Some people are just not ready to learn a given topic, others need a lot of help -- and with that help can make a good "go" of it -- and some "breeze through" easily.
|Vygotsky's Zone of Proximal Development (ZPD) which shows what has been mastered, what can be learned with some help ("scaffolding") from More Knowledgeable Others [MKO's], what would need much help to be learned, and what could not reasonably be learned in the current period even with much help|
That brings up the organisation -- the architecture -- of Benjamin Bloom's Mastery learning curriculum approach:
|Bloom's mastery learning approach uses small successive units [1 and 2], which have "formative assessment" modules [A and B] that allow for enrichment and corrective study, so that students master one stage before they move on to the next. This allows us to inject a degree of individualisation, interactivity, feedback and flexibility, but requires a considerable investment in developing the specific structure in light of the challenges that students may have and the opportunities for enrichment.|
It was brought to my attention some years ago by Stieg Mellin-Olsen, of Bergen University, that there are in current use two meanings of this word. These he distinguishes by calling them ‘relational understanding’ and ‘instrumental understand-ing’. By the former is meant what I have always meant by understand-ing, and probably most readers of this article: knowing both what to do and why. Instrumental understanding I would until recently not have regarded as understanding at all. It is what I have in the past described as ‘rules without reasons’, without realising that for many pupils and their teachers the possession of such a rule, and ability to use it, was
why they meant by ‘understanding’ . . .
|The Knowledge Dimension||The Cognitive Process Dimension|
|Meta-Cognitive Knowledge||Appropriate Use||Execute||Construct||Achieve||Action||Actualize|
|Even in "advanced" countries, only a fairly small proportion of the population achieve Piaget's formal operations stage of development, perhaps up to 30 - 35%|
According to Piaget, two major principles guide intellectual growth and biological development: adaptation and organization. For individuals to survive in an environment, they must adapt to physical and mental stimuli. Assimilation and accommodation are both part of the adaptation process. Piaget believed that human beings possess mental structures that assimilate external events, and convert them to fit their mental structures. Moreover, mental structures accommodate themselves to new, unusual, and constantly changing aspects of the external environment.
Piaget's second principle, organization, refers to the nature of these adaptive mental structures. He suggests that the mind is organized in complex and integrated ways. The simplest level is the schema, a mental representation of some physical or mental action that can be performed on an object, event, or phenomenon.
Piaget identified four stages in cognitive development:
- Sensorimotor stage (Infancy [~ 0 - 2 yrs]). In this period (which has 6 stages), intelligence is demonstrated through motor activity without the use of symbols. Knowledge of the world is limited (but developing) because its based on physical interactions / experiences. Children acquire object permanence at about 7 months of age (memory). Physical development (mobility) allows the child to begin developing new intellectual abilities. Some symbollic (language) abilities are developed at the end of this stage.
- Pre-operational stage (Toddler and Early Childhood [2 - 7 yrs]). In this period (which has two substages), intelligence is demonstrated through the use of symbols, language use matures, and memory and imagination are developed, but thinking is done in a nonlogical, nonreversable manner. Egocentric thinking predominates
- Concrete operational stage (Elementary and early adolescence [7 - 11 years]). In this stage (characterized by 7 types of conservation: number, length, liquid, mass, weight, area, volume), intelligence is demonstarted through logical and systematic manipulation of symbols related to concrete objects. Operational thinking develops (mental actions that are reversible). Egocentric thought diminishes.
- Formal operational stage (Adolescence and adulthood). In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts. Early in the period there is a return to egocentric thought. Only 35% of high school graduates in industrialized countries obtain formal operations; many people do not think formally during adulthood.
[W]ithout re-introducing general cognitive ability into the educational discourse, efforts at theorizing educational goals, setting standards and conceptualizing assessment frameworks cannot be successful . . . .
If you ask teachers anywhere in the world what counts to them as a ‘clever’ or ‘smart’ response from a student, you get a remarkably consistent sets of characteristics (Adey, 2007). High in the lists of common answers are a set of essentially convergent abilities (‘thinks logically’, ‘applies knowledge from one context to another’, ‘demonstrates deep understanding of a concept’) and also a set of generally divergent abilities (‘creative’, ‘asks surprising questions’, ‘goes beyond the given’). These responses explicate professional, intuitive, experienced-based conceptions of general ability . . . .
[P]ractitioners’ conceptions of general ability always include features which require that the student make some sort of connection in their mind between, for example, one context and another, actuality and possibilities, or between data items such that a pattern is perceived. The professionals who are in the business of teaching and learning therefore see connectivity as a central characteristic of smart behaviour. ‘Connectivity’ here is used in the simple sense of the conscious or unconscious making of connections in the mind between one idea and another. In this sense, making comparisons, extrapolating, relating causes to effects, or the elucidation of any relationship between variables all involve connectivity . . . .
[The ideas and findings of recent researchers] converge to a common denominator . . . From the point of view of development, it reﬂects the brain’s and thus the mind’s increasing ability to construct and use goal-relevant and appropriate mental structures and skills and the responsiveness of the developing brain to environmental stimuli. From the differential point of view, it reﬂects the fact that differences between individuals in plasticity will eventually demonstrate themselves as differences in actual intellectual constructions and achievements.
|Demetriou's model of developing conceptual schemes in domains of learning and capacity|
a: I can "see" with my mind's eye not only the three-dimensional tree trunk, but I can make it become coloured and transparent, i.e. I am moving from the projection to a three dimensional mental image. (Someone I discussed this with said how for her, it could "pop" out of the page for a moment, then it would collapse back into being simply lines on paper.)
b: I can see the three-dimensional model in a vague back and over the head position normally, but I can move it in front of me, rotate it, watch it grow or shrink, etc.
c: I can bring up other elements such as the Vygotsky window diagram or the Bloom taxonomy, and compare and make connexions. (But usually just to one or two such at a time. Beyond that, details get fuzzy, though I can zoom. I can suspend several fuzzy related diagrams then zoom in on each in succession.)
d: I can make up a mental movie, that seems to float in a chiaroscuro, black background world with vivid colours. I suspect the colours bear a suspicious resemblance to the palette of an LCD computer screen
e: I suspect that from this I can make a storyboard, and could either make a video animation, or could use the pattern to construct an interactive simulation, with help in the detailed coding.
f: As a related exercise, I think about how I struggled at first to understand how the inverting operational amplifier circuit worked. For this, I found it very helpful to get a sheet of paper and draw the diagram, then play out the calculation that yields the gain expression. That tells me a limit on my personal mental chalkboard. Here, courtesy Wiki [I feel too lazy to draw it this morning], is the inverter circuit, where the triangle is the amp, the lines are wires and the zig-zags are the resistors:
g: I then put the Bloom taxonomy next to it. Recalling and recognising and correctly placing the elements in the diagram was simple, and so was recalling that Ohm's law, the summing point restraints that say that the feedback network drives the voltage between the differential signal input terminals to zero, while the terminals do not draw any current. Here is a model of the idealised innards of the amp, also showing power supply terminals but leaving off offset and compensation terminals or the like:
h: I then saw that the derivation that moves from that to deriving the gain expression requires Bloom's analysis. That analysis requires significant algebra and ability to interpret electronic circuit symbols in light of a graphical-algebraic model. So, probably there were weaknesses there when I first met this diagram and the related analysis.
i: Going on to understand how to "take out" the idealisations requires evaluation of the weaknesses involved and a fair understanding of the innards of the op amp, which is usually based on some sort of long tail pair differential amplifier driving gain and power stages. (The linked diagram is for the classic 741.) In Integrated Circuit (IC) op amps -- and to follow the pinouts of the chip package is itself a little exercise in several levels of reasoning -- there are often current mirrors, which requires some understanding of IC creation techniques. The realities of these techniques drive the behaviour of real world as opposed to idealised op amps. (This in turn leads to the value of chopper stabilisation and other modifications that improve behaviour.)
j: I then could generalise the derivation, to deduce integrators, differentiators, logging and antilogging amps, filters, oscillators etc. Real world considerations then ground design of practical circuits that have to work in the practical world, e.g. I remember being puzzled at first by how we move from having a ratio of resistances etc to a good practical value from the various series of standard values. (As a couple of rules of thumb, a reasonably high resistance will not load input transducers or stages over-much, but too high a resistance brings to bear noise and parasitic filter effects that can undermine a design. If you want precision, count on having to pay for high quality, low tolerance components. Don't expect to find wound components that will be precise. But, there are op amp circuits that can simulate their behaviour; if you absolutely need such in an instrument.)
k: This shows me that capacities from earlier learning do make a difference to how one can progress in a new area, and that analysis and evaluation do often precede design or genuine creation. (Canned cookbook designs based on following the usual cookbooks or magazines etc don't count. Of course, all too often often cookbook designs may not work with the particular cluster of tolerances in a given circuit and it has to be fiddled with through debugging or troubleshooting.)
6 --> When our schemes are accurate and useful models we can say we have formed good and insightful understandings.
7 --> As a bonus, we have a handle on why models, simulations and theories are so important in science, and a glimpse onward at the significance of worldviews.
The creation of interactive supportive learning structures with the ability to adapt to the needs of individual learners, also emerges as a vital capacity for educational transformation. This points straight to the importance of the multimedia, web connected computer as a major educational instrument. Indeed, maybe many of the more knowledgeable others in the learning activity space will be coming to us digitally!