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Batanero, C. (1997). Cuestiones metodológicas en la evaluación de los conocimientos matemáticos de los alumnos y de su evolución. Quadrante, 6(2), 25-43.

In this work the problem of the inferences is analysed on the mathematical knowledge of the pupils, his structure and evolution, from the answers observed in assessment situation.
Keywords: Assessment in Mathematic; knowledge structures; teaching variables; research methodologies.

Barreira, C. & Pinto, J. (2005). The research in Portugal about pupil's assessment for learning. (1990 - 2005). Investigar em Educação, 4, 21-105.

In this article it is done a investigation review made in Portugal about the assessment of pupils' learning, between 1991 and 2005, in several teaching levels not high school. As main conclusions stand out: The assessment practices focus on a summative perspective, in spite of the teachers to join conceptually to the formative evaluation; The tests keep on being the commonest form of assessment, in spite of the use some support instruments to the practices of formative assessment; The training is still relatively scarce and it takes not into account normally the institutional and cultural dimension of the assessment.
: Pupil's assessment for learning; basic teaching assessment law analysis; assessment practices; assessment representations and practices; formative assessment; summative assessment.

Bondoso, T.; Santos, L. (2009) Portfólios... and other discoveries. Educação e Matemática, 101, 3 - 9 (download)

In the context of the project AREA a set of teachers has been carrying out several studies in order to construct of knowing about the assessment as a learning device regulator of learning. This project is developed in a group of work in which they are shared and study experiences developed in several levels of teaching. In individual, the experience that we will be going to board develops in a kid garden room with 5-year-old children.
Keywords: Pre-school education; assessment; learning; portfolio.

Carvalho, Mª; P. (1992). Formative assessment by computer. Quadrante, 1, 139-147.

This paper describes a first phase of database use that allows to develop a formative assessment with the pupils, in individual to supply them feedback, and then, the pupil can learn by itself. The developed experience comprised the subject  Numerical Equations of the 1st degree in Q of the 7th year Mathematic program.
Keywords: Assessment in Mathematics; Mathematics learning in computer-assisted; formative assessment.

Cuadra, F.; Romero, L. & Martinéz, E. (2003). Concepciones y creencias del profesorado de secundaria andaluz sobre enseñanza-aprendizaje y evaluación en matemáticas. Quadrante, 12(1), 75-101.

This work describes and characterizes the conceptions and beliefs on mathematics assessment sustained by Andalucian secondary teachers. It is an exploratory study. Survey technique is used by means of a closed multiple-choice scale questionnaire applied to a sample (n=163) of the studied population (n=412). The factor analysis of the data let us detect general factors that establish the global conception sustained by the teachers on this topic. The general factor is articulated by means of 16 and 13 partial factors respectively, which show several teachers’ beliefs. The cluster analysis of the teachers factor scores allows to establish thought tendencies with regard to the beliefs in teaching and learning and assessment already detect and to interrelate them.
Keywords: Teacher's thought; conceptions and beliefs; assessment in Mathematics; teaching and learning process; mathematic pedagogy.

Cyrino, M.; Soares, Mª T: & Buriasco, R. (2003). From the evaluation of the school profit to the assessment for learning in Mathematics: a research in Paraná/ Brazil. Actas do XIV SIEM (Seminário de Investigação em Educação Matemática) (pp. 381-412). Lisboa: APM. 

This paper concerns a study that analysed the pupils' productions of a sample of 729 obtained proofs of the universe of the proofs of Mathematics carried out by the pupils of 4th and 8th series of the Paraná Education System. In individual, it tried to identify if the pupil chooses or not a proceeding that resolves the question, if it tests the alternatives, if it registers the data of the question and which the type of used notation.
Keywords: Assessment in Mathematics; pupil's works; solving process problems.

Dias, P. & Santos, L. (2008). Reflect before acting. The regulated assessment in Mathematic B. In L. Menezes; L. Santos; H. Gomes & C. Rodrigues (Eds.), Avaliação em Matemática: Problemas e desafios (pp. 163-171). Viseu: Secção de Educação Matemática da Sociedade Portuguesa de Ciências de Educação. (download)

In the research it tried to understand if the pupils' answer to a task can be helped by the understanding and anticipation of its resolution. In other words, reflect before acting will be able to contribute to a good performance and consequently to remove errors and difficulties. It was concluded what this kind of regulated assessment may: to develop the written understanding skills; to allow the passing of errors and difficulties; to contribute to the task resolution successfully. The deep reflection about the task can help the pupil to exceed errors and difficulties, as soon as it will develop the necessary mechanisms to value the feasibility or otherwise of a particular strategy for resolution.
Keywords: Assessment in Mathematics; reflections; resolution strategies.

Dias, S. & Santos, L. (2008). Why is it important to identify and analyze the pupil's errors and difficulties? The regulator feedback. In L. Menezes; L. Santos; H. Gomes & C. Rodrigues (Eds.), Avaliação em Matemática: Problemas e desafios (pp. 133-143). Viseu: Secção de Educação Matemática da Sociedade Portuguesa de Ciências de Educação. (download)

This paper presents the research of the feedback (Mathematics) results done with a group of 7th grade. The study central question was "what the pupils think of the comments / annotations that the Mathematics teachers write when they assess the first version of the works?" The outcomes showed that the pupils are not got used to an assessment in two parts and have a tendency to distrust the real teacher intentions. Additionally, most of pupils wrapped in the research reveal to attach little importance to the comments written by a teacher. For last, this research demonstrated to the difficulty that the teacher feels in giving the certain feedback of form to lead the pupil to identify his error, to correct it and, ideally, not to correct it again.
Keywords: Assessment in Mathematics; learning in Mathematics; feedback.

Graça, M. (2003). Assessment Solving problems: What relation between the conceptions and the teachers' practices? Quadrante, 12(1), 53-73. 

This research is developed in a Mathematics Education Msc dissertation and it focuses on the study of the relationship between the conceptions and the practice of three mathematics teachers in problem solving assessment. The methodology is qualitative and interpretative, based on case studies. Data gathering was done through interviews, observations, and notes. Data analysis was based in a set of categories previously defined. These teachers revealed coherence between its beliefs and practices in problem solving, although they also showed some discrepancies in their own assessment practices. The main direction of the conclusions of this research points to the necessity to understand more deeply the relationships between the conceptions and the practice of the teachers in problem solving. The need to teacher training programs in this domain was another aspect referred by the teachers of this study. This study also emphasises that assessing problem solving is a complex task, even to experimented teachers.
Keywords: Assessment in Mathematic; solving problems; conceptions; teaching practice.

Heuvel-Panhuizen, M. (2003). Paper-and-pencil assessment that provides footholds for further instruction needs to break with a number of taboos in assessing mathematical knowledge. Quadrante, 12 (1), 21-36.

The question that remains is how written assessment methods could be improved in such a way that they can help teachers to make informed instructional decisions. The present article will address this question within the framework of Realistic Mathematics Education (RME) - the approach to mathematics education in the Netherlands - and will show some of the didactical alternatives for the traditional written assessment that have been developed within this approach. The focus will be on the alternatives developed for primary education.
Keywords: Assessment in Mathematic; numerous; solving problems.

Martins, C. (2004). The portfolios in initial Mathematics' teacher training. Quadrante, 13(1), 63-89.

This paper reports a study whose aim main was to check whether the use portfolios during the pedagogic training of the Mathematics and Sciences of the Nature initial formation teachers contributes to the development of reflection on Mathematics teaching and learning and, specifically on the assessment by encouraging the pupils personal and professional development. Developed in the context of initial teacher training, the study ends that the portfolio can be an important tool in developing the capacity for reflection and its use encouraged communication between the trainee and supervisor to improve the quality of supervision.
Keywords: Assessment in Mathematic; reflection; initial teacher training; stage; portfolio.

Morgan, C. (2003). Criteria for authentic assessment of mathematics: Understanding success, failure and inequality. Quadrante, 12(1), 37-51.

The activity of assessment and the associated criteria, whether explicit or implicit are an essential part of any educational practice. Indeed, Bernstein's theory of pedagogic discourse identifies the evaluative rules as constitutive of pedagogic practice - without them there would be no transmission or acquisition but only conversation. The differences in assessment between traditional and reform pedagogies are not only related to the types of tasks used for assessment purposes but also to their orientation and the types of criteria they involve. It is this aspect of authentic assessment practices - the nature and application of their criteria - that I intend to explore in this paper.
Keywords: Assessment criteria; authentic assessment; Bernstein's model of pedagogy; social justice; pedagogic discourses.

Oliveira, I. & Pereira, J. (1993). Diagnostic assessment: development of a test of Mathematics performance. Quadrante, 2, 113 - 126.

This paper takes as a base the assessment for learning test development with diagnosis function, particularly up the problems solving processes and several techniques are explored for its analysis. The items design took referring the new Mathematics program of the 5th and 6th grade which was in force for the first time in the 6th grade. This test was applied to a group of 100 pupils. By the outcomes it can be said that the pupils had the worst performance in the problems resolution. Also showed weak performance in the activities that were demanding to communicate reasoning, to validate conjectures and to formulate arguments to justify their opinions.
Keywords: Assessment in Mathematic; diagnostic assessment; problems solving;  mathematic communication.

Paula, I. (2005). The portfolios as a integrating learning process and assessment in Mathematics. In GTI (Org.), O professor e o desenvolvimento curricular. (pp. 191-215). Lisboa: APM.

This paper reports the work developed in 2003/04 with 6th grade pupils. This research had as aim main to investigate whether the portfolios models the curriculum developed in the classroom.
Keywords: Curricular development; Mathematics; portfolio; learning process; school practice.  


Pinto, J. & Santos, L. (2006). Modelos de avaliação das aprendizagens. Lisboa: Universidade Aberta.

This book discusses the pupils' assessment for learning, taking it as a social relationship whose main purpose is to help the teachers and the pupils to build a formative interaction in terms of teaching / learning. Besides doing a short review this on the evolution of the concepts and of the functions in assessment, the book focuses mainly on the problems that emerge from a framework of formative assessment, namely: the importance of the work around the assessment criteria, the rule's paper leaning process, and still the potentialities and limits of some tools of assessment.
Keywords: Assessment models; assessment of learning; formative assessment; assessment tools.

Rogers, L. (1996). Social factors in the assessment of mathematics: Students, teachers, schools and political institutions. Quadrante, 5(2), 61-74.

This paper considers the broad purposes of assessment of mathematics with regard to the demands of society, the requirements of the institutions evolved, and the perceptions of the teachers and the students. The relationships between these different agencies are discussed, and it is demonstrated that there is a danger that assessment can be regarded as the principal means of control and regulation of the educational system.
Keywords: Assessment in Mathematic; assessment purposes; assessment arrangements; assessment dilemma.

Santos, L. (2009). Diferenciação pedagógica: Um desafio a enfrentar. Noesis, 79 pp. 52-57(download)


Since the late 80's that several learning assessment standards and many curriculum documents give special emphasis to the training component of the assessment. But one thing is to prescribe, another is to put it into practice. Developing an assessment practice with regulatory intent isn’t always easy. This process involves several types of changes in the classroom work, such as the nature of proposed tasks, the work method, and rethinking the role of teacher and pupils. Increasingly, teachers are confronted with pupils' diversity who have not only diversity in the learning place, but also in the think and learn forms, not to mention the different cultures, values and Portuguese language, in presence. Thus, the creation of pedagogical differentiation situations becomes increasingly a teaching imperative. In this paper we discuss the concept of pedagogical differentiation and it’s different achievement levels. Will be presented and discussed concrete examples of practices to develop differentiated teaching in the Mathematics classroom.
Formative assessment; pedagogical differentiation; Mathematics for all; teaching and learning of Mathematics.


Santos, L. (2008). Dilemmas and challenges of regulatory assessment. In L. Menezes; L. Santos; H. Gomes & C. Rodrigues (Eds.), Avaliação em Matemática: Problemas e desafios (pp. 11-35). Viseu: Secção de Educação Matemática da Sociedade Portuguesa de Ciências de Educação. (download)

In this text there shows up the evolution of the formative assessment conception along the last decades. Then, several assessment practices of regulated nature possible of being developed in the daily life of the work of the classroom are described and discussed, in Mathematics individual. For last, there are discussed some of the dilemmas and obstacles with which the teachers are confronted about this assessment kind. This discussion is supported in research carried out in this area and, very much in individual, in some first results obtained in the work developed in the context of the Project AREA.
Keywords: Formative assessment; oral questioning; feedback; self-assessment.

Santos, L. (2003). Assessment with competences: an impossible task? Educação e Matemática, 74, 16-21.(download)

This article tries to discuss the competence concept according to the documents published in the context of curricular reorganization  for the primary education and the understanding of assessment in this new reference picture. In particular, there are presented some case-studies of Mathematics classroom that intend to show the correct practice of assessment.
Keywords: Assessment; Mathematic; competence; school training; observation; questioning; feedback.

Santos. L. (2003). Assessment for learning in Mathematics. Quadrante, 12 (1), 1-5. (download)

This article presents and discuss possible reasons why the assessment was and a problematic question keeps on being today.
Keywords: Assessment in Mathematic; assessment nature; the teaching work and the assessment; the assessment and the society.

Santos, L. (2003). The assessment in guiding documents for the Mathematic teaching: a brief analysis. Quadrante, vol. XII 1, 7-20. (download) 

This paper analyses the way how assessment is considered in several documents that influenced mathematics curricular development in Portugal over the last 25 years. This analysis allows the identification of different conceptions of assessment which follow the major trends in this field, although the discourse underlying this evolution is not always coherent in terms of the principles enunciated and action guidelines proposed. There is also the tendency to view assessment as a process that is closer to the direct interveners in pedagogical action. The evolution of the types and tools of assessment is marked by the diversity.
Keywords: Curriculum assessment; mathematics curricular guidelines; assessment of learning; assessment guidelines; assessment tolls and ways.

Santos. L. (2002). Self-assessment regulated: why, what and how? In P. Abrantes e F. Araújo (Orgs.), Avaliação das Aprendizagens. Das concepções às práticas (pp. 75-84). Lisboa: Ministério da educação, Departamento do Ensino Básico.  (download)

It is discusses in this article how self-assessment can be understood as a metacognition process, i.e. as a internal mental process through which one takes conscience of the different moments and aspects of his cognitive activity. There are presented and discussed some practices ways  that are favourable to the development of pupils' self-assessment abilities.
Keywords: Assessment; regulated assessment; self-assessment; feedback;  perspective to the mistake; assessment criteria.

Santos, L. (1997). Exams: a way to proceed? Educação e Matemática, 43, pp. 5-12.

This paper discusses the preconditions that can explain the importance held by some against the existence of the national examinations. In particular, it presents arguments that contradict the conditions relating to: promotion of equity, quality of education, credibility, and means to meet the programs.
Keywords: Assessment; equity; homogeneity; quality; credibility; practice teaching.

Santos, L. (1997). Portfolio or the folder's pupil. Educação e Matemática, 42, pp. 11-12.  (download) 


This paper presents the portfolio as an alternative assessment tool. In particular, discusses the possible content, the ways of implementing it, the teacher and pupils role in this process and any potential.
Keywords: Assessment; portfolio; reflection process; learning process; relationship teacher-pupil.  


Santos, L. & Pinto, L. (2009). Self-regulating assessment in Mathematics: saying before doing. Bolema, 33, 51-68. (download).


This article reports on a study involving secondary students in mathematics, where formative assessment was introduced throughout the school year. In this teaching strategy students had to interpret a task, write the resolution strategy and then solve it. The teacher gave feedback to the pupils on their outcomes, changing on the feedback over the year. An interpretive methodology was chosen, using observation, with audio recording and document analysis of students' work, logbooks and teacher reflections. The study pointed out that this strategy contributes to the learning process, helps students to reformulate incorrect reasoning and presents persistent errors and misunderstandings. It is the opposite of standard practice, in that it raises difficulties for the students and challenges the teacher to reflect and interact with others.

Keywords: Mathematics education; self-regulated assessment; mathematics learning; assessment interaction; formative assessment.


Santos, L. & Pinto, J. (2003). What pupils think about assessment? Educação e Matemática, 74. (download)

This paper presents and analyzes the assessment students' views Portuguese pupils terminals of years of schooling cycle, with the exception of the 12th grade that was replaced by 11th grade, given their conditions. In the selection of students were taken into account different performances and different geographical contexts, including cities of the interior and the coast.

Keywords: Assessment; Mathematic; classification final grades; conceptions of pupils.

Semana, S. & Santos, L. (2008). Assessment and Mathematical reasoning. Educação e Matemática, 100, 51-60. (download)

Within the AREA project was developed a study aimed at understanding the role of the written report as a tool for regulatory assessment of the pupils mathematics learning  in the 10th, 11th and 12th grade, in particular with regard to mathematical reasoning. The development of the written report was supported with a report script with the negotiation of assessment criteria and written feedback, supplemented when necessary by oral feedback. The results of this study indicate that use of the Reporting Guidelines and assessment criteria to allow a growing realization, by pupils, of the objectives. The two types of feedback provided, a combined action, help build the mathematical reasoning' s ability.
Keywords: Mathematical reasoning; written report; regulatory assessment.

Vieira, J. (1994). Formative assessment: an experience. Quadrante, 1, 149-161.

This article describes an initial phase of using a database to develop a formative assessment with the students, in individual supply them feedback, and then the pupil can learn by himself. The developed experience comprised the subject Numerical Equations of the 1st degree in Q of the 7th grade Mathematics program.
Keywords: Assessment in Mathematic; 7th, 8th and 9th grade; Mathematics learning in computer-assisted; formative assessment.