Learning hierarchies

Much of today s instruction, either explicitly or implicitly, has its organizational roots in learning hierarchies, as put forth many years ago by Robert Gagne (1970). Learning hierarchies have been extraordinarily valuable to us as researchers and teachers because they help us to understand at a very detailed level the many subordinate skills and prerequisites of a task. However, they have less value as models of learning, and they often fail when used as guides for instructional development. 

The premise behind the learning hierarchy is that one can analyze a task into its many constituent parts, building a hierarchy of skills and establishing prerequisites for each one. The general form of a learning hierarchy contains several levels, ordered from top to bottom in terms of their prerequisites. To achieve the skill at the topmost level, one must be able to do all the subordinate skills at the level immediately below it. To achieve the skills at the next level down, one must first master its necessary subordinate skills. And so on. One ends up with a tree structure that has the desired skill at the top and as many branches as needed to represent subordinate skills. At the bottom of the hierarchy are those elements for which there are no prerequisites. Instruction aimed at the target skill (the task at the highest level) commences with the elements at the lowest level of the hierarchy, and the skills at each level are addressed in order such that instruction about the prerequisites always precedes instruction about any skill. Eventually, all necessary prerequisites will be satisfied. 

One of the features of the learning hierarchy is that by definition the elements at the bottom of the hierarchy are independent. That is, each may stand alone and does not depend upon the others. If this were not the case, one or more would serve as a prereq-... 

Finally, the structure of the hierarchy itself is at odds with schema theory. The schema is a network of related knowledge, not a branching tree with distinct levels. A great deal more connectivity exists in the schema than in the learning hierarchy, and instruction based on schemas focuses directly on these connections. 

Developing learning hierarchies is but one of several ways that instruction may be carried out. Consider for a moment how one... 

Lay out everything according to some measure of difficulty, then start with the easiest and move to the most difficult. This strategy builds directly on the principles of task analysis that underlie learning hierarchies. The approach has some advantages, but the danger is that the most difficult topics will never get covered because earlier topics take more time than expected. Moreover, it obscures the fact that difficulty is not a unidimensional characteristic. What is difficult for one student may not be difficult for another. This approach is seen quite often in elementary school textbooks. 

As individuals acquire skills, they develop rules that govern these skills. Recall the learning hierarchy described in chapter 4, and consider that the acquisition of the hierarchy corresponds roughly to the formation of the appropriate sets of rules for carry-... 

Knowledge can be represented by a hierarchy if some pieces of it are prerequisites for others. One imagines a tree structure, with various branches, as in Figure 10.2. This representation is reminiscent of Gagne s learning hierarchy described in chapter 4. The elements at the end of each branch are the individual skills required to carry out the composite skill formed at their juncture. A hierarchy would exist, for example, for the case in which one wished to evaluate a composite skill, SI, that was composed of two separate elements, El and E2, as shown in Figure 10.2. 

As far as an external aesthetic is concerned, we do have two important clues to help guide us (1) chaos theory, from which we learn that natural processes that appear complicated can often be well understood using relatively simple rules, and (2) complex systems theory, from which wc learn that interesting phenomena often emerge on higher levels from parts that are mutually interacting on the lower levels of a hierarchy. 

Novak, J. D. (2002). Meaningful learning The essential factor for conceptual change in limited or inappropriate propositional hierarchies leading to empowerment of learners. Science Education, 86, 548-571. 

The solution to the learning problem should provide the flexibility to search for the model in increasingly larger spaces, as the inadequacy of the smaller spaces to approximate well the given data are proved. This immediately calls for a hierarchy in the space of functions. Vapnik (1982) has introduced the notion of structure as an infinite ladder of finitedimensional nested subspaces ... 

Materials Characterization. Regarding education in the characterization or analysis of materials—a central topic of materials chemistry—there is a similar hierarchy of importance of subjects that chemistry students (and faculty) will need to have learned. Reference 7 treats this topic systematically, and Roy and Newnham (11) presented a comprehensive (albeit somewhat outdated) presentation of the architecture of materials characterization. Thus Rutherford backscattering and extended X-ray absorption fine structure (EXAFS) are excellent characterization research tools, but in the sequence of tools used every day on every sample, they are insignificant. Thus for structural characterization, X-ray powder diffraction reigns supreme, yet the full power of the modern automated search routines that can be universally applied are taught only to a minuscule fraction of even the materials science student body. 

Moody, J., Fast learning in multi-resolution hierarchies," Research Report, Yale University, YALEU/DCS/RR-681 (1989). 

Learning on social responsibility needs to be embedded at ah levels within the organic supply chain (producers, processors, exporters, wholesalers, retailers) and certification hierarchy (standard-setting organisations, certification bodies, farmer organisations, farmers). Ideally the standards and regulations hsted under a hierarchy are - temporary... 

As noted by Baer et al. [197, 198] important lessons concerning nanoscopic engineering may be learned from biology. Just as the hierarchical structures in bio-composite systems exhibit (a) various scaling levels (dimensions) (b) possess very specific connectivities (interactions) between these levels and (c) have appropriate architectures to manifest a desired spectrum of functional properties (see Fig. 51), so can one envision a hierarchy of abiotic, nanoscopic complexity. 

It is attractive to imagine that all we have to do to ensure learning in students is to analyze the task we wish to teach, which is a top-down process, and then teach the various pieces of the hierarchy, which is a bottom-up process. Under this model of instruction, we start with the elements in the lowest boxes at the bottom of the hierarchy, provide instruction on all elements at this level, move up a level and do the same thing, and continue until we reach the top of the tree. 

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