1
“Spiral Curriculum”
American schools follow a “spiral curriculum” in mathematics; that is, they spend such a substantial proportion of time on review each year that only limited progress can be made with new material… American students who perform poorly in arithmetic are subject to a special form of the spiral curriculum, which might be termed the “circular curriculum”: they repeat arithmetic over and over until they stop studying math” (Gamoran, 2001, p. 138)
Gamoran, A. (2001). Beyond curriculum wars: Content and understanding in mathematics. In T. Loveless, Ed., The Great Curriculum Debate, pp. 134-162. Washington, D.C.: Brookings Institution Press.
The Process of Education 1
The process of education
The Process of Education (1960) was a landmark text. It had a direct impact on policy formation in the United States and influenced the thinking and orientation of a wide group of teachers and scholars, Its view of children as active problem-solvers who are ready to explore 'difficult' subjects while being out of step with the dominant view in education at that time, struck a chord with many. 'It was a surprise', Jerome Bruner was later to write (in the preface to the 1977 edition), that a book expressing so structuralist a view of knowledge and so intuitionist an approach to the process of knowing should attract so much attention in America, where empiricism had long been the dominant voice and 'learning theory' its amplifier' (ibid.: vii).
Four key themes emerge out of the work around The Process of Education (1960: 11-16):
The role of structure in learning and how it may be made central in teaching. The approach taken should be a practical one. 'The teaching and learning of structure, rather than simply the mastery of facts and techniques, is at the center of the classic problem of transfer... If earlier learning is to render later learning easier, it must do so by providing a general picture in terms of which the relations between things encountered earlier and later are made as clear as possible' (ibid.: 12).
Readiness for learning. Here the argument is that schools have wasted a great deal of people's time by postponing the teaching of important areas because they are deemed 'too difficult'.
We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development. (ibid.: 33)
This notion underpins the idea of the spiral curriculum - 'A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them' (ibid.: 13).
Intuitive and analytical thinking. Intuition ('the intellectual technique of arriving and plausible but tentative formulations without going through the analytical steps by which such formulations would be found to be valid or invalid conclusions' ibid.: 13) is a much neglected but essential feature of productive thinking. Here Bruner notes how experts in different fields appear 'to leap intuitively into a decision or to a solution to a problem' (ibid.: 62) - a phenomenon that Donald Schön was to explore some years later - and looked to how teachers and schools might create the conditions for intuition to flourish.
Motives for learning. 'Ideally', Jerome Bruner writes, interest in the material to be learned is the best stimulus to learning, rather than such external goals as grades or later competitive advantage' (ibid.: 14). In an age of increasing spectatorship, 'motives for learning must be kept from going passive... they must be based as much as possible upon the arousal of interest in what there is be learned, and they must be kept broad and diverse in expression' (ibid.: 80).
Bruner was to write two 'postscripts' to The Process of Education: Towards a theory of instruction (1966) and The Relevance of Education (1971). In these books Bruner 'put forth his evolving ideas about the ways in which instruction actually affects the mental models of the world that students construct, elaborate on and transform' (Gardner 2001: 93). In the first book the various essays deal with matters such as patterns of growth, the will to learn, and on making and judging (including some helpful material around evaluation). Two essays are of particular interest - his reflections on MACOS (see above), and his 'notes on a theory of instruction'. The latter essay makes the case for taking into account questions of predisposition, structure, sequence, and reinforcement in preparing curricula and programmes. He makes the case for education as a knowledge-getting process:
To instruct someone... is not a matter of getting him to commit results to mind. Rather, it is to teach him to participate in the process that makes possible the establishment of knowledge. We teach a subject not to produce little living libraries on that subject, but rather to get a student to think mathematically for himself, to consider matters as an historian does, to take part in the process of knowledge-getting. Knowing is a process not a product. (1966: 72)
The essays in The Relevance of Education (1971) apply his theories to infant development.
Bruner 1
Jerome Bruner (1915-).
Jerome Bruner was born in U.S.A and his influence on teaching has been important. He was possibly the leading proponent of discovery approach in mathematical education although he was not the inventor of the concept (Romiszowski.,A.J.,1997).
Bruner describes the general learning process in the following manner. First the child finds in his manipulation of the materials regularities that correspond with intuitive regularities it has already come to understand. According to Bruner the child finds some sort of match between what it is doing in the outside world and some models or templates that it has already grasped intellectually. For Bruner it is seldom something outside the learner that is discovered. Instead, the discovery involves an internal reorganisation of previously known ideas in order to establish a better fit between those ideas and regularities of an encounter to which the learner has had to accommodate.
His approach was characterised by three stages which he calls enactive, iconic and symbolic and are solidly based on the developmental psychology of Jean Piaget. The first, the enactive level, is where the child manipulate materials directly. Then he proceed to the iconic level, where he deals with mental images of objects but does not manipulate them directly. At last he moves to the symbolic level, where he is strictly manipulating symbols and no longer mental images or objects. The optimum learning process should according to Bruner go through these stages.
1. Enactive mode. When dealing with the enactive mode, one is using some known aspects of reality without using words or imagination. Therefore, it involves representing the past events through making motor responses. It involves manly in knowing how to do something; it involves series of actions that are right for achieving some result e.g. Driving a car, skiing, tying a knot.
2. Iconic Mode. This mode deals with the internal imagery, were the knowledge is characterised by a set of images that stand for the concept. The iconic representation depends on visual or other sensory association and is principally defined by perceptual organisation and techniques for economically transforming perceptions into meaning for the individual.
3. Symbolic mode. Through life one is always adding to the resources to the symbolic mode of representation of thought. This representation is based upon an abstract, discretionary and flexible thought. It allows one to deal with what might be and what might not, and is a major tool in reflective thinking. This mode is illustrative of a person’s competence to consider propositions rather than objects, to give ideas a hierarchical structure and to consider alternative possibilities in a combinatorial fashion, (Spencer.K.,1991, p.185-187).
The association of these ideas of manipulations of actual materials as a part of developmental model and the Socraterian notion of learning as internal reorganisation into a learning by discovery approach is the unique contribution of Bruner (Romiszowski.,A.J.1997, p.23).
In 1960, Bruner (then a professor of Harvard University) proposed a “spiral curriculum” concept to facilitate structuring a curriculum ´around the great issues, principles, and values that a society deems worthy of the continual concern of its members´ (Bruner, 1960). The next decades many school system educators attempted to implement this concept into their curriculum. Bruner (1975) described the principles behind the spiral curriculum in the following way:
”…I was struck by the fact that successful efforts to teach highly structured bodies of knowledge like mathematics, physical sciences, and even the field of history often took the form of metaphoric spiral in which at some simple level a set of ideas or operations were introduced in a rather intuitive way and, once mastered in that spirit, were then revisited and reconstrued in a more formal or operational way, then being connected with other knowledge, the mastery at this stage then being carried one step higher to a new level of formal or operational rigour and to a broader level of abstraction and comprehensiveness. The end stage of this process was eventual mastery of the connexity and structure of a large body of knowledge”…(p.3-4).
It was in the 1980s, that a body of literature had accumulated in support of individual components of a spiral curriculum model. Reigeluth and Stein (1983) published the seminal work on “ The Elaboration Theory of Instruction”. It proposes that when structuring a course, it should be organised in a simple-to-complex, general-to-detailed, abstract-to-concrete manner. Another principle is that one should follow learning prerequisite sequence, it is applied to individual lessons within a course. In order for a student to develop from simple to more complex lessons, certain prerequisite knowledge and skills must first be mastered. This prerequisite sequencing provides linkages between each lesson as student spirals upwards in a course of a study. As new knowledge and skills are introduced in a subsequent lessons, they reinforce what is already learnt and become related to previously learned information. What the student gradually achieves is a rich breadth and depth of information that is not normally developed in curricula where each topic is discrete and disconnected from each other (Dowding, T.J. 1993).
Bruner suggested that cognitive process precede perception rather than the other way around, that a person may not perceive an object until he or she has recognised it. These cognitive theories of perception emphasise the role of knowledge in how we interpret the world.
Howard Gardner (1987,p.6) defined cognitive science as “a contemporary, empirically based effort to answer long-standing epistemological questions- particularly those concerned with the nature of knowledge, its components, its sources, its development, and its deployment. ”The theories of the constructivist are originated from this school of thought.
The beginning of the 1950s and maintaining through the 1990s, educators drew on rising insight of communications specialists, learning theories, and systems engineers. The 1990s have been marked by the challenge of constructivism.
Two Types of Curriculum 1
Two Types of Curriculum
There are two types of curriculum widely used today. Spiral curriculum is where a wide number of topics are taught in the early grades. The topics are cycled throughout the years, developing deeper understanding through the later grades. The United States uses spiral curriculum. Mastery curriculum covers a smaller set of topics and focuses on students' deep understanding of each topic. Students who are in a mastery curriculum program score higher than those who are in a spiral curriculum (school reform). Our country might need to lean toward mastery curriculum if they want to keep up with other countries. This shift might also cause a decline in student boredom. It seems possible that a solution to boredom could be an alteration to mastery curriculum. Mastery curriculum could help the problem of teacher burn out, which could be a cause of student boredom.
A movement to mastery curriculum might help with teacher burn out as well as boredom. If the curriculum focused on a more centered set of topics, there would be less repetition on the part of the teacher and the students. The students would learn what was needed, and move on. There would be no going back in later years to dig deeper. That would have been done initially. The students would not be wasting their time relearning, and the teachers would focus on a smaller set of topics. Mastery curriculum seems like a good way to eliminate teacher burn out, which causes student boredom.
Changing something as drastic as the curriculum from spiral to mastery may also force changes in other problem areas, such as teacher burn out and dropout levels. If students are given a smaller range of things to study, looking at them in greater depth, they would probably be more inclined to stay in school, simply by eliminating boredom. Changing a curriculum to a more specialized one would also focus the teachers' attentions, allowing them to implement more exciting and engaging projects. The mastery curriculum could be taught with project-based learning and still work effectively. These changes needed to make high schools a better place might not be effective, but reducing boredom is certainly a step in the right direction.
Things don’t add up… 1
Things don't add up in B.C. math classes
By Bill Hook and Karin Litzcke
Vancouver Sun
Editorial Section, Issues & Ideas Page
Friday March 04, 2005
Reading and math are the two crucial elementary school subjects required for high school and life beyond, but British Columbia's elementary math curriculum is crippling learning, especially among disadvantaged students.
B.C. has used what is called a "spiral" curriculum since 1987, following a tradition of emulating U.S. educational practice.
A spiral curriculum runs a smorgasbord of math topics by students each year, the idea being that they pick up a little more of each with every pass. In reality, the spin leaves many students and teachers in the dust.
Ideally, the curriculum should cover fewer topics per year in more depth.
Presently, teachers face having Grade 4 classes who still cannot add 567 + 942 nor multiply 7 x 8 because the Grade 1, 2, and 3 teachers were forced to spend so much time on graphing, polygons and circles, estimating quantity and size, geometrical transformations, 2D and 3D geometry and other material not required to make the next step, which is 732 x 34.
And because elementary math fails to provide a solid foundation, many basically capable students simply give up when faced with the shock of high school algebra, which would be the doorway to advanced technical training at all levels. High school math teachers cannot make up Grades 1 to 7 while teaching Grade 8.
Alarm bells about the math curriculum have been ringing in B.C. since the United States, which used spiralling almost exclusively, registered a dismal performance on the Third International Mathematics and Science Study (TIMSS), a test that comparatively evaluated more than 500,000 students from 15,000 schools in 40 countries, first in 1995 and again in 1999 with the same results.
The B.C. ministry of education, to its credit, realized right away in 1995 that the U.S. performance on TIMSS suggested weaknesses in B.C.'s curriculum.
Also aware of some then-emerging data indicating that students in Quebec -- which had retained a sequential curriculum when B.C. went to the spiral -- were outperforming other Canadian students in math, Victoria commissioned researcher Helen Raptis, now a University of Victoria professor, to compare B.C. and Quebec test results and curricula.
In her report, submitted to the ministry in late 2000, Raptis showed that the average B.C. student was more than two years behind the average Quebec student in math by Grade 10, and explored the extent to which curriculum might be responsible.
Her report did not flatter B.C.'s curriculum, reading in part:
"The range of skills and operations within a specific topic area is deeper in Quebec, moving constantly between the abstract and concrete properties of mathematics concepts and maintaining a place for mental as well as rote processes.
"The B.C. curriculum is inconsistent in its treatment of abstract and concrete concepts.
"Objectives and notes throughout Quebec's curricula highlight the view that mathematics learning is interrelated and cumulative.
"These conscious links are not evident in B.C.'s mathematics curricula. Instead, learning objectives from prior years are repeated outright."
In 2002, the U.S. National Research Center for TIMSS published similar conclusions, finding that the curricula of virtually all the U.S. states had too many topics that were introduced too early, repeated too often, and covered too superficially.
The U.S. TIMSS report noted, too, that the spiral curriculum "favoured the children of well-off or sophisticated parents who could provide supplementary tutoring, and was terribly unfair to the disadvantaged. The learning of the luckier students snowballs while that of the less fortunate ones -- those dependent on the incoherent U.S. curriculum -- never begins to gather momentum."
To date neither the Raptis nor the TIMSS reports have generated any action in B.C., but California, also alarmed by the 1995 U.S. performance on TIMSS, implemented a redesigned curriculum with far fewer topics per year in 1998.