Table of contents
Technical opinion on the Mathematical Syllabuses for Secondary Schools
Coordinating Committee of the Mathematics and Education Section of the SPA
New mathematics syllabuses (A and B) for grade 11, as well as the syllabus for mathematics applied to social sciences, have recently been proposed for discussion. As it has done on previous occasions, the Education and Mathematics Section of the Portuguese Society of Educational Sciences welcomes the willingness of the authors of the programs to debate their ideas and wishes, through this opinion, to contribute to it.
In order not to disperse our efforts, we decided to focus on an assessment of the Mathematics A syllabus and the Applied Mathematics for Social Sciences syllabus.
Mathematics A Program
Although only the Year 11 syllabus was put up for discussion, it seemed to us that its discussion would be more fruitful if considered together with the proposals for Year 10, paying special attention to the introduction that frames the entire Mathematics A curriculum for secondary education. This introduction has been considerably expanded both when compared with the current curricula and with the first proposals for Year 10, which were the subject of a previous opinion.
We wish first of all to express our agreement with the concern expressed by the authors in developing a modern program, adapted to the demands of today’s world, and which seeks to incorporate many of the ideas that run through the concerns of mathematics educators in other countries. The valorization of a student-centered approach, to the detriment of an emphasis on language, the search for diversified methodologies to study mathematical topics, rejecting the vision of a monolithic mathematics based on logic, as well as an incorporation of technology that sees it as an opportunity to improve the quality of learning, and not as a threat to mathematical knowledge, are characteristics of this program, as they were of the previous one, and deserve our full support.
There are, however, some points that need to be reworked, which we will indicate below.
About the Purposes of the Program
Section 2.1 on the aims of the syllabus should contain general objectives which explain the ways in which mathematics can contribute to the more global aims indicated for the whole school cycle and which clarify the essential contribution of mathematics to the education of young people at this age level who have chosen these General Courses. We therefore disagree with the inclusion of purposes which, although meritorious in themselves, do not express a relationship with the discipline, a situation which covers the latter two.
We also believe that an effort should be made to condense this, which could, for example, involve defining two broad areas.
The first, centered on the idea of “Thinking mathematically”, would encompass the first and second aims with minor changes to the wording (the changes we propose are uploaded):
1 – Develop the ability to use mathematics as a tool for representing, interpreting and intervening in reality;
2 – To develop the ability to reason mathematically, namely by formulating and solving problems, communicating, drawing up and testing conjectures and carrying out deductive reasoning.
The second area would establish links between mathematics and scientific culture and society, and would include the third, fourth and sixth aims, albeit with changes:
3 – Promote the deepening of scientific, technical and humanistic culture (the reference to the importance of further studies is already made in the decree law that institutionalizes the new curricular framework, Art. 3, 1).
4 – To contribute to a positive but critical attitude towards science and particularly mathematics (similar to what is in program C).
6 – To support the development of critical awareness and intervention in areas such as … (complete).
The fifth purpose must be eliminated.
About the Cross-Cutting Themes
Although there was a reference to these topics in the draft program presented last year, the authors have considerably expanded their description. Although we recognize a remarkable effort of elaboration, we would like to express some reservations about this.
A first objection concerns the characteristics of these themes. We believe that they are very different in nature and can hardly be classified under the same name. Some relate to didactic procedures, others to the quality of learning, and others can hardly be distinguished from syllabus content. Although it is a good thing not to reduce the syllabus to the study of mathematical topics, this plurality of cross-cutting themes makes it difficult for teachers to grasp them.
A second objection focuses on the lack of articulation between the themes, the syllabus and the methodological suggestions. For example, the three pages of the introduction to section 2.4 make proposals that in some cases overlap with the themes, while in others they go in other directions. How should the teacher manage this contradiction? Does the program have a third dimension, which is also partially independent of the other two?
A third objection relates to the scarcity of examples of the use of cross-cutting themes interlinked with the syllabus itself, and we also get the feeling that the cross-cutting themes/content interconnection is being blurred throughout the text.
Although the idea of designing a curriculum that allows for non-linear readings, using, for example, the crossing of two dimensions, seems interesting to us, the ambiguities contained in the text, aggravated by the lack of exemplary cases, compromise its effective application by teachers. The programs in force in the 2nd and 3rd cycles also contain references to “transversal themes”, which, in the absence of effective articulation with the program topics, and in conflict with dominant teaching practices, are forgotten.
We therefore propose that the reference to cross-cutting themes be simplified (no more than three or four), emphasizing only those related to mathematical processes to be developed by the students. We also propose enriching and clarifying the relationship between the themes, the syllabus and the methodological suggestions.
About an occasionally prescriptive style
In our educational tradition, school curricula have the function, among others, of informing the various parties involved in education (teachers, textbook authors, parents, etc.) about the content that must be taught. However, there is no single path to successful teaching, nor is there one teaching method that proves to be effective for all students. It follows that, even in an educational system with a centralizing tradition such as ours, there must be (and indeed there has been) some room for manoeuvre so that each teacher can adapt their methodologies both to the diverse characteristics of their students and to the way in which they specifically appropriate the syllabus.
However, the syllabus under analysis occasionally adopts an overly prescriptive style. We are referring in particular to p. 18, where there are several sentences that aim to over-condition teaching choices: “methodological options … must be followed”, “methodological indications … are not simply indications” (are they indications or not?), “although this does not constitute a rigid (sic) instruction … it must be an obligatory reference … and must limit …” (it is not rigid, but it is obligatory and it does limit!). In this case, we suggest rewriting these paragraphs.
The implementation of the program
As we have already pointed out in interventions in the Monitoring Committee, the effort leading to proper implementation of the program should be focused on developing and disseminating quality teaching examples at classroom and assessment level. The role played by projects developed by schools can be fundamental here in incorporating the programs into the professional culture of mathematics teachers.
Applied Mathematics for the Social Sciences Program
We don’t have a tradition of diversified mathematics programs in Portugal. Developing programs of this type (and critiquing them) is therefore a difficult task. We will therefore limit our opinion to a few general aspects.
About the program’s aims
The aims proposed for this program are practically the same as those proposed for Program A. As we have argued before with regard to the latter, we believe that the aims should be linked to the general aims of the cycle, as well as explaining the essential contribution of mathematics to the education of the young people who have opted for these General Courses. Although they may therefore have some points in common with those of Mathematics A, these aims should contain something specific, because that is the only way to understand why this program is different from Mathematics A. If the aims were the same, would that mean that they could be achieved with a simpler program, and what would Program A be for?
The two differences between the proposed aims lie in the exchange of the third in Program A for the fourth in this program, i.e. while the students in the first program should deepen their scientific culture, those in the second program should improve their illiteracy. We think this is too little, and that a more stimulating wording could be found for the second program. (The La Palisse-style wording “reducing innumeracy … increasing literacy” could also be improved).
About the contents
In general, the contents seem adequate. However, the inclusion of Decision Theory should be better justified. This is an unfamiliar subject, the sources of which are in English, and which should deserve more substantial development. Shouldn’t we opt instead for the study of Game Theory, which is better known?
On support for teachers
We anticipate the need for strong support for teachers who are about to start teaching this program. In this sense, we think that the methodological suggestions should be expanded, following, for example, what was done in the previous programs, which contained a development that helped clarify the authors’ intentions.
It will also be necessary for official structures to be willing to provide in-depth support, otherwise this program will be devalued.
February 2nd, 2001