Plenary lecture: Michèle Artigue

Title: Didactical Design in Mathematics Education

Current context
Increasing interest in design issues. Reflection on the value of the outcomes of didactical research, and impact of research on educational practices.
Motivation: external and internal

• math education is a sensitive domain for our societies
• increasing pressure of international evaluations, tests, etc.
• increasing debates about curriculum reforms and the supposed influence of didactical research on these

External side (Burkhardt and Schoenfeld, 2003)

• Start from evidence that educational research does not often lead directly to practical advances
• Development of “engineering research”
• Design experiments – promising model of interaction

Internal side (Cobb, 2007)

• Multiplicity of theoretical frame
• Two criteria proposed
• Multi-level vision of design
• Experimental design has to be its unique methodology

Didactical design – mathematics education
Diversity of perspectives

• Didactical design as research tool
• Didactical design as development tool
• Math education as design experiment

Didactical engineering (emerge in the early eighties)
Initial distinction between phenomenotechnique and didactical engineering

Didactical engineering as a research tool, shaped by theoretical foundations
– Influence of the theory of didactical situations (Brousseau)
    Learning processes as adaptation processes (Piaget)
    Focus on situation and milieu
    Distinction between different functionalities of mathematics knowledge (acting, expressing and communicating, proving)
    The teacher role

Didactical engineering – the predominant research methodology in the French didactic culture (esp. in the eighties)

Relationships between research and practice

• Relationship that is not under theoretical control
• Products communicated in different arenas (publications, teacher formation, etc.)

Relationships between research and practice

• Relationship that is not under theoretical control
• Products communicated in different arenas (publications, teacher formation, etc.)
• Results reproduced, used in textbooks, etc.

Subsequent evolution

• Better understanding of teachers’ practices
• Development of less invasive research methodologies
• New theoretical constructions
• Substantial body of research that impacts the vision of didactical design

Didactical design today

• Still a tool widely used
• Same epistemological sensitivity
• Importance of interaction with the milieu, more sophisticated vision of the teacher role
• Same importance to the a priori analysis
• (Differences on the view of the teacher in France and Italy, for instance)
• Didactical engineering still a research tool

Praxeology

• Practical part – type of task (technique)
• Theoretical part – technicological discourse (theory)

A bit late, but here are my notes from the plenary lecture from the third day at Norma08:

Plenary lecture – Eva Jablonka

PART 1 – “Mathematics for all. Why? What? When?”

Math as a core subject in compulsory education (empirical fact). Industrialised countries provide basic maths for all (in school). BUT – many children don’t go to school in several countries around the world. It varies between countries when children can stop taking mathematics courses.

Mathematics for all, beyond primary level – why?
Goals as an apologetic discourse.
Common list of justifications:

• Skills for everyday life and activities for workplaces (useful)
• Sharing cultural heritage
• Learning to think critically (formative goal)

Examples of critical thinking in classrooms (Harols Fawcett, 1938)

• Selecting significant words and phrases, careful definition
• Require evidence to support conclusions
• Analyzing evidence
• Recognize hidden assumptions
• Evaluate the argument itself
• etc.

“Everybody counts” (National Academy of Sciences, 1989)
Help develop critical habits of mind, understand chance, value proof etc. (p. 8).

The notion of “thinking critically” – what is it?
Fawcett – precision of language
Swedish example – relation to environment, etc. (global view, more vague)

Is there an epistemic quality of mathematics that is linked to thinking critically? (interesting question!)

Recent descriptions – renaissance of formative and methodological goals
– Communicating mathematically (discuss, advantages, disadvantages, etc.)

    Communicating freely and critical thinking takes place in some sort of an ideal democratic environment.

    Are mathematics classrooms ideal speech communities?

– Learning to model and solve problems mathematically
    Danger of overemphasizing utility (connections with engineering, social science departments, etc.)

– Recruitment into the mathematics, science and engineering pipeline as justification (economic development in a country, etc.)
    There has to be a “critical mass” from which to select future mathematicians. (similar argument to sports, being successful in sports)

How successful are the students in compulsory mathematics courses for all?
International tests (PISA, TIMSS, etc.) – only a small percentage will reach the highest level. Discussions of “average achievements”, comparisons between countries.

Conclusions
Compulsory mathematics, not for all. Global failure of math education?
Which groups of students are successful/less successful? (interesting question)

PART 2 – “Mathematics for all!” (mission statement)
Challenges:

• Demographic development (declining number of students, in many industrialised countries)
• “Learning to leave?” – Successful students often end up moving away (from their country, local area, etc.) – How can a mathematics curriculum serve the local needs of local communities?
• Organization of participation – students’ choices. Why do so many students choose not to pursue further studies in mathematics after the compulsory course? To what extent should we “force” them to choose mathematics?
• Changes in social contexts
• Increased stress on instrumental knowledge and of the marketability of skills. Danger of oppositions between rationales for mathematics and liberal arts for instance.
• Professional groups fighting against the “contamination of mathematical knowledge”. Consequence of shift towards process skills in the curricula. (Back to basics movement, math wars, etc.)
• A widening gap of mathematical knowledge between constructors and consumers of mathematics (Skovsmose, 2006) – threat to democracy (you have to rely on the experts).
• The “de-mathematizing” and restricting effects of mathematical technology. Use of technology liberates us from the details of mathematics.
• Confrontations of local knowledge and mathematical knowledge acquired at school. (Students don’t appear to use the mathematics they learned in school outside the classroom)

Research is addressing some of these challenges:

• Classroom research looking into these speech communities
• Concern about “mathematical literacy”
• Empirical studies of local mathematical practices at work-places (and local communities)
• Students’ goals and motives
• Consequences of changes in students’ backgrounds
• Problem of transition between different tracks of mathematics education

Jablonka doesn’t think there will be a universal curriculum for all.

Plenary lecture – P. Drijvers
Title: “Tools and tests”

Drijvers starts off giving some introductory notes about the Freudenthal Institute.
“Tools” = technological tools in this connection.
Why use tools and tests? The teaching and learning should be reflected in the assessment, and assessment should be driven by teaching and learning.
What are we actually assessing? Tools skills or mathematical skills?

Tests with tools, why would we do it?

• Prepare students
• Allows for different types of questions
• Assessments should reflect learning
• etc.

Drijvers goes on to present some examples from other countries (France, Germany, etc.) of tasks where technological tools are involved. The use of tools in the tasks is often questionable (or non-existent). In some examples, graphing calculators are allowed, but the tasks do not indicate any usage of these tools. Drijvers also presents some examples that are interesting to discuss from the point of view of “realism” and “authenticity”, and he takes up this discussion in a few cases. Ends the section of examples with an example from the Netherlands, and he makes a humorous comment about this being the perfect example of a really good task. In discussing this example, Drijvers continually come back to the issue that this is something that you can imagine. And in the Dutch vocabulary, “realism” means something that you can imagine. Within a Dutch context, a realistic task is therefore a task that the students can imagine.

He then brings the discussion to a meta-level, introducing concepts like artifacts and instruments, and goes on with a presentation of what is called instrumental genesis.

Conclusions so far:

• Assessment with technology is an issue in many countries
• Reasoning, interpretation and explanation is also asked about (not just ICT-output)
• Different ways of dealing with technology (discusses some trends)

Tools for digital assessments. Why digital assessment?
Discusses some of the limitations of software, types of feedback, etc.

All in all, an interesting presentation with several important issues being raised.

Plenary – J. Skott
The Norma 08 Conference takes place in Copenhagen this week, and I am attending. I will therefore have a focus on this conference this week. The first plenary lecture was presented by Danish researcher Jeppe Skott, and here are my notes from the presentation (which was very interesting by the way). I also plan on covering the conference on twitter, so take a look there as well for live reports!

Title: “The education and identity of mathematics teachers”
Research on mathematics teachers has grown tremendously during the past 20-30 years. Skott starts with a presentation about publications, journals, monographs, etc.
Two main concerns:

• Teachers’ knowledge
• Teachers’ beliefs

In the 1980s – a shift in the view of learning, mathematics, etc. changed the whole field of school mathematics (fallibilism, social constructivism). Teachers placed in a new role, as opposed to before. Teachers supposed to understand what students are doing, and to guide their learning. New role: planned unpredictability (interesting concept!)

Teachers’ knowledge
Displays a couple of examples from the literature that displays teachers’ (lack of) knowledge about mathematics (for teaching). Perhaps pre-service education is not what it should have been?
The importance of Shulman’s work. The article “Those who understand…” A main idea: content matters! Two of Shulman’s concepts important:

• Content knowledge
• Pedagogical content knowledge

What is it that teachers’ should know about? (content knowledge)
What is it that makes a topic difficult? (pedagogical content knowledge)

The mathematics of the classroom – the mathematics of the mathematician.
Liping Ma – asked teachers in China and the US lots of questions concerning basic mathematics. Many teachers (esp. the US teachers) weren’t able to solve the problems. A basic question for her – What is the relevant knowledge needed by teachers? American teachers – list of disconnected procedures. Chinese teachers – alle these procedures were related. “Understanding with bredth.”
D. Ball, H. Bass et al. Classroom based approach. Mathematical challenges from the classroom. (Elements from the LMT measurements) D. Ball calls it “unpacking mathematical knowledge” – digging deeply into the conceptual issues.

A shift in the area of developing a knowledge base for teaching:

• From – number of courses
• to – knowledge of school mathematics (L. Ma)
• to – knowing in action (D. Ball)

Beliefs research in math education
In order for any reform to have an impact there needs to be a change in the teachers’ beliefs.
Developing and changing beliefs. Several suggestions and attempts (see points in slide).
Relationship between beliefs and practice.

A moral so far: There is a need for contextualizing mathematics education to the act of teaching.

Discussion of the relationship (or expected relationship) between development of curriculum and curriculum material and teaching practice.
As researchers, a main issue is the one of theorizing practice.

Poses an interesting question: In what sense is mathematics education an applied field?
Points at an interesting quote by P. Cobb about the issue of mathematics education (research).
Interesting model about the dimensions of research (by Stokes).

A main issue for research in math education is maybe not about theorizing, but about having impact on practice.

The end of the talk filled with intriguing questions and interesting metaphors. (Thaetetus’ ship – if you replace a plank, and then another plank, when is it no longer Thaetetus’ ship, but a new one?)

All in all a very interesting presentation! Hopefully these notes could be deciphered by others as well…