Key words: Teaching-learning, mathematics, elementary education, teachers’ training.

Even though the reform proposed a significant change in the focus on the teaching of mathematics in elementary education, which even led to the design of new free text books, we know, as shown by Llinares (1992), that the changes in the teaching of mathematics are not originated only by a normative curriculum, because teachers filter this curriculum through their mental framework, which includes mathematical knowledge, concepts and beliefs concerning mathematics as a discipline and its teaching process, as well as other aspects related to its role within the classroom. In this same sense, according to Ernest (2000), the empirical studies have confirmed that teachers’ ideas, beliefs and preferences concerning mathematics influence the way they teach.

On the other hand, context plays an important role in the construction of mathematical concepts and procedures by the apprentice. Their importance lies, according to different researchers, such as Brousseau (1994), Charnay (1994), Carraher, Carraher & Schliemann (1995), Nunes & Bryant (1996), in giving those concepts and procedures meaning and, sense. For Brousseau (1994), for example, the teacher should first carry out the work in the reverse sense to the order followed by scientists, a re-contextualization and re-personification of knowledge: to search for situations which give meaning to the knowledge that is to be taught. According to those authors, many of the difficulties in teaching mathematics are caused because of the use of irrelevant contexts with little significance for the student.

We have chosen a basic subject from the mathematics curriculum for elementary education, the subject of subtraction, as a means to begin to the identify teachers’ concepts on the teaching of mathematics. Teaching subtraction seems to us to be a mathematics subject of great interest for the purpose of our investigation, because it constitutes a subject where we can observe the school’s resistance to the changes brought about by the different reforms in the mathematics curriculum for elementary education. Besides, it is one of the subjects in basic mathematics teaching which has been discussed the most in the permanent teacher training programs and which, however, continues to be a source for didactic conflicts.

To get a general overview of] the studies made about conceptions, beliefs and professional knowledge we reviewed the work of Thompson Teacher’s beliefs and conceptions: a synthesis of research (1992) and the work of Fennema & Loef Teacher’s knowledge and its impact (1992). In the same way, we have gone through the works of authors like Llinares (1996) and Llinares & Sánchez (1989); Carrillo (1998), Contreras (1999), Flores (1998), who have carried out studies about teachers’ beliefs and conceptions regarding mathematics and its teaching.

The first stage of this research, reported on Martínez Silva (2001), is the most recent antecedent of this in paper. In that stage we were able to progress with the construction of a conceptual framework about the subject, as well as in the design and the validation of certain tools for information gathering that we have recovered for this second stage of the study.

On the whole research the main objective has been to study the elementary school teachers’ conceptions regarding the teaching and learning of subtraction, particularly concerning the role of contextualization within this process.

The questions we wanted to answer through this study are the following: What conceptions do teachers have concerning the teaching of subtraction? What conceptions do they have about the role of contextualization in the teaching and learning process of subtraction? Which content or aspects of subtraction do they teach? Which aspects do they emphasize more? Why? What kind of situations/contexts do teachers use to give sense/meaning to the teaching of subtraction? In which way do teachers’ conceptions relate to the aspects of subtraction they emphasize and to the didactic situations they propose?

Our position on the idea of conceptions takes the definitions used by authors like Ruiz (1994), cited in Flores (1998), Ponte (1994), Thompson (1992), Carrillo (1998), Contreras (2000), Fischbaun & Ajzen (1984), Moreno (2000), Furinghetti (1999).

For the purpose of this research we will understand the word conceptions as the whole of the internal representations evoked by a concept. They are the implicit organizers of the concepts, essentially cognitive in nature. They describe the nature of mathematical objects and their various images in the mind, regardless whether these are symbolic, graphic, etc.

The conceptions do not only refer to the nature of mathematics and mathematical objects, but also to the teaching and learning of mathematics. In order to achieve our research goals we will understand conceptions on the teaching of subtractions as the position taken by teachers regarding the purpose, objectives and teaching contents of subtraction in elementary education, the teachers’ and students’ role, the most appropriate type of teaching activity or instructional process, the role given to contextualization within the teaching and learning of subtraction, etc.

We will understand as contextualization of teaching mathematics, the process through which the teacher tries to establish relationships between the knowledge they have to teach and the situations in society where this knowledge will be used. In other words, the process through which a series of “real” situations of children’s everyday lives, are transformed in mathematics problems in order to illustrate the mathematical concepts or procedures that have to be taught.

• They work in public schools in the suburbs of Monterrey, Nuevo Leon, Mexico, with a low socio-economic standard of living.
• They teach 2nd or 3rd grade pupils in elementary school.
• The majority works a double shift.
• The majority has not participated for the last three years in a training course for the teaching of mathematics.
• The group’s average age is 45.
• The average teaching experience is 24 years.

• The solution of subtraction problems.
• The use of informal and non-conventional procedures to solve the subtraction problems.
• The difficulties met in the learning of the conventional algorithm for subtraction.

a) For the teachers, contextualization of the teaching of subtraction should be done through the posing and solving of problems presented in written sentences.

In the analysis made of the open questionnaire answers, we found that 26 of the 29 situations suggested by the teachers to approach the subject of subtraction can be found in the category of problems posed in written sentences which is equivalent to 89% of the presented problems. The three remaining situations were written numerical calculation exercises (11%).

It is interesting to observe that all the situations suggested by the teachers refer to written problems and numerical exercises, and how the approach to problems and exercises through other means of representation, as for example orally, graphic, through drawings or in a concrete manner, are absent. The previous results are similar to the ones found in Martínez Silva (2001), where the majority of the suggested problems were presented in a written form (85.5%), while the problems presented orally, through drawings, tables, graphs or concrete material were very few.

In the CPT course, the problem set forth in writing appears as the prototype of the “reasoned” problem:

[...] Something happened yesterday: I gave the children a reasoned problem: “There is space for 500 pupils in the school. On Monday, 220 people enroll in school and another one 170 on Tuesday. You are asked: “Are there still places left in the school? How many are missing?”. There we have two operations. They don’t subtract; only one got the right answer [J-DE2.77].1

[...] In the first and second year, it is done with objects they can manipulate... [Ma-DE2.20].

But in third grade, you have to take that away from them, because the children start to draw all over their notebooks and that can’t happen in third grade… you have to stop this bad habit [C-DE2.21].

And use concrete material… at the beginning [C-DE2.33].

The concrete material is essential for the smallest children, so they can acquire knowledge [N-DE2.61].

–[...] You sort of give them a clue, like “How much money does he have now?” to indicate it’s a subtraction [C-DE6.13].

–The keyword should be in the question, so they know that it’s a subtraction or an addition, right? [Ma-DE6.14].

–Change the question, right? to “How much money does he have left?” Right? [Ma-DE6.15].

–Or “How much money has he got left?” [C-DE6.16]. –No, he has left over [Ri-DE6.17].

–So he knows they are going to give him back... that there is change left. Right? [Ca-DE6.18]

–Or “How much change does he get back?” But that would even be longer. That expression of he has left over is… [C-DE6.19].

–[...] Look. You use words like missing, take away, remain. Those are words that a kid can understand [Ri-DE6.35].

–Yes. They understand that something is left over. I have realized that, at least in my group, they understand better if I say, “How much have I left over?” They already know that that means after they have taken away the money, right? [Ma-DE6.36]

–Well, for us maybe not, but maybe for the children it’s easier to understand “How much do you have left over? [Ro-DG6.5].

–I say yes [it’s appropriate]. That it is handled like this in the groups. Yes, both concepts are used with the children. For example, at home, we simply say to them “How much do you have left?” “How much money do you have left?” [N-DG6.6].

–Besides those two questions, what else could we ask? Or rather, what could we ask? [M].

–How much money did you get back? [Ma-DG6.7].

–Any other way of saying that? [M]. –How much change did you get back? [N-DG6.8].

[...] Luis has 185 marbles and he lost 62 in the game. How many marbles has he left?

[...] Luis has 36 marbles and Antonio has 36 marbles, but they lost 24 marbles between the two. How many marbles do they have left all together?

[...] Luis has 25 marbles. If he lost 13 during the game, how many has he got now?

[...] In a vase there are 15 flowers; in another one there are 7. How many flowers are missing in the second vase in order to have the same number?

[...] I want to buy a toy that costs $48. I have saved $25. How much do I need to be able complete the amount?

[...] Luis had 25 marbles. If he lost 13 during the game, how many marbles does he have now?

[...] Luis had 25 stickers. If he gave 8 away, how many does he have left?

[...] There are 25 marbles in a tin and 15 were taken out. How many marbles are left in the tin?

[...] Ruben wants to buy a ball costing $30 and he has saved $20. How much is he still missing?

[...] Pepe collects matchboxes. He has 12 and he wants to have 25. How many is he missing?

[...] Juan has 8 marbles and Pepe has 3. How many is Pepe missing to have the same number as Juan?

[...] Hugo is 18 years old and Sergio is 6 years younger than Hugo. How old is Sergio?

–[...] The problem could have been that the child hasn’t read the problem attentively. He didn’t concentrate on the problem [C-DE2.1].

–This is a problem of attention or reasoning. The child knows the procedure; he/she arranges the units under the units and the tens under the tens, but the only trouble is that he added instead of subtracting them [Ma-DE2.5].

–[...] The problem could have been that the child didn’t read the problem carefully. He/she didn’t concentrate on the problem [C-DE2.1].

–The child didn’t understand it. He/she didn’t read it correctly [C-DE2.29].

–But in this case, why do you think that the girl used an addition to solve the problem instead of using a subtraction? [M-DG2.7].

–Because she didn’t read the problem, because she didn’t understand the problem. A lot of children are already solving the problem, before they have even read it. They get the numbers orally [Ma-DG2.8].

–[...] I say it’s not a teaching problem; you are not going to teach the children that they have to subtract or add here. The child has to know through logic, what he/she is supposed to do [N-DE2.42].

–I insist that it is not a teaching problem, but a reasoning problem. I am seeing that the child hasn’t understood [N-DE2.46].

–[...] We were saying that it was precisely that, that the teacher was not making use of the student’s previous knowledge, that the teacher doesn’t allow the students to create and solve subtraction problems and that the teacher doesn’t use concrete material. And the recommendation is precisely that, that the teacher should use concrete material as a basis to teach mathematics. There lies the basis so the pupil can learn to solve problems [I-DG2.2].

–I say it is a problem, because it is a problem of knowledge that the child doesn’t have, or that he/she doesn’t have and it shows up in the linguistic area when the pupil cannot interpret the text and therefore another procedure is applied here. What does this make you do? You have to go back, you have to interrupt your teaching plan and when you reach the higher levels, the problem is even bigger. We can go to the fifth or sixth grades and we can see that they are struggling a lot [J-DG2.11].

In relation to the type of situations that should be set forth to the children during the process of teaching subtraction we have found certain inconsistencies. In some cases, the teachers consider that they have basically to provide the children with real problems or situations represented through concrete material. In a second approach, they consider representing problems and situations through drawings. Up to this point, this position coincides with Carraher et al. (1995) point of view; they point out the importance of providing pupils with everyday situations as a context for the teaching of mathematics. However, all the situations suggested by the teachers for teaching subtraction to children refer to problems set forth in written sentences and to numerical exercises. The setting forth of problems and exercises through other means of representation, either oral, graphical, through drawings or concrete material, is non existent.

The insufficient use of mathematical problems to be solved by using other means of representation has serious didactic consequences, because, as is pointed out by Verschaffel & De Corte (1997) and Vergnaud (1991), representation plays a fundamental role in the process of solving problems. This is why it is necessary on the one hand, to expose the children to mathematical situations or problems represented through oral, written or graphical narration or with drawings or concrete material. On the other hand, they should be encouraged to use different forms of representation in the process of solving problems. These two aspects complement each other and allow the children to learn to face very different types of situations and problems and to develop more flexible strategies for problem solving.

The information obtained concerning the type of problems suggested by the teachers to teach children how to subtract, as well as the use of “keywords” as indicators for the type of operation the children have to use to solve the problem, also have important educational repercussions, since the setting forth of problems plays a fundamental role in the teaching and learning of mathematics in elementary school. It is clear that there is little variety regarding the order and presentation of the information of the problems, as well as regarding the type of related calculation involved. The fact that pupils are exposed only to one or two of the categories of problems suggested by Vergnaud (1991), to the use of “keywords” and that the place of the unknown quantity is generally the search for the final measure in the case of transformation problems, limits the development of related calculation in the pupils and focuses them only on the learning of numerical calculations.

The teachers’ conceptions about the difficulties in the learning of subtraction coincides with two of the three great perceptions pointed out by Carraher et al. (1995) that have been used to explain the pupils’ failure to learn mathematics. For them, the pupils’ difficulties to learn vary in nature: difficulties related to the pupil, to the didactic process, and hose linked to contextual aspects of practice. However, there is a tendency to explain the pupil’s learning difficulties in solving the subtraction problems, based on causes pertaining to the pupil.

In a more general way, we can point out that we have found the coexistence between teachers with a traditional conception and those with a constructivist one, concerning the teaching of subtraction and mathematics in general. Besides that, there have been inconsistencies in the teachers’ conceptions: for example, between the importance they give to contextualization and the type of situations they propose where didactic intervention is needed.

The inconsistencies found, as pointed out by Thompson (1992), seem to be the result of a complex relationship, whose influence springs from many sources:

• The teachers do not have the necessary skills and knowledge to apply the changes or reforms to the teaching of mathematics.
• The social context in which the teaching of mathematics takes place, imposes restrictions and offers opportunities.
• The teachers belong to a pedagogical group culture and tradition in which syntactic aspects are still favored over the semantic ones in the teaching of mathematics.
• The teachers adhere to teaching ideals that they cannot reached.

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