Issue 1: National Curriculum, Standards, and Teaching Practice

What is the relationship of national standards and national curriculum to teaching practice in classrooms in your country?

France: Antoine Bodin and Catherine Sackur

Opening Statements

Antoine Bodin
Université de France Compte IREM

The main question I am supposed to answer in this short presentation is: How does your education system function, both in theory and practice? And more precisely: What is the relationship of national standards and national curriculum to teaching practice in classrooms in France? I will try to answer part of these questions from a research and curriculum development point of view. Later, when Catherine speaks, she will tackle these questions more from a teacher's perspective.

Concerning the question regarding the relationship between national curriculum/ standards and teaching practice, an answer that immediately comes to mind is that in France, indeed, this relationship should be both direct and obvious. France is a centralized country, so standards and curriculum are decided at the national level and apply for all in the country. It follows that the whole process should be a hierarchical and compulsory one. The reality, however, is much more complex.

An influential network of university institutes for mathematics teaching (e.g. the 26 IREM¹ institutes across the country), along with the Professional Association of Mathematics Teachers, makes it difficult for the Ministry to fully control teaching practice. Practice is actually influenced by both the centralized, hierarchical system, as well as by strong teacher individualism associated with a host of non-official influences². Surprising though it may seem, the impetus for some of the non-official influences comes from the ministry itself.

In an effort to maintain a balanced and unified education system, France actually tries to keep the best of its centralization heritage while delegating some national responsibilities to the Regional Education Boards (30 Academies in France). For example, the Ministry has assigned the task of writing the official curriculum to a group known as the National Mathematics Expert Group³. While the ministry is primarily responsible for the development of the national curriculum, its development is best viewed as a recursive process involving teachers and policy-makers rather than as a process of delivering regulations.

To encourage good teaching practice and teachers' personal involvement in the process, the ministry relies on several features and devices. These include:

• The National Curriculum designed in the Ministry
• National Examinations at grades 9 and 12 (especially the Baccalaureate)
• Diagnostic National Assessments at the primary level (grade 3), lower secondary level (grade 6), and upper secondary level (grade 10), and
• Incentives for teaching innovation.

The Power Point slides: Regulation of the French Educational System for Mathematics found in Appendix D present a system analysis of the French mathematics curriculum design and development process.

To conclude my presentation and open discussion and comparisons with what is at issue in other countries, I offer several questions for consideration. These include:

• How can we promote a student-centered, activity-based, constructivist teaching approach without negatively impacting students' proficiency?
• How can we set national standards without reducing teaching and learning activities to a collection of insignificant basic and procedural skills?
• How can we maintain national unity in the mathematics curriculum while allowing for openness, teacher initiative, and innovative practices?
• How can we preserve the value and reliability of the baccalaureate without interfering with curricular and teaching reforms?
• How can we take advantage of the links between mathematics and other topics, and facilitate integrative teaching without sacrificing mathematical rigor?
• How do we determine what mathematics content is more relevant for general education (i.e. mathematical literacy) and what is more relevant for those preparing to study advanced mathematics?
• Conversely, how do we identify which present mathematics topics can be safely removed from the curriculum?
• How can we maintain high standards without discouraging students from learning mathematics? And how do we choose which standards to uphold?

Catherine Sackur
IREM de NICE-UNSA

I am a high school math teacher. I teach students who are in the last three years of secondary school (age 15-18). Recently I've been teaching the very last year for students who have chosen an advanced course in mathematics. I am a researcher in mathematics education, and I participate in teachers' education by supervising student teachers in their last year of training⁴.

I have been asked to speak on the relationship between national standards/curriculum and teachers' practice in the classroom. First of all, I feel the need to clarify what we mean by national standards and national curriculum in our country. In France, a curriculum is a text written by experts chosen by the Education Department. A curriculum has three parts:

1. motivation for choices
2. a rather detailed list of the notions in the different chapters, and the expected competencies
3. comments on these competencies (e.g. "Students should know..." "It is possible to have students work on...but no specific knowledge on this point is expected..."). As the final examination is the same for all the schools, these comments are important. The basic knowledge tested by the examination is supposed to be the same everywhere.

I would like to approach the problem of relationship between national curriculum and practice in class from two different angles. In an effort to be a little concrete, I'll begin with an example on the use of pocket calculators. Around 1984, graphic calculators became less expensive and more accessible to students. Since then, the education department has emphasized the use of pocket calculators in elementary and secondary schools. There have been many efforts to integrate them into math education, and they became an active part of the math teaching at those levels. Teachers have been required to integrate pocket graphical calculators at all levels from primary school to the end of high school. In grade 10, students are expected to know how to trace the graph of a function. In grade 12, they use some of the software and programming features to study sequences, or some algorithms in number theory.

On the contrary, at the university level there has been no pressure to introduce pocket graphical calculators, and they are not used at all. So, we observe two things with regard to policy and practice:

1. the demands of the institution are really taken into account, but
2. the possibility exists that the policies conflict with preferred practice. For example, many teachers believe that number theory is a place for logical reasoning, not for algorithms and software.

Two questions may be used to organize the discussion. To what account do the demands of the curriculum influence practice in classrooms? And to what extent does classroom practice vary in our very centralized system? With regard to the first question, one can distinguish three trends in content and teaching:

• Long lasting practice
      Use of calculators
      Use of software
  
• Medium (These are no longer relevant. Many teachers still pretend to rely on these practices, but they have been totally changed in the long run. Sort of ritual.)
      Pre-lesson activities
      Modules
      Experimentation in mathematics
  
• Short-lived practice
      Scientific discussion
      Open-ended problem-solving
  

To understand these phenomena, consider:

• Innovative practice in the classroom places new demands on teachers.
      It is difficult to anticipate how the situation will unfold. Teachers must be prepared to improvise depending on students' reactions.
  
• The reactions of students and the society are very influential.
      Students are very reluctant to accept new ways of teaching. Also, there is an idea in society of what teaching mathematics should be, and it is very difficult to go against that notion.
      Old versus new dialectic: New practice must not be too far away from old practice.
  
• Problem of preservice and inservice training.
      At a high level, people who decide about the curriculum have a constructivist model of learning arising from the influence of researchers on math education.
      At the university, the model of teaching generally is that if you are a good mathematician, you will teach well. You learn a theory, you apply it, and that's all. The teachers have not learned to behave in a different way.
  

Although we have a very centralized system, there is still variability in classroom practice. While the contents and timetable are the same for all schools, teachers are supposed to have a certain level of autonomy on different matters. So, how do teachers take into account the expectations of the curriculum and education officers who visit them, and how do they use their autonomy in the classroom.

Variation within an individual teacher's practice

Teaching practice varies on two levels: within and between. Although national standards are provided, each teacher is more or less a creator of curriculum. Teachers vary their practice from year to year. They may choose to present the chapters in a different order or vary the weight given to different chapters. They may also choose to vary their approach to teaching the content based on the characteristics of students in the class.

I have noticed this in my own practice. I may read about something in a review for teachers and decided to experiment with it in my classroom. Sometimes my choice depends on the other teachers in the school. We may decide to collaborate and agree to emphasize a topic on which I do not usually spend more than the minimum expected time. Sometimes I choose to emphasize a particular notion if I think it is more useful than another in helping my students overcome their difficulties or if they seem particularly interested in a concept. For example, my decision to emphasize group structure is partly dependent on the students I have in my classroom. We have many opportunities to study group structure in geometry⁵, number theory⁶, and in the theory of complex numbers⁷. This year I've decided to emphasize this point. In another year, I might choose to spend more time on space geometry or on integrals and approximate calculation of integrals (e.g. the rectangle method).

Variation between teachers

Teachers' beliefs both about the nature of mathematics learning and the purpose of certain content influence their practice. Teachers are often reluctant to change their beliefs about these. For example, in the new curriculum, number theory is seen as an opportunity to introduce algorithmic thinking and the use of software, but as I mentioned earlier many teachers view number theory as a place where students can learn logical thinking (e.g. implication and logical equivalence).

Earlier I mentioned teachers' preservice and inservice training. This issue of beliefs is one where teacher training is fundamental. Aside from the official preservice and inservice training, we have many opportunities to meet other teachers who develop rather different ways of teaching and who experiment with new practices in their classroom (e.g. "math en jeans," mathematical rally, and narratives on some research activities such as open-ended problem solving). Some of these experiments have also influenced the experts who decide on the curriculum.

Considering these issues, we are led to ask two important questions about the relationship between policy and practice in classrooms: 1) how does a centralized system benefit students? and 2) how does variability in practice benefit students? Some additional issues we may want to consider during the discussion include:

• Increased access to school and the influence of this on practice and content
• The role that university teachers and society play the in determining the content of the curriculum⁸
• Recent emphasis on practice over content in the curriculum
• Adapting math teaching to the evolution of mathematics⁹ and to the evolution of students.

Participants identified the similarities and differences represented in Table 2.

Table 2 France: Similarities and Differences Among Participating Countries with Respect to National Curriculum, Standards, and Teaching Practice

Observer Commentary

Hyman Bass
University of Michigan

Your comments raised many questions in my mind. I speak mainly from my experience here in the U.S. We are struggling with the same problems, but we have a system that is quite different. Except for the U.S., most of the countries have a national curriculum that is formed by somebody in a Ministry of Education. This is very different from the U.S. where there is a long tradition of local control. National forces make practice somewhat similar anyway. For example, commercial textbooks and tests make practice rather centralized. But this is not organized by a ministry of education. Even though you have a centralized curriculum, you have access to many textbooks. This is surprising for Americans. We would like to know how variable these texts are, and how are they determined to be consistent with the national curriculum. Does anyone evaluate available texts? Do individual teachers make independent decisions?

There was a very interesting contrast in the use of calculators. Some used no calculators, and others used them widely. France had relatively open use, but calculator use is much less common in Japan. They are even available to students in India. Why is this? It would be interesting to learn more about this.

It was interesting to hear how much freedom teachers have even in systems that are centralized. This would be good for U.S. teachers to learn, because they resist centralized curriculum, thinking it means that they would not have professional authority. Even in your systems, teachers have much autonomy. This is an area that would be interesting to explore. What is the nature of teachers' creative work even in nationalized systems?

I have questions about how these centralized curricula and standards are produced and how they are received. Who writes them, and what are their professional credentials? This is a special issue in the U.S. because we have no system for this. For example, mathematics standards were produced by the National Council of Teachers of Mathematics (NCTM), the professional organization of teachers, while the science standards were produced at the National Academy of Sciences. Who should do this work? We have a lot to learn, but we have no system for this. .Also, how are these documents received and used by practitioners (people who work in schools with students)? How much do these documents guide the practice of teachers? How do they influence or constrain teachers' autonomy? Do teachers see these documents as "friendly" to their work or do they see these as coming from a community distant from their work, seeking to control their practice?

If we wanted to create such a system in this country, from what source we would take the people? What would qualify them to do this?

Hiroshi Fujita
The Research Institute of Educational Development, Tokai University

I was interested by the presentation of the French people and also by your discussion. At this moment, I want to present my impression of the formulation of the problem. A national curriculum and the educational system can be a means of controlling education. The implementation of a national curriculum must be done by a teacher with appropriate freedom. National curriculum set by the government and local initiatives set mostly by teachers modify these national curricula. Local educational authorities also play a role in this modification.

At the Ninth International Congress on Mathematics Education (ICME-9), a speaker from the ministry in Singapore explained that in Singapore there is a national curriculum instituted by the government. Still, a great deal of freedom is given to the schools. In Japan, local initiatives are maintained mostly by teachers. Schools and local committees work for better implementation of the national curriculum.

For a better national curriculum, there needs to be a feedback process. What I heard from France and what I see in Japan is that the government or ministry sets a national curriculum, and they set up councils or committees to help implement the curriculum. Money is spent to disseminate the curriculum and train the teachers for the change. Teachers give input to the ministry about the curriculum, and organizations such as universities, educational institutes, and projects become involved in making these adaptations. In Japan, the Academic Society for Mathematics, Applied Mathematics, and Science Education give input. Our ministry is not very tied to this Society at this moment, which is asked to cooperate and advise the councils and committees. The government is very strong over schools, but they are very weak over mass communication. Industry seeks a strong influence over the ministry. There is a tendency to underestimate the role and value of mathematics and science.

Hashimoto said that calculators are not frequently used in Japanese classrooms, but actually the national curriculum encourages the use of calculators. However, the entrance exams to university do not permit calculators, so this has an influence. The value of calculators in education differs according to the level of pupils. In elementary school, we should be very careful. For upper elementary we can use them carefully. In the lower secondary school, calculators can be useful and should be encouraged. The value of calculators cannot be discussed without description of how they will be used. I support their use because they can help students develop in pupils an appreciation of mathematics.

Themes That Emerged From the Discussion
Theme 1: Centralization and autonomy

Educational systems in different countries vary in terms of how centralized and regulated they are and in terms of how centralization is achieved. A variety of mechanisms are employed in an attempt to control what happens in mathematics classrooms. These include the establishment of national curricula and national (high-stakes) examinations, the choice of approved textbooks, and regulations regarding acceptable instructional practices, such as the use (or not) of calculators in the mathematics classroom.

Illustrative quotes

"There is a general tendency for decentralized systems of education to attempt some centralized procedures (e.g. national aims, curriculum, standards committees, national tests and vice versa). Centralized systems may attempt to extend authorities to local units." (Mina)

"One of the factors [in the success of centralization] is the degree of freedom that the teachers enjoy in teaching in a country. The issue of centralization might not be as such the major issue responsible for the outcomes of mathematics education in a particular country." (Mina)

[One of the important issues involves] how to keep in focus both the small scale local situation on an everyday basis, and also to look at the global situation and consider how things will work on the large scale. (paraphrase of Lins.)

Theme 2: Standards, goals and mathematical content

Another central theme that arose from the discussion was related to the scope of national standards and curricula. On the one hand, standards that are too narrow or prescriptive can result in a mathematics curriculum that is less challenging than what was intended. On the other hand, standards that are too broad or "loose" can be ineffective, in that they enable any type of mathematics instruction to fit under the "umbrella" of standards. There is concern about this in the United States.

"In India, there is an effort to work out minimal levels of learning (MLL). The idea is that at a certain stage the child will acquire these skills by certain grades (lists of skills at end of grades). The MLL influences curriculum framing and classroom teaching. The negative aspect is that many teachers consider this as a maximum level of learning and don't go beyond." (Agarkar)

National guidelines can only set content recommendations. There are many arguments because people try to use broad goals such as "thinking algebraically" as content. Feedback says that standards such as this are difficult for others to understand. (Paraphrase of Lins)

Key Questions

Key questions about national standards and curriculum include those related to:

• What does it mean for an educational system to be centralized (or not)? Are we working from a shared definition? In the discussion we saw great variation across our countries.
• How does centralization influence what teachers do?
• How is centralization of an educational system achieved? And who are the players?
• How might research influence policies aimed at centralizing mathematics teaching and learning?
• What are the implications of centralization policies for classroom practice? For example, what are the implications of approving only certain mathematics textbooks on instruction? On student achievement? How much autonomy do teachers actually have in a centralized educational system, and how is it exercised?
• Given that most mathematics teachers have considerable autonomy to make their own decisions regarding what happens in their classroom, even in settings with very strong regulations and mandates of their country/state/local agency, what are the implications for the development and nature of policies aimed at centralizing mathematics teaching and learning? Some centralization policies are conceptualized with the assumption that teachers maintain autonomy; what are their characteristics? What roles should mathematics teachers play in the establishment of policies aimed at centralizing mathematics instruction (i.e. development of a national curriculum and national exam)?
• What is the underlying view of mathematics teachers and their competence/efficacy in highly centralized and in decentralized educational systems? What assumptions about teaching and learning are being made?
• To what extent is mathematics instruction influenced by mandated examinations?

The nature of standards

• What are the implications of content goals that are so highly specified that they are measurable?
• Can standards allow for focus on "knowledge with understanding" and also be specific enough to define assessment expectations?
• What is the dynamic of moving to high standards or to more inclusive standards in terms of student access and learning?

The nature and purpose of math instruction

• What mathematics should students learn? Why should students learn certain mathematical content? Education through mathematics and education for mathematics: what are the implications of each?
• How much freedom is there for content choices?
• What is the role of technology in learning mathematics?
• What are the implications of massification of mathematics education?

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