International Panel on Policy and Practice
in Mathematics Education: 2001 Report

Issue 3: Content in the Mathematics Curriculum

Describe the role of algebra in the middle and secondary mathematics curriculum in your country. Similarly, how are ideas from probability and statistics currently configured in your system?

Kenya: George Eshiwani and Beatrice Shikuku

Opening Statements

George Eshiwani
Kenyatta University

Education in Kenya consists of one to two years of pre-primary education (ages 4-6), eight years of primary education (ages 6-14), four years of secondary education (ages 14-19), and four years of university education leading to a bachelors degree. Two major examinations are set by the National Examinations Council; the Kenya Certificate of Primary Education (which takes place at the end of primary school), and the Kenya Certificate of Secondary Education (which takes place at the end of secondary school). Each exam determines whether or not students progress to the next level of education.

In Kenya the curriculum is controlled by the Kenyan Institute of Education, which draws its representation from a wide range of teachers and experts from universities. The secondary school mathematics syllabus is very demanding on the majority of students, many of whom find certain topics extremely difficult to comprehend. Algebra, as well as probability and statistics, are included in the Kenyan curriculum. In the first year of secondary mathematics education, algebra content emphasizes coordinates and graphs and simplifying expressions. In the second year, students encounter linear equations; quadratic expressions and equations; linear inequalities; and basic statistics11. In the third year of secondary mathematics education, students continue their work with quadratic expressions and equations and are introduced to binomial expansion, matrices, sequences and series, and probability. In the fourth and final year of secondary mathematics education, students study matrix transformations, statistics involving variance and standard deviation, time series and trends, and indexed numbers including weighted averages. In addition to these topics, students in the fourth year also study linear programming, differentiation, and integration.

A number of examples taken from the secondary examination illustrate some of the difficulties students have with various concepts in Algebra, Probability, and Statistics.

Two examples are:

1) Simplify

Although this item was targeted towards average students, 80% of students received a score of 0, and 10% received the maximum score of 3.

2) The volume of an object is given by . Express c in terms of p, r, s, and V.

Although this item was targeted towards average students, 80% of students received a score of 0, and 5% received the maximum score of 3. To receive 1 point (which was attained by 3% of students), the students needed to arrive at the following:

Although the curriculum is well spelled out in the policy statements, student performance was nowhere near expectations. Reasons for this range from poor teaching to acute shortages of mathematics textbooks.

Beatrice Shikuku
Booker Academy

In Kenya about 10% of the children like and are willing to study math. The rest have to be persuaded or forced to study mathematics because it is compulsory. They have a completely negative attitude towards the subject (especially the girls), and therefore, as a classroom teacher I find that teaching mathematics in Kenya has been, and still is, an uphill task. The main reason for these problems is that up to the late 1970s, nobody chose to go to the university to study education as a profession. The good mathematics students studied engineering, medicine, accounting, or any other course but teaching. Many of those who failed to meet the minimum requirements for their preferred careers became teachers. Such mathematics teachers tended to scare the learners to cover up their lack of content knowledge and their inadequate preparation to teach the lessons.

Children seem to find the learning of mathematics difficult and painful. It took a very bright and brave child to accept the pain and learn mathematics. It was even worse for girls as they often could not withstand the fear. The situation got even worse in the 1980s as those educated in this manner became the educators. In addition, many students came to school having heard horror stories about mathematics learning from their parents. These factors gave mathematics a monstrous face, and to date, we are still trying to change this image to one with a more friendly face.

As one who has willingly chosen to be a math teacher, I have gone out of my way to address my colleagues in an effort to convince them to put a smile on our subject. Since I am also an examiner at the national level, I meet over 400 mathematics teachers once every year, and I believe we are making a breakthrough. We are making mathematics smile. I have also gone out of my way to talk to parents and convince them that mathematics is not as bad as they thought but that it was taught badly. I tell them that the ugly monster of their day is no more, and that it is now very enjoyable to learn mathematics in many schools.

I have made mathematics very popular by eliminating punishment to the slow learning and taking more time with them instead. I always wear a smile on my face, try to be fair but firm, and ensure that they complete their homework. I use real life situations and objects that can be seen and touched during the lessons. As a result, I have registered 100% pass rates in my class, and this has given me the courage to visit other schools in the country and speak to students on how to make mathematics very easy.

In the early eighties the government restructured the mathematics syllabus, which previously had options to take care of varied potentials in mathematics. Now there is a common syllabus for all. Allowing different options of mathematics had a very negative effect on learners who ended up with the option considered to be for weak students, while encouraging arrogance in those who took the option for stronger students. This arrogance developed at an early stage in life and unfortunately spilled over into the teaching of mathematics by those students who ended up being mathematics teachers. The common syllabus used now is appropriately designed to take care of students with different potential. While there are many textbooks, they have the same basic content. The Kenyan Institute of Education approves books, and teachers have a vote in approval. These steps have helped to improve the image of the subject. Many people now appreciate the value of mathematics.

Also, since March of 2001, the Kenyan government has banned corporal punishment in schools. This should go a long way toward making mathematics acceptable and, therefore, easy to teach. The good news is that in the year 2000, only 12% failed mathematics at the KSCE (secondary) level. Four years ago the failure rate was 38%. This is a great improvement.

Participants identified the similarities and differences represented in Table 4.

Table 4 Kenya: Similarities and Differences Among Participating Countries with Respect to Content in the Mathematics Curriculum

Country Similarities to Kenya Differences from Kenya
Brazil Teacher knowledge strongly influences curriculum enactment

Curriculum content is similar

Statistics content involves descriptive statistics and some data analysis

Similar problems with teachers and teaching approach
Different syllabi for different students. Math is not for everyone.

Many believe that math is boring and there is nothing teachers can do about that.
Egypt Algebra is central in secondary curriculum

Problem is with teaching approach, not with the curriculum
 
France All students take algebra

Gap between expectation and student performance on exams
 
India Algebra is central in secondary curriculum, which includes statistics

Students perform poorly on data analysis items

Instruction is calculation based, not interpretation based
Most students perform adequately on assessments of algebra, statistics, and probability knowledge
Japan Algebra is central in secondary curriculum, probability is taught to all students, and most high school students take calculus

Teacher knowledge strongly influences curriculum enactment

Effort to connect math to real life; instruction is focused on interpretation and meaning

Students have a negative attitude towards math
Few students study statistics

Different curricula for students based on college plans. Non college-intending and non-science students are required to take fewer core courses.
Sweden All students take algebra Most students have a positive attitude towards math (Half to three-fourths say that they like and value math)
United States Algebra is central in secondary curriculum, and statistics and probability are becoming more popular

Effort to connect math to real life

Some curricula are strongly interpretation-based

Teacher knowledge strongly influences curriculum enactment

Gap between expectation and student performance on exams
Different curricula for students according to ability. Not all take algebra

Statistics and probability not central in secondary curriculum

Calculus not taught to all students

Access to texts is a problem in some rural and inner-city areas

According to TIMSS, students value math and say that they like it

Students do not do as well as expected on exams

Observer Commentary

Hiroshi Fujita
The Research Institute of Educational Development, Tokai University

About algebra in the secondary school:

  1. Algebra is the main transition from numeracy to mathematics.
  2. Algebra is the core language of mathematics. Because of this, many other topics need knowledge of algebra (e.g., probability, geometry).
  3. Algebra is a basic component of mathematical literacy.

About the content of algebra in the secondary school:

  1. The content is rather stable over time and across countries.
  2. The concept of "algebra for all" depends on the enrollment ratio of the age cohort to secondary education. In Japan and in the U.S., the enrollment ratio to secondary education is almost 90 percent, so it is impossible to institute algebra for all overnight.
  3. The concept of "core algebra" in high school makes sense.

My understanding of the content of algebra:

  • Number systems (rational numbers, irrationals, complex)
  • Use of symbols
  • Symbolic manipulation of expressions (expansion, factorization up to binomials)
  • Binary relations (=, <, >, . . .) binary equations
  • Unitary operations ( x -> |x|, 1/x, xn, . . .)
  • Sequential operations

Hyman Bass
University of Michigan

We talked about algebra, statistics, data, but most of our conversation was about algebra. One issue was "what is algebra," and another was "why do we teach algebra?" Lins said that up to algebra, we teach math of the street, but algebra is not the math of the street. Hiroshi made an interesting assertion with which I am not sure I agree. He said algebra for all is maybe idealistic in countries where all students are in secondary education. Is algebra essential enough to teach all students? I believe that it is both relevant and possible to do so. But we will have to see over the next few decades.

The curricula of the different countries were remarkably similar except for the U.S. Here algebra has been reinterpreted. The portrait of algebra from most of the countries is like what Professor Fujita put on the overhead. In the U.S., there is a movement to give greater emphasis to the notion of function as the unifying concept. Function has a distinguished history in mathematics. Felix Klein said this at the turn of the century. Lins said it doesn't matter. He is right-this is a matter of convention and definitions.

Recall the video when Shea spoke of the number 6 being odd. The tension in that discussion derived from the fact that kids were using the same term in different ways. It became necessary to reconcile the different definitions. Like those students, we better have a common definition for what we mean by "algebra." We should recognize that we mean different things by "algebra." We might need more terms.

The rationale is that in the modern world, people are inundated with data and have to process quantitative information, and so some notion of patterns, functions is appropriate in the school curriculum. To do this takes a lot of time, time that is taken away from a traditional emphasis on skills and notation. Rational expressions are the expressions needed in calculus. If students enter the university and cannot work with them (the expressions), it will be a problem. But we need a more intrinsic rationale. Many students are not going to go on for more mathematical study.

I think we should ask what the algebra of the street is. Why do we require kids to study so much mathematics for 12 years? Many of us are convinced that mathematics enables people to function more rationally as adults, but it is difficult to specify this in ways that would define a curriculum. It is hard to make a direct connection to everyday life. But without this, adults are impoverished in everyday life. For example, in algebra one learns to name things. If you wrote a novel you would not call all the characters he and she and he and she. The act of naming things, giving them a compressed expression, is central to algebra so that you can manipulate them more efficiently, and communicate more clearly.

We tend to think of learning mathematics as learning some particular topics, but mathematics offers learning beyond its specific topics. I am not prepared to argue that some particular topic has to be in the curriculum. Lists of topics are easy to make. The impediments to children learning these lists is that there are things to know to do, certain habits of mind that mathematicians do. But these things are not explicitly taught. When instructors ask students to prove something, this is a highly developed skill, and we assume that children can just do it. Mathematical practices are just as important as topics, and we don't teach them except abruptly. We need to make mathematics practices an integral part of the curriculum, even in the early grades. Algebra is a subject where many of these practices arise.

Themes That Emerged From the Discussion
Theme 1: The mathematical content of the curriculum

A central theme that arose during this session was what considerations are important in making decisions about mathematics content in the school curriculum and syllabus. In particular, the group focused on algebra, statistics, and probability and attempted to spell out the purposes for requiring students to learn these subjects. Some important factors to be considered in selecting mathematics content include helping students develop certain "life skills" that will enable them to become informed citizens, empowering them to study further mathematics, and preparing students for examinations. Discussions also highlighted the tension between learning mathematical skills just for the sake of skills versus learning mathematical skills for the purpose of applying them to situations and contexts.

Illustrative quotes

"In the U.S. we have the same questions about the content of algebra as that described by the Kenyan curriculum. We have been trying to move the foundation of some of the algebraic ideas down to primary grades. The new Standards push ideas about patterns down to second grade. We have been more successful in developing data and statistics than we have with probability. Our students don't understand it. Except for simple things, they have trouble. We separate combinations and counting problems away from probability." (Burrill)

"In India the algebra content is pretty good. There is more than what is expected from a traditional curriculum. Most schools have average students comfortable with algebra, but for below average students algebra is one of the great stumbling blocks. Algebra, logs, and proofs are the three big stumbling blocks for below average students. Word problems are also difficult. Those preparing for the competitive exams have high level of skills. Student performance in statistics and probability is very good because it is algebraic in nature. In data analysis their performance is poor. Their computations are not based on valid interpretations-not on what things actually mean. Another area of weakness is the visualization in the curriculum (e.g. reading and interpreting graphs, and using guide maps)." (Shirali)

"Many adults find algebra relatively useless and don't see what it is good for. Our parents and our public don't see the value." (Burrill)

Theme 2: : Assessment/examinations and inadequate student performance

National exams were seen as having a strong influence on teaching. In many countries, teachers' effectiveness is based entirely on how well their students do on exams. A common concern was that the exams do not necessarily reflect student understanding or their true levels of achievement.

Illustrative quotes

"The results of exams especially in developing countries might not have anything to do with actual level of student achievement." (Mina)

"In France there is a gap between official expectations of students and results. We cannot tell what students are really able to do. Exam results very often do not mean anything. When we try to assess true learning, we also find gaps, but most exam items relate to skill testing. Where do you try to learn to teach students how to use the skills in resolution of situations? We should teach basic skills, but students should learn to use them in context." (Bodin)

Theme 3: Algebra - its purpose, content

In general, algebra as it is reflected in the syllabi of the various countries was similar across the countries. Algebra tends to include equations, expressions, binomials, and complex numbers, and serves as a tool for modeling, and expressing relationships in symbolic forms. In most countries, function and related concepts are not considered as part of algebra but rather of analysis.

Illustrative quotes

"Many people said they had some of the same topics in your curriculum. Algebra is a very big subject. For example, Hashimoto said that they treat cubic equations in Japan. What about polynomial functions of various degrees - linear, quadratic, and higher order? Are general theorems about polynomial functions included - for example, the binomial theorem? Most of you made reference to exponential and log. Do you have the natural ex? Suppose we only consider 2x? This is introduced first for an integer variable x. Is it treated when x is a continuous variable and, if so, how is it done? What is the boundary of this subject in the curriculum?" (Bass)

Theme 4: Motivating students to learn mathematics

Different strategies have been used across our countries to motivate students to learn mathematics. High stakes national examinations that have serious consequences for students are one means. Sometimes in the past, punishment has been used. Making mathematics interesting, meaningful, and useful to students was seen as a way to motivate their learning.

Key Questions

Key questions that arose during the discussion related to:

The nature of the content

  • Should students learn mathematical skills just for the sake of skills? Or should the emphasis be on applications to the real world?
  • Given the power of technology today, what mathematics is important for students to learn? What mathematics should become obsolete?
  • Can algebra be thought of both as basic skills and also basic skills that can be used in context?
  • What level of manipulation of symbols is necessary in algebra, and what is the role of technology in general?
  • What is really important for students to learn and know in algebra?
  • What are effective strategies for determining the similarities and differences in the algebra (or probability or statistics) content that students are taught in different countries? Is it enough to compare lists of topics? To compare examination items?

Relationship between policy and content

  • How do educational policies send messages about what mathematical content is valued? And how is this translated into practice?
  • How can educational policies effectively address equity issues, such as accessibility to technology, that impact mathematics instruction?
  • Have the purposes of learning algebra been adequately established and is there consensus (among policy makers, teachers, the public) about this? Why is algebra in the mathematics curriculum?
  • Why is algebra a big stumbling block for below-average students?
  • Do assessments actually reflect what students learn?


11 Basic statistics involves the use and interpretation of data organization tools such as histograms and pictograms, and the calculation and interpretation of measures of central tendency.


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