It is a clear educational vision that distinguishes top-performing countries from others, says researcher Donald MacNab, who argues that performance on international tests such as the Third International Math and Science Study (TIMSS) is closely linked to countries’ belief systems regarding the value and purpose of mathematics education.
There are no easily identifiable differences in curricula that can account for the large discrepancies in math performance between high- and low-performing countries, he says. He cautions against simplistic reactions and “quick-fix” strategies, warning that mandating curricula or methods that copy those of high-performing countries may not achieve the desired improvements. MacNab suggests that we must look deeper to discover the differences that affect our students’ learning and performance in mathematics. He believes that the reasons for the performance differences between high- and low-scoring countries almost certainly lie in deep-seated beliefs about the teaching and learning process. He urges societies to look at what they value in mathematics education.
Cultural beliefs influence education
Translating curricular objectives into pupil learning experiences is a complex process strongly influenced by a country’s cultural assumptions about teaching and learning, he says. These assumptions form a kind of hidden curriculum that educators, consciously or unconsciously, bring to teacher training, curriculum development and teaching. The curricula that are implemented by teachers and experienced by students may differ from the intended curricula because of these underlying beliefs. When a clear and well-understood educational vision is lacking, these largely unarticulated cultural assumptions and beliefs may lead to a less effective math curriculum that is detrimental to students’ learning, he says.
A study of English schools, for example, found that English schools are characterized by “the use of complex pedagogy, the complicated nature of the teaching role, [and] the multiplicity of goals — academic, social, behavioral, cultural — that can be pursued and that make a common mission difficult to achieve.” These complexities and multiple goals create an environment in which topics may be presented in insufficient depth or inadequately interwoven with related concepts. The sequencing of topics may not be developmentally appropriate, and the relationship between real-life contexts and the logical structure of mathematics may be inadequate for a full understanding of the underlying math principles.
Educators in the United States have drawn similar conclusions about U.S. schools from the disappointing TIMSS results. In the report “Attaining Excellence” American educators conclude that “there is no single, coherent, intellectual vision underlying our efforts” and that “fragmented ideas, concepts, and skills are often weighted by tradition, inertia, false assumptions, and lack of an articulated intellectual conception that would help these pieces coalesce into a focused goal-directed action.” The report goes on to say that we should not blame teachers or schools, but must examine common beliefs and choices that shape what educators do.
Pitfalls of constructivism
MacNab states that the dominant theory of learning in Western cultures for the last 30 years is constructivism, the central message of which is that “children construct math meaning within their own internal schema and that external attempts to impose schema by the teachers are for the most part doomed to failure.” He believes the core problem of constructivism is that it is not easily implementable in typical school classrooms.
In contrast, MacNab asserts, high-scoring countries have developed a more effective and easily implementable vision within which the aims are prioritized and interrelated to produce effective teaching strategies. In Taiwanese schools, “the presence of simple pedagogies such as time, opportunity to learn and whole class instruction, combined with goal certainty and clarity, creates a system of homogeneous quality.” Studies all point to the clearer sense of purpose and organization driving mathematics education in high-scoring countries. High-scoring countries have developed a more powerful answer to the question, what is the purpose of mathematics education?
What is math education for?
The evidence from studies cited by MacNab suggests that a more productive approach is one that invites children to explore math ideas and processes from the combined perspective of understanding and use. This approach recognizes that functional use does not always or necessarily require full understanding, but that functional use can help develop understanding.
Although such an approach does not ignore the constructivist perspective or psychological aspects of how learning is attained, its underlying vision is less complex than that of constructivism. Its overriding purpose is to enable children to use mathematics. It places the teacher in her traditional role at the center of the learning process, charged with analyzing and structuring math into effective learning sequences.
The child explores, gains understanding, and develops the ability to use mathematics, but the child does not create or construct mathematics. In MacNab’s view, children should not necessarily be expected to be self-motivating or keenly interested in everything they do in class. The role of the teacher is to ensure that pupils are effectively involved in the activity so as to acquire and be able to apply appropriate mathematical knowledge and processes. The pupil must be cognitively engaged rather than passively receiving information. A key task of the teacher is to foster this active involvement.
A major ingredient of mathematics education in high-performing countries is pupils’ motivation to perform well. In the United Kingdom and the United States, our goal seems to be to ensure that students like mathematics. One of the characteristic features of late 20th century Western society and culture is the emphasis placed on liking and emotional engagement, the absence of which is seen as a major stumbling block to learning.
In contrast, MacNab states that the most important component is the active will to succeed. Mastery of mathematical ideas, concepts, and processes, and the ability to put them into practice, is what society needs from its mathematical education. Once that is accepted, matters of approach and teaching strategies can be analyzed with this single aim in mind. MacNab points to the California Department of Education’s introduction to its new math standards: “These standards are based on the premise that all students are capable of learning rigorous mathematics and learning it well, and all are capable of learning far more than is currently expected. Proficiency in mathematics is not an innate characteristic; it is achieved through persistence, effort, and practice on the part of students and rigorous and effective instruction on the part of teachers.”
Different responses to improving rankings
Western countries have differed in their responses to disappointing TIMSS results. Scotland has called for more whole-class teaching, more active involvement of students, less emphasis on calculators, more emphasis on mental arithmetic. In New Zealand, the response has been to identify factors responsible for poor performance such as poor understanding of math by teachers, classroom disturbance and bullying, lack of challenging learning materials and ineffective implementation of the intended curricula and assessment procedures. MacNab concludes that the heart of the problem is the lack of a single purpose to guide our intentions and focus our goals for mathematics education.
He suggests that countries answer a number of practical questions before they consider the goals of their mathematics education:
- What should be the balance between the cultural, intellectual and practical
aspects of mathematics?
- What degree of attention should be paid to developing students’
understanding of the nature and purpose of mathematics?
- How should the subject be related to other curricular areas?
- What should be the balance between the inductive and deductive aspects of
- What should be the balance between understanding of math processes and
- How should motivation be cultivated?
“Raising Standards in Mathematics Education: Values, Vision, and TIMSS” Educational Studies in Mathematics Volume 42, Number 1, 2000 Pp. 61-80.
Published in ERN February 2001 Volume 14 Number 1