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Technical opinion about the document “Curricular Goals for Primary School – Mathematics
The Portuguese Society of Research in Mathematics Education (SPIEM) strongly recommends that the Ministry of Education and Science (MEC) should withdraw the proposal of Curricular Goals for Primary School – Mathematics, currently under discussion. This recommendation is the result of a careful analysis of the aforementioned proposed goals, scientifically supported by the results of national and international research in mathematics education, from which it is concluded that the proposed goals reflect globally a poor and reductive conception of what mathematics is and what students should learn about mathematics. In reality, it resumes curricular guidelines that are already outdated and that were at the root of the low performance of Portuguese students revealed in previous decades but which have been consistently improving in recent years. A possible implementation of the proposed targets now presented by MEC would be very serious for the country, as it would correspond to a significant regression in the mathematical learning of Portuguese students.
SPIEM considers that the proposed new targets are clearly inconsistent with the current Mathematics Programme, which is aligned with the current curriculum guidelines for school mathematics in countries with good levels of achievement. As specified below, the main teaching purpose highlighted in each theme of the Programme is not translated into the proposed goals, which do not value mathematical understanding (“understanding” or “comprehension” are words that do not appear once in the proposed goals), nor the progressive formalisation of mathematical knowledge, They prefer to assume formalisation as a given from the outset, even in situations where this does not make the slightest sense (for example, the emphasis given to mathematical formalisation goes as far as indicating that algorithmic and/or formal representations should be used to perform the integer division of two numbers up to 10, p. 16, descriptor 8.3). Much of what the targets indicate in each content ‘subdomain’ is new vis-à-vis the Programme – for example, axiomatisation of mathematical theories, set theory, ‘knowing the Greek alphabet’. There are also contents of the Mathematics Programme which are not included in the proposed goals or which are drastically reduced, such as for example algebraic thinking in the 1st cycle, the statistical investigation process, probabilities.
SPIEM points out that the document presented completely ignores what national and international research has identified as relevant in terms of the progression of students’ mathematical knowledge in the various mathematical topics and skills. In fact, this document, contrary to the learning goals that exist in several countries, is not based on the results of research in Mathematics Education. Priority is given to formal representations rather than understanding, complex formulations are indicated (such as the definition of a plane, a statistical variable or the axiomatics of a theory), which hardly any student understands, and initial approaches are indicated to introduce concepts that do not translate into the aspects on which they should be anchored (such as assuming the fraction as a measure as the starting point for understanding the concept of fraction, …).
Overall, the proposed curriculum goals constitute a body of indications that are neither articulated nor grounded, outdated and formulated with language that is not always adequate and clear. It is not understandable, for instance, (i) how “counting to one hundred”, “decoding the numbering system” or “knowing the Greek alphabet” can be stated as general objectives; (ii) why the targets ignore a wide field of mathematical problems, focusing attention on problems that are solved by one or more arithmetic operations; (iii) why abstract definitions are highlighted and are difficult for students to make sense of. We present below more detailed comments on the proposed curriculum goals organised by the mathematical themes defined in the Mathematics Programme in force in primary education.
Numbers and operations
The main teaching purpose stated in the syllabus is “to develop in pupils a sense of number, an understanding of numbers and operations, and mental and written calculation skills, and to use this knowledge and skills to solve problems in a variety of contexts” (p. 13). However, this main teaching purpose is not reflected in this proposed targets presented.
Estimation disappears completely and mental calculation is practically ignored (the expression “mental calculation” does not appear in the proposed objectives and only three formulations related to it appear: “Mentally add a two-digit number with a one-digit number …” (p. 5), “Mentally add or subtract 10 and 100 …” (p. 9), “Mentally perform multiplications of one-digit numbers with multiples of ten …” (p. 16)). The whole organization of the proposed goals is focused on the quick preparation of the structuring of the standard algorithm and its use in certain types of problems (only step problems are mentioned, ignoring other types of equally important problems). This option is completely inadequate and it is now known that it is at the root of the failure of pupils in terms of the development of their calculation skills, as numerous studies have shown.
The proposed targets reflect an appreciation of aspects that have the effect of diminishing pupils’ calculating power. It is not clear, for example, that only decimal decomposition is indicated when there are situations where other types of decomposition are much more effective and powerful or that students are required to add, subtract and multiply two natural numbers whose sum is less than one million using the algorithm. This guidance is very undemanding as pupils should know how to mentally calculate, for example, 9998 + 2; 345 + 20; 9998 – 2 or 25×16.
Several “descriptors” correspond to an inadequate indication, both at the level of what the student is required to know and at the level of what is proposed for the progression of his or her knowledge. For example, to consider that a first-year student should “(…) recognise that in the representation ’10’ the digit ‘1’ is in a new position marked by the placement of ‘0’” (p. 4) is to focus on aspects that are too abstract that a student of that grade cannot understand; moreover, such emphasis is not productive from the point of view of learning the concept in question. Also the indication to introduce non-negative rational numbers by “measure” (the proposed goals even consider the subdomain “measuring with fractions”) is an example of how to opt for what no expert on the subject recommends. In the target proposal, there is a preconceived idea completely refuted by research on this topic: that pupils’ knowledge of rational numbers develops according to the mathematical formalisation of the underlying concepts. Descriptors 7 and 8 that make up the objective “Measuring with fractions” (p. 17) for Year 3 are an example of this preconceived idea. What are the students expected to highlight as a learning objective: To “fix the line segment”? Orally reproduce that the number straight line is the support line of a ray? Such goals are totally misplaced for 8-year-old pupils.
The Mathematics Programme in force indicates as specific objectives the understanding of fractions with the meanings quotient, part-all and operator and also the location and positioning of non-negative rational numbers on the numerical straight line, having as a starting point the resolution of problems, namely equitable sharing and the use of area models. The knowledge that students are developing about rational numbers, particularly the fractional representation, is very contextualised, unlike what appears in the proposed Goals where the “descriptors” are based on very formal and abstract formulations, unsuited to the level of cognitive development of students at this level of schooling.
It should also be noted that one of the representations of rational numbers, the percentage, is practically absent from the proposed targets, when the Programme and current research highlight its importance, in association with the idea that students should establish connections between the various representations of these numbers. In the 2nd cycle there is a wrong choice to treat all four operations with rational numbers in grade 5, which will be too demanding for 10-year-old children, as well as to introduce operations with relative rational numbers in grade 6. Given the already ambitious objectives for this cycle of education, the proposed targets are unrealistic, compromising students’ learning.
Regarding the 3rd cycle, new objectives are also introduced concerning real numbers that are not present in the Mathematics syllabus. Operations with irrational numbers are not part of the programme that only suggested the simplification of some simple expressions with radicals (nor of the programmes in other countries). These are introduced in the 7th grade, whereas in previous programmes, going back to the 1980s, they only appeared in the 9th grade. This is another of several examples of integrating learning objectives without an understandable purpose and not taking into account the students’ level of mathematical development.
Still with regard to the goal of “Operating with square and cubic rational roots”, in the 7th grade, the draft goals totally ignore the technology that is so accessible nowadays and that it is the school’s obligation to prepare students for its proper and critical use. Thus, the goal of getting students to construct tables of perfect squares and cubes and to use them to “determine the decimal representations of square (respectively cubic) roots…” (p. 59) is totally anachronistic. Without the presence of technology it will also be of little interest to refer to the representation of rational numbers in scientific notation that will emerge in the eyes of students as just another content to learn without much personal or social meaning. The use of a calculator is only mentioned in the subdomain Trigonometry, which is clearly insufficient in a highly technological society such as ours.
The main teaching purpose that is stated in the Programme is “to develop in students a spatial sense, with emphasis on visualisation and understanding of properties of geometric figures in the plane and in space” (p. 20). Again, this main teaching purpose is not reflected in the proposed targets presented. Spatial visualisation and understanding the properties of geometric figures are missing from this proposal. In fact, the development of spatial visualisation, which includes perceiving the three-dimensional world around us, is summarised in the proposed curriculum goals to issues of location and orientation in space centred on the formalisation of specific concepts, such as the concepts of direction and angle. Regarding the understanding of properties of geometric figures in space and in the plane, one of the objectives of the Programme, the curriculum goals never use in the descriptors the verb “to understand” and, therefore, there is a real contradiction between these and the Programme.
In fact, the proposed curriculum targets correspond to inadequate indications, both in terms of what students are required to know and in terms of the progression of their knowledge. For example, in the introduction to the subject of Geometry and Measurement, it is stated in the proposed curriculum goals that one starts “by visually recognising elementary objects and concepts such as points, collinearity of points, directions, lines, rays and line segments, parallelism and perpendicularity, from which more complex objects such as polygons, circles, solids or angles are constructed” (p. 2). Besides contradicting the methodological indication on the teaching of Geometry conveyed by the Programme, it also contrasts with methodological guidelines at international level, which suggest an approach to Geometry starting from space to the plane and not starting from elementary concepts of the plane such as point, line, ray, line directions, etc. With regard to the formalisation of some concepts, the school year to which they refer and the way in which this formalisation is presented are considered inadequate. For example, goal GM2.2.1. (p. 12) on the identification of a ray starting at O contradicts the stipulations of the syllabus and the rigour of the definition is excessive for the 2nd year of schooling.
Regarding the progression in learning, it is considered that the proposed curriculum goals do not take into account internationally accepted guidelines on learning geometry, such as Van Hiele’s learning levels. For example, in the 1st cycle, students are between level 1, Information, and level 2, Analysis. Hence, targets such as GM2.2.7. “Identify and represent rhombuses and recognise the square as a particular case of the rhombus” (p. 12) are necessarily inadequate since they place students at level 3, that of Sorting.
Data organisation and processing
The main teaching purpose stated in the Mathematics Programme for primary education is “To develop in pupils the ability to understand and produce statistical information and to use it to solve problems and make informed and argued decisions, and also to develop an understanding of the notion of probability.” (p. 59). However, the draft goals do not reflect this teaching purpose in a balanced way, nor do they meet what it calls for as essential. It should be noted that the Mathematics Programme in force has assumed a strong commitment in this domain, extending the complexity of the data sets to be analysed, the measures of central tendency and dispersion to be used, the forms of data representation to be learnt and the work of planning, carrying out and analysing the results of statistical studies to be carried out, as well as the work with ideas related to probabilities, thus placing itself alongside the international curriculum guidelines that stress the importance of students’ statistical education.
The proposed curriculum goals point to a clear devaluation of this theme, contrary to the option taken by the Programme. This devaluation is visible right from the start by the reduced number of general objectives indicated, but also by the late entry of many topics that the Programme points to earlier (for example, the proposal of the goals only introduces ideas and concepts of probabilities in the 9th grade) and also by the limited perspective of many of the concepts related to the subject Organisation and Processing of Data.
A key point of criticism is the way in which the proposed targets approach statistical ideas, ignoring the fact that their fundamental aim is to develop pupils’ statistical literacy. In reality, instead of valuing the study and interpretation of real and relevant situations, it places the emphasis on the execution of techniques. With regard to measures of central tendency or location, it indicates procedural rather than conceptual definitions that only reveal whether students can obtain values, regardless of whether they understand what those values correspond to. For example, the proposed targets call for students to identify “the arithmetic mean of a set of numerical data as the quotient of the sum of the respective values (repeating each portion a number of times equal to the absolute frequency of the category in question) and the number of data” (p. 38), thus introducing the mean as the result of applying a complex algorithm and only useful in calculating a weighted average, rather than as a representative value of a set of data – note that no algorithm may even be needed for the determination of the mean. This proposal of targets thus ignores the fact that the main difficulties of students in Statistics do not lie in the execution of procedures, but rather in the interpretation of measures and in drawing conclusions about the situations under analysis – which is, in fact, the essence of statistical studies.
With regard to graphs, the proposed targets reveal a total lack of insight into how students construct graph meaning and a coherent sequence for their learning. For example, with regard to the bar chart, its introduction in the 2nd grade is not envisaged as an evolution of the point chart, an approach that research has shown to be effective; without any scientific basis, the bar chart appears alongside the Venn diagram or the Carrol diagram. The reference to the Cartesian frame of reference (p. 38), which the goal proposal associates, without necessity or relevance, with line graphs, is also forced, unreasonable and absolutely formalised, resulting in a completely artificial entry into the domain of functions.
Another perplexing observation has to do with the incomprehensible inclusion of set theory topics in the subject of Data Organisation and Processing. In fact, set theory is not even mentioned by the current Mathematics syllabus.
It should also be noted that the proposed targets contradict the Mathematics syllabus in all that concerns Probabilities. In fact, it proposes that students explore random situations involving the concept of chance from primary school, that they make appropriate use of concepts related to the possibility of a given event, and that random experiments provide suitable contexts for data collection and analysis. However, the proposed goals refer the approach to Probabilities to the 9th grade only, in condensed format, and in a formal way, reneging on the research recommendations about what students can and should learn in this subject.
It is also strange that in a subject like Organisation and Processing of Data no reference is made to the use of technology, neither as a tool for calculation or representation, nor as a tool for analysis. It is recalled that most of the experience that students will have with the concepts of this subject during their lives will be through the computer; therefore, the competence to deal with software dedicated to the organisation, processing of data and their representation is something fundamental but which this proposed targets neglects.
The main teaching purpose stated in the Mathematics Programme is as follows: “To develop in students algebraic language and thinking, as well as the ability to interpret, represent and solve problems using algebraic procedures and to use this knowledge and skills in exploring and modelling situations in a variety of contexts.” (p. 55). The proposed curriculum goals do not meet this purpose, presenting particularly critical features that compromise the desired mathematics education for students at the end of their basic education in this domain, as presented below.
Firstly, the absence of a developmental perspective of algebraic thinking from primary school, internationally advocated by research in mathematics education and present in the curriculum revisions of most countries with high levels of success in mathematics, should be highlighted. The proposed curriculum goals seem to start from the wrong assumption that algebraic language and procedures are transparent and that it is enough to present them to students for them to use them. It is paradigmatic of a misplaced view of how algebra learning develops the fact that the first “descriptor” of the first general objective of the 3rd cycle (“Multiplying and dividing relative rational numbers”, p. 50) consists of proving a property concerning the addition of rational numbers in algebraic language seems to assume that students should master this type of language by the beginning of 7th Grade.
In the introduction to the proposed curriculum goals, it is stated that it presents “a sequence of general objectives and descriptors, within each sub-domain, that corresponds to an appropriate teaching progression” (p. 1). However, it is not at all clear how pupils will progress in the mastery of algebra. For example, the (supposed) use of the algebraic language appears in the first subdomain of the 3rd cycle, as well as the resolution of equations in the 7th grade, but recognising and operating with monomials only appears in the 8th grade, which indicates a perspective of learning formal algebraic manipulation without understanding on the part of the student.
In fact, the existence throughout the curriculum goals proposal document of an inadequate view of how students develop the ability to understand and use symbolic language is evident when presented, for the 5th grade, descriptor 11, “Translate mathematical statements expressed in natural language into symbolic language and vice versa, knowing that the multiplication sign can be omitted between numbers and letters and between letters…” (p. 37; Algebraic Expressions). Targeted and meaningful work in a variety of situations is needed for students to use algebraic language with understanding, which will not be the case in 5th grade in the terms presented in the goals. The current version of the Mathematics syllabus in advocates the beginning of the development of algebraic thinking from the 1st cycle onwards, but this perspective is not present in the proposed goals.
Additionally, there is no reference in the proposed curriculum goals under analysis to the ability of the student to “interpret and represent situations in various contexts, using algebraic language and procedures” or to “interpret formulas in mathematical and non-mathematical contexts” (p. 55), objectives of the Mathematics Programme. In contrast, in the statement of the general objectives and descriptors of the curriculum goals verbs such as “Define”; “Designate” and “Identify” predominate, which correspond to statements of learning objectives at a more basic level that are not in line with a broader purpose of understanding algebraic language and procedures and their use in different contexts.
Secondly, the predominance of objectives that focus on definitions and formal language, without a defined purpose, is highlighted. Unlike the Mathematics Programme, which emphasises the understanding and interpretation of algebraic concepts and language, the present objectives present descriptors full of definitions and formal designations, as is the case of the subdomain Functions in the 7th grade. SPIEM seriously questions the inclusion of ten descriptors for “Defining functions” in the 7th grade (which includes, for example, “numerical function” (p. 56)), and whether it disregards the understanding of the concept of function (as a relationship between variables) and the ability to use it in various situations, as recommended by the Programme. It is not by memorising definitions and terminology that students will develop the concept of function, as the extensive research on this topic shows. Therefore, the goal “Define functions” is very reductive in relation to what the Programme intends for students in the 3rd cycle.
Taking into account the orientation of the Mathematics Program in force that advocates the work with sequences since the 2nd cycle and, in particular, at the beginning of the 3rd cycle for the introduction of the general term of the sequence through the symbolic language, it is inappropriate that in the curricular goals now proposed the fourth objective “Define sequences and successions” appears only after “Define functions”; “Operate with functions” and “Define functions of direct proportionality”. SPIEM also questions here, once again, the formality required of 7th grade students in distinguishing the definition of sequence and succession. What is its purpose for students in general? Thirdly, the introduction of new objectives in addition to those in the current programme should be highlighted. Since the algebra topics covered in the syllabus are already quite extensive, the introduction of new topics will make the task of meeting learning objectives very difficult for teachers and will contribute to increased student failure. The objective “Operate with functions” is one of the new topics introduced and its relevance is not seen. In fact, SPIEM strongly questions the inclusion of descriptors relating to properties of operations with linear and affine functions, especially since students are required to demonstrate and prove (descriptors 6 and 7 in the 7th grade) these properties. Also, the formality required in the descriptors relating to the objective “Identify the equations of the straight lines in the plane”, in the 8th grade, goes beyond what is pointed out in the Mathematics Programme, for example, by requiring the student to “Demonstrate, using Tales’ theorem, that the non-vertical straight lines in a given plane passing through the origin of a Cartesian referential fixed in it are the graphs of linear functions” (p. 67). Also the ten descriptors of the subdomain “Properties of functions and their graphs”, in the 9th grade, due to the formality with which they are presented, constitute learning objectives that go beyond what is presented in the Programme in force.
The proposed curriculum goals begin by stressing that problem solving, mathematical communication and mathematical reasoning are “indispensable to the achievement of the listed objectives” (p. 1), so that, according to the authors of the proposal presented, they are contemplated “explicitly or implicitly in all descriptors” (p. 1). A reading of the document shows, however, that this situation does not correspond to reality. Quite the opposite. There is no explicit reference to mathematical communication and verbs that translate actions related to this activity (e.g. explain, argue, debate, question…) are practically non-existent. There are numerous studies, particularly in the fields of education, psychology and philosophy, which show that the practice of communicating and discussing ideas are essential activities for the structuring of thought and, therefore, fundamental to understanding.
At the same time, the analysis of these objectives and descriptors reveals a strongly reductive perspective of the role and place of problem solving and mathematical reasoning, whether we look at these two capacities in the light of the nature of mathematics or in the light of what research says about mathematical learning processes.
In fact, analysis of the wide range of descriptors presented shows that problem-solving clearly has a secondary role. For example, in the 1st year of schooling, 31 descriptors are presented within the theme “Geometry and Measurement” without a single one related to problem solving. Also in the 7th grade, 18 descriptors are presented for the objectives “Define Functions” and “Operate with functions” without any reference to problem solving. The mathematical modelling present in the general teaching purpose of the Mathematics Programme for Algebra is totally absent in this proposed curriculum goals. Moreover, problems arise exclusively from a perspective of applying concepts and procedures. A paradigmatic example of this situation can be found in the 1st cycle and in the subject “Numbers and Operations”. Here the problems arise only after the direct teaching of arithmetic operations and the typology of problem classification used by the authors of the proposed goals is based merely on the number of “steps” used, that is, on the number of operations performed. Now, problems in mathematics are not just a field of application of knowledge. They have many other functions including justification, motivation, recreation and vehicle. They are all important, an idea well highlighted by Ian Stewart when he states that “problems are the driving force in mathematics”. The restricted perspective conveyed by the curriculum goals on problem solving limits students’ development of the ability to elaborate strategies to deal with unfamiliar situations and, in this way, their intellectual maturity.
After the introductory note, in the draft curriculum goals there are only five explicit references to reasoning. One highlights the need to prevent circular reasoning (p. 75). The others are exclusively associated with deductive reasoning (pp. 21, 45, 49, 66). At the 3rd cycle of basic education, descriptors are beginning to appear whose wording includes the words “prove” or “demonstrate”, two important aspects of mathematical reasoning but which also refer to deductive reasoning.
However, in mathematical activity there is not only deductive reasoning. Why limit students’ mathematical experience to learning this kind of reasoning? Without denying the importance of deduction, to exclude the possibility for students to also learn to reason inductively or by analogy (for example) is to eliminate the possibility for them to learn to conjecture and to understand the meaning and role of counterexamples in mathematics. As both national and international research shows, if students do not engage in activities that require the formulation and testing of conjectures, if they are not confronted with mathematical tasks whose resolution appeals to plausible reasoning, they will hardly understand the need to demonstrate or attribute importance to this activity, i.e. they will hardly learn to demonstrate. Mathematical reasoning is closely associated with a habit of thought related to understanding why “things” happen. Without the understanding of this why, one may even “know how” (an equation, a count, an algorithm) but will not be able to use knowledge in a criterious, critical and flexible way. One will hardly succeed in activities that go beyond the trivial or routine. Understanding why one does what one does inevitably leads back to the activity of justifying which, therefore, should have a prominent place in students’ learning from the beginning of their schooling. However, in the curriculum goals, it seems that this activity only begins to be truly important in the 3rd cycle of basic education. In fact, it is only in this cycle that the authors of the proposed targets include the verb ‘to justify’ in the set of those they enunciate in the paragraphs entitled ‘Reading the curriculum targets’. In the cycle prior to this one, the activity of justifying is subordinated to the action of “Recognising” what is not intelligible, especially since justifying statements by evoking already known properties (meaning attributed to the verb to justify in the 3rd cycle) is a desirable and accessible activity for 2nd cycle students. In the 1st cycle there is no reference to the need for students to justify mathematical ideas, which will lead to a significant decrease in the quality of their learning both in this cycle and in future studies.
From the analysis presented, SPIEM stresses the need for MEC to withdraw the proposed curriculum targets under discussion. It should also be noted that there are other learning goals under experimentation and that there is no evaluation of them, despite their harmony with the Mathematics Programme in use in basic education. Thus, SPIEM recommends that the Ministry of Education and Science, instead of proposing “new” curriculum goals, channel its efforts and investments into a scientifically sustained action that allows Portuguese students to continue to improve their mathematical learning.
Portuguese Society of Research in Mathematics Education
Lisbon, July 23rd 2012