Jones, G., & Ritter, F. E. (1997). Modelling transitions in children's development by starting with adults. Full paper appeared in Proceedings of the European Conference on Cognitive Science, 62-67, Manchester, UK.
Gary Jones & Frank E. Ritter
Psychology Department
University of Nottingham
Nottingham NG7 2RD
England
E-mail: gaj@psychology.nottingham.ac.uk
26th November, 1996
Abstract
Determining what processes occur during cognitive change is difficult
because transition mechanisms can only be inferred from the data. We
present a task in which cognitive change is apparent, and propose that
computational modelling can help in examining how transition occurs. Young
children's behaviour tends to be very complex because of their lack of task
knowledge, and so we initially examine the performance of adults on the
task. We have used the results of this experiment to build both an adult
model and a simulation of the task. This model will be able to predict how
behaviour changes when task characteristics (or perception of the task)
change, and when domain knowledge is removed. Manipulating the model in
these ways will allow us to see to what extent children's task performance
can be attributed to knowledge and to what extent it can be attributed to
developmental processes such as perception.
The need to look at transitions in cognitive development has been known for quite some time (e.g. Simon, 1962). However, most literature in children's development has tended to concentrate on describing children's behaviour at each performance level with little regard to how progression from one level to another occurs. Siegler and Shipley (1995) see this as being problematic because their studies suggest that the one-to-one correspondence is misleading. They also believe that if children's development is thought of in terms of direct one-to-one correspondences, then this presents a large gap for theories of transition to fill. They therefore put forward the need to examine transition in conjunction with performance.
The solving of physical problem solving puzzles would seem an ideal method to examine both children's performance and the transitions therein. This is because a detailed analysis of the task behaviour is possible via videotape. This should mean that strategies are readily visible, whereas tasks that involve a lot of mental work may require the experimenter to infer what specific strategies are being used. For this reason we consider the use of a physical problem solving puzzle, the "Tower of Nottingham", in which to study transitions in children's performance.
The task
The Tower of Nottingham is a puzzle which consists of 21 blocks, with a
goal to produce a pyramid structure. There are six layers to the pyramid,
the lower five consisting of four blocks each, with a single block as the
top layer. The blocks in the lower five layers all share the same
characteristics, differing only by size. Two of the blocks have half-pegs,
with one block having a hole and the other a peg, such that placing the peg
in the hole brings the half-pegs together to form a peg. Similarly, the
other two blocks have half-holes, and placing the peg of one block into the
hole of the other produces a hole. A square shaped layer is then produced
by inserting the newly created peg into the newly created hole. The
described block features also permit the construction of a layer in a
variety of other ways.
Two further features exist: each block has a quarter circle indent on top and a quarter circle depression underneath. When a layer is created, the quarter circles form circles in the centre such that layers can be stacked on top of each other by placing the circular depression of the upper layer onto the circular indentation of the lower layer.
In studies involving this task (e.g. Wood & Middleton, 1975) children show a progression in performance with age, such as a reduction in errors and time taken, and an increase in the correct operations accomplished. Since the task was designed to examine the effects of instruction, all of the studies involved this to some extent (e.g. tutoring, being shown pictures of the stages of completion). These also showed differences with age, with regard to how often the tutor had to intervene, and how much reliance the children placed on instructional aids.
These studies show a variety of strategies that are used by the children, as well as possible developmental issues (e.g. young children have difficulty selecting blocks by absolute size). Children also become more efficient at the task as they get older, such that they no longer use some of the inefficient strategies that younger children use (e.g. no longer sampling with replacement). This shows that there are different performance levels in the task, and therefore transitions must occur in order to transcend these performance levels. It is both of these issues which we are interested in characterising.
Examining Transitions
Whilst it is relatively simple to characterise behaviour at each
performance level, defining how, why and what transitions take place is
problematic. Siegler (1995) puts forward the microgenetic approach as a
method by which transitions may be studied. In this approach, behaviour is
observed as much as possible during the period of transition. However, we
see two problems with this method. First, the exact period when transition
takes place is likely to be vastly different across individuals. Second,
the method only offers more data from which to infer the mechanisms of
transition. That is, it can tell us in more detail what is occurring during
the transition stage, but it does not explain how it occurs. Therefore
further methods are necessary in order to study cognitive change.
Computational modelling is one method that can help explore theories of transition because models enable learning and knowledge to be independently and directly manipulated. This means that models can predict what initial knowledge is required to produce the behaviour seen at each performance level, and predict how transitions may occur. Such predictions can be tested against the thorough analysis of change that the microgenetic approach provides. We will therefore be looking to use computational modelling as a tool in which to study and summarise transition.
The use of computational modelling involves being able to define the behaviour that is occurring at each performance level, since the model requires the knowledge and procedures that children may be using. To the extent that these cannot be defined, the model can make predictions as to what the missing elements could be. Therefore modelling task behaviour can provide a method for examining to what extent task performance can be attributed to knowledge and what extent can be attributed to developmental processes.
There would appear to be two clear ways to go with regard to creating a model. One method is to model a less advanced performance level and see if that model can then progress to the more advanced performance levels that we see on the task. The other method is to begin at the most advanced performance level (that of adults), and then see if reduced versions of this model show behaviour that looks like less advanced performance levels. In both cases, the model should make clear what predictions it makes, both in terms of hypotheses as to what the missing elements are, and in terms of task predictions that have not been examined yet (this is a test for any computational model).
The problem with beginning at the lower performance levels is that young children on the Tower of Nottingham often generate complex behaviour due to lack of knowledge. It can therefore be quite difficult to ascertain what strategies and what initial knowledge they may have. On the other hand, adults on the task can be considered as being at the highest performance level, and are also able to give verbal protocols. This will help to provide a clearer picture on what strategies and knowledge they have when starting the task. We have therefore decided to start by examining adult behaviour in the Tower of Nottingham, which should provide a baseline for the level of performance that children will eventually attain.
Adult Behaviour on the Task
We had ten adults attempt to build the Tower of Nottingham whilst giving
verbal protocols. In the first stage, half of these were shown a picture of
the completed tower prior to beginning the experiment (the "goal"
condition), with the other half being told to "build something special"
(the "non-goal" condition). Once the tower had been completed, it was
dismantled by the experimenter (out of view of the subject) and subjects
were asked to re-build it (the "repeat stage").
The results of this experiment can be described as follows. Subjects in the goal condition had a distinct advantage over their non-goal counterparts in the time taken to assemble the tower in the first stage of the experiment. This advantage almost disappears in the repeat stage. All subjects completed the task quickest when doing it for the second time, and for all subjects there was a trend towards taking less time per layer the further into the task they were. In the first stage, subjects in the non-goal condition show much more errors than those in the goal condition, whilst all subjects are virtually error-free in the repeat stage.
These results provide evidence that some form of learning is occurring throughout the task for all subjects. What we can ascertain from video and protocol analysis is that adults start with the basic knowledge that is required to complete the task, such as pegs can go into holes, half-pegs can fit to other half-pegs to make pegs, pyramids are made from items of different sizes, four quarter circles make a circle etc. What subjects must be learning, therefore, is how to apply this knowledge to the task at hand. In the first stage, the goal subjects must first learn how to build a layer, and the non-goal subjects must learn both this and that the blocks make a pyramid. This can be seen as the main reason for subjects being quicker in the repeat stage. The within stage learning that is evident occurs after this new knowledge is in place, and therefore must come from experience with the task.
An examination of the strategies used reveals a variety of strategy use in both the selection of blocks and in the application of those blocks to produce layers. For example, most subjects produce a layer by fitting together blocks using at least two of the possible methods by which layers can be assembled.
Strategy selection can be influenced by the spatial location and orientation of the blocks. For example, two blocks adjacent to each other with the peg of one block facing the hole of the other invites the subject to try fitting them together. If the blocks are at different orientations, subjects are less inclined to try fitting them, and even less so when the blocks are distant from each other (though spatial location is less important once subjects have learnt to select blocks of the same size).
These analyses enable us to produce a cognitive model of adult performance on the task. This model must be able to learn the various strategies that adults use on the task, as well as being quicker to build each layer the further into the task it is. Since strategy selection can be influenced by the physical environment, this presents the need for a simulation of the blocks. Such a simulation must include orientation and spatial location.
The model
A preliminary model of the task is complete, and is achieved by a cognitive
model in ACT-R (Anderson, 1993) and a graphical simulation of the task in
Garnet (Myers et al, 1993). These interact with each other to accomplish a
significant amount of the task. The complete model will perform the whole
task, and will therefore be able to learn which task features are relevant
and which strategies are most efficient.
The simulation environment includes all of the blocks in a graphical display window provided by Garnet. Presently this is only two-dimensional, so the simulation can only rotate blocks on one plane, though blocks can be turned upside down. This is not considered a great limitation since we will be able to capture most of the task behaviour without the need to simulate three dimensions. All block features are represented.
The simulation also includes an eye, whereby information in the fovea, parafovea and periphery is represented. The eye is able to attend to any task feature. We also include a simple hand simulation, which is currently able to pick up, drop, rotate and fit together blocks.
At the moment, the model is able to submit requests to the simulation to scan the table in order to view what is on it. Following this it analyses the task features it has seen, and attempts to apply domain knowledge to the task in order to assess how or what to build. Based upon these analyses, requests are sent to the simulation to select blocks relating to those features the analysis has deemed important. Once the model is notified that blocks have been selected, it attempts to put them together. The simulation then returns the results of this attempt. All of this behaviour is consistent with most of the adult subject's behaviour on the task.
Timings are provided for both cognitive and simulation aspects of the model. Whilst the need for timings is not as important as the need for the model to cover qualitative aspects of the task, it does provide a benchmark that performance is in realistic timescales, and is especially useful when predicting behaviour. It is often the case that models perform a task too quickly, mostly because they fail to account for timings related to physical interaction with the environment. Since our model provides timings for this, we expect to be able to estimate how long the task should take under different conditions. Presently, the model is too slow because of the time taken to analyse the task features.
Summary
The complete model will enable distinct predictions to be made regarding
performance when task characteristics are altered, such as how well
perception works. It should also make predictions regarding task difficulty
and time to complete the task when subjects are given different amounts of
task knowledge. Both of these may well influence cognitive development on
this task.
The first area that we wish to investigate is how behaviour declines when domain knowledge is removed (either declarative knowledge, procedural knowledge, or both). This should help determine to what extent children's behaviour may be due to lack of knowledge and to what extent it may be due to developmental influences. This work is fully expected to be complete by March, 1997.
References
Anderson J.R. (1993). Rules of the Mind. Lawrence Erlbaum Associates.
Myers B.A., Giuse D., Mickish A., Vander Zanden B., Kosbie D., Landay J.A.,
McDaniel R., Parthasarathy R., Goldberg M., Dannenberg R.B., Marchal P. &
Pervin E. (1993). The Garnet Reference Manuals, Version 2.2. Carnegie
Mellon University.
Siegler R.S. (1995). Children's Thinking: How Does Change Occur? In
Schneider W. & Weinert F. (Eds), Memory Performance and Competencies:
Issues in Growth and Development. Lawrence Erlbaum Associates.
Siegler R.S. & Shipley C. (1995). Variation, Selection and Cognitive
Change. In Simon T. & Halford G. (Eds), Developing Cognitive Competence:
New Approaches to Process Modeling. Lawrence Erlbaum Associates.
Simon H.A. (1962). An Information Processing Theory of Intellectual
Development. Monographs of the Society for Research in Child Development,
27 (2, Serial No. 82).
Wood D.J. & Middleton D. (1975). A study of assisted problem solving.
British Journal of Psychology, 66(2), pp181-191.