Sustainable Concepts

sketchometry - Smartphone

sketchometry is a digital sketchpad for mathematics teaching. But software alone is not sufficient enough to make teaching sustainable. In order to foster and improve teaching and learning, suitable concepts are necessary that describe how software and devices can be integrated into the classroom in a meaningful way.

For many years the development of such sustainable concepts has been a research focus of the Chair of Mathematics and Mathematics Education and the Research Center for Mobile Learning with Digital Technology at Bayreuth University.

Digital media can contribute to a perceptible improvement of mathematics teaching. Thereto these media have to be

  • integrated into the teaching-learning situation in the classroom,
  • combined and linked with traditional media such as worksheets, textbooks, and compass & ruler constructions,
  • included in the training of future teachers and in the in-service teacher training.

Using sketchometry on a tablet or smartphone enables interactive constructions directly in the classroom. Using sketchometry allows new and challenging approaches to inquiry-based mathematics education. Mathematics becomes an experimental science.

Tablet or Smartphone

sketchometry - Smartphone

Students autonomously create constructions with sketchometry and manipulate them interactively. By doing these experiments they discover geometric relationships, make assumptions, and try to verify them. Thus they are actively involved in the learning process.

Classroom work allows the combination of the digital tool with traditional media. For example, an exercise from a textbook is elaborated with sketchometry. An added value results from the possibility to modify the construction interactively and to share it with other students. Results are manually recorded in a study journal or an exercise book.


sketchometry - Smartphone

Specially prepared sketchometry worksheets guide the students through the learning process. We distinguish two stages, a construction stage and an exploration stage.

Construction Stage

At first a sketchometry worksheet contains some construction orders. These are short instructions for drawing an own sketch or construction with sketchometry. The students have to play an active part.

Exploration Stage

sketchometry - Smartphone

In the directly following exploration stage instructions on the worksheet encourage the students to experiment with the construction they have created, to make observations and assumptions, and to make handwritten notes and sketches. In addition to the mere generation of a construction, now the intensive study of the mathematical problem plays an important role.

Study Journal

sketchometry - Smartphone

The students write down their observations, ideas, sketches, assumptions, and findings in their study journals. Constructions with compass and ruler are possible, too. This kind of autonomous handwritten documentation supports a sustainable learning process. To facilitate the first steps into autonomous documentation – especially for younger students – well-structured result sheets are highly recommended. The items of the result sheets are related to the exercises of the corresponding sketchometry worksheets.

Worksheet Videos (only in German)

Particular learning videos take up the contents of sketchometry worksheets. The videos focus on different topics. In some videos the development of the construction with sketchometry is explained step by step:

  • Which sketchometry gestures are necessary?
  • What has to be considered during the construction?
  • Which fore knowledge is required?

In this way the students get to know not only the mathematical content but also the handling of sketchometry.

Other videos emphasize exploration and investigation and immediately involve the students by asking them to do the construction and exploration by themselves. The videos are suitable for classroom demonstration by the teacher, for autonomous learning by the students, and for independent repetition (at home).

Think – Pair - Share

If the students work alone on the tablet or smartphone with sketchometry, a multi-stage procedure is recommended for the lesson. The following three stages divide the learning process into three sections. If students share a tablet, the first two stages can also be combined.

Think – Individual Work

The students create their own dynamic constructions and perform experiments (constructing and exploring). They investigate, observe, assume, look for solution strategies, and develop ideas of their own for a solution.


If the constructions are dragged and modified, sketchometry automatically adjusts all dependent objects. Thus mathematics turns out to be an experimental science. The tablet pc is the laboratory; a dynamic configuration on the screen represents the experiment.

Pair – Learning with a Partner

The students mutually exchange their ideas. They show each other their discoveries and discuss their findings.


The students learn to argue mathematically. This active discussion leads to a deeper and sustainable understanding of the mathematical topic.

Share – Presentation and Communication with the Whole Class

The results are presented and discussed within the whole class. A common result is worked out from different students’ contributions.


The role of the teacher has changed, he/she more and more becomes a moderator. If necessary, additional contents are provided by the teacher.

The advantage of this triple stage method is that the students have to deal with the same problem three times but each time from a different point of view. So we have no boring repetition and therefore the students get a deeper understanding of the problem and the solution process.