Key Words

Introduction

In previous studies (see e.g. Bonotto, 1995) I have analyzed some difficulties regarding the understanding of the structure of decimal numbers. These include conceptual obstacles that elementary and middle school students encounter in mastering the meaning of decimal numbers and in ordering sequences of decimal numbers (Nesher and Peled, 1986; Resnick et al., 1989; Irwin, 2001). Results from two questionnaires, each involving elementary and middle school Italian teachers and each concerning the way they teach the topic of decimal numbers in class shed light on the way the usual instructional practice seems totally extraneous to the richness of the experiences students develop outside school (Bonotto, 1996). Many teachers introduce decimal numbers by extending the place-value convention. They tend to spend little time allowing children to understand the meaning of decimal numeration or reflect on decimal number properties and relationships. As a consequence, children learn to carry out the required computations, but have difficulty in mastering the relationship between symbols and their referents, and between fractional and decimal representations.

In agreement with other researchers (e.g. Hiebert, 1985; Irwin, 2001) I believe that the decimal numbers concepts need to be anchored in students’ existing knowledge; and, one way to do this is to help students integrate their everyday knowledge with school mathematics.

The study presented in this paper involves a teaching experiment based on a sequence of classroom activities in upper elementary school aimed at enhancing the understanding of the structure of decimal numbers in a way that was meaningful and consistent with a disposition towards making sense of numbers (Sowder, 1992). As in other studies (Bonotto, 2003; 2005 and 20072007a), the classroom activities are based on an extensive use of suitable cultural artifacts – in this case some menu of restaurants and pizzerias. The classroom activities also are based on the use of a variety of complementary, integrated, and interactive teaching methods, and on the introduction of new socio-mathematical norms, in an attempt to create a substantially modified teaching/learning environment. This environment is focused on fostering a mindful approach toward realistic mathematical modeling and a problem posing attitude.

Theoretical and Empirical Background

The habit of connecting mathematics classroom activities with everyday-life experience is still substantially delegated to word problems. However, besides representing the interplay between in- and out-of-school contexts, word problems are often the only examples that are provided to students to cultivate basic sense experiences in mathematization and mathematical modeling. Yet, word problem rarely reaches the idea of mathematical modelling; and, they often promote in students a “suspension” of realistic considerations and sense-making.

Rather than functioning as realistic contexts that invite or even force pupils to use their common-sense knowledge and experience about the real world, school arithmetic word problems have become artificial, puzzle-like tasks that are perceived as being separate from the real world. Thus, pupils learn that relying on common-sense knowledge and making realistic considerations about the problem context – as one typically does in real-life problem situations encountered outside school – is harmful rather than helpful in arriving at the ‘correct’ answer of a typical school word problem, Verschaffel etal. (1997, p.339).

Several studies point to two reasons for this lack of use of everyday-life knowledge: (a) textual factors relating to the stereotypical nature of the most frequently used textbook problems: “When problem solving is routinised in stereotypical patterns, it will in many cases be easier for the student to solve the problem than to understand the solution and why it fits the problem” (Wyndhamn and Säljö, 1997, p.364), and (b) contextual factors associated with practices, environments and expectations related to the classroom culture of mathematical problem solving: “In general the classroom climate is one that endorses separation between school mathematics and every-day life reality” (Gravemeijer, 1997, p.389).

Finally, in my opinion, another reason for not using realistic considerations is that the practice of word problem solving is that they have meaning only within the school. Rarely will students encounter these activities in this form outside of school (Bonotto, 2007a).

I claim that an early introduction in schools of fundamental ideas about modelling is not only possible but also desirable even at the primary school level. Mathematical modelling has received much more curricular attention over recent decades (e.g., Blum et al., 2007b). A particularly sustained and theoretically highly developed program has been carried out by Lesh and his colleagues (Lesh, 2003; Lesh and Doerr, 2003; Lesh and Zawojewski, 2007). In this contribution the term mathematical modeling is not only used to refer to a process whereby a situation has to be problematized and understood, translated into mathematics, worked out mathematically, translated back into the original (real-world) situation, evaluated and communicated. Besides this type of modeling, which requires that the student already has at his disposal at least some mathematical models to use to mathematize, there is another kind of modeling, wherein model-eliciting activities are used as a vehicle for the development (rather than the application) of mathematical concepts. This second type of modeling is called “emergent modeling” (Gravemeijer, 2007). Although it is very difficult, if not impossible, to make a sharp distinction between the two aspects of mathematical modeling, it is clear that they are associated with different phases in the teaching/learning process and with different kinds of instructional activities (Greer et al., 2007). However, in this contribution the focus will be more addressed to the second type of mathematical modeling.

In elementary schools, to introduce fundamental ideas about realistic mathematical modeling, and lay foundations for developing a “mathematization disposition”, I believe that we must create more realistic and less stereotyped problem situations which are more closely related to children’s experiential world. In particular, an extensive use of suitable cultural artifacts, with their incorporated mathematics, can play a fundamental role in bringing students’ out-of-school reasoning experiences into play, by creating a new tension between school mathematics and everyday-life knowledge (Bonotto, 2007b).

The cultural artifacts that we introduced into classroom activities (see for example Bonotto, 2003; 2005 and 2007a) are concrete materials, real or reproduced, which children typically meet in real-life situations. In this way we offered students the opportunity of making connections between the mathematics incorporated in real-life situations and school mathematics. These artifacts are part of their real life experience, offering significant references to out-of-school experiences. In this way we can enable children to keep their reasoning processes meaningful and to monitor their inferences. Finally, I believe that certain cultural artifacts lend themselves naturally to helping students with problem posing activities.

Problem posing also is an important aspect of both pure and applied mathematics, as well as being an integral part of modelling cycles which require the mathematical idealization of real world phenomenon (Christou et al., 2005). For this reason, problem posing is as important as problem solving (Silver et al., 1996; Ellerton and Clarkson, 1996; English, 1998 and 2003; Christou et al., 2005) and is of central importance in mathematical thinking (e.g. NCTM, 2000). Some of the preceding studies provided evidence that problem posing has a positive influence on students’ ability to solve word problems and those activities also provide a chance to gain insight into students’ understanding of mathematical concepts and processes. In particular, it was found that students’ experience with problem posing enhances their perception of the subject, provides good opportunities for children to link their own interests with all aspects of their mathematics education, and can prepare students’ to be intelligent users of mathematics in their everyday lives.

Problem posing is seen here as a classroom activity which is important both from the cognitive and the metacognitive viewpoint. Children’s expression of mathematical ideas through the creation of their own mathematics problems demonstrates not only their understanding and level of concept development, but also their perception of the nature of mathematics (Ellerton and Clarkson, 1996) and their attitude towards this discipline.

Problem posing has been defined by researchers from different perspectives (see e.g. Silver et al., 1996). In this paper I consider mathematical problem posing as the process by which students construct personal interpretations of concrete situations and formulate them as meaningful mathematical problems. This process is similar to situations to be mathematized, which students have encountered or will encounter outside school. According to English (1998) “we need to broaden the types of problem experiences we present to children … and, in so doing, help children “connect” with school mathematics by encouraging everyday problem posing (Resnick et al., 1991). We can capitalize on the informal activities situated in children’s daily lives and get children in the habit of recognizing mathematical situations wherever they might be”.

The Study

The Basic Characteristics of the Teaching/Learning Environment

The basic characteristics of the teaching/learning environments that we use have been described in other recent publications [see e.g. Bonotto, 2003, 2005, 2007b]. Our design principles include the following.

  1. 1.

    Create more realistic and less stereotyped problem situations based on the use of suitable cultural artifacts – e.g. labels or supermarket receipts – which can provide a didactic interface between in and out-of-school knowledge, and encourage out-of-school reasoning experiences to come into play.

  2. 2.

    Use a variety of complementary, integrated and interactive instructional techniques (involving children’s own written descriptions of the methods they use, individual or in pairs working, whole class discussions, …).

  3. 3.

    Establish a new classroom culture also through new socio-mathematical norms, for example the norms about what counts as a good or acceptable response or solution procedure, in order to undermine some deeply rooted and counterproductive beliefs such as mathematics problems have only one right answer or there is only one correct way to solve any mathematical problem.

Participants/Materials/Procedure

The study was carried out in two fourth-grade classes (children 9–10 years of age) in a lakeside resort in the north of Italy by the official logic-mathematics teacher, in the presence of a research-teacher. As a control, two fourth-grade classes were chosen in the same town. Most of the local population are involved in tourism and catering, and most of the children’s parents either owned or worked in restaurants, bars, ice-cream parlors or pizza shops; for this reason it was found that the pricelists and menus of pizza shops, fast food and normal restaurants were part of the children’s experiential reality.

The teaching experiment was subdivided into six sessions, each lasting two hours, at weekly intervals. The first session was devoted to the administration of the pre-test and the introduction of various kinds of artifacts which the children, divided into groups, had to analyze, by reading and interpreting all the data present, whether numerical or not. Sessions 2–6 concerned five experiences involving different opportunities offered by the artifacts. The sixth session was also devoted to administration of the post-test. Sessions 2–6 were divided into two phases. In the first, each pupil was given an assignment to carry out individually or in pairs. In the second phase, the results obtained were discussed collectively and the various answers and strategies compared.

Data

The research method was both qualitative and quantitative. The qualitative data consisted of students’ written work, field notes of classroom observations, and mini-interviews with students after the experiences. Quantitative data were collected using pre- and post-tests which administered to two experimental classes and two control classes. The two tests were constructed by taking items normally used in the bimonthly tests utilized by the same teachers or usually present in the textbook.

Research Questions and Hypotheses

The first general hypothesis was that the teaching experiment class would foster the understanding of some aspects of the multiplicative structure of decimal numbers in a way that was meaningful and consistent with a disposition towards making sense of numbers (Sowder, 1992). This would be expected due to the opportunity children had to refer to a concrete reality (via the cultural artifact), explore their strategies and compare them with those of their schoolmates, and use estimation and approximation processes, as well as both problem posing and problem solving abilities.

A second hypothesis was that, contrary to the common practice of word-problem solving, children in this teaching experiment would not ignore the relevant, plausible and familiar aspects of reality, nor would they exclude real-world knowledge from their observations and reasoning (hypothesis II). Furthermore, children also would exhibit flexibility in their reasoning, by exploring different strategies, often sensitive to the context and quantities involved, in a way that was meaningful and closer to the procedures emerging from out-of-school mathematics practice (hypothesis III).

Finally I wanted to evaluate the impact on problem solving of problem posing activities and the use of suitable cultural artifacts.

Some Results

By presenting the students with activities that were meaningful and that involved the use of material familiar to them, motivation was increased even among less able students. A good example is the case of an immigrant child with learning difficulties related chiefly to linguistic problems. For her, as for many others, being confronted with a well-known everyday object with “few words and lots of numbers” acted as a stimulus. Indeed, it led her to say “It’s easier than the problems in the book because we already know how things work at a restaurant!”

An analysis of class discussion showed that a process of problem critiquing (English, 1998) was set up whereby the children attempted to solve, criticize and make suggestions or correct the problems created by their classmates. Here is an example taken from the second session in which the children read and interpreted the data and information in the various menus (products on offer, prices, ingredients, cover charges etc.). Working individually on one of these menus, the children compiled a hypothetical order, which they themselves chose according to their experience outside school. In so doing, they had to follow the structural features of a blank receipt (description of goods, quantity, cost etc.) provided by the teacher. Finally, the children had to calculate how much they would have to pay, by adding up the bill.

  • I. Look at this. Do you think it could be a kind of problem?

  • P 737. It’s not written…

  • I. You’re right… It’s not written, but all the data is there. We could write the text … Shall we try?

  • P 734. A man goes to a restaurant and orders 2 dishes of sea-food, 1plate of escallops in lemon sauce, 1 mixed salad, 1 fresh fruit, 1 still mineral water, 1 medium Coke. How much does he spend?

  • I. Does everyone agree?

  • P 740. You can’t do it like that because there are no prices … We have to put a menu underneath the problem or put the prices in the text.

  • P 725. And then… how do we know how many people were eating if they didn’t put the cover charge?

The idea of “pretending to be at a restaurant” and “acting like grown-ups” [“grown-ups don’t take a calculator or a pencil and paper with them to see if they can afford to order this or that …. They work it out in their heads” said one child, P725] helped the children, including those with greater difficulties, to reason more freely and adopt calculation strategies they had never used before.

Another example, taken from the fifth session, shows impact of using a more complex menu where students were asked: (i) to analyze the menu and to read and interpret all the data and information contained therein; (ii) to choose what to order knowing that they had only 15 euros to spend; (iii) to make a mental estimate of what they would have to pay and whether it would come within their budget; (iv) to write out the bill in full to check their estimate. Here is an example, P734, of a child considered “low level” who had been placed in the “extra help group”.

First of all, I take away the money for the cover charge, which is obligatory. So € 15.00 – € 1.25 is like doing € 15.00 – € 1.00 which makes 14.00 but it’s a bit less because it was € 1.25… so we can pretend that I’ve still got € 13.50 for example. Then I decided to have the cheapest pizza, the pizza marinara, so I have to take away € 3.35. A quick way is to take away € 3.50 and so I’d still have about € 10.00 and I can add something on the pizza, for example ham which costs € 1.44. Let’s pretend it’s 1.50, so € 10.00 – € 1.00 makes € 9.00 and then taking away another € 0.50 it makes € 8.50. In the end, I can have a Coke as well which costs € 3.00 and that makes € 5.50… (a brief pause)… But I wanted a dessert and now I can’t afford it… I have to leave something out… (he thinks for a few seconds)… I need another euro because the desserts cost € 6.50…. Perhaps it’s better to have water instead of a Coke. That way instead of € 3.00 I only spend € 2.00 for drinks and now I’ve got € 6.50 which is the price of a ‘semifreddo’ dessert.

As can be seen, P 734 is unable to calculate mentally € 15.00 – € 1.25 but by approximating upwards all the prices, he is able to find a solution to the problem after his first failed attempt. Like P 734, all the other children, without exception, managed to carry out this exercise. Most of them preferred to use subtraction and calculate the amount they had left in order to decide if they should order something else and what to order. Only two children preferred to use addition and calculate as they went along the amount of the final bill. Here is an example:

I started with the cover charge which is € 1.25 then I added the pizza siciliana which is € 5.94… but to do it quickly I pretended it was €6.00 and so the total was € 7.25. Now I can get something to drink. I decided to have water which costs less so that I could perhaps manage to have a dessert afterwards. So I say € 7.25 + € 2.00 = € 9. 25. All the desserts cost € 6.50 each. If I do € 9.25 + 6.50 = 9.00 + 6.00 = 15.00 but then I have to add on the 25 cents and the 50 cents and so it’s to much because I’ve only got € 15.00 to spend. So, I think I’ll have a cheaper pizza, for example a margherita which costs € 3.61 (say 3.50). So, I have to start again by doing 1.25 + 3.50 = 3.00 + 1.00 = 4.00 + 25 cents = 4.25 + 50 cents = 4.75 cents. Now I add on the water which is 4.75 + 2.00 = 6. 75 and try again adding the dessert doing 6.75 + 6.50 = 6 + 6 = 12.00 + 50 cents = 12. 50 + 75 cents which is about one euro so it’s more or less € 13.50.

The less confident children (as in the case reported above) made an estimate that was less precise, whereas the more confident ones attempted a more careful estimate down to the number of cents. Here is another example, P 740:

I first took away the price of the cover charge, so I did € 15.00 – 1.25 = 14.00. Then I took away another € 0.25 which is half of 0.50: € 15.00 – 0.50 = € 13.50 + 0.25 = € 13.75. Then I decided to take away the price of a dessert which is what I like most, so I do € 13.75 – € 6.50. Here, I can do what I did before and divide 75 cents in two and do 50 and 25. So 13.50 – 6.50 = 7.50 €. But I’ve got to add on the 25 cents and so € 7.50 + 25 = € 7.75. Now I can have a Coke which is easy because I have to take away 3 euros and that makes € 4.75. With that money I can have a pizza Napoli which costs € 4.64 and still have some money left.

These examples clearly demonstrate that the children exhibited flexibility in their reasoning, by exploring different strategies, and that they often were sensitive to the context and quantities involved, in a way that was meaningful and consistent with a sense-making disposition and closer to the procedures emerging from out-of-school mathematics practice (hypothesis III confirmed). Many other examples of written works also demonstrated that the children did not ignore the relevant, plausible and familiar aspects of reality, nor did they exclude real-world knowledge from their observations and reasoning.

Conclusion and Open Problems

This paper presented results from a teaching experiment characterized by a sequence of activities based on the use of suitable cultural artifacts, interactive teaching methods, and the introduction of new socio-mathematical norms. An effort was made to create a substantially modified teaching/learning environment that focused on fostering a mindful approach towards realistic mathematical modeling and problem posing. The positive results of the teaching experiment can be attributed to a combination of closely linked factors, in particular the use of suitable cultural artifacts and an adequate balance between problem posing and problem solving activities.

Regarding the use of cultural artifacts, the implementation of this kind of classroom activity requires a radical change on the part of teachers as well (for an analysis sees Bonotto, 2005). These tools differ from those usually mastered by the teacher that are quite always highly structured, rigid, not really suitable to develop alternative processes deriving from circumstantial solicitations, unforeseen interests, particular classroom situations. The teacher has to be ready to create and manage open situations, that are continuously transforming, that can be mastered after long experimentation and of which he/she cannot foresee the final evolution or result. As a matter of fact, these situations are sensitive to the social interactions that are established, to the students reactions, their ability to ask questions, to find links between school and extra-school knowledge; hence the teacher has to be able to modify on the way the content objectives of the lesson.

In future research, we will take a further look at the role of cultural artifacts not only as mathematizing tools that keep the focus on meaning found in everyday situations and as tools of mediation and integration between in and out-of-school knowledge, but also as possible interface tools between problem posing and problem solving activities.