## Sóley Benediktsdóttir

Upper Secondary Teaching

**Development of projects on differentiation with emphasis on understanding and discovery**

**Advisors: ** Benedikt Steinar Magnússon and Bjarnheiður Kristinsdóttir

**Examiner:** Kristín Bjarnadóttir, Professor Emerita

**Abstract**

Mathematics in upper secondary schools is a challenging subject for many students, and many even become convinced they are unable to learn the subject. In order to counter such beliefs, it is important that teachers refuse to accept that sort of thinking. There are ways to help students overcome their pessimism and, according

to research, one of the best ways to help them is to guide them towards discovering their own connections between different elements of mathematics and using these connections to better their understanding. The objective with this thesis is, therefore, to create and develop projects for students in upper secondary schools

with this idea in mind. The projects focus on differentiation because the author is interested in that field and because studies conducted among upper secondary school graduates have shown they struggle most with differentiation and integration of all the mathematical fields.

Three projects are presented here in which students have to deliberate the mathematical ideas which are being focused on, and as a result, contrive their own understanding of these ideas. The first project inspires a graphical interpretation of differentiation with the use of the slope of a tangent. The second project centres on the derivative of power functions and discovering a graphical pattern to find the derivative. The third project is on optimization, where the students are supposed to find the largest volume of a gift box, both graphically and through calculations.

Sixteen upper secondary school students worked in groups on the first project. Their answers and ideas were used to improve all three projects. The conclusions showed that the project was of reasonable difficulty for the groups and there was an interest in knowing more about differentiation after having completed the project. Other conclusions are that the students have yet to adequately master the use of mathematical language and reasoning, and they need to have more faith in their own reasoning, enough to be confident with their answers.

In addition to updating the projects in accordance with the study, an assessment rubric was developed, which can be of use for teachers for formative assessment. Additionally, summarization projects derived from the main projects were added as well as a detailed project description and teaching guidelines with connections to the main components of the Icelandic curriculum for upper secondary schools. Lastly, the projects were analysed from the perspective of the Teaching for Robust Understanding (TRU) framework. That framework is supposed to aid teachers in creating an environment where students can learn with a focus on understanding. In the appendices of this thesis, the three projects can be found as well as the summarisation projects, the teaching guidelines, the project descriptions, and the assessment rubric.Upper Secondary Teaching

**Development of projects on differentiation with emphasis on understanding and discovery**

**Advisors: ** Benedikt Steinar Magnússon and Bjarnheiður Kristinsdóttir

**Examiner:** Kristín Bjarnadóttir, Professor Emerita

**Abstract**

Mathematics in upper secondary schools is a challenging subject for many students, and many even become convinced they are unable to learn the subject. In order to counter such beliefs, it is important that teachers refuse to accept that sort of thinking. There are ways to help students overcome their pessimism and, according

to research, one of the best ways to help them is to guide them towards discovering their own connections between different elements of mathematics and using these connections to better their understanding. The objective with this thesis is, therefore, to create and develop projects for students in upper secondary schools

with this idea in mind. The projects focus on differentiation because the author is interested in that field and because studies conducted among upper secondary school graduates have shown they struggle most with differentiation and integration of all the mathematical fields.

Three projects are presented here in which students have to deliberate the mathematical ideas which are being focused on, and as a result, contrive their own understanding of these ideas. The first project inspires a graphical interpretation of differentiation with the use of the slope of a tangent. The second project centres on the derivative of power functions and discovering a graphical pattern to find the derivative. The third project is on optimization, where the students are supposed to find the largest volume of a gift box, both graphically and through calculations.

Sixteen upper secondary school students worked in groups on the first project. Their answers and ideas were used to improve all three projects. The conclusions showed that the project was of reasonable difficulty for the groups and there was an interest in knowing more about differentiation after having completed the project. Other conclusions are that the students have yet to adequately master the use of mathematical language and reasoning, and they need to have more faith in their own reasoning, enough to be confident with their answers.

In addition to updating the projects in accordance with the study, an assessment rubric was developed, which can be of use for teachers for formative assessment. Additionally, summarization projects derived from the main projects were added as well as a detailed project description and teaching guidelines with connections to the main components of the Icelandic curriculum for upper secondary schools. Lastly, the projects were analysed from the perspective of the Teaching for Robust Understanding (TRU) framework. That framework is supposed to aid teachers in creating an environment where students can learn with a focus on understanding. In the appendices of this thesis, the three projects can be found as well as the summarisation projects, the teaching guidelines, the project descriptions, and the assessment rubric.