“KWARC was!”

Theme: Technology changes the teaching situation. Students have other intention than their teachers when using mathematical tools.

Do we need to change the why we teach or should we stick to “old-fashion” skills and methods and rather carefully use software in (early) math education?

Presentation by Mary Billington, University of Agder at the 4th European Workshop on Mathematical & Science eContents; see abstract

Theme: Student can build up their own mathematical knowledge, starting with meaningful problems instead of learning algorithms and theories without connection to prior knowledge.

Mathematical Software (Java Applets) for children and secondary school students. For example to teach addition rather then counting objects. They aim at making applications more user friendly by implementing small tools, which only cover one topic, and have self-explaining interfaces.

Presentation by Peter Boon, Freudenthal Institute University of Utrecht (Education Software Designer)
at the 4th European Workshop on Mathematical & Science eContents

More Details

Applying an Incremental Approach: From small exercises towards an online curriculum
* organizing activities
* capturing students’ work
* customizing the activities to make it fit in learning resources
* more authoring facilities to generate digital activities with less effort
* digital learning trajectories instead of single activities

Notes on my Discussion with Peter Boon
(Please not all the below is my personal opinion and understanding of Peter’s explanations.)

The DME can be used to create applets of interactive problems. Peter showed me how to export these and how they can be easily integrated in an HTML page. We could use DME to make our precourse system more interactive and interesting for our students. However, DME is not open source, but for schools and university a rather small amount is charged. The DME system is fully web-based and available online. I’d be happy to give a demo in one of our group meetings.

In contrast to MathDox, DME is a client-based application, i.e. all computation is done on the client allowing for large number of users. However, database access are of course still on the server side. MathDox (Jelly) and our system (PHP) are server-side implementation, i.e. all computation is taken place on the server – multiple user access at the same time might become a problem. However, our system is not used in the classroom but is an additional discussion platform next to the course. Multiple access to the same time is less likely.

Portability of DME across operation systems is not a problem (in contrast to us who have to consider different Browser specialities, DME is JAVA-based and thus has less problems to support different browsers, however there are some issues on running it on Mac).

DME allows to create problems only, no complete course materials. These problems can be exchanged between teachers of one school. Peter is working on allowing an exchange across schools.

BTW: Setting up a central repository for mathematical eLearning content seems a very interesting challenge. During the discussion with Peter, I got a much better idea how this repository is different from our understanding. So far we have been focusing on rather “dead” mathematical materials. We aim at making these more active, by e.g. allowing flexible elisions and dynamic notations. However, these documents are still not interactive in the understanding of the eLearning community. Here interactive problems include e.g. the applets Peter Boon generates, i.e. content that includes user-specific (interaction) data. In this sense, active content can be understood as forms that need to get and set user data via and interface of an eLearning environment (which stores the user data in its database).

In the SCORM context these interactive contents are managed by the SCORM runtime environment, which handles the communication between interactive context and an eLearning environment. We need to distinguish between content providers and system designers. Content providers do not want to know about the details of an eLearning environment (such as Moodle or DME) but simply want to provide their content. These contents are only able to call a standardized API of an eLearning environment. They get and set user-specific data via this API. In particular the user-specific data is transferred as text. The interactive content can decode and encode these text files and e.g. initialize its applet. In constrast, the eLearning environment generates the text file from its database.

Providing a central repository for these eLearning contents seems challenging but really interesting. A very interesting conversion. Thank you!

Theme: What do mathematics education researchers do, what methods and theories do they apply their work? How could research be used by mathematics teachers/ lecturers? How can education researchers and mathematics teachers work together?

An interesting introduction into education research in mathematics and how it differs from research in mathematics/ other science.

Plenary lecture by John Monaghan, University of Leeds
at the 4th European Workshop on Mathematical & Science eContents

Details on the talk:

Education researchers

• generally work in a team
• lead or support in a team
• at any time several projects will be ongoing

Education research: lots of different types …

1. action research (problem solving)
2. jobbing research (getting a government contract)
3. own interest research

1+3 tend to have research question; RQs in 1 change more often/ develop over time
2 tends to require outputs (someone else decided the focus, can be frustrating)

Research Questions
Getting the RQs right can take a lot of effort …

• setting them in practice
• getting the logic right
• ensuring feasibility (finishing at some point in time)
• requires several iterations

Example RQs in education research:

• How can research access the self-knowledge of students when using technology for learning?
• What learning results from the use of technology by students?
• How does technology use in the lives of students change and develop over time?

Methods: Data collection, analysis, interpretation

• data collection: tests, questionnaire, interviews, observations
• analysis: qualitative, quantitative (descriptive, inferential)
• no real method for data interpretation (Jeffrey Saxe’s model)

Ph.D. often do not answer: Why are you doing a questionnaire? What is the aim of the survey? How does it relate to your research questions?

Theories:

• theories in maths education have nothing to do with theories in maths/ logic (think “frameworks” or “paradigm” or “world views”)
• implicit theories: we must have a “world view” to make statements e.g. in the statement “High ability students are more attentive in class?” the world view is: “it is assumed that students can be categorized as high or low ability”
• explicit theories
• rather wide theory: activity theory
• educational theories: pedagogic codes
• narrow theories (restricted to math education) e.g. TDS

Examples of education research:

• Issues Arising When Teachers Make Extensive Use of Computer Algebra in their Mathematics Lessons
• Investigation the move from occasional to regular use of ICT in mathematics classes
• Implicit aspect of pencil and paper mathematics assessment that came to light through the use of computers: kids did lots of problems on pen&paper and using ICT

How to use education research in mathematical teaching?
see slides

see JEM Network

Download from here.

Tutorial by Antonio F. Costa, Miguel Delgado, Estrella Gómez (Universita Nacional de Educación a Distancia, Spain)
Notes from the 4th European Workshop on Mathematical & Science eContents

This is collaborative work with Dominique Archambault (université Pierre et Marie Curie, Paris), who is currently working on interaction of braille and sighted display – e.g. to allow blinds to ask question on formulae parts they select via their braille display, which can then be highlighted for the professor or tutor (see pdf).

The main input requirement for their applications is providing semi-structured documents with Presentation MathML. However images are still a problem. But improving documents for blind people also improves the adaptability and usability for other users. Very interesting work in the scope of adaptable/ interactive documents.

Tutorial by Cristian Bernareggi (Università degli Studi di Milano)
Notes from the 4th European Workshop on Mathematical & Science eContents

More Details:

• methods for accessible documents: braille display, speech synthesizer, screen magnifier
• requirements: structured text with presentation MathML (or even OpenMath/ content MathML)
• requirements for TeX-based formats: human readable; preserve two-dimensional layout at least for inherently two-dimensional notations
• challenges: images (alternatives: tactile printing, text description)

Further Readings/ Events

• Adaptive Content Processing Conference 2008 (ACP 08): Adaptive content processing for educational environments
• Collecting information about Braille Mathematical Notations; and Use MathML in online contents! (Dominique Archambault, Blog entry)
• Quality issues regarding Accessibility (Dominique Archambault at the 3rd JEM Workshop 2008, Barcelona)
• Access to Mathematics and Science – @Science Workshop in Paris
• A multimodal interactive system to create and explore graph structures

Hand-on tutorial of an assessment system.

I think this system would be valuable for our precoure evaluation. We could set up questionnaires with random math problem and get a better idea on the math background of our students. It doesn’t seem possible to start from scratch and provide a good preparation course. We should get in contact with the project leaders and try using this system.

Tutorial by Matti Pauna, Mika Seppälä (University of Helsinki)
Notes from the 4th European Workshop on Mathematical & Science eContents.

More Details

• focus: high school and undergraduate students
• exercise types: multiple choice, open answer, fill in blanks
• questions are automatically generated (questions may include graphics)
• 2000 questions types from which (a virtual unlimited number of) specific problems can be generated
• answering by formulae, automatic verification
• exercises are algorithmic
• problems include hints and complete solutions
• automatic grading
• purpose: assess and practice mathematical skills
• purpose: illustrating graphs and images
• questions are arranged according to a standard taxonomy (classification of exercises)
• usage: questions are used by high school teachers as homework (giving bonus points)
• usage: for diagnostic testing at the beginning of university courses; this allows to divide students into groups according to their individual learning needs
• benefits: immediate feedback to students and automatic grading for teachers

Student Mode

• Multiple Choice quiz: returns my answer, correct answer, comment and a visual button for correct/ incorrect answers.
• It would be nice if I could ask for hints or an example for incorrect problems. Some questions provide hints, but for learning and understanding these might not sufficient
• Student have to adapt to the maple syntax for entering formula, they have to adapt to the notation of the system e.g. (4;3) is incorrect but (4,3) is correct; however this was never a problem – at the beginning of a course they do test quizzes to get familiar with the system …
• Students can take the quizzes as often as they wish, the system remembers their attempts and reduces the grade respectively. But this can also be ignored, i.e. taking quizzes becomes a game in which the best score counts.
• For high school math this seems to work; e.g. for questions on graphs of function (intersection points) or simple calculations; for more complex problems student can print the quiz and then entering their final answer. But then the teacher looses the way of solving the problem – here we sometimes give extra points even the result might be wrong.
• It is a better approach to model question types and generate concrete problems from that: this reduces cheating as each student can get different problems and answers can not be copied so easily.
• Student has a grade book: can see his answers and final results

Teacher Mode

• teacher get an overview of the students results and can manually change points, give extra credits, and enter comments.
• teacher can edit assignments: adding/ deleting question types (during runtime these types are instantiated) and modify their points; setting a passing score (students have to take the assignment until a certain number of points)
• creating anonymous practice that are used for grading.

Programmer Mode

• writing questions is more difficult, you actually write little programs (issue with debugging, careful editing)
• one has to write the algorithm using maple; preview is available; from that the problems can be generated
• Question creation is based on the Maple TA System, which provides an editor
• enter formulae in Maple Syntax to generate Presentation MathML or enter Presentation MathML using the editor
• Webalt started in 2005, about 12 people create these question programs in the maple programming language
• any type of question can be modelled (e.g. using free text, but then automatic verification is not be possible)

Presentation Issue
The system is generating Presentation MathML, however, for the presentation it is converted into an image. But their have been critics on this as e.g. enlargement of the fonts will not enlarge the image (for visual impaired people) and conversion to braille will fail.

Further questions:

• What about alternative solutions? Can you support all problem solving strategies or do you enforce certain practice? Do you really practice all possible mathematical skills and the way of doing math? Is this not to result-focus?
• Which skills do you want to train; have you ever modelled the competency students gain by solving these exercises?
• Is the taxonomy public? Have you linked exercises to mathematical concepts, e.g. if teachers is talking about concept “A” you can retrieve the best suited questions?
• Are questions interlinked/ related/ have you modelled dependency?
• are you sure that: “students gain understanding of their learning process and can concentrate on the weak points and practice more”