High School Teachers Across Denmark Prepare Students for Post-secondary Education Using Maple

Challenge

After assisting with mathematics courses at Technical University of Denmark (DTU), several high school teachers realized how big of a gap there was between high school mathematics courses and university courses. DTU, in conjunction with the Danish Ministry of Education, wanted to find a way to help students bridge that gap and be better prepared for post-secondary education.

Solution

Professors from DTU worked with various high schools to integrate Maple into Gymnasium math courses, thereby putting STEM students on the same page as university students, and making the transition into university much smoother.” Maple was eventually adopted into 110 of 160 high schools in Denmark to ease the switch from high school to university mathematics. Students are also required to complete a major project using Maple in their final year of high school and submit a detailed report.

Result

The result of this initiative is that students are better prepared for entering into post-secondary education. Their awareness of mathematics concepts and applications has increased, facilitating a greater retention of knowledge and the ability to apply what they’ve learned to advanced topics.

The initiative also led to the development of a video series to showcase visualizations from Maple, and to encourage students to pursue mathematics.

Mathematics has always played a central role in secondary school curriculum in Denmark. The Danish Ministry of Education continues to emphasize its importance as it mandates reforms and new standards that students are expected to meet in order to graduate. The country’s high standards of mathematics have led to the adoption of Maple in 110 of 160 Denmark high schools. Maple is a software tool from Maplesoft that makes it easy to explore, visualize and solve problems in mathematics.

Maple’s entry into the high school scene in Denmark first began when a group of mathematics professors at the Technical University of Denmark (DTU) hired a few high school math teachers to help them with one of their largest math courses. The professors benefited from more than just their help in class, though. The experience also opened their eyes to how students learn mathematics in the years leading up to the time they enter the University, and how different that was compared to the learning methods at DTU. “At that time, there was disconnect between how students were used to learning math, and how we taught math at DTU,” noted Professor Steen Markvorsen, one of the DTU professors. “We used to spend a lot of time and effort each year bringing students up to speed so all the students were on the same page.” This led to the idea that high schools, known in Denmark as Gymnasiums, and DTU could work together to bridge the gap between secondary and post-secondary schools. In doing so, students would be better equipped for their studies at the university level.

To get the initiative off the ground, Markvorsen and his colleagues made presentations to high schools around Denmark to gain support for the idea. “As we discussed ideas with school representatives, what became clear is that high school teachers endorsed the concept that using mathematic tools to help students would be a great way of preparing them for higher levels of education,” explained Markvorsen. “As they got to know Maple, they agreed it would be an ideal tool to help bridge the gap. Integrating Maple into Gymnasium math courses would put STEM students on the same page as university students, and they would find the transition into university much smoother.”

*Figures: Steen Markvorsen uses Maple to develop an intuitive proof of Euler’s theorem for the normal curvatures of surfaces*

As DTU is a technical university that graduates engineers exclusively, math plays a central role in all its degree programs. For years, Markvorsen has been using Maple to teach mathematics in a first-year compulsory introductory course. Equipping incoming students with Maple would ease the transition into DTU for both the instructor and the students, and students would benefit significantly more from all courses at the university, especially those that use Maple.

The result of this initiative has culminated in bringing Maple to classrooms across Denmark, as well as enhancing a major project that students must complete in their final year of secondary school. The project requires students to create a detailed report about an application of mathematics in an area of their choice. Students manage the entirety of the project, including finding a topic, and seeking teacher inputs. It is in the consultancy stage that DTU plays a big role in assisting students. Students can seek input and direction from DTU’s mathematicians about their specific applications of research. DTU provides kick-starters to warm students up to possible topic choices, and has created videos to encourage and motivate students to start thinking about their projects. The university also supplies Maple exercises for Gymnasium students to practice.

Using Maple, students have access to a number of resources that could be applied to their projects in mathematic as well as in other areas such as Physics, Statistics, Chemistry and Engineering. “The project has been very successful so far,” explained Markvorsen. “It’s quite technical in nature, and requires a great deal of research and planning by the students. They use Maple, an industry tool, to solve practical problems, and benefit greatly from being able to bounce ideas and questions off an expert in the field. It’s an excellent opportunity for them to think about mathematics and its applications, very differently from what they are used to.” The project also fulfills the Ministry mandate to incorporate applications of mathematics into the Gymnasium math curriculum. Once the project report is written and finalized, it is sent to a Ministry examiner who, in collaboration with teachers, assigns the student a grade that contributes to their final mark in the course.

Now that the initiative has been in place for a few years, Markvorsen has noticed a difference in students. “Since introducing Maple at the Gymnasium level, students are more aware of the applicability of mathematics than they were previously. It has caused them to be more open-minded about learning math. They retain more and think more deeply, and apply their knowledge to advanced topics.”

The initiative to promote mathematics among high school students continues to gain momentum in Denmark. Inspired by the Gymnasium teachers, Markvorsen and other Danish professors have teamed up to be a part of a film project called “Ten Danish Mathematicians” - ten videos that spotlight Denmark’s leading mathematicians. The films, which often showcase visualizations from Maple, are produced with the goal of encouraging students to pursue mathematics, and make them aware of the fact that “the world is full of mathematics”. Teachers can access supplementary material that corresponds to each video so students can get practical exposure to each application as well. Two videos are being produced every 6 months until all 10, and their associated project materials, are complete.

With the successful integration of Maple at the high school level, Markvorsen and the are turning their attention to primary schools. The DTU wants to introduce students to Maple at a young age, to build competency and better prepare them for math education at the high school and university levels. There is great potential for using Maple with primary school students, as Danish industry and society are continually calling for improved mathematical competencies, Markvorsen said. “We, at DTU, and our society as a whole, have a clear interest in high school and primary school students gaining greater competencies in mathematics,” he said. “The development of these skills in primary school, with a view to more advanced and complex training at the other end of the educational spectrum, is so much easier with a uniform, flexible and powerful tool like Maple.”

Maple offers a uniform environment, with built-in engines for advancement through all levels of education, and allows for the integration with other software, Markvorsen said. “This latter possibility carries important and strong assets for building up and strengthening the bridges from primary school to gymnasium and from gymnasium to university,” he said. “In the case of DTU, we are already experiencing great benefits from the students’ prior knowledge of Maple from their respective gymnasium studies.”

*Professor Steen Markvorsen discusses foam structures and minimal surfaces, including mean curvature surfaces using Maple (video is in Danish)*

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