Introduction
The structure of information in a course is a crucial component for both student learning and teaching strategies. Teaching effort can be coordinated through a scaffolding approach that attempts to provide learning resources to the student and help them build an understanding of the concepts. Students achieve specific learning objectives or clusters of specific learning objectives (major concepts) in an order that is usually dictated by the course sequence and in relationship to what they already understand. This assumption, that students build on previous skills and knowledge, can be addressed through causal analysis. My previous work addressed how causal analysis can be facilitated by using databases and how this helps the instructor and student understand what a student has learned.
Causal analysis looks back at prerequisite skills and knowledge, but what we really need is the ability to understand the forward moving learning process. In my courses, the goal of learning is for students to be able to understand the concepts and give performances of that understanding. This is a highly individual process, with students starting at very different places and using the resources in different combinations and sequences.
Link back to papers on structure of courses. What did the data show?
This paper addresses the importance of structure in a course (or curriculum) by posing the question, "If structure is important then different structures should lead to both different patterns of learning and different learning outcomes." The structure of the course in this case will be the sequence and syncrhonicity of the presentation of concepts to students. The structure of this presentation will also be determined by how different concepts are presented to depend on previous or concurrent concepts. Additionally, idle periods can be important either positively (such as leaving time for ideas to sink in) or negatively (pushing a needed concept out of immediate recall).
Example Unit
For the sake of clarity, let's consider a concrete example consisting of five concepts that the students are supposed to learn over a four week period. This might constitute some portion of a unit in biology of environmental science.
The five concepts are presented in the following order and the instructor has assumed the following interrelationships between them (with respect to student learning):
| concept | order in interrelationship |
| A. the increase in organisms in any time period is related to the number of organisms (the more adult rabbits, the more baby rabbits will be born) | student expected to know this at the beginning |
| B. the increase of a population of rabbits will approach exponential growth if there are enough resources to sustain that growth |
these two both depend on the student knowing A they are presented together in the first lesson if they learn one they will can learn the other easily |
| C. as the population grows, the resources will be depleted | |
| D. resource depletion will cause a slowing of growth rate | this is presented in the second lesson and depends on the students knowing both concepts B and C |
| E. a population of rabbits will reach a carrying capacity depending on the level of resources | this is presented in the third lesson and the students need to understand concept C and remember concept A to put it into context |
I might have data on how students learned these concepts or a similar set of scaffolded concepts. There should be assessment data for the first several steps and problem set or quiz questions for the final steps. The initial student preparedness would be indicated on the pre-quiz. Even if I have student performance data, it is probably at the wrong time scale to directly use in this approach.
Abstraction of this into logic and sequence
The five concepts and their sequence can be represented abstractly using a graphic and logical statements.The flow of the concepts looks like figure 1.
The relationships can be simplified to logical statements based on a time sequence from time 0 to time 2 that represents two time steps.
Table 1
| understanding of concept | A | is high at time 1 | the students learned A before |
| understanding of concept | A | is high in time 2 and later | if the students consolidated that understanding by learning B or C in time 1 |
| understanding of concept | B | is high (any time) | if students just previously understood either A or C |
| understanding of concept | C | is high (any time) | if students just previously understood either A or B |
| understanding of concept | D | is high (any time) | if students just previously understood B or C |
| understanding of concept | F | is high at time 1 | if they already new E at time 0 |
| understanding of concept | F | is high in time 2 and later | if the students either already just understood E or they had just previously learned C and yet still remember A |
These logical statements will be translated and formalized below. Later the logic for what the students must do to remember concept A (in bold) will be changed slightly to address how structure is important to teaching and learning.
Modeling approach: Boolean NK networks with multistate links
The learning will be simulated using Boolean logic that connects these five components. The state of each node will be determined by logical statements that link not only to the previous time step but to time steps before that. In this model we will only link back two time steps.
This will be modeled in Excel, using Boolean logic operators AND, OR and NOT. For a description of setting up a Boolean NK network in Excel please see this reference.
Constructing a Boolean NK Model
http://web.pdx.edu/~rueterj/esr202/unit3/nk.htm
The logic statements from Table 1 are given below in Excel format.
Table 2: Excel format for logic statements.
Results and comparison to change in logic
The logic for learning of concepts A through E can be simplified into the following equations. This logic is different than normal Boolean NK models because the outcome of a state can depend on the state two time steps back. For example, leaning concept E depends on whether the student didn't know C two time steps ago, but learned it last time step. In addition, this multistate logic makes the progression through states depend on how you got to a particular state.
| Logic 1 | ||
| initial | time1 | time2 and beyond |
| A0 | A1=A0 | *A2= B1OR C1 |
| B0 | B1 = A0 or C0 | same |
| C0 | C1 = A0 or B0 | same |
| D0 | D1= B0 and C0 | same |
| E0 | E1 = E0 | E2 = E1 or (notC0 and C1)and A1 |
A convenient short hand for denoting combinations of A,B,C,D and E with values of either 1 or 0 is to write a five digit binary number and then express that as a base ten number. For example, A=1, B=0, C=0, D=0 and E=1 becomes 10001base two and 17 base ten. This allows us to look at the progression of states, starting with all possible states of student preparedness and learning for concepts A through E.
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The point of analyzing these learning states is to determine how students progress through the different states and where they eventually end up, i.e. simulate what combinations of concepts they learned. The goal of this teaching unit is to have all students learn all of the concepts up to and including the most difficult concept "E". I have simplified the possible starting points in this example by leaving out any initial state that represents the student already knowing concept "E". Since E is the last digit in the binary code, that means that I have left out all of the odd-numbered initial starting states. In addition, this short hand makes it easy to see when a student acquired concept "E", the first occurrence of a negative state value. In the initial logic and sequence that is presented here:
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The importance of the structure and sequence is apparent if we compare this first logic "Logic 1" to a slightly modified "Logic 2". Logic 2 differs only in that the concepts are presented in a manner that requires the students to acquire concepts B and C in order to maintain A. For example, they might have to be able to apply A to both situations to remember the fundamental concept and not get it confused with either B or C.
Logic 2 initial time1 time2 and beyond A0 A1=A0 *A2= B1 AND C1 B0 B1 = A0 or C0 same C0 C1 = A0 or B0 same D0 D1= B0 and C0 same E0 E1 = E0 E2 = E1 or (notC0 and C1)and A1
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This slight change in logic results in an easily noticeable difference in the overall student outcome and pattern of learning.
Thus, fewer original combinations of learning lead to successful outcomes and some original states (where students don't know A, but know B and C) lead the students into a confusing situation where they are unable to move ahead. In a real classroom, an example of this could be negative transference, where the students think they understand a concept from some other course, but are actually confused and can't learn the course material at hand. In particular, I have seen this happen to students who only knew how to solve a problem algorithmically but didn't really understand how to deconstruct and solve a problem. |
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Analysis of impact of structural change on learning outcome
The comparison of Logic1 and Logic2 in the above examples demonstrates how the construction of resources and the sequence of learning could have an impact on the student learning outcome. In this case the switch to from an OR to an AND represents steps in the learning process that can either stand alone or must be mastered simultaneously by students. Although the OR logic leads to more conditions that lead to full learning (state31), the AND condition might be useful as a more stringent test of understanding.
If this type of modeling can be shown to be related to actual student progressions, then the interpretation of conditions that lead to blockages of progression might actually be more valuable than understanding the conditions that should lead to full learning.
I should look at student data from Bi101.
Conclusion
In most other fields, simulations are helping people with rapid prototyping and continuous improvement processes. Learning is a very complex task that takes place in an obscure medium. Complex in the sense the initial conditions and changes during the process can lead to very different outcomes. Obscure in the sense that it is probably impossible to probe for the level of understanding without disturbing that level of understanding.
The model presented here could be used with a scaffolding approach to teaching in which learning resources are made available to students in a deliberate sequence. Pre-quizzes could be used to prepare students for the unit and to determine their starting "state" Classroom assessment techniques or short assignments could be used as both teaching strategies and to monitor the progress of each student through the unit. The unit could end with a performance of student understanding that would build on what each student learned. Even a crude learning model (such as presented here) may help us improve teaching and learning by helping us appreciate the importance of initial student readiness and the possibility for multiple paths.