## Abstrakt

Abstract This assignment is our final report that sums up our three semester course for math-counselors at Roskilde University from fall 2014 till now. The report is made up of the three semester reports one for each of the semesters and an introduction that sums up our work and puts forward important perspectives we have found highlights what we find essential when helping our students overcome their learning difficulties. The semester reports contains both theoretical reflections on why students encounter problems when trying to learn math and a number of empirical studies we have completed on the different types of learning difficulties students can have with respect to the three semester topics: mathematical concepts and concept making, mathematical reasoning and mathematical modeling. During the three semesters we have worked with students understanding of variables, scaffolding of students work on deductions and mathematical proofs and finally students ability to solve exam assignments that involved a degree of mathematical modeling. The semesters have had roughly the same layout with our case studies being based on three pillars - the phases: detection, diagnosis and intervention. The detection phase being an analysis of the test results from a student test made by our professors, followed by a diagnostic phase where we have selected a small number of students for an interview to be able to diagnose their learning difficulties and finally the intervention phase that explores what we as counselors can do to help the students deal with and overcome the different problems we have diagnosed them having. The effects of the different types of interventions have been evaluated in different ways, sometimes with tests and sometimes with interviews. We conclude that our intervention had measurable effect in all three semesters. The focus on variables gave our students a better symbol- and formalism-competence, STARUD gave them a much needed mathematical language about proofs, and the our focus on explaining the modeling process gave the students a much better understanding on how math is used when solving problems mostly outside math. One of the driving ideas though the three semesters have been, that to understand how mathematical ideas are learned we have to look in great detail on how students express their knowledge. This analysis of the exact way a student formulate his or her understanding of a particular mathematical concept is essential if we want to reach the Zone of proximal development of that person to help the student to construct the knowledge we want them to have. To find the appropriate level of information to give the student and deliver this information in an appropriate fashion can seem a straightforward thing for a teacher to do, but as Brousseau mentions in his paradox of devolution, this is all but simple. It is vital that we as teachers give the right degree of devolution or we end up in a situation where the students try to guess what we expect them to answer instead of trying to answer the questions from through their own reasoning.

Uddannelser | Matematik, (Bachelor/kandidatuddannelse) Bachelor el. kandidat |
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Sprog | Dansk |

Udgivelsesdato | 26 jan. 2016 |

Vejledere | Mogens Allan Niss & Uffe Thomas Jankvist |

### Emneord

- begrebsdannelse
- beviser
- modellering
- Matematikvejleder
- uddannelsen