• A series of numbers constructed according to an algebraical rule is to be continued by one or more numbers. Subjects are required to infer the rule from the number series presented and to calculate the missing number with the help of this rule.
  
• The following example demonstrates a very simple task:

30 32 36 44 60 .... ?

• One possible rule is: x_n = x_n-1 + 2n.
  
• Other formulas that correspond to the example: x_n = 3x_n-1 - 2x_n-2, where x_n is the number, we are trying to find, x_n-1 is the preceding number (here: 60), and so on.
  
• We call the number of immediate predecessors that are used for the solution of the problem the level of recursion.
  
• Krause(1985): investigation of mental processes and rule detection.
  
• The level of recursion is one suitable property for component-based method of problem construction
  
• Cognitive demands covered by number series problems:
  1. The subject has to recognize properties and regularities of the presented sequence (e.g. the level of recursion)
  2. Ahypothesis concerning the underlying rule has to be established, applied, and tested.

Problem construction and hypothesis

• Number series problems are extremely variable. What types of components can be combined in which ways?
  
• Three distinct components: M₁, M₂, M₃
  
• Attributes are shown in Table 1.
  
• Concerning attributes b₂ and c₂: Note that the definition of the factors f = 1 and g = 0 is included for technical reasons: although a recognition of a multiplicative or additive factor is not necessary for a solution of the problems which are characterized by b₂ or c₂, giving such zero values is appropriate for a complete problem definition by elements of a Cartesian product.

Table 1: Number Series: Problem Components

	Components	Attributes
	M1	a1 Level of Rec.: 3	a2 Level of Rec.: 2	a3 Level of Rec.: 1
	M2	b1 Multiplicative Factor:	b2 Multiplicative Factor:
	M3	c1 Additive Factor:	c2 Additive Factor:

• A linear order is defined on the attributes of each component.
  
• This assumption means that, for example, recursion level 3 makes a problem more difficult than recursion level2, or the existence of a multiplicative factor that is greater than 1 provides more complication than factor 1.

Figure 1: Number series: Orders of attributes and problems.

• Problem construction rule for these components: product formation
  
• The product provides 12 combinations of attributes of the type (an, bn, cn).
  
• We call the set of these combinations problem set Q_t.
  
• Next step: Application of the componentwise ordering rule.
  
• Structure of the twelve problems, where problem (a₁, b₁, c₁) is assumed to be the most difficult and (a₃, b₂, c₂) the simplest.

Method

• Problem properties:
  
  • multiplicative constant is either 2 or ,
    
  • additive constant is always a single- or two-digit element of ,
    
  • maximal recursion level is 3.

• All solutions are numbers greater than 100 (prevent guessing)

Table 2: Number Series: Calculation Rules and Problems

Attributes		c1	c2
a1	b1	xn = 2xn-3 + xn-2 + xn-1 + 4	xn = 2xn-3 + xn-2 + xn-1
		1,5,9,20,43,85 172	6,6,7,25,44,83 177
	b2	xn = xn-3 + xn-2 + xn-1 + 1	xn = xn-3 + xn-2 + xn-1
		16,16,17,50,84,152 287	26,34,41,101,176,318 595
a2	b1	xn = xn-2 + 2xn-1 + 2	xn = 2xn-2 + xn-1
		1,4,11,28,69,168 407	5,11,21,43,85,171 341
	b2	xn = xn-2 + xn-1 + 5	xn = xn-2 + xn-1
		12,17,34,56,95,156 256	25,34,59,93,152,245 397
a3	b1	xn = 2xn-1 + 1	xn = 2xn-1
		7,15,31,63,127,255 511	4,8,16,32,64,128 256
	b2	xn = xn-1 + 13	xn = xn-1
		33,46,59,72,85,98 111	113,113,113,113,113,113 113 (not presented in investigation)

 

• Two investigations:
  
  • Investigation I: Psychological Institute of the University of Heidelberg, Germany; 18 subjects; trivial problem (a₃, b₂, c₂) not used
    
  • Investigation II: Psychological Institute of the University of Graz, Austria; 30 subjects; complete set of 12 problems presented
• Structure ₁ for 11 problems (as used in Investigation I) corresponds to 49 knowledge states. This is 2.4% of 211 = 2048 possible solution patterns.
  
• 12 problems (Investigation II): Structure ₂ with 50 knowledge states (i.e. 1.2% of 212 = 4096 possible patterns).
  
• Problems were presented in a randomized order. Subjects had to write down the solution on a sheet of paper. If the solution was not given within 7 minutes, the next problem was presented.
  
• Subjects who either gave the wrong solution or wanted to give up before the 7 minutes had expired were asked to go on thinking about the problem. After the last problem, the subject was asked for the rules he or she used for solving the problems.

Results

• Only knowledge space ₁ considered for the comparisons between data and hypothetical structure.

Table 3: Number Series
Minimal Distances and Mean Distance between the Solution Patterns and the Hypothetical Knowledge Space for Investigation I and Investigation II and the whole Group of Subjects.

	Distances	d
	0   1   2   3
Investigation I (18 subjects)	16  2    -    -	0.11
Investigation II (30 subjects)	18  7   3   2	0.63
Both Investigations (48 subjects)	34  9   3   2	0.4

• Investigation I: Solution pattern of 16 out of 18 subjects is identical with a knowledge state in ₁. Only patterns of two subjects have a distance of 1 to a state in ₁.
  
• Investigation II: Solution patterns with distances of 2 and 3 occurred. All in all, 14 different solution patterns with a distance of 0 have been observed.

Discussion

• Hypothetical conclusions about the componentwise ordering rule were rather accurat.
  
• An alternative and more economical theory for the data could be stated by a lexicographic order on the problem set
  
• Lexicographic order: Component M₁ (recursion level) is the `most important' component, M₂ (multiplicative factor) the second most important, and M₃ the least important component.
  
• Lexicographic order: Only 4 solution patterns of Investigation I and 6 patterns of Investigation II agree with a state. (Table 4)

Table 4: Number Series
Minimal Distances and Mean Distance Between the Solution Patterns and the Knowledge Space that is Based on a Lexicographic Order for InvestigationI and InvestigationII and the Whole Group of Subjects.

	Distances	d
	0   1   2   3
Investigation I (18 subjects)	4   11 2   1	1.00
Investigation II (30 subjects)	6   12 7   5	1.37
Both Investigations (48 subjects)	10  23 9   6	1.23

• Only 12 knowledge states (consisting of 11 problems) are assumed to exist--these are about .6% of the potential response patterns.
  
• Although the lexicographic order is much more restrictive than the componentwise order, these results may also be a product of the assumption concerning the importance of the components.

Ambiguity of Number Series Problems (Alternative Solutions)

• For every number series problem, alternative solutions can be found.
  Assumption that a subject who is able to solve a problem, will use one particular calculation rule is not always realistic.
  

• Calculation rule for the series 5,11,21,43,85,171 ..... is?
  
  • xn = 2x_n-2 + x_n-1. This is problem (a₂, b₁, c₂) in our problem construction scheme.
    
  • The rule x_n = 2x_n-1 + (-1)ⁿ will also provide a correct solution.
    
• Although subjects were told that only positive constants are to be added in the problems, we cannot exclude the possibility that a subject will use such an alternative rule.
  
• Construction and ordering of number series problems must be based on an exact analysis of the uniqueness of the problems.
  
• Korossy(1990): Method, that allows the uniqueness of the solution to be determined.
  
• Main result: only heavy restrictions on the domains of the recursive formulas lead to less ambiguous ranges for the solutions. In the case of different rules for a problem, a generalized model using surmise systems and knowledge spaces and the respective principles for constructing these structures may be appropriate.
  
• Overall conclusion:
  
  • it is impossible to construct a number series problem that can be solved by only one rule.
    
  • it is possible to minimize the number of alternatives to a degree that allows one to work with this type of problem.
    
  • if the manner in which an ambiguous problem has been solved is known, it may be possible to infer which of the assumed cognitive demands has been mastered by the subject.

References:

• Albert, D. & Held, T. (1999) Component-based Knowledge Spaces in Problem Solving and Inductive Reasoning
  In Albert, D.(Ed.) & Lukas, J.(Ed.), Knowledge Spaces: Theories, Empirical Research, and Applications (pp. 15-40).
  Mahwah, New Jersey: Lawrence Erlbaum Ass., Publishers
  

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