About Daniel Romão

Have a Ph.D in Education. Textbook author. Researches on (mathematics) Education and Discourse Analysis.

Understanding Formative and Summative Assessment

Pieces of scrabble board game with the word assess formed in the center.
To assess: evaluate or estimate the nature, ability, or quality of.

Assessment is an essential part of the teaching-learning process. It is from it that it becomes possible to monitor the trajectories developed by each student, as well as to replan the teaching actions. In this sense, assessment should be understood as a learning tool that focuses on improving the teaching-learning process.

Some assessment practices assume a strict focus on results, invisibilizing the path taken by the student throughout the learning process. In this sense, by restricting themselves to the assessment of grades and concepts, such practices create an erroneous or even distorted picture of what the student learned. Moreover, by focusing on the final results, this understanding of assessment hinders its use as a way to reflect on and improve the organization of the pedagogical work.

When discussing assessment, is important consider it according to a broader definition; not limited to the student, but also focused on the teacher and the school. Moreover, it is essential avoid consider it from a negative standpoint, related to performance measurement or according to a classificatory bias. Instead, it should be considered in a positive way, i.e., serving as a basis for improving the teacher’s work, as well as an instrument at the service of the learning process.

Assessment should create a “biography” of student learning, thus providing a basis for improving the quality of instruction

National council of teachers of mathematics (nctm)

In a sense, any assessment that acts directly in the service of student learning and development is formative. From another perspective, an assessment which focus on evaluate students learning, skill acquisition, or achievement scores may be considered summative.

Formative versus Summative Assessment

Broadly speaking, it is possible to differentiate the two perspectives according to the following aspects:

  • What its purposes;
  • How it obtains information;
  • When it takes place.
Infographic comparing summative and formative assessment. On the left, aspects of sumative assesment: What? To measure how much has been learn. When? At the end of the learning process. How? Exams, tests and reports. On the right, aspects of formative assesment: What? To monitor the learning process. When? During the learning process. How? Portfolios, self-assessment, concept maps, conversations.

Although the remarkable human and pedagogical value of the formative assessment, the summative perspective also presents valuable aspects, especially in pragmatic terms.

Summative assessment it’s used for:

  • Achievement tests;
  • Benchmark assessment tests;
  • School accountability.
  • Placement tests;

Such use of assessment results can either positively contribute to the improvement of schools, curriculum standards, and pedagogical practices, or negatively stigmatize low-performing schools, teachers, and students.

However, even according to the positive view cited above, we should consider the following criticism. Since the assessment results can influence public policy or private funding of schools, isn’t difficult to expect that administrators, teachers (and even students’ families) will propose an educational practice focused on preparing for the exams. This leads to an undermining of the teaching and learning processes.

In the following we’ll discuss the theoretical perspective of formative assessment.

Formative assessment: a theoretical framework

The concept of formative assessment owes a lot to the psychologist and educator Benjamin S. Bloom (1913-1999). He is historically considered one of the precursors in the theoretical field of formative assessment.

Analyzing the common strategy of teachers and schools to organize concepts, content and skills in instructional units, Bloom verified that was reasonable to assess student learning at the end of each unit. However, from this analysis, he could also verify that this form of assessment was restricted to exposing the lessons and explanations of that unit, were or weren’t adequate.

In this sense, Bloom considered another approach, one which the assessment could work as instruments of learning.
Through continuous feedback, it would be possible to identify the difficulties of each student and help solve them throughout the instructional unit, not only at the end.

Based on these premises, formative assessment should focuses both creation and exploration of contingency moments in order to regulate the teaching-learning process. These moments establishes in the teacher’s personal relationships with an individual student or with the class.

In this sense, a formative interaction would be established when such interaction stimulates the student’s cognition. This type of interaction can be verified in genuinely dialogical situations, i.e., the teacher formulates answers that provide the student with reflections on a certain topic. By doing this, students can formulate another answers, hypotheses, and also questions that continue this dialogical movement.

From a formative perspective, learning occurs by articulating different feedbacks: teacher to student, student to teacher, and student to student. In this sense, the effects of one’s feedback mark the beginning or the way other’s feedbacks are forwarded. Such effect, in turn, would be produced by the interventions made by the teacher, as well as the observations and representations that guide them.

A classroom practice is formative to the extent that student’s achievement evidence is elicited, interpreted and used by teachers, students or their peers in order to make decisions and take actions on the very learning-teaching process. With this in mind, since assessment lies at the heart of the organization of education, a change of perspective in terms of assessment also corresponds to a profound change in a society’s own conception of school and education2.

Formative assessment in practice

Usually, the starting point of that chain of interactions occurs when the teacher presents to the students a theme, a discussion, or a problem. Considering that all educational situations require intentionality, when a teacher starts a chain of interactions he or she can anticipate most behaviors, attitudes, effects and reactions from the students.

It is worth emphasizing that teacher’s observation shouldn’t be characterized by informality. As with the design of any pedagogical situation, the observation must be intentional. Knowing the intentions of situation presented and antecipating some attitudes and behaviors of the students, teachers can establish an observation routine asking themselves:

  • Did the students understand what is to be done?
  • How do students mobilize their prior knowledge to deal with the situation?
  • How do students express their strategies to their peers?
  • What knowledge students mobilize?
  • Are the students up to the task? Is it too hard or too easy?

By doing so, teachers obtain precious informations on how their students learn, what are their difficulties, aims and objectives. This sistematic observation enables teachers to understand student’s personal trajectories. In this way, the teacher can establish individualized analyses about what the student knows and what directions are most appropriate for his or her personal trajectory. However, this is only possible from systematic and careful annotation.

The ways in which each teacher will do this are personal. It depends on the profile of the students, their ages, social context and school curriculum. Regardless of the choice made, the most essential thing is to create a personal routine of note-taking and recording of the situations experienced in the classroom. Moreover, understanding that this type of assessment focuses on both the individual and the group of students as a whole, it is necessary to organize the notes in such a way as to fulfill this dual purpose. In this sense, one suggestion is to take notes individually throughout the class and, at the end, systematize in a summarized way the main questions, difficulties and advances found by the class.

One way of doing so is by utilizing anecdotal records, i.e., a descriptive narrative of an specific classroom event. This type of tool is often used when negative things happen, such as behavioral problems or indiscipline. But why not use it in a broader perspective? One that also includes positive aspects? Looking at it this way, it’s possible to observe good ideas from the students, unique and creative problem-solving strategies, important discussions among peers, as well a tool for teachers understands students’ difficulties in order to redesign activities or offer specific support.

Anecdotal record sample, designed by Daniel Romão.

However, in most school realities found around the world, making this kind of annotation throughout the lesson is an arduous task. In this sense a plethora of formative assessment tools may be used. To mention a few:

  • Student portfolio (comprising student productions, problem solving, notes, projects, drafts, and self-assessments).
  • Student conference (individual conversation with the student to understand their perceptions, doubts and ideas about the topics studied).
  • Misconception check (proposition of problem situations about key concepts in which students need to verify their veracity or not. Thus, it is possible to understand how much the students have already appropriated the concepts and processes in question).
  • Concept map (by creating a graphical organization of the concepts learned, the students may perceive multiple relationships between them and identify missing concepts or even misconceptions).
  • One minute essay (a focused question about a topic that can be answered within a short amount of time).

Finally, it is important to point out another great tool for formative assessment: self-evaluation.

Although it has an overall positive impact both on learning and development of student autonomy and responsibility for their own learning trajectory, self-evaluation can be a challenging assessment tool for teachers, students and their families.

Therefore, formative the assessment extends into everyday school life according to a diversity of practices also taken according to a formative perspective. There is a common misunderstanding when talking about self-evaluation because in most people’s minds a summative perspective of assessment still prevails. Thus, it is common to think that self-evaluation means giving grades to oneself.

The essence of self-evaluation is to allow the student to realize his/her mistakes (alone or with the help of peers) and to become the protagonist of his/her own development as a learner. In this sense, self-assessment can focus both on specific content (such as math, language, and science) and on the ability to interact, explain, argue, represent, and communicate with peers.

For example, the teacher may organize students in pairs or small groups and propose a problem-situation. Then the students jointly evaluate their strategies and resolutions, verify what they did get or not get right, and also get a glimpse of his own progress. By comparing their strategies with those of their peers, they learn both new ways of solving a problem and reinforce their own ways of solving it by exposing them to their peers.

Group of teenager girls in a classroom enthusiastically discussing a problem.
Students working with their peers.

With this in mind, we would like to suggest one last reflection: is it possible to give a formative approach to summative assessment?

One way to offer a formative approach to tests and exams that have a summative nature is to use feedbacks and specific comments that aim to make the student reflect. Instead of just giving “correct” or “wrong” marks, it’s more interesting to describe the possible misconceptions, points to improve and think more about.

In the same way, it is also interesting to think about self-assessment and peer-assessment as a way to analyze summative assessment instruments. Thus, a student could analyze a colleague’s test or exam, see where he or she went wrong, and try to explain how to correct it.

Another possibility doesn’t launch this self/peer assessment only at the end of an exam, but during its application. The idea is that students would solve as many questions or problems as they can solve autonomously. Then students would be grouped with peers according to the questions they have the most difficulty with. After a period of discussion, the students would be separated again and invited to resume solving autonomously.

Finally, the same strategy can be adapted not around questions that present difficulty, but the opposite. One could choose questions that were easier and organize the students in groups to discuss the various strategies and ways of solving to produce a common answer that was better or more complete than the answers produced individually.

References & Further reading

Black, P.; William, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability.

Block, J.H. (1971). Mastery learning: Theory and practice. New York: Holt, Reinhart & Winston.

Anderson, L.; Krathwohl, D.; Airasian, P.; Cruickshank, K.; Mayer, R.; Pintrich, P.; Raths, J.; Wittrock, M. (2000). Taxonomy for Learning, Teaching, and Assessing, A: A Revision of Bloom’s Taxonomy of Educational Objectives, Complete Edition. Pearson

Klenowski, V. (2002) Developing Portfolios for Learning and Assessment: Processes and Principles. ‎ Routledge.

Mathematics: discovery or invention?

Lamp bulb on a wooden table. Numbers and mathematical symbols in spiral from the lamp.
Is math a discovery or a product of human creativity and inventiveness?

For some, the question of the title may suggest an evasive adventure by the meanings of words; a mere semantic reverie. There still those for whom the actions of “discovering” and “inventing” are equivalent, and thus the discussion becomes irrelevant. 

However, considering mathematics in one way or another, it changes its status before other sciences and society and has direct implications on its teaching.

The difference between discovery and invention 

When talking about discovering “something”, it is admitted that such a “thing” has an existence a priori. An example of this was the discovery of America in the 16th century. The continent has always existed, it wasn’t created by Europeans. For them, what happened was a process of discovery. Similarly, we can consider that the original inhabitants also discovered the existence of these newcomers to their lands. 

And an invention? In this case, the “thing” invented did not exist at all before whoever invented it. For example, the magnetic compass. The Earth’s magnetism preexists to any human action. However, the creation of an artifact that uses this phenomenon to help locate it on the surface of the planet is an invention! 

Bringing this discussion to mathematics scope, we will resume our initial question: discovery or invention? If mathematical entities such as numbers and geometric figures were discovered, this means that they preexisted in some way. If, on the contrary, they were invented, it means that their existence began as the end of process of invention. 

In the following we will present some constitutive elements of these two perspectives.

Mathematics as discovery

The origins of this perspective are diluted along its own historical construction. However, we can identify strong traits that ground such a perspective in classical Greek thought, particularly in Plato and Aristotle.

Plato (left) and Aristotle (right) in detail of the work “School of Athens”, by Raphael.
Plato (left) and Aristotle (right) in detail of the work “School of Athens”, by Raphael.

According to Plato, ideas (eidos) would exist ‘in and of’ themselves, independent the actions of any subjects. Thus, ideas would inhabit a dimension of their own, of purity and immobility called the intelligible world. Such ideas would manifest themselves in the world which we live in, the sensitive world

The access to the intelligible world would be reserved for philosophers, holders of reason and true knowledge. Nevertheless, it would be in the intelligible world that mathematics entities would inhabit. In this way they would exist in a full and autonomous way, endowed with perfection. Thus, knowing mathematics would mean the ownership of true science and knowledge. 

In what way, then, would people have access to mathematics? Plato presents an possible answer his dialogues ‘Meno’ and ‘Phaedo’: the theory of reminiscences (anamnesis).

Reminiscence means recollection, or remembrance. Since, for Plato, souls would inhabit the intelligible world before materializing in the sensitive world, accessing the first would mean an exercise in recollection of the soul itself. In Meno, Socrates (as Platonic character) proposes to demonstrate the theory to the other character, Meno. 

Socrates asks Meno to choose one of his servants and questions him at length on an classic Geometry problem known as ‘the duplication of square’. Throughout his conversation with the servant, Socrates problematizes the answers given and creates new questions leading the servant’s reasoning and putting him in doubt or in apparently insoluble situations (aporia).

Socrates: — You realize, Meno, what point he has reached in his recollection. At first he did not know what the basic line of the eight-foot area square was; even now he does not yet know, but then he thought he knew. He answered confidently, as if he knew, and he did not think he was at a loss, but now he thinks he is at a loss; and so, although he does not know, neither does he think he knows. […] Look, then, at how he will emerge from his perplexity while searching together with me. I will do nothing but ask questions, not instruct. Watch whether you find me instructing and explaining instead of asking for his opinion. […] So these opinions were in him all along, were they not? […] These opinions have so far just been stirred up, as in a dream, but if he were repeatedly asked these sorts of questions in various ways, you know that in the end his knowledge about these things would be as perfect as anyone’s. […] If he always had it, he would always have known. If he acquired it, he cannot have done so in his present life.

Once mathematical entities would preexist in intelligible world and would be accessed from recollection of a previous life, this perspective places mathematics as discovery — i.e. Mathematical truths are therefore discovered, not invented. Perhaps not a discovery in the same sense previously exemplified. It isn’t a search for something external, such as a continent or a natural phenomenon. On the opposite, it’s a search inside the soul itself.

According to Michael Foucault, such a conception coated the mathematics with ideality and occurred throughout history, being questioned only to be repeated and purified. This metaphysical view of mathematics has spread its influence till modern days. Nevertheless, this conception has also been criticized by a number of intellectuals, mathematicians and scientists. The main arguments of this criticism are founded on the conception that mathematics would be a human invention.

Mathematics as invention

For Platonism, mathematical entities were eternal, i.e., they had no beginning. However, to contest such an idea also means to discuss a possible origin for mathematics. 

Some historians support the origins of mathematics in the observation of nature and the surrounding world by early humans. According to this approach, the human ability to recognize and compare sizes, shapes and quantities constitutes the core of mathematical knowledge.

Another approach points out, through the effects of labor. This is not antagonistic to the first approach. We understand that both complement each other. As hominids began to walk erectly, their hands were free to make tools, construct artifacts. Thus, notions such as symmetry, measures, proportionality and counting gradually developed according to the development of the labor itself. Added to this approaches, there are also signs that link the development of mathematics to religion, ritualistic doing, and even in games and toys.

In this sense, objects seen in nature are never exactly geometry forms, for example. There are no perfect straight lines, circles, squares in nature. The creativity, inventiveness and analytical spirit of human beings have created ideal models of these natural forms. An exercise of transformation from concrete to abstract. And even more, by using such abstract models to unravel the mysteries of nature and solve everyday problems, the human being resigns this knowledge by using it in the real “concrete” world.

South American Indigenous people identified the equilateral triangle as an optimal solution for the creation of an apparatus for sifting manioc flour (Gerdes, 2013, p. 49).

In this sense, Mathematics isn’t a collection of objects and concepts endowed with an a priori existence. It’s a invention of the humankind in response of its observation and action on the world.

References & Further reading

Foucault, Michel. (1982). The archaeology of knowledge: and the discourse on language. Vintage.

Gerdes, Paulus. (2013). Ethnogeometry: awakening of geometrical thought in early culture. Lulu.

Plato. (2002). Meno.

Courant & Robbins (1996). What is mathematics?

Hersch, Reuben (1997). What is Mathematics, really?