International Panel on Policy and Practice
in Mathematics Education: 2001 Report
Issue 6: Excellence and Access
How does your country and culture deal with the challenges of excellence and accessibility in mathematics education? What is the balance of power and input into the system between the various educational constituencies?
Sweden: Gerd Brandell and Susanne Gennow
Before explaining how we deal with excellence and accessibility in mathematics education, I would like to describe some aspects of the larger school system. As shown in the table below, the Swedish school system consists of pre-school, compulsory basic school, and non-compulsory school (See Table 7 below).
Table 7. School System in Sweden
In Sweden, national goals govern all subjects in Compulsory School and all courses in Upper Secondary School. These national goals are defined according to two types of objectives-aspiration objectives and objectives that must be achieved. While aspiration objectives describe the qualities of pupil attainment that the schools must seek to develop, objectives, which must be achieved, are those all pupils must be provided with the opportunity to attain. These objectives frame efforts to address issues of excellence and accessibility in the Swedish education system.
Mathematics Education in Sweden
A primary method of promoting accessibility in mathematics education involves requiring that all students in compulsory school take mathematics. By the end of year nine, students have completed a total of 900 hours (60 minutes at a time) of mathematics education. A primary method of promoting excellence involves setting standards and defining goals for student achievement. The standards that relate to mathematics education set goals to attain in years five and nine. Progress toward the achievement of these goals is measured by national tests. While the national test that students take in year five is optional, in year nine students are required to take the Compulsory National Test. The results of this test influence admission to Upper Secondary School.
Students are eligible for upper secondary school in a national or local program if they attain the degree goals set for Swedish, English, and mathematics. Ninety percent of all students attained that eligibility in 1999. This number is steadily decreasing, though. It was less than 90 percent in 2000. Despite this, 98 percent of students entered upper secondary school in 1999/2000. This 8 percent difference is due to policies designed to increase access to upper secondary education. This policy makes it possible for students who fail to attain the degree goals, to still enter upper secondary school. Students who gain access in this manner follow an individual program where they complete their basic education and in parallel take courses within a national or local program.
Once admitted to upper secondary school, students enroll in a program consisting of various subject courses. There are 17 national programs and a great number of local programs. Three out of the 17 national programs prepare students for further studies. The others are vocational, but at the same time provide general eligibility for higher studies. This provision of general eligibility is another way that the Swedish system increases access-in this case, access to higher studies.
Access to mathematics in upper secondary school is increased by the policy that makes Mathematics A- the first course in mathematics-compulsory for all students regardless of program. Upon completion of this course, students take the Compulsory National Test in Mathematics A, then proceed through their program. The natural science program is one of the three national programs that prepare students for further studies. Students who enter this program take the most mathematics. There are seven mathematics courses available. Mathematics A, B, C, and D are compulsory. Mathematics E, Discrete Mathematics and Extension Mathematics are optional (See http://www.skolverket.se/pdf/english/natsci.pdf for details).
In each course in the degree program, students receive grades that show the extent to which pupils have achieved the objectives set down for each subject. Grades are given in three steps: Pass, Pass with distinction, and Pass with special distinction. The criteria are decided on a national level for each grade. Upon completion of upper secondary studies, students receive a leaving certificate that partially fulfills the requirements for entrance into higher education. By 2000, 73 percent of the students who entered in 1996 earned a leaving certificate (most of them after three years). In order to enter higher education students must pass at least 90 percent of subject courses in the degree program. In 2000, 80 percent of students with leaving certificates fulfilled the requirements for higher education and were eligible to enter universities. This means that approximately 57 percent of that age group (entering upper secondary in 1996) attained general eligibility for higher education before the age of 20 years. This figure demonstrates the outcome of Sweden's policies for increasing accessibility.
Strengths, Weaknesses and Problems
The primary strength of the Swedish approach to accessibility is that there are no insurmountable impasses in the system. The compulsory components in Compulsory Basic School and Upper Secondary School ensure access to a relatively high level of mathematics for "all." Alternative options for admission to Upper Secondary School increase access for students who do not fit the standard mold, and Municipal Adult Education gives everybody opportunities to study courses in mathematics at the upper secondary level (or compulsory level) and complete their education. For students who are interested in furthering their mathematics education, it is always possible to choose more mathematics courses.
The primary strength of the approach to excellence is that there are two types of standards. The aspiration objectives that frame teaching help to minimize tendencies to translate minimum expectations for competency into maximum expectations for achievement. The objectives that must be achieved send a message to teachers, students, parents, and society that all students are capable of learning at a high level. Coupled with the lack of insurmountable impasses, the Swedish approach recognizes and respects diversity in pace and circumstance and gives students opportunities to soar as high as their motivation can take them.
The primary weakness of the Swedish approach to accessibility is that efforts to promote accessibility divert some resources from promoting excellence. While there are new programs aimed at enriching the experiences of gifted and strong students, there is concern that some very gifted students do not get appropriate material or challenges. Research indicates that very gifted students do not learn much during compulsory school, especially during years seven through nine. As a result, there is concern that we may lose some potential science students who choose other programs in upper secondary because they do not receive adequate stimulation in science subjects during years seven through nine. Even among those who chose to enter science, there is a great deal of concern that by the age of 18, when they leave upper secondary school, they may have developed their ability much less than could have been possible if they experienced more challenges earlier.
The primary weakness in the Swedish approach to excellence is that despite our efforts, many students do not attain the goals. In the compulsory school 5-10 percent of students do not pass when finishing compulsory school in year nine. In upper secondary school, approximately 30 percent of students fail the Compulsory National Test for Mathematics A. These figures suggest a need to adjust our approach, especially at the upper secondary level, so that more of the students who have access to higher level mathematics also achieve a higher level of excellence.
The primary problem with the Swedish approach relates to student performance on assessments. The general tendency is that students entering the upper secondary or the tertiary level do not perform well on different diagnostic entrance tests. This is a general problem experienced at many schools and universities and may also apply to subjects other than mathematics. If 98 percent of students leaving compulsory school and 80 percent of students with leaving certificates are able to pass 90 percent of their courses, poor performance on diagnostic entrance exams suggests that there may be a problem somewhere in the system. Whether this is a problem within the curriculum, with teaching, with the assessments, or the result of a combination of these is a topic for research. As this problem is not specific to Sweden, we are interested in learning what people in other countries do about it.
I have interpreted the topic for this introduction as a general issue, not related to one specific problem within mathematics education. In one of the examples I will touch upon the question of excellence and accessibility in mathematics education, the other topic for the session. The question could be reformulated as "How are power and influence distributed among various authorities, professional groups, and other identifiable groups on the development of the school system and specifically on mathematics education?" The basis for an understanding of this matter is knowledge about the steering system and it's functioning.
The steering of the school system in Sweden is based upon the principles of a goal-result-oriented steering process. The central steering is realized through national goals for the education. The results of the implementation in teaching and learning are evaluated and estimated through a system of national assessments and evaluations. The means to attain the goals are on the other hand in principle delegated to the local level (i.e. the local municipalities). There are several arguments for such a decentralized system. The main one is the conviction that an effective system requires that decisions be made by those who have close knowledge about the conditions in every specific case. At the same time this steering system guarantees equity on a national level through centrally defined goals and national evaluations. Hence the system ideally combines equity on a national level with local power to shape the content of schooling.
There is a general consensus among most involved - for instance all political parties - about the principles and ideal functioning of the steering system. But it is apparent that there are many obstacles for an ideal functioning of the system.
Description of the influencing bodies and parties
A comprehensive description of the steering system and other bodies that exert influence upon mathematics education would take up too much time and leave no room for problemizing and discussion. Therefore, I will build my presentation upon a general structure into which the various "constituencies" fit. The structure is presented below. In Appendix F the authorities, bodies and lobby-groups are shortly described according to the structure. Examples of importance for mathematics education are given in the next section. In a closing section I will then discuss strengths and weaknesses of the system related to the general question about influence and power. Finally I will put forward some questions for discussion.
The three sectors are the following:
The borders between the three groups are not completely clear. Teacher education is for instance closely related to the academic world in some respects, but more to the school system in others. Within each of three sectors I distinguish between two levels:
The structure is tentative and does not work in all cases. The list presented in the sections to follow is not complete. The intention is to give a general idea of what kind of "constituencies" there are (See Appendix F).
Examples of processes and reforms, illustration of input and power
Reformed teacher education
The reformed teacher education will be starting in 2001/2002 and fully implemented by 2005. A simplified and very short analysis of the process is interesting as it shows the complex pattern of influencing groups active during the process. The reformed teacher education was preceded by not less than three investigations during the years 1994 - 1995. These were initiated at different levels, one by the Parliament, one by the National Agency for Higher Education, one by the teachers union. All investigations were initiated on the presumption that the quality of teacher education needed to be improved, but the efforts proceeded from very different perspectives. Some conclusions pointed in opposing directions. One main goal for the government, the parliament, and other groups has been to integrate teacher education more fully into the academic world and connect more closely to research in education and didactics. Another goal is to connect closely to the profession as a teacher and give high priority to teaching practice. These goals are not easy to adjust. There also exists a historically-rooted discrepancy between teacher education for younger and older children that creates many problems for the united structure desired by the government and some other groups but far from all.
Hence, the group commissioned in 1997 by the government to work out a new teacher education structure had a very delicate task. They presented a new structure in a 1999 report. The suggested model was criticized by many parties during the procedure preceding the government decision, mostly from universities and from representatives for the existing teacher education programs. Mathematicians and mathematics educators were worried about lowering entrance requirements. They also saw a risk for a reduction of the mathematics content within different options of the programme and possibly also of mathematics education. Finally the decision taken by the parliament in 2000 was very close to the original suggestion and showed no evident revisions due to the critic from the mathematics community. However, the structure leaves great freedom for universities to shape the programs.
One interpretation of the outcome is the following. The issue was very controversial. In order to satisfy several parties the decision did not really settle all of the controversial questions. The conflicts were thereby moved to the local level where the solution will depend on the relative strength of different interests within each university and among other local or regional influencing groups. On the other hand, a united structure is now adopted, which was one main political goal for the government; teaching practice has been strengthened and possibilities for universities to demand higher scientific quality of the degree paper have been created. In this sense a compromise has been reached.
The outcome of the future implementation is very difficult to predict. Several of the universities had great difficulties to negotiate among themselves and make a decision on the implementation. Some university administrations found this so difficult they even demanded an extra year for preparing their programs, a demand that was not accepted by the ministry.
A revision of curricula of mathematics at compulsory and non-compulsory school took place in 2000. It was adopted in 2000 and will be fully implemented in 2002/2003. I will not go into the whole revision but only choose a couple of questions related to the mathematics syllabus for upper secondary level. In the last big reform in 1994 the school system at upper secondary level was unified, and 17 national programs introduced. Almost all Swedish young persons (96 percent of an age cohort) attend one of the programs offered at upper secondary level - whether it is vocational or primarily preparing for further studies. There is a common core for all programs.
One basic course in mathematics, called mathematics A, is part of the common core. Many mathematics teachers find it difficult for such a diverse group of students to take a common course at this level (at 16 years of age). The student group is diverse in different respects: in aptitude, interest, mathematical competence, self-confidence and beliefs about mathematics. For some Mathematics A is their final mathematics course, for others it ought to create the basis for a series of other mathematics courses, i.e. in the science program. There are insufficient results among students at some programs on the national tests on mathematics A, with an extremely high failure rate. Therefore, many teachers felt a need for a revision of this course, as it was not challenging enough for students in the science program and at the same time difficult to grasp for some of the students in vocational programs.
One suggestion put forward to the National Agency for Education from teachers was to introduce two or three different courses, each designed for a group of programs. Another was to let some vocational programs leave out the mathematics A course and instead integrate mathematics into the other subjects. From several different parts of the mathematics community there was a manifest support for the idea of revising the Mathematics A course in some way. Other constituencies, such as some universities and professional organizations, also supported the idea.
The problem with the Mathematics A course touches upon the issue of excellence and accessibility. Mathematics A is required for access to higher studies. The same is the case with the rest of the common core. The current system gives anyone who has successfully completed a program (any program) at upper secondary level the right to enter some university courses. If Mathematics A was not compulsory it would be difficult to uphold this system, and students would have to supplement their skills before being admitted at university. This is a political issue of great ideological importance for the government (social democrat) and neither the minister nor the Agency for Education were willing to listen to the suggestions about a revised course A.
Another issue was more successful for the mathematics community. In the reformed program starting in 2000 a new branch of the science program is introduced, namely a mathematics-computer science branch. It is one of three options, the others being a "science-science"-branch (emphasis on mathematics, physics, chemistry and biology) and an environmental branch (emphasis on chemistry, biology and ecology). The computer science/mathematics branch was introduced during the process due to input from the mathematics community and some universities. This illustrates the other side of the accessibility/excellence coin. The new branch will give students a new possibility to specialize in mathematics already at the upper secondary level. There was no considerable opposition towards this idea, a sign of the fact that most parties realize the growing importance of some students getting a more advanced mathematics competence at this level.
Other examples of recent reforms and initiatives are the following. In each case it is possible to make a similar analysis of the influence of several different constituencies.
In both cases the mathematical community, especially some university departments, the ICME-SE, and the NCM have been very active promoting investigations and putting forward suggestions.
Strengths, Weaknesses, Opportunities and Obstacles
The steering system is consistent with the principles of a goal-result-oriented steering process. The central steering of schools takes place by national goals and national assessment that guarantee equity on a national level while the means to reach those goals are in principle delegated to the local level with close knowledge about the conditions in every specific case. There is a general consensus among most involved about the principles and ideal functioning of the steering system. There is much room for creative initiatives at every level. Teachers have great responsibility and freedom.
Some schools have a special teacher appointed by the head to act as a leader of the group of mathematics teachers and to support colleagues with a competence development program. If the group of teachers work well together and have support from the head and the municipality, these schools have every chance of succeeding with their mathematics program.
Municipalities are responsible for the competence development of teachers, and there is an agreement on the extent in time every year. This agreement leaves room for competence development in mathematics and mathematics education that can also take the form of sharing ideas with teachers at other schools.
There is no systematic evaluation of curriculum reforms and of current pedagogical practice. If this was established much of the input from different bodies could be handled in a positive way through such a process. There is no systematic influence from teachers or other parts of the system on textbooks and no feedback or control of the pedagogic quality and the agreement with the curriculum and syllabuses in textbooks other than those initiated by authors and publishers. There are no national guidelines or manuals for school mathematics, which may or may not be a weakness.
A strong municipality may challenge the system and stretch the limits further than the government actually is prepared to accept. Such conflicts exist between Stockholm (with a liberal majority) and the social democrat government. (These controversies may be also viewed as a strength!).
The great majority of schools lack teachers with qualifications as researchers in mathematics or mathematics education. For upper secondary schools this should be the case according to the law, but schools do not succeed in recruiting these teachers, and many heads probably do not give priority to the problem.
Mathematics is often associated with natural science in general, and programs for enhancing mathematics, science and technology are based on what scientists and engineers find useful. Very little input from university mathematicians finds its way into the system. For decades university mathematicians have kept out of the debate about school mathematics with few exceptions.
Most big mathematics departments now support the idea of developing graduate programs for research education in mathematics, specializing in mathematics education. Eight departments at different universities will take part in a new research education program within the national graduate school starting later this year. This marks a new sense of responsibility and readiness to get involved with school mathematics from university mathematicians.
Stockholm (the largest municipality) and several other municipalities are prepared to support teachers substantially to get training as researchers or participate in a masters program in mathematics or in mathematics education (or in other subject areas).
Subject area didactics tend to attract relatively more interest from politicians and from some important funding bodies compared to general education that has until now heavily represented research related to schools and education.
Since 1991 teachers in public school no longer are employed by the state but by the municipalities (or by private bodies running schools). Many teachers still deeply regret this reform and find their conditions deteriorated and their influence weakened. There is a serious lack of confidence between teachers and school politicians in many municipalities.
The informal division of the ministry into two parts - each represented by a minister - hampers reforms with common and simultaneous action in school and universities. There is also a marked discrepancy between the goals expressed by the school-side and the university-side of the ministry. The two agencies represent different cultures. The national agency for education is dominated by general educationalists with a view on skills and knowledge that differs in some respects from the demands of the universities and the view that is dominating in the national agency for higher education.
No expertise in mathematics or mathematics education is to be found on a regular basis in the ministry or the national agencies. Personnel representing such competency may be recruited to a position in any of the agencies, but in those cases, it often happens more or less by chance. Most municipalities also lack specific expertise in mathematics education on a central level.
Almost no bodies or networks exist that cover pre-school and school or school and university. This is both a local and a national problem. The barriers between the different levels are very real and have a negative influence on the system. The creation of ICMI-SE is an effort to address this lack of a common place for discussion.
Many schools do not have access to a qualified and experienced teacher who may act as a driving force at the local level. Many heads of schools are not sensitive to the need of such a teacher.
Participants identified the similarities and differences represented in Table 8.
Table 8 Sweden: Similarities and Differences Among Participating Countries with Respect to Issues of Excellence and Access
Themes That Arose from the Discussion
Mathematics is compulsory at primary level in all countries and at secondary level in many countries. Many countries have also implemented "second-chance" and, where relevant, financial support policies designed to promote access to secondary mathematics. Countries have large differences in terms of percentages of people going through the system. Some countries are considering how to reduce streaming/tracking practices that limited access to secondary mathematics . A very relevant issue seemed to be the focus of the system as a whole: What does the system steer against or aim at - passing a final examination, achieving entrance to the next level, or educating the whole population?
"A lot of research has been done on grouping. One result is that grouping does not make any difference if the teaching, contents, materials, etc., are not improved." (Brandell)
"Tracking is unfamiliar to Japanese society, which has an emphasis on equality. In school system there is no tracking. In classrooms at all levels, students are heterogeneously grouped." (Hashimoto)
"In the U.S., there were earlier systems with different mathematics courses for different student groups. For instance shop classes did some mathematics that was mainly reinforcing arithmetic and not introducing new concepts. This system is now largely abandoned because it meant very small possibilities for these students to continue at the next level." (Eddins)
"Teachers have different goals with their teaching depending on their views but also on the examination system. In some systems the teacher finds his/her most important goal to be that all students pass an exam or reach a certain level; in others to help as many as possible to pass some entrance exam; in others that some/many students reach high results." (Brandell)
Theme 2: Challenges to increasing access
Efforts to promote access led to different outcomes in different nations. The success of policies for promoting access to secondary mathematics for all students is tied to variations in participation at the secondary level. In some countries, such as the U.S., Japan, and Sweden, participation is very high. This is not the case in countries such as Kenya, India, Egypt, and Brazil. Also, policies that tie secondary school admission to examination results present challenges to promoting access. The demographics of students who sit for exams may not reflect the demographics of the population. As a result, increases in access may not be observed among students from particular ethnic groups or socioeconomic classes. With the exception of Sweden and Japan, in nations where admission tests are used, these variations significantly influence the outcomes of efforts to promote access without reducing expectations for achievement.
In addition, rates of failure vary considerably across demographic groups. Students dropping out to work or because of repeated failure was a common concern.
"In Kenya many talented students are lost by poverty; the family cannot afford the schooling." (Shikuku)
"Any country wants to have excellence in any profession; similarly any country would like to have good mathematicians. At the same time, they have a concern for all." (Agarkar)
"It used to be that students going on to secondary education would begin different kinds of study; those going to university more formal courses; students intending vocational areas, or not strong in arithmetic - might review arithmetic, apply it, in shop classes, consumer classes. This was a concern, because it meant very different opportunities for students who did not do well. Students were put into a track (streaming) that was very difficult to move from. If students began in vocational track, they couldn't move to university track. Too often, these decisions were not necessarily made on academic grounds, and instead students from minority groups, low socioeconomic status, etc. were put into the low tracks. But, like France, we have very strong commitment to equity and equal opportunity for all." (Eddins)
Theme 3: Challenges to promoting excellence
Both across and within participating nations there are significant differences in approaches to defining and assessing excellence. In some of the participating countries no special effort has been made to define excellence. In others, there are ongoing debates over the meaning of excellence, how it can best be achieved, and how it is best measured.
"Most systems are very eager to handle the accessibility problem. Few or no systems can cope with the excellence problem. Most teachers are not capable of addressing both questions inside the classroom. It is difficult to reach both the low attaining and the talented students in a classroom teaching situation involving 30 students or more." (Shirali)
"In Brazil, it would be inappropriate to stress excellence too much when access to education is such a problem. Ten percent of the population holds 90 percent of wealth, and many students leave the system in order to work. Many are in poor health or have other problems related to poverty. It is more important to focus on addressing this than on these talent issues. Excellency is only for the middle and upper class. If we stress excellence too much, there will be higher rates of failure and even more students would drop out." (Lins)
"We have new attention to applications, uses of mathematics, use of situations. This is accepted by some schools, not by others. Have found that many students thought not to be talented in math had not found the mathematics being taught worthwhile. Some of them are now doing very well. Some of the students who did well when all we taught were algorithms aren't doing as well because now they have to think more deeply. Still a turmoil - there are disagreements about this 'integrated' approach. Mathematicians worry about whether there is enough preparation." (Eddins)
Theme 4: Special provisions for struggling and talented students
In some of the participating countries, addressing the special needs of struggling and talented students is the responsibility of the family, not the schools. In these countries, although there may be some special programs or schools, most students who receive extra help or enrichment obtain it during private lessons from tutors. In countries where addressing these needs is primarily the responsibility of the schools, many challenges exist in identifying these students and addressing their needs. In most of these nations, exams results are used as a guide. This is relatively effective in identifying students having academic difficulty but is often less effective in identifying talented students. Some students who could perform well do not gain access to special classes, and some students who gain access do not perform as well as expected. Also, some students who gain access do not have experiences that encourage them to pursue higher education in math and science fields. These outcomes raise questions about both the identification process and the content of the classes.
The very existence of these programs has conflicted with views on equity in many of the participating countries, and the concerns raised by identification processes have exacerbated these concerns. In many of the countries, special programs for struggling students have been implemented and maintained, but many programs for talented students have been abandoned for equity reasons. There are many concerns about how to provide for students who have the potential to become scientists but emerge from the regular system unmotivated and/or unprepared to do so.
"In Sweden the experience from the mathematics special class in upper secondary is that tests are not a perfect way of identifying the students. We miss some students this way." (Gennow)
"In France, there is an experience of finding the "school-talented" by using tests, but excellence is not the same as well-achieving. Quite often when a test is given, students who can do procedures are identified, but later we learn they aren't as creative as we expected. Talented pupils should be identified based on more input, such as interviews, than based on just an aptitude test or performance in mathematics." (Sackur)
"Previously, the talented students were accelerated. Most recently, we have realized this didn't solve the problem; students would take as much math as required to get into universities; two or three years." (Eddins)
"In Brazil, as well as in the U.S., there is a focus on applications and 'every-day-mathematics' as a means to interest the weaker students." (Lins)
"In compulsory school, talented students may become bored and lose interest. Many are not challenged enough in grades 7 through 9." (Gennow)
Key questions about access and excellence include those related to:
Policy and Practice
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