#### Table of contents

#### Technical opinion on the document “Mathematics – Basic skills”, DEB

Coordinating Committee of the Mathematics and Education Section of the SPA

In this technical opinion we will highlight three issues. Firstly, the general framework aspects that relate to the relationship established between this document and other actors with curricular influence. Secondly, we will discuss the document in general, ending thirdly with some specific curricular issues.

##### 1. General framework

1.1. This is a positive action

We think it is positive the intention carried out by the DEB to promote a reflection on the curricula, establishing relationships between the three cycles of basic education, which have rarely been made explicit. The concern for dialogue on these issues with other educational actors (teachers, schools, researchers, etc.) is equally meritorious and can only lead to an enrichment of the final documents.

1.2. Concern for dispersion factors

We would, however, like to express some concern about the consequences inherent in the appearance of yet another programme document. In Portugal we have a tradition of single centrally defined curricula and the existence of several documents containing curriculum guidelines can be an element of disorientation in schools. In particular, the problem arises of how to resolve any discrepancies between these documents and the programmes in force. For instance, what to do when the brochures do not refer to a theme that the curricula contain? Is that topic no longer part of the programme? While we assume that this is not the DEB’s intention, there is the possibility of confusion.

We assume that the DEB has an integrated view of the various brochures in preparation. However, considering, for example, the differences in style between the documents on Portuguese and Mathematics, it is not clear to us that this view will be shared by the authors of each of the booklets. This integration difficulty may also exist in relation to teachers in schools. We are thinking in particular of primary school teachers, who will be confronted with a multiplicity of documents, all of which are intended to have consequences for their teaching activity and contain possibly distinct pedagogical visions.

We therefore recommend that the DEB understand the launch of these brochures as an interim step towards a more comprehensive curriculum review.

1.3. Suggestions for deepening and implementing the documents

We understand that brochures should be concise and written in a straightforward language. Although they are condensations of more in-depth reflections (expressed in books published by the DEB), the recipients are unlikely to have access to these insights. In the United States the launching of curricular reflections (the Standards) was associated with the presentation of multiple practical examples. Similarly, the implementation of the recent reform of the secondary school mathematics curriculum benefited from the contributions made by a collection of short books that developed and concretised many ideas expressed in the curriculum.

We therefore recommend that the DEB look into the feasibility of launching small books to support the brochures.

##### 2. General comments on the Mathematics booklet

We consider overall that the Maths booklet is informative and clearly written and fit for purpose. We think, however, that some aspects should be changed.

2.1. Appreciation of the general aspects of each mathematical topic

The graphical layout chosen to distinguish between the competences mentioned for all cycles and those mentioned for each cycle is unfortunate, as it values the latter while overlooking the former. In the drafting of this opinion we were constantly confronted with this problem and we strongly recommend that this graphical aspect be changed.

2.2. Inclusion of recommendations for all cycles

The fact that specific competences for all cycles are not mentioned in the topic “Statistics and Probabilities” and in the topic “Algebra and Functions”, may lead to the idea that these topics should not be addressed in the omitted cycles, despite the fact that valid competences for all cycles are mentioned in these topics. We think that this situation should be changed and further on we propose some solutions.

2.3. Promoting an integrated view of the brochures

We think it is DEB’s intention that the booklet devoted to mathematics be considered both in relation to the other subject booklets and to the “General and transversal competences” booklet. We suggest that there should be cross-references in the various brochures as a way of prompting the reader to access the other brochures.

2.4. Reference to early childhood education

We understand the criteria why the DEB has not included references to early childhood education. However, this is becoming increasingly widespread and may be integrated into basic education in the short term. We think it is feasible to include some specific recommendations for this age group, although safeguarding their non-mandatory nature.

##### 3. Recommendations on specific curricular themes

3.1. Numbers and Calculus

The key competences indicated in the area of Numbers and Calculus, throughout all cycles and in the different cycles, are, in general, in agreement with the current programmes. However, taking into account its centrality to the exercise of citizenship, as well as the relationship of this topic with technology, it is suggested that:

Percentages are explicitly placed at the level of key competences, namely describing and comparing parts of the whole and recognising the equivalence between fractions, decimals and percentages, as a way of understanding numbers and their forms of representation and relationships between them.

A critical attitude towards technology, namely through the development of estimation strategies, is developed and integrated, from an early age, at the level of essential competencies.

3.2. Geometry

The key competences indicated in the area of Geometry at the various cycles are generally in agreement with the current programmes. However, one notes a more explicit reference to visualisation and spatial reasoning to be developed at all cycles and no reference is made to trigonometric ratios in the 3rd cycle.

With regard to the comparative analysis with other countries, in some cases the introduction of what is referred to as Coordinate Geometry and graph theory, normally included in representation systems, is noted from a very early stage.

Although our tradition points to work on coordinates associated with functions, the way they appear in the documents analysed is very much linked, in the early years, to relative position, and later to representation in a referential and the distance between points.

We generally agree with the recommendations contained in the document in relation to Geometry and do not feel that they should be amended.

3.3. Statistics and Probabilities

The key competences indicated in the area of Statistics and Probabilities reveal a positive concern to go a little beyond what is indicated in the current syllabuses, which, in this area, are clearly insufficient. Even so, the competences indicated in this document are too limited in view of the imperatives of a basic training for the exercise of citizenship.

Firstly, these competences do not clearly relate Statistics to the process of carrying out an investigation using quantitative data, which begins with asking questions, goes through the preparation of a plan for collecting and analysing data, using various computational and conceptual tools, and ends with making inferences. The document refers to some of these aspects but does not address them in a coherent and properly articulated manner. In fact, it is significant that this area is called “Statistics and Probabilities” instead of “Statistics and Data Analysis”, as would be more appropriate.

Secondly, the mentioned skills leave something to be desired on a number of specific points:

are not very explicit when it comes to studying the general characteristics of data sets, which requires more than “informal” ideas about data dispersion involving the use of concepts such as range, mean deviation, variance, standard deviation, symmetry and outlier;

do not mention the important notion of sampling and the sampling process, which is essential for making and critically analysing inferences;

do not indicate the need for students to be able to use a variety of ways of representing and analysing data, making use of new technologies.

Thirdly, as we mentioned in 2.2, the competences indicated do not include specific objectives for the 1st cycle, unintentionally giving the message that this is not a very important subject in this cycle. In fact, the 1st cycle would be the appropriate time to introduce the notions of mean and mode, as well as absolute frequency, in addition to initiating the learning of forms of data representation such as pictograms and bar charts.

3.4. Algebra & functions

The skills to be developed throughout the cycles are in line with similar themes found in foreign curricula. The emphasis on the study of regularities and their representation through multiple processes, the manipulation of equations and the use of the notions of correspondence and transformation are very important aspects of this subject and are made explicit in the proposal.

1) A generic issue. As we mentioned in 2.2, the fact that only one set of skills is presented for the 3rd cycle gives the message that these aspects are difficult to apply in the other cycles. In this sense, we present two suggestions for the inclusion of some aspects concerning the 1st and 2nd cycles.

1a) Functions. The two curricula analysed propose the study of patterns and relationships from an early age. Although this aspect is also referred to in the section on Numbers (improving the concept of number) I think there should be a further reference here, emphasising the propaedeutic importance of a pre-functional approach to this study of patterns and relationships.

1b) Algebra. Very simple “equations” are already solved very early on, as an extension of the properties of arithmetic operations. Formulas (which can be thought of as functions) are also used from an early age. For example, both the Americans and the Britons see these two uses as an extension (an anticipation in time) of the use of algebra and functions at lower age levels.

2) Some foreign curricula emphasise the idea that algebra relies on a complex codification of arithmetic properties implicitly suggesting that the teaching of the former should establish strong links with the latter. This recommendation is in line with the research findings, which point precisely to the need for a number-equation connection as a way to avoid a purely instrumental learning of algebra. This perspective, however, is not referred to in this proposal.

February 22nd 2000