How does statistical reasoning compare with evolutionary knowledge in terms of its impact on evolution acceptance? The answers to these questions will help to illuminate the degree to which statistical reasoning and evolutionary reasoning are interconnected, and provide insights into how the emerging field of statistics education could help to advance evolution education efforts. Learners in any biology classroom make sense of their experiences using their personal conceptual ecologies, which in turn affect their knowledge and acceptance of evolution and their ability to experience conceptual change e.

Therefore, the state of a learner's conceptual ecology influences if, when, and how conceptual change will take place. The components of a learner's conceptual ecology develop ontogenetically and include cognitive factors e.

Thus, it is clear that many factors comprise a learners' conceptual ecology and contribute to their cognitive processing. Abstract concepts, in particular, are understudied in terms of their impact on conceptual change but appear to be connected to evolution understanding e. Abstract concepts include randomness, probability, temporal scales, and spatial scales. Probability in mathematical and scientific contexts is often described as the likelihood of a particular outcome and is measured by the ratio of trials in which that outcome occurs over the total possible number of outcomes.

Probabilities are assigned a numerical value between 0 and 1 Feller, ; the closer a probability value is to one, the more likely the outcome. Students in evolution education engage with probabilities and probabilistic equations e. In addition to these simple mathematical representations, students must also engage with probabilities in the context of evolutionary change.

Specifically, probability undergirds the three principles that are necessary and sufficient for explaining evolutionary change by natural selection i. For instance, the probability of reproduction increases when individuals have traits that are well matched to their environment. Depending upon the number of possible outcomes, the aggregate of outcomes of multiple but few trials can also be unpredictable and without an obvious pattern. However, over many trials, the aggregate of random outcomes eventually generates predictable patterns that can be described in terms of probability.

When evolutionary biologists talk or write about randomness, they use the concept to describe the outcome of individual events that are independent of an organism's requirements, environment, or evolutionary trajectory Eble, ; Millstein, Overall, randomness and probability are complex but essential features of evolutionary reasoning. Randomness is a common term in everyday discourse, and it has a variety of meanings c.

However, although random processes operate without purpose or direction, beneficial outcomes can and do emerge from such processes. However, understanding randomness is closely connected to understanding probabilities i. But this connection between randomness and probability might confuse students, and thus, could hinder a successful understanding of these concepts. This might be even more problematic when the context changes from a clear mathematical context to a biological context cf.

## Journal of Statistics Education, V4N3: Sahai

Collectively, these factors are widely considered to impact the learning of evolutionary concepts, but large scale, empirical studies of the magnitudes of these effects remain unknown. The course content is aligned with the five core concepts of biological literacy listed in the American Association for the Advancement of Science's Vision and Change policy document AAAS, Evolution is one of these core concepts and a constant theme throughout the course.

Randomness and probability were discussed briefly in so far as they relate to evolution but these topics did not receive focused attention. The course was taught by two evolutionary biologists. Enrollment in spring semester was undergraduates, of which Of the consenting students, Asians Most students were sophomores English was their first language for This study was approved by the university's Institutional Review Board.

To investigate the relationships among statistical reasoning, evolution understanding, and evolution acceptance, we asked students to complete a suite of postcourse online diagnostic tests consisting of instruments that assess conceptual knowledge of randomness and probability, levels of evolution knowledge, and levels of evolution acceptance.

The instruments we used are outlined below see also Table 1 , and item examples of each instrument are available as Supporting Information accompanying the online article see Table S1. The RaPro instruments are conceptualized as two distinct components that are designed to measure students' statistical reasoning in the context of evolution RaProEvo and mathematics RaProMath; Fiedler et al. Student answers are scored as the number of key concepts KC; i. In our sample, students were asked to reason about a the gain of a familiar trait i.

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We then calculated the mean numbers of different KC and AC that students produced in their responses to the two items respectively: key concept diversity [KCD] and alternative conceptions diversity [ACD]. The Conceptual Assessment of Natural Selection CANS is a multiple choice instrument that also seeks to determine how well students understand the basic process of natural selection Kalinowski et al. The instrument is divided into four animal or plant contexts i.

The Conceptual Inventory of Natural Selection CINS is one of the most widely used instruments to assess students' understanding about natural selection in a broad sense Anderson et al. The instrument is divided into three item contexts that are based on evolutionary events studied by scientists such as Galapagos finches, Venezuelan guppies, and the Canary Islands lizards. Face validity evidence was verified by independent content experts, student interviews, and statistical analyses based on classical test theory CTT; Anderson et al.

Although the CINS has been shown to have some psychometric problems e. The Generalized Acceptance of Evolution Evaluation GAENE is a new instrument designed to assess students' acceptance of evolutionary theory as the best currently available scientific explanation Smith et al. The items were designed to measure only acceptance, and not related concepts such as knowledge or belief.

Smith et al. Further work examining validity inferences was conducted by Sbeglia and Nehm The final sample size for this study was students. Person abilities are generated by a weighted maximum likelihood estimation WLE of item difficulties. Rasch analysis can be used to evaluate the psychometric functioning of instrument data within a given population. To do so, we evaluated the internal structure of the instruments, the separation reliability of the items, the fit of the items to the Rasch model, and the alignment of person and item measures.

We discuss each evaluation method below. We compared the fit of the two nested models using likelihood ratio testing, which can be used to evaluate whether the goodness of fit of a multidimensional model is significantly better than a unidimensional model e. Additionally, support for a multidimensional structure would suggest that the nature of the response patterns differs based on whether the items ask about randomness and probability in an evolution context or a mathematics context, such that scores from each dimension would not covary in a predictable manner across participants.

For example, a subset of participants could receive high scores for RaProEvo and low scores for RaProMath, but a different subset in the same population could show the reverse pattern. A low correlation between the scores on the two instruments would be observed and would indicate that randomness and probability in an evolution context and in a mathematics context represents distinct constructs, suggesting that RaProEvo and RaProMath target different competencies.

## How strongly does statistical reasoning influence knowledge and acceptance of evolution?

We evaluated response reliability to determine if the model could predictably separate items and persons. Reliabilities above 0. Additionally, we calculated the WLE person separation reliability that represents a measure for the differentiability of the persons regarding their abilities. Overall, the instruments' reliabilities were sufficient e.

### Course Description

Similarly, the WLE reliabilities are sufficient except for the RaProEvo that seems to be lower than desired but still acceptable. To determine if items behave as expected by the Rasch model, we generated weighted mean squares item fit statistics WMNSQ. Greater values indicate the presence of more variation than expected by the Rasch model i.

In such a map, the distributions of person abilities and item difficulties are plotted on the same scale and can be directly compared, which can be used to glean insights about the relative difficult of items, the probability of particular participants answering correctly, and the extent to which the instrument's difficulty matches to the ability of the participant sample.

An instrument that is well matched to a sample will show item difficulties that span the abilities of the entire population. The authors also reported that the RaPro instruments supported a 2D structure indicating that they measured two distinct competencies, one related to statistical reasoning in the context of mathematics and the second related to statistical reasoning in the context of evolution Fiedler et al. Nevertheless, more research is needed with other cohorts to define which items might be removed after all.

In this article, we replicate these psychometric analyses on a larger population of North American students to test for generalization validity of the RaPro instruments. To evaluate the relationship of statistical reasoning with evolution knowledge and acceptance, we first evaluated how the seven instrument measurements covaried with one another by conducting pairwise correlations between their Rasch person abilities or—in the case of the ACORNS—raw scores.

This analysis allowed us to compare our correlation coefficients with those reported in other published studies to contextualize the importance of statistical reasoning. Nevertheless, there are substantial weaknesses with the use of correlation analyses, including that they do not allow for the prediction of outcome variables, and cannot control for the impact of other variables on the relationship between the two that are of interest. Thus, the regression analysis explained below were conducted to address these limitations.

To assess the relationships among statistical reasoning and evolutionary knowledge and acceptance, we performed a hierarchical regression using a forward selection approach. Next, we added student demographic and background variables i. Finally, we added the CANS measure of evolutionary knowledge to the model as a predictor variable for evolution acceptance Step 4.

To examine and verify the internal structure of students' conceptual knowledge of randomness and probability i.

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RaProMath items; see Table 4. Thus, students' conceptual knowledge of randomness and probability in the context of evolution and in the context of mathematics appear to be empirically separable competencies. Therefore, separate person abilities were generated for each RaPro instrument and included individually in the regression models.

Given the superior fit of the 2D model, we generated a joint Wright map of the person and item scores resulting from this model see Figure 1. In this map, items of equivalent difficulty are located at the same position on the scale e. Specifically, most students could answer items concerning the process of natural selection broad probabilistic aspects of the process , while items regarding the origin of variation were distributed along the scale, indicating a large range in difficulty. Nevertheless, the patterns in the mathematical context RaProMath indicate that most students could answer questions concerning probability as ratio , while only high performers correctly answered items regarding the probability of events.

In contrast, single event items i.

However, nearly no item targeting a specific topic were of redundant difficulty. The only exceptions are items of the topic probability as ratio in the RaProMath i. As expected, RaProEvo and RaProMath were significantly and positively correlated with levels of evolutionary knowledge and acceptance, and significantly negatively correlated with the ACORNS alternative conception diversity scores see Table 5.

Comparing these correlation coefficients with those reported in the literature reveals that a correlations of evolutionary knowledge and acceptance light gray bars in Figure 2 are of similar magnitude as published correlations dark gray bars in Figure 2 and b the correlations between statistical reasoning and acceptance black bars in Figure 2 are not substantially different to the correlations between evolutionary knowledge and acceptance.

Thus, using correlations, statistical reasoning appears to be as associated with the acceptance of evolution as is knowledge of evolution. The addition of RaProMath to the model significantly increased the variance explained to In contrast, the inclusion of background and demographic variables decreased the variance explained to Overall, statistical reasoning can be regarded as a strong and significant predictor of evolutionary knowledge, and it explains a substantial proportion of variance.

Moreover, the addition of RaProMath to the model increased the variance explained to The addition of demographic and background variables in Step 3 significantly increased the explained variance by 7. Finally, the inclusion of CANS as an independent variable in Step 4 significantly increased the variance explained to Given the significant coefficient for the RaPro instruments in the final model Step 4, Table 6 , statistical reasoning also explained a unique portion of the variance in evolution acceptance beyond what can be explained by evolution knowledge.

Acceptance of evolution has been associated with multiple variables in the literature including understanding of evolution e. We suggest that statistical reasoning might be another important contributor to evolutionary knowledge and thus also to evolution acceptance, but little empirical work has examined its role. Based on the results, we can conclude that the recently developed RaPro instruments, which measure statistical reasoning in the contexts of mathematics and evolution, held up to validity and dimensionality testing in our large population of North American undergraduate students see also Tables 3 and 4.

Importantly, we found convergent evidence that statistical reasoning in the context of mathematics was a different competency than in the context of evolution in both the German and North American populations see also Table 4. Establishing the internal structure of these instruments for a given population is key to precisely measuring their statistical reasoning. However, both studies also indicate possible ways to improve the instruments. Specifically, the joint Wright map of Fiedler et al. For instance, items concerning accidental death i. Other items that seem to be too simple were items concerning origin of variation i.

http://pro.bannon.ie/9503-cell-phone-spy.php It could be argued to exclude these items or revise them i. However, it should be kept in mind that the American sample already received evolution instruction and we do not know how the items would function prior to instruction. In contrast, items of the RaProMath were designed to focus on randomness and probability in the context of mathematics, and we would not anticipate that students' responses would change substantially during an introductory biology course.

Thus, we would argue that most of the items concerning probability as ratio i.