more effective.

Learning can be intended as a multiple-step process in which children’s everyday experiences can be originally connected to the classroom practice.

However, we do not want to suggest that our research is a prototype for all the classroom activities related to mathematics (but not just limited to it). In educational practice, innovative experiences must be joined by more traditional activities, of reinforcement, of computation, standardized exercises.

Given their paradigmatic value, we do believe that by enacting some of these experiences in classroom, the teachers and the children are offered an opportunity to change their attitudes towards school mathematics. These innovative practices involve children’s intentions and attention, stir up dilemmas, give meaning to mathematical symbols introduced. Furthermore, offer children the opportunity to tune their understandings towards the real world situations they face in their everyday experience, as Resnick, 1994, has noted.

Some constraints in the innovative practice must be highlighted. In their school practice, elementary teachers usually are encoureged to utilize rigidly structured manipulatives, that are not well suited to develop school practices that incorporate unforeseen interests, circumstantial solecitations, particular classroom situations.

Teachers need to be prepared to create and manage continually evolving classroom situations in which the process and the final outcome cannot be decided in advance, rather than to plan and implement rigid curricola. Any classroom environment is sensitive to the social interactions that are established, to the students atttitudes, questions, and ways of connecting classroom practice to everyday experience, and therefore the teacher is expected to master both the mathematical content and manage the classroom life in order to fullfill the educational objectives.

Interactivity means that both teachers and learners are agents as well as being acted on. The very hard task for the teacher is not to completely plan the lesson sequence in advance, but to expect ramifications and convergences.

The teacher has to promote the institutionalization of the children’s built up knowledge, that is the process of sharing knowledge, in order to appropriately fulfill the school aim.

Even in the higher school grades, the teachers should foster the students ability to overcome the simplifications of the mathematics that is embedded in the cultural artifacts, to grasp mathematics as a cultural activity in itself (abstraction, generalization, formalization, etc.).

Why should the teachers accept this new, more responsible role? We suggest that in innovative curricola, the teachers can fulfill new educational goals. Usually, mathematics education is full of failures and frustrating. Innovation is "a learning process for the Landscape as a whole and for its particular agents", Freudenthal, ibid.

Many teachers who participated to our activities confirm our expectations, they assert how it is impossible for them to return to more traditional activities, made up by rule packages, codified didactical paths, and so on.

They state that innovative practices foster their own interest and reflection along with students’ ones.

The teachers have progressively modified their attitudes and practice in mathematics education, and have proposed new artifacts in classroom activities. During their teaching activity, the teachers have reconceptualized the classroom as a community of learning both for children and for the teachers themselves.

In order to learn "how to lead the children’s discourses, and to manage the interactive didactical approach, and acquire the pleasure of teaching", Pontecorvo, ibid., the teacher cannot loose the pleasure of learning. In this case, she/he is able to foster the pleasure of learning in their pupils. The pleasure of teaching is an obvious consequence.

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