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Tone Dalvang, Forum for mastering mathematics at Srlandet Centre for Special Needs Education
Title: The compass model a possible tool for dialogue, reasoning and understanding of situations in which learners experience difficulties in their mathematical education
The presentation has four parts: An introduction about the official Norwegian support system in special needs education, and a case concerning a 12 yearold girl presented with learning difficulties in mathematics. The compass model is then applied as a tool for dialogue and reasoning about the present situation, and to make plans for future interventions. In the end purposes of the model are discussed.
When difficulties in learning cannot be handled by the school or the parents Norway has a support system for special needs education. The law of education states the parents right to have a report of the situation worked out. The pupil is usually referred to the Educational Psychological Service (hereafter EPS), which is a service of all municipalities in Norway, and a case is then established. Complicated cases often undergo both psychological and medical examinations. This might include that EPS turns to Medical institutions for assessment concerning specific difficulties in mathematics or dyscalculia.
When the results of the examinations are available the school normally invites to an interdisciplinary meeting with the school psychologist, the special teacher, the school nurse, the class teacher and parents. In the meeting a discussion takes place amongst the participants. The pupils situation is presented and decisions are made for the future. The participants bring in different information and experiences, which they share to the benefit of the pupil.
The medical/psychological categories are important and play a central role. They often serve as the meeting ground for the participants and form the discourse of the meeting.
Specific difficulties in mathematics or dyscalculia are often applied categories for classifying these children as results of certain tests. Implications of the classifications often lead to particular organisation of the teaching and for the distribution of economic resources through an individual educational plan, which often means that an assistant is added to the staff.
In Norway acalculia is estimated to 12%, and dyscalculia 36%, 715% are estimated having difficulties in mathematics, whereas approximately 2025% do not pass the final examination in lower secondary education (age 1516) (Lunde, 2008, p. 100).
Unfortunately, we have no exact numbers how many persons have been examined or reported with diagnosis of learning difficulties in mathematics. This is partially due to lack of consensus both national and international when it comes to causes, tools, descriptions and terminology, and partially the fact that the situations are both complicated and complex, often involving various learning difficulties.
A case
To illustrate a possible use of the model an available case is first described. A girl age 12 has been reported with learning difficulties in mathematics. Examinations and tests have been conducted both by EPS and Medical Institutions over a period of six years. This means that the medical, psychological and neuropsychological discourse plays an important role in offering a diagnosis to the pupil. The existing documentation is presented in short:
Report from an ergonomist concerning motor skills:
The test indicates reduced tempo/speed (fine motor skills) and a sometimesweak concentration
Report from a psychologist concerning:
Intelligence resources and learning style reported by Leiter Performance Scale  Revised (Visualisation and Reasoning battery)
WISCIII/ Wechsler Intelligence Scale for Children (Third Edition)
Functional test by Vineland Adaptive Behavior Scale
WISC III shows a result all together below her age level (significant discrepancy)
Leiter shows an intellectual capacity within normal variation, but in lower section
Conclusion: Without doubt general learning difficulties. Insufficient cognitive competence (attention and memory)
Medical report about epilepsy:
Generalised (normal EEG and Cerebral CT) epilepsy of absence type (access might turn out as loss of consciousness, cognitive disturbance, create learning difficulties and emotional problems)
The child is not retarded due to the epileptic conditions
Description from the speech therapist:
The pupil is struggling with reading ability (can read, but do not always comprehend what is read)
Recommendation from EPS concerning special education. Program of actions:
Assistance in learning good strategies
Experience mastering or to manage
Build up basic skills through experience
Dynamic approach as charting method
Partially same book as the rest of the class
Computer, calculator, mobile with calculator training
Intensive teaching for a period
Level based training, slow progression and a lot of repetition
Increase working speed through programs that correct or mark answers
Practice independency and selfmanagement
Adjusted working plan (for the pupil)
Predictability and clear routines (individual messages)
Support in getting started, complete and finish tasks
Concept training (method Magne Nyborg)
The compass model
The model is introduced as a possible tool for dialogue, reasoning and understanding of situations in which learners experience difficulties in their mathematical education. In this paper the situation is described briefly through the documentation above.
The model consists of three layers: An inner circle, a middle circle and an outer circle. It has two arrows: A thin arrow pointing in the direction of one particular mathematical competence at a time, and a thick arrow marking the way from the actual situation with former aims (C1) towards future arranged situations with new possibilities and new aims (C2).
Context 1(C1) forms the background for the actual situation. The complexity of the case is evident. Many different experiences, persons and descriptions are presented at the meeting. What are the identified/experienced challenges of the situation for the individual pupil and for the current pedagogical practice? What does everyone need to know about the situation to understand the current challenges? The language in the documentation provided for the meeting is mostly medical and psychological. How is this translated into an educational form? How is the information used in the school setting, and which categories are relevant for supporting the child? What impact will they make on didactical decisions?
The structure of the model invites an interdisciplinary meeting to focus on possibilities, moving from difficulties to possible resources that the student can bring into a learning situation (C2). When it comes to a better understanding of a difficult learning situation the content and the organization of the learning situation are as important as the students resources and limitations. These three perspectives form the middle circle, which can be rotated. The participants of the meeting can do the rotations in order to prioritise one element at a time.
The students resources and limitations in mathematics are often given four different frames of interpretation (Engstrm, 1999):
Neurological/medical
Psychological
Sociological
Pedagogical/didactical.
The organisation of learning situations also includes different perspectives, as for example: Individual work or cooperation.
Teacher centred or pupil centred.
Inside or outside the class.
Inclusion or segregation.
Even when it comes to the subject matter choices can be done, as for example:
Big ideas or topics.
Open or closed tasks.
The model aims at the relations between the content and organization of the learning situation and the pupils resources and limitations. To understand the challenges of the case presented above the participants of the meeting can pose several questions about possible relations between the different dimensions, as for example: Are there priorities when it comes to organizing the classroom and the learning situation for a pupil described by reduced tempo/speed (fine motor skills) and concentration and motivational problems? Is the current practice based on individual learning or collaborative learning? What are the pupils interests/wonderings and questions, and are they part of the topics in mathematics? The intellectual recourses and learning style of the girl is described as all together below her age level, how does this influence the mathematical content? What situations enlarge the problems for the pupil, and what situations minimize them? What does the meeting think about the program for action recommended by the EPS? Will an assistant be a good solution?
We have allowed ourselves to put the mathematical competencies (Niss et al., 2002) into the outer circle of the model, though interpreting them as different capacities that might be involved when the pupil participates in learning mathematics. The competencies vary in weight at different levels at school:
Thinking mathematically (mastering mathematical modes of thought)
Posing and solving mathematical problems
Modelling mathematically (i.e. analysing and building models)
Reasoning mathematically
Representing mathematical entities (objects and situations)
Handling mathematical symbols and formalisms
Communicating in, with, and about mathematics
Making use of aids and tools (IT included)
The competencies demonstrate that factual knowledge and technical skills are necessary, but certainly not sufficient when it comes to learning mathematics. What competencies are stimulated through the program for action recommended by the EPS? The competencies can function as topics for the interdisciplinary meeting discussing how the mathematical education can function best for the pupil in relation to all the different elements in the model. The thin arrow can be rotated and set at the competencies most relevant for the particular case. Niss (2003) emphasizes:
Finally, by being explicit instruments of characterisation they can also be used as metacognitive support for teachers and students by assisting them to clarify, monitor and control their teaching and learning, respectively. (p. 124)
Why this model?
Forum for mastering mathematics (Forum for matematikkmestring) is situated in a centre of special needs education. In our work within special needs education in mathematics we are sometimes invited to interdisciplinary meetings as a second line service for the EPS. We have developed this model because we have experienced the interdisciplinary meetings as often onesided focused on presumptions of a pupils shortcomings or defects.
Our experience has been strengthened by a case study by Eva Hjrne and Roger Slj (2004). In this research they write about how classifications and diagnosis are widely used by multi professional teams as interpretations of what constitutes the background of school difficulties of a growing number of children.
Thus, there is almost no professional pedagogical discussion involved in the meetings that we have recorded. For instance, there are few, if any, analyses of the kinds of situations that elicit the problematic behaviour () In other words, there are no attempts to understand the child and his or hers problems as they surface in the attempts to adapt to the role of being a pupil and acting in an educational setting. Also, there are no discussions of the kinds of measures that have been taken to deal with the problems, and what the effects have been. This is confirmed by the abstract nature of the written documentation, which does not document in any detail the analyses made or the progress of the work with the child. In this sense, there is no systematic and cumulative problem solving process that focuses on the child and his or her problems in school. (p. 21)
We find that the conceptualisations of mathematics in the interdisciplinary meetings are quite narrow, represented mostly by arithmetic, factual knowledge and technical skills. The medical/psychological perspective most often dominates the explanations and gives a direction for the programme of action. The medical/psychological categories are not neutral, but need to be discussed in relation to the school context. Reflection is necessary.
The components of the compass model can offer a structure to a meeting where the pedagogical discussion plays an important role. The concrete events in the classroom, the attitudes and expectations have to be brought into the discussions as well as sociological and social questions. The main aim is to use the model as a tool to map out existing learning situations and to arrange for alternatives emphasizing students mastering of mathematics.
The case presented entails challenges and tensions for all involved. A focus on socio cultural perspectives (Dysthe, 2001) on learning and development suggests particular interpretation of what might be involved in each element and how these elements overlap and mutual interact with each other. The model is a tool to support continuity in the programme of action, due to dynamics and complexities in learning situations. What each participant reads into the different terms applied in the model is though of great importance.
The model is developed as a possible tool. As an experiment it has been presented and discussed at several occasions, for small and large groups of teachers and advisers in the EPS. Especially the advisers have been important participants in the discussions of the model, because they often have the role of leading the interdisciplinary meetings. Several advisers have reported that they bring the model with them to interdisciplinary meetings. Some find it useful as an introduction of the meeting showing the complexity of the situation. Some say they use it as a structure for the meeting. Some find it difficult because it involves so many different aspects, and some have reported that they consider the competencies as very difficult to understand. The advisers in EPS in Norway are not often familiar with the contexts of mathematics.
The presentation will aim at discussing the limitations, potentials and further developments of the model.
References
Dalvang, T. & Lunde, Olav. (2006). Med kompass mot mestring et didaktisk perspektiv p matematikkvansker. Nordisk matematikkdidaktikk, 11 (4), 3764.
Dysthe, O. (2001) Dialog, samspell og lring. Oslo: Abstrakt forlag.
Engstrm, A. (1999) Specialpedagogiska frgestllningar i matematikk. Arbeidsrapport ved Pedagogiska institutionen. rebro: rebro universitet.
Hjrne, E. & Slj, R. (2004) There is something about Julia Symptoms, categories, and the process of invoking ADHD in the Swedish school: A case study Journal of Language, Identity and Education, 3(1), 124, 2004
Lunde, O. (2008) Matematikkvansker. I Rygvold, AL. & Ogden, T.: Innfring i spesialpedagogikk, Gyldendal Akademisk, Oslo.
Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM project. In A. Gagatsis, & S. Papastavridis (red.): 3rd Mediterranean Conference on Mathematical Education  Athens, Hellas 345 January 2003, 116124. Athen: Hellenic Mathematical Society.
Niss, M. & Jensen, T.H. (eds) (2002): Kompetencer og matematiklring. Uddannelsesstyrelsens temahfteserie, nr. 18, 1334, Undervisningsministeriet (Ministry of Education).
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