Paradigms extended: how to integrate behaviorism, constructivism, knowledge domain, and learner mastery in instructional design

Abstract

The pedagogical paradigms of Direct instruction (behaviorism/objectivism) and Constructivism are often seen as opposing paradigms at the ends of an instructional design continuum. Unfortunately, this view makes the two approaches mutually exclusive. Designers must use the one at the expense of the other. A previous study proposed that the two approaches should be integrated along two axes, producing a matrix of four quadrants of learning, called “injection”, “immersion”, “construction” and “integration”. This model maps directly onto the Cynefin framework of knowledge management that contains four knowledge domains, “known”, “chaos”, “complex” and “knowable”. The combined models allow for a mix of direct, or constructivist teaching and learning. The problem is that the model addresses only two elements of the pedagogical triangle of teacher, content, and learner. It integrates knowledge domains (learning material) with direct and generative approaches to teaching but is silent on the needs of learners. In this paper we propose an extension of the model to add learner mastery to the domains of knowledge, and teaching and learning methods to extend the current decision matrix by adding the learning curve. We consider the metrics for deriving a learning curve, and then fit the required- or expected level of competency and the learner’s level of general competency to the learning curve. We then map the curve to the four quadrants to establish when a particular quadrant is appropriate at a specific phase of the learning curve. This improved model may assist teachers, (lecturers, instructional designers) and learners to select appropriate pedagogical paradigms and methods, based on the domain of knowledge and learners’ mastery of learning outcomes. Finally, we propose a sequential approach to selecting learning paradigms, based on how much of the material has been mastered.

7. Appendix: The mathematics

This section describes the logic behind the development of the learning curve and is presented as additional information.

External view of learning process

Central to the first phase was the development of tools to assist in the measurement and collection of data. As it is currently nearly impossible to directly measure the changes that occur in the brain during learning, alternative measures need to be developed. It is a well observed fact that the brain responds perceivably different to different ‘strengths’ of stimuli (Weber, 1834). According to the works of Weber and Fechner described as the Weber-Fechner Law (Fechner, 1907, 2012),

$$P=K\log I$$

where $P$ is the perceived magnitude, $K$ is Weber’s constant and $I$, the stimulus impulse or strength.

Based on work of Weber and Fechner, it was decided to measure the brain’s response in how it well it reacts when confronted with solving tasks involving the stimulus, the new knowledge component (both knowledge and skill). Four assignments, tasks involving “separable” knowledge components, were selected from the course, Examples 1–4, and the students’ performance on these tasks measured. This involved Time-on-Task (ToT) measurements to reach a preset proficiency. The Time-on-Task measures were measured for each student individually for the four different knowledge components (Examples 1–4): two tasks that are closely related and may be presented in any order (Example 1 and 2) at the start of the course; two tasks that need to be presented in order (Example 3 and 4) presented in the latter half of the course.

The total ToT measures were taken over specified periods; for the delivery of the new content, Tuition phase (T₀–T₁) the time spent to complete the delivery of theoretical components and Demonstration phase (T₁–T₂) the time spent to complete the demonstrations of practical applications of theory, the Reproduction phase (T₂–T₃) the time spent by the students before they are able to demonstrate their ability to duplicate results obtained in the demonstration phase, the Application phase (T₃–T₄) the time spent by the students before they are able to demonstrate their ability to apply the new knowledge component, and finally, the students enter into the Expert phase at time T₄, the phase where students are ready to be assessed for mastery of the knowledge component under review. The Time-on-Task (ToT) measures, T₀–T_1–T₂–T_3–T₄, are presented as colored bars at the bottom of for the different phases of teaching and learning.

The ToT measures were taken for each student individually and averaged for all the students in the study. summarizes the averaged ToT measures for a combination of different time intervals for the four tasks. It is represented as the time that elapsed since the start or introduction (T₀) of the specific knowledge component and the set outcome level is reached (T₂, T₃ or T₄), as an example the Tuition and Demonstration phase (T₀—T₂), for each of the Examples 1–4. From this data, it was observed that there was a plateau in the learning curve (Aylward, 2018) (Table 5).

Table 5 Time-on-Task measures

Using regression and the data from the learning curve can be constructed using two interchangeable models:

Model 1:

$${\text{P}}_{{1}} (t) = \alpha (1 - e^{\beta t} ) + (1 - \alpha )\left( {\frac{{e^{(mt + c)} }}{{1 + e^{{\left( {mt + c} \right)}} }}} \right)\;with\;0 < \alpha ,\;\beta < 1.$$

Model 2:

$${\text{P}}_{2} (t) = \alpha (1 - e^{ - \beta t} ) + \left( {1 - \alpha } \right)(1 - e^{{ - \eta (e^{\gamma t - 1} )}} )\;with\;0 < \alpha ,\beta ,\gamma ,\eta < 1$$

where $\alpha$ represents the level of the plateau. The plateau represents the fundamental knowledge needed to understand the knowledge component, its purpose and function. This is a percentage of the total knowledge to be gained for the whole knowledge component to be mastered, usually $\alpha =0.1$ of the total (1) but could also be $\alpha =1$ for the exponential learning curve. β represents the learning rate for the first phase and γ the learning rate for the second phase. $\eta$ represents the time when the student reaches the Expert phase at time T₄ (Figs. 15, 16, 17). The formulas produce similar results and describe a whole family of curves that models different rates of progression for mastery of knowledge components. This includes the exponential learning curve (Top left–black curve) through to the newly defined learning curve with the plateau (Bottom right–red curve). The learner’s rate of progression (R) is a function of the learner/student/trainee themselves (l), the environment (e) and the current knowledge-skill level (k)–R is thus a function the current belief and value system within a set environment: $R=f\left(l,e,k\right).$ This relates back to the driving and opposing forces to learning presented in Fig. 3 (Table 5).

Fig. 15

External view of learning curve (Extened from Aylward et al. 2019)

The different teaching and learning intervals, T₀–T_1–T₂–T_3–T₄, have been introduced above and the measured results are presented in a graphical format. Both graphs, black and red, are derived from either model equations. Black represents the case where the material is apparent, easy, and thus quickly mastered without going through the plateau phase–the standard exponential learning curve. The red graph represents learning that first goes through the plateau phase, slow progress or learning–I don’t know what I’m doing or even supposed to do phase, before rapidly mastering the material, the ah-ha I’ve got it phase. There exist a whole family of curves ranging between these two curves. These curves describe the external view of learning progression.

Internal view of learning process

In consolidating the various levels of mastery and the learning curve as discussed above Aylward proposes an eight-level taxonomy of mastery that can be fitted to the learning curve (Table 3). This is an expansion of the levels defined and measured above to reflect the difference between theoretical practice and real-world practical application, thus adding three levels to the proposed 5 level structure of Dreyfus & Dreyfus (1980, p. 50). The proposed levels relate problem difficulty (well defined, broadly defined and ill defined), competence level and confidence level as compiled in. This level descriptor resonates with the Council of Europe’s (Council of Europe, n.d.) which proposes a six-level description of mastery of language, ranging from basic user, through independent user to proficient user. Each of the three categories has an upper and lower range. Ours, however includes a level 0 (unaware) and a level 7, which goes beyond proficient, bringing it to eight.

The eight stages as presented in are now plotted onto learning curve as is shown in for the various stages of mastery.

Fig. 16

Knowledge-skill level

It was further observed by Aylward et al. (2019) that stimuli in close proximity to subsidiary knowledge, Fig. 7, had a much shorter interval, Example 2 in, than larger stimuli, more difficult concepts, Examples 3 & 4. During the evaluation of mastery of all knowledge components at the end of the semester it was found that many students inexplicably struggled, even with knowledge components already mastered. To explain this observation wider literature study was required to build a robust measure.

Stevens expanded on the works of Weber and Fechner to derive a log-based scale (Stevens, 1957, 1961):

$$\log b-\log a=\log c -\log b=\log d-\log c$$

Fig. 17

Logarithmic interval scale

It was observed, (Aylward, 2018) that the total time needed to master the knowledge components also showed a logarithmic relation. Following the Weber-Fechner Law by taking the log of the ToT measures of, we arrive at a linear measure for the perceived difficulty, forming the base measure for the perceived difficulty, as proposed by Steven,and used by Aylward to measure the perceived difficulty experienced by students when confronted by learning stimuli.

Fig. 18

Log scale calculations

Now we can make statements such as: one knowledge component is twice or half as difficult when compared to another knowledge component. Similar results were also found in the way the brain responds to stimuli in other species (“Numerosity Representations in Crows Obey the Weber-Fechner Law,” 2006) leading the researchers to believe that the brain is hardwired in the way it responds to stimuli when confronted with different levels of stimuli. Known knowledge can be masked by ‘large’ stimuli, unfamiliar question or style of question, during evaluation of knowledge or skill obtained.

Using the measures developed and the literature surveyed a new internal view of the learning process was proposed by Aylward (2018). The Dreyfus brothers defined five stages (levels) of learning: novice, advanced beginner, Competent performer, proficiency and expertise (Dreyfus & Dreyfus, 1980, p. 50). Based on the works of Dreyfus & Dreyfus, Aylward et al. expanded the different stages into levels of mastery including three additional levels in order to distinguish between purely theoretical and application-oriented mastery. These and other learning competency level descriptors are overlaid on the learning curve. In Fig. 6 the four internal phases of learning are shown, including the mastery levels overlay. We call these four phases phases: Introduction–first contact with subject matter; Germination–development of fundamental understanding through observation and duplication—behaviorism (objectivism); Digest–integration of new knowledge into believe system—scaffolding; Regenerate—generation of new recipes, procedures and algorithms. This is true for both the mastery of a single knowledge component or the mastery of a wider field knowledge, i.e. ranging from preschool through to the highest tertiary level–spanning from simple mathematical concepts to advanced mathematics (Figs. 18, 19).

Fig. 19

Internal view of the learning process

In this section it was shown how a learning participant ‘moves’ between different levels of learning–the internal view of learning progression. The study points to the brain being hardwired in the way it responds to stimuli. If one understands this process one can develop measures to support learning in a more predictable way.