Formative assessment opening mathematics: Dialogical Learning

1. Country: Switzerland
2. The ages of the learners who can be assessed using this method: 12-13 years. The dialogic learning method can be adapted for all age groups.
3. Who can conduct this assessment: learner, teacher, peer
4. Educational contexts: Mathematics and German. Elements of it flow into other subjects and it is applicable to all fields.

This example was collected during Erasmus+ project “Personalised and Formative Assessment Practices Supporting School and Learner Development” of the European Council for Steiner Waldorf Education, Learning for Well-being Foundation and the Hungarian Waldorf Federation and published in their developed book “Assessment as Dialogue: Twenty Inspiring Practices from Classrooms and Schools Across Europe”. The stories, included in this collection of good assessment practices, were contributed by courageous, inventive, and open-minded teachers, parents, school leaders, experts, and students from 12 countries around Europe.

5. The assessment method: Opening Mathematics - Dialogical Learning

Type of assessment:
Formative: personalised feedback is given for every task the pupils engage with. A collective oral feedback process also takes place based on the pupil’s work at suitable moments.
Self-assessment: pupils record their thoughts and feelings regarding the task they are working on in their journals and are also regularly asked to summarise what they have learnt. Self-assessment is key for developing a deep connection with the subject.
Peer: pupils are regularly asked to read and comment the work of their peers.
Ipsative: assessment of the learning journals is purely based on the individual progress and learning journey.

Values:
Contextualised: pupils are introduced to the process at the beginning of the year.
Individualised: the tasks are open and allow for individual creativity, also in the way pupils approach them. The formative assessment is based on this individual connection with the subject.
Participatory: this method is based on pupils’ active engagement with every task. Pupils’ answers to the tasks are also what determines the next step in the programme.

Dialogic learning is a teaching and formative assessment method that enlivens the way that understanding arises. Pupils engage deeply and methodically with the subject and are given space to write down their thoughts and feelings, describing their engagement and how they specifically approached the task they were given.

It generally includes a cycle of three:
1. an open task given to the pupils by the teacher,
2. individual and collective feedback on what the pupils produced and
3. planning of the next step taken by the teacher, mainly based on the concrete feedback obtained from the pupils and where the class needs to make progress.

Applied to mathematics, it approaches concepts through the language of understanding before the language of the understood is developed together, giving room for all pupils to engage in the process in a way particular to each one of them and possibly involving their daily lives. Formal concepts are thus not introduced at the beginning but arise as results of a joint undertaking that builds on awakening self-evidence.

Timeline and preparation:
The main preparation time goes into writing the open questions for the tasks and then reading all the journals to prepare the next step.
This method is clearly time-consuming for the teacher but there are resources available to facilitate the process.

Form of documentation:
The learning journals that the teacher gives feedback on with checkmarks and comments.

The Teacher
Patrick Kolb has been a primary teacher since 1991. He teaches all main subjects except for foreign languages: mathematics, German, biology, history, geography, sports, and music. He had always wanted to work with children and started his teacher training course for primary school at the age of 16 in parallel with his Matura (end of secondary school certificate in Switzerland). Today, teacher training courses start after school for all levels, but in Patrick’s case, he was qualified to start teaching at the age of 21. At first, he did not have much experience and had a lot to learn, 30 years later he is still teaching and has become a reference point as a practitioner of the dialogic learning method. Interest in this method is also finally growing around him. Patrick started his career at a public school in Zug, he then moved to Mettmenstetten for two years, before starting at the primary school of Cham in 1999.

The Practice
During his teacher training Patrick came across the ideas of Martin Wagenschein, a science educator who worked in mathematical and scientific didactics. Martin is one of the precursors of modern teaching techniques, such as constructivism, inquiry-based science and inquiry learning. Patrick was fascinated by his ideas on how concepts rose genetically, and how knowledge comes into being, and this was the beginning of his adventure with alternative teaching methods. Patrick recalls that one of the first real life experiences he organised was a discussion with 24 pupils about how and why a candle burns. He began using the Socratic method, and fairly quickly transferred the questions and answers to paper. The advantage of the written form is that everyone gets a chance to express their thoughts independently. Patrick was moved by how the children wrote their stories about their experience.
He then discovered the books by Peter Gallin and Urs Ruf (for example Gallin and Ruf: ich, du, wir: 4.-5. Schuljahr Sprache und Mathematik, Zürich 1999; Gallin and Ruf: Furthering Knowledge and Linguistic Competence: Learning with Kernel Ideas & Journals; Gallin: Dialogic Learning: From an educational concept to daily classroom teaching; the last two articles are available at www.ecswe.eu/wren/researchpapers_assessmentevaluation.html), outlining the dialogic learning method and how to apply it. The books contained practical help and examples for every step, including how to formulate open questions for assignments. He did not have a dialogic teaching partner in his school but found a friend from the Mettmenstetten school he had taught at before, and organised an exchange of good practices with him. Patrick then convinced the school director that it would be useful to work in this direction. He was given authorisation to experiment. However, the director never decided to make it a part of the school assessment policy.
The dialogic learning method embeds the learning process within the formative, selfand peer-assessment process that in turn feeds into the determination of how and what content follows in subsequent lessons.
The method consists of three key steps. The first one is to engage the pupil in a creative process with a task (“Auftrag” in German, literally translated as assignment, but an “Auftrag” will generally have a wider scope than typical school assignments). It is important for this task to be open, that is, not have a right or wrong answer to it. An example of an open task in mathematics concerning fractions might be: “collect or draw pictures that represent fractions”; “where do you find fractions when you go shopping?”; “where do you find them in real life?”. Patrick applies this dialogic learning method in the two main subjects for class 5 and 6, which are mathematics and German. One reason for this is that the books on the method give guidance for those two subjects, but the main reason is that they have the most dedicated time each week and are therefore more conducive for this approach to teaching. The philosophy, of course, flows into the other subjects, notably into subjects like practical arts. An example of an open task to start a German lesson on describing a person from a given text might be: “Choose a person and describe what you think is special about them”, or “think about what you have read: what came to your mind while reading? Pay attention to your feelings, thoughts. What did you feel while reading?”. It is important to note that pupils are encouraged to not only work on the task but also to share their thoughts and feelings on how they are doing it. This self-assessment component is a part of the task itself. Patrick furthermore regularly asks them to write about what they have learnt. These summaries are often very interesting and include thoughts on how others have done things differently.
Tasks are often worked on in the classroom during lesson time. In such cases, the peer-assessment method of “musical chairs” can be practiced: pupils put their work down on their desk, walk around, choose someone else’s work and make comments on it. Patrick has observed that pupils get into this very naturally. Often pupils seek the positive elements and have a harder time being critical.
Once the tasks are completed, the following step consists of collecting the notebooks with whatever the pupils have produced, and to give them some quick feedback in the form of checkmarks and short comments. The checkmark system works as follows: one, two or three check marks are given as ipsative feedback, i.e. not against a normative standard, but seen within the context of the development of each pupil individually, i.e. depending on the depth that any given pupil has reached, and their previous level of understanding and abilities. Coherence is more important than giving a “correct” answer: a pupil may go off on a tangent based on a mistaken thought, yet still write a coherent text that may be more interesting than “a correct answer”. Three check marks indicate coherency and depth that Gallin and Ruf characterise as a “Wurf”, indicating an argumentative coherence in entertaining and wrapping up thoughts in a way that can pique the interest of the reader. One checkmark means “OK”, two somewhere in between, and a crossed-out checkmark indicates insufficient work. Giving checkmarks is not meant to be a scientific undertaking but rather a quick, more gut-like reaction of the teacher to the work of the pupil against the backdrop of this individual. Thanks to this regular, personalised feedback, this stage of the process is part of the formative element that also enables the teacher to understand what has happened in the classroom, and to develop a sense of what to do next.
Based on what the teacher has discovered in this stage two, the third step is to plan the next lesson. The content of the lessons is thus determined by the pupils’ needs to progress, and not by a pre-made programme. During the class Patrick delivers some targeted explanations and selects excerpts from the pupils’ work to share with others in the class. The principle behind this is that we can all learn from what the others have done. Generally, pupils feel elevated when their work is shown to everyone. Inevitably, the work of some is selected more often but a conscious effort is made to vary and give everyone a chance to contribute to this collective process. At the end of this third stage, it is time for a new task and the cycle begins again.
The focus of the method is enabling pupils to connect with the subject - and develop an understanding out of that. Not the other way around, as is often the case in school teaching, where finished knowledge is first presented by the teacher, expecting pupils to connect to largely finished forms, which tends to literally turn many pupils off, simply because they lack the ability to connect. In the dialogic learning practice, pupils are brought into motion from the very beginning, having to write something for every task, giving them time to engage with the subject deeply and methodically, and on their own terms. They do not just learn something by rote but engage as an actor: it can grow within them rather than being thrust upon them. In this sense, assessment is not a goal Patrick pursues in its own right, and it is not even primarily in his consciousness when he is teaching, but rather a naturally embedded dimension within the process of each and every pupil connecting to a subject individually. A direct effect of this dialogic learning practice is that pupils generally enjoy the subjects much more, including the subject of mathematics.
Patrick has been practicing dialogic learning with mathematics for over 20 years. He explains that the way to generate interest and engagement is to engage the language of understanding before developing the language of the understood (a distinction first proposed by Martin Wagenschein). Pupils may initially use words that are not usually considered adequate within the subject, but that is not a problem, as the construction of concepts comes afterwards, in a joint effort of the class, with the guidance of the teacher. It can happen that certain pupils will introduce unusual words out of their prime engagement with the subject, some of which may stick. These should not be seen as aberrations from the normative form agreed upon by society, but rather as singular expressions arising from a singular learning group at a singular point in time. An example of engaging students in mathematics would be to list all numbers from 1 to 100, then cross out all the multiples of 2, 3 and 5, and ask pupils to observe what is left over. The quality of the remaining numbers appears through the exercise, and the concept of prime numbers is only introduced once they have reflected and commented intuitively. For other topics Partrick lets the children invent tasks themselves, within a topic. He says that pupils often come up with questions that are much richer than those found in textbooks, and in fact he even uses some of them in future tests. Another important element in mathematics is to work with language and encourage pupils to form complete sentences to describe what they observe. He takes the example of proportionalities that can be explored by describing everyday situations such as a birthday party and the link between the size of the pieces of cake and the number of guests. Starting with simple experiences and examples makes it possible for connections, later named as concepts, to become self-evident.
The formative assessment practice that is embedded in the dialogic learning method can be completed by a summative element. In Patrick’s case there are some tests throughout the year, at the end of each subject cycle, that is, four or five per semester. The tests are always adapted to the content that was covered during the period and before each test, he takes time to reflect and summarise what the pupils have learned. If necessary, they also have the opportunity to do some traditional preparatory exercises. He notices that there is no particular nervousness around these tests, as they build on what pupils have already engaged with, and because the test gives them another opportunity to show what they are able to do.
The tests are the same for everyone but should contain questions that speak to different levels of competence. Moreover, the grades for the tests and the “grade” that arises from the check marks given to the journal work, which is purely ipsative, reflecting their personal progress and efforts, are amalgamated into one final grade that reflects both.
As Patrick has many years of practice behind him, he has received feedback from numerous pupils during and after their time in his class and has noticed some trends. He distinguishes the feedback he receives when they are still pupils, as opposed to when they have a few years of perspective. Right at the beginning of engaging with the dialogic learning process for the first time, there is an adaptation phase, as they are suddenly asked to write much more intensively than before. Once they have gotten used to the routine of being set tasks, the phase of active engagement starts. Patrick explains that even those who had doubts at the beginning find something to write about every time and develop a personal relationship with the subject. After two years of practice, they are often happy to move on, as it is hard work to keep the journal up-to-date, so they are open to try other ways. A few years down the line, however, pupils often remark that they noticed how much competencies they gained from their learning journals. They are often very conscious of the life skills it has brought them. One of Patrick’s former pupils recently confided in him that he had kept all of his learning journals as they are the most personal documents of his whole school life.
Colleagues who teach in the upper classes have commented that the pupils taught by Patrick are autonomous, used to working with any topic they are introduced to, and able to produce something out of their own thinking.
In terms of communication with parents, Patrick introduces the dialogic method once at the beginning of the year every time he starts with a new class. During the year, there is no specific communication around the practice, except for informal communication and specific requests. Parents can also consult the journal of their child. This process could be improved if the necessity for more parental involvement is felt. Each year there is a parent-teacher meeting during which the final grades are communicated. During these meetings Patrick also tells them about the journals. Patrick explains that twenty years ago, parents were skeptical about his alternative methods, as the tendency back then was to have very high expectations and to put a lot of pressure on pupils. Today there is greater understanding and support. Patrick attributes this change to the fact that he is more experienced, has support from the school and that expectations from the school system are not as narrow as in the past. Generating more pupil engagement is even encouraged in the latest national curriculum reform.
Looking back at the strengths and limitations of this practice, Patrick underlines that over the years he has learnt much about mathematics, German, and the children who participated in the process. Although he has been teaching the same age group for all these years, working in this way brings diversity and gives the pupils more opportunity to show who they are and what they can do. Some may not perform so well in tests but they are able to show what they can do in their learning journals. Patrick recalls a pupil who often struggled passing the tests but was extremely strong in analysing her mistakes and wrote very interesting content in her journal, which was taken into account in her final grade. The only real difficulty he sees for the teacher relates to the time factor. It takes a lot of time to constantly be reading what the pupils write, although it can be very instructive. He also mentions that you sometimes have to resist pressure from the curriculum. From his experience it does not take more time to cover all the topics in this dialogic way, and delving into the depth according to where the pupils are leading you is essential, even if it seems to be leading you away from the programme temporarily. As Patrick engages the dialogic learning process for mathematics as well as German, this requires staying connected to both threads simultaneously and switching back and forth in his head.
According to Patrick, these challenges are worth overcoming for the depth of the engagement and insight that results from the process both for pupils and teachers.
When you adopt this method, the way that knowledge comes into being is enlivened. Teaching is then no longer a one-way street of “passing on” knowledge, but entails becoming a midwife for the rebirth of knowledge in each individual student. The subject becomes a base for dialogue between all parties, integrating a robust formative assessment practice within it.

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