A basic definition of scientific reasoning includes: isolation and control of variables, combinatorial reasoning, correlational reasoning, probabilistic reasoning, proportional reasoning, etc. (e.g., Lawson 1978 , 1985 Lawson et al. 2000).
For the CTSR data set, models (3) and (4) have four content areas: control of variables, combinatorial reasoning, proportional reasoning, and probabilistic reasoning.
Table 4 Average Rasch difficulty estimates for the CTSR items by tier and content Tier Content Tier 1 Tier 2 Average by Content Control of Variables −1.23 −1.22 −1.22 Combinatorial Reasoning −0.16 0.02 −0.07 Probabilistic Reasoning −3.08 −3.32 −3.20 Proportional Reasoning 0.57 −0.40 0.08 Average by Tier −0.88 −1.02 Model Fit statistics: AIC AICc BIC 12558.08 12572.13 12820.07 Note.
Among the questionnaires was Lawson's (2000) revised CTSR that consists of 12 two-tier items (i.e., a total of 24 items), organized into four aspects of reasoning: Control of Variables (8 items); Combinatorial Reasoning (8 items); Proportional Reasoning (4 items); and Probabilistic Reasoning (4 items).
Thus, we conclude Sara shows a difficulty in identification of treatment groups and combinatorial reasoning (Table 2, area of difficulty 2, d and f).
This fact has been known for 50 yr, since Karplus first demonstrated that students have problems with formal operational reasoning patterns like combinatorial reasoning, or the simultaneous manipulation of two independent variables in a study (Fuller, 2002).
A new formal framework for Stringology is proposed, which consists of a three-sorted logical theory S designed to capture the combinatorial reasoning about finite strings.
Scientific reasoning ability is defined as the ability to demonstrate the patterns of formal operations including proportional reasoning, identifying and controlling variables, probabilistic reasoning, correlational reasoning, combinatorial reasoning, and hypothetico-deductive reasoning (Lawson 1982).
We categorized the four levels of understanding combinatorial thinking as investigating some cases, how am I sure that I have counted all the cases, systematically generating all cases, changing the problem into another combinatorial problem and understanding combinatorial reasoning.
Combinatorial Reasoning follows the pattern similar to the LPDI items in data set 1: tier 1 items are somewhat easier (average difficulty estimate of −0.16) than the tier 2 items (average difficulty estimate of 0.02).
