Complex Learning• Designs for Knowledge Evolution
People are notoriously adaptive. This instructional model borrows biological metaphors to conceptualize how students can evolve adaptive forms of knowledge. It conceptualizes learning as a process of adaptation to an increasing range of contexts, where the adaptation occurs through the co-evolution of different cognitive resources needed to handle the features and selective pressures of each context. Though a rather coarse borrowing of biological metaphor, it does lead to some prescriptions for instruction as follows.
The key features of DKE are the following:
1. Students generate symbolic models to differentiate between contrasting cases. This permits students to simultaneously perceive what is significant about the cases and their contexts while developing a structured account of what they perceive.
2. Students test their models across contexts of contrasting pairs. As they confront new contexts, some models fail and students notice properties of the new contexts and their models that “selected” against survival.
3. Students mutate new models that can survive in the context. The new models evolve from the understanding developed from previous models, even if the students need to abandon the form of their earlier model and try a new “genetic” line.
Across the multiple contexts and opportunities for co-evolution, the learner comes to perceive important features of the problem domain and evolves models that can adapt to those and future features. We can best illuminate DKE with a brief study that helped students learn about the statistical concept of variability. We presented college students with a sequence of tightly focused contrasting cases, and we asked the students to invent symbolic models (formulas) that capture what is different about each pair of cases. For example, the first pair of contrasting cases presented the two distributions: {1 3 5 7 9} versus {3 4 5 6 7}. We pointed out that the two distributions have something in common, namely, the average. We explained that the average is a convenient way to characterize what is common about the distributions. It is much easier to communicate the averages than the complete distributions, especially when the number of items gets very large. We then asked the students to notice that there is also a difference between the two sets of numbers, called the “spread,” and to invent a formula that can capture what is different. The students typically invented a range formula that subtracted the largest from the smallest number. We then presented a new contrasting case: {1 3 3 3 9} versus {1 3 5 7 9}. Students saw that their range formula did not work for this case. They came to perceive that “spread” is not simply defined by the end values; it involves density as well. They had to evolve their original model to handle the next context of contrasting cases. They frequently invented a model where they subtracted each number from the largest number in the list. As the process of contrasting cases plus invention continued, students noticed additional features and developed models that were robust to those features. For example, we presented the contrast: {1 3 5} versus {1 1 3 3 5 5}. This helped the students notice that distributions also have different sample sizes and that their formulas needed to accommodate this possibility.
Students rarely evolve the conventional solution agreed upon by experts; namely, the variance formula. However, as we have demonstrated, inventing models over contrasting cases prepares students to understand the variance formula at a deep level when it is presented to them, for example, in a lecture. For example, they appreciate that dividing by “n” elegantly solves the problem of different samples sizes. We have found that the DKE approach is fruitful for developing students' understanding in the domains of statistics, trigonometry, and rational number, and that it can be readily used by teachers to complement their traditional methods of instruction.
