Informal Inferential Reasoning Research Articles

Promoting students’ informal inferential reasoning through arts-integrated data literacy education

PurposeArts-integration is a promising approach to building students’ abilities to create and critique arguments with data, also known as informal inferential reasoning (IIR). However, differences in disciplinary practices and routines, as well as school organization and culture, can pose barriers to subject integration. The purpose of this study is to describe synergies and tensions between data science and the arts, and how these can create or constrain opportunities for learners to engage in IIR.Design/methodology/approachThe authors co-designed and implemented four arts-integrated data literacy units with 10 teachers of arts and mathematics in middle school classrooms from four different schools in the USA. The data include student-generated artwork and their written rationales, and interviews with teachers and students. Through maximum variation sampling, the authors identified examples from the data to illustrate disciplinary synergies and tensions that appeared to support different IIR processes among students.FindingsAspects of artistic representation, including embodiment, narrative and visual image; and aspects of the culture of arts, including an emphasis on personal experience, the acknowledgement of subjectivity and considerations for the audience’s perspective, created synergies and tensions that both offered and hindered opportunities for IIR (i.e. going beyond data, using data as evidence and expressing uncertainty).Originality/valueThis study answers calls for humanistic approaches to data literacy education. It contributes an interdisciplinary perspective on data literacy that complements other context-oriented perspectives on data science. This study also offers recommendations for how designers and educators can capitalize on synergies and mitigate tensions between domains to promote successful IIR in arts-integrated data literacy education.

EXPLORING INFORMAL STATISTICAL INFERENCE IN EARLY STATISTICS: A LEARNING TRAJECTORY FOR THIRD-GRADE STUDENTS

Recent research suggests the benefits of supporting a progressive understanding of concepts of inference prior to the teaching of procedures and formal calculations through the study of informal statistical inference (ISI). To contribute to the growing knowledge about the early learning and teaching of statistics, particularly regarding the development of informal inferential reasoning (IIR), we designed a learning trajectory (LT) that addresses ISI in K–4 students (ages 5 to 9 years). This article describes part of the LT in detail, in which third-grade students (n = 12) were introduced to sampling, frequency distribution, randomness and sampling variation as well as to developing a data sense in online lessons. The results of this type of teaching show that the creation and collection of authentic data in a playful context, together with an exploratory analysis of the data as a precursor to utilizing aspects specific to IIR, promoted the integration progress of IIR components in the oral and written informal inferences of students.

YOUNG CHILDREN’S PROBABILISTIC AND STATISTICAL REASONING IN THE CONTEXT OF INFORMAL STATISTICAL INFERENCE

This paper reports the statistical and probabilistic reasoning of young children in terms of randomness, variability, and data representations in the context of informal inferential reasoning (IIR). Using the IIR approach, a task was designed and conducted one-on-one with 28 children aged 5 to 6 years old, in a case study setting. The researcher used a voice recorder during interviews, took photos, and recorded field notes. The data were analyzed according to the principles of informal inferential reasoning, which are generalizing beyond the data, using data as evidence for generalizing, and using probabilistic language whilst being aware of uncertainty. The findings indicate that young children are capable of making informal inferences from a sample space, describing event probability, and constructing bar graph and pie chart data representations.

“Your truth isn’t the Truth”: Data activities and informal inferential reasoning

This paper investigates data activities in an afterschool setting, offering a deeper understanding of the social nature of students’ informal inferences by investigating how informal inferences are negotiated in group interactions, influenced by social norms, and how statistical concepts come into play in learners’ informal inferential reasoning (IIR). Analyses take up a multi-sited orientation to investigate how youth used quantitative and contextual resources during a research activity to make meaning of data and negotiate emergent social tensions. Findings show how data activities that are part of informal inferential reasoning, such as collection, interpretation, generalization, inference, and representation unfolded as social, political, and personal. Implications call for designs for learning that better support working with data and understanding real-world phenomena and sociopolitical issues in ways that leverage youths’ experiences, enabling them to take part in social action as critical community actors.

A FRAMEWORK FOR THINKING ABOUT INFORMAL STATISTICAL INFERENCE

Informal inferential reasoning has shown some promise in developing students’ deeper understanding of statistical processes. This paper presents a framework to think about three key principles of informal inference – generalizations ‘beyond the data,’ probabilistic language, and data as evidence. The authors use primary school classroom episodes and excerpts of interviews with the teachers to illustrate the framework and reiterate the importance of embedding statistical learning within the context of statistical inquiry. Implications for the teaching of more powerful statistical concepts at the primary school level are discussed. First published May 2009 at Statistics Education Research Journal: Archives

Early statistics in kindergarten: analysis of an educator's pedagogical content knowledge in lessons promoting informal inferential reasoning

PurposeThis study aims to describe the pedagogical content knowledge (PCK) of a kindergarten educator who implements a lesson plan about informal inferential reasoning designed in a lesson study group.Design/methodology/approachTo this end, we analyzed teaching interventions in two kindergarten lessons focused on the playful task of tossing two coins, associated with inferential statistical reasoning. The study highlights the importance of arguing and promoting this reasoning to develop statistical thinking. It is crucial to recognize how early students can be subject to learning experiences that promote a language of uncertainty, assess the evidence provided by the data, and make generalizations.FindingsThe results reveal that while the educator demonstrated knowledge and skills relevant to the curriculum and conceptual teaching strategies, the understanding of the content by the students and the integration of the PCK components still present a challenge.Practical implicationsThe lesson study collaborative teaching practices that promote PCK have proven effective for informing the design and implementation of instructional practices supporting the development of early statistical thinking in young children.Originality/valueThe study enriches the knowledge regarding the potential of the lesson study (LS) in the professional learning of kindergarten educators. It also contributes to a comprehensive approach based on authentic playful experiences in grade K that supports the development of early statistical thinking in young children.

Inferential Reasoning of Secondary School Mathematics Teachers on the Chi-Square Statistic

Statistics education has investigated how to promote formal inferential reasoning from informal inferential reasoning. Nevertheless, there is still a need for proposals that explore and progressively develop inferential reasoning of students and teachers. Concerning this, the objective of this article is to characterize the inferential reasoning that secondary school mathematics teachers show in the practices that they develop to solve problems regarding the Chi-square statistic. To achieve this, we use theoretical and methodological notions introduced by the onto-semiotic approach of mathematics knowledge and instruction. In particular, we have taken a theoretical proposal of levels of inferential reasoning for the Chi-square statistic. Based on the results, the main conclusion was that the proposal above effectively predicted the teachers’ practices, allowing us to distinguish characteristic elements of the levels of inferential reasoning.

A Case Study of Promoting Informal Inferential Reasoning in Learning Sampling Distribution for High School Students

Drawing inference from data is an important skill for students to understand their everyday life, so that the sampling distribution as a central topic in statistical inference is necessary to be learned by the students. However, little is known about how to teach the topic for high school students, especially in Indonesian context. Therefore, the present study provides a teaching experiment to support the students’ informal inferential reasoning in understanding the sampling distribution, as well as the students’ perceptions toward the teaching experiment. The subjects in the present study were three 11th-grader of one private school in Yogyakarta majoring in mathematics and natural science. The method of data collection was direct observation of sampling distribution learning process, interviews, and documentation. The present study found that that informal inferential reasoning with problem-based learning using contextual problems and real data could support the students to understand the sampling distribution, and they also gave positive responses about their learning experience.

점그래프와 상자그림의 교수학적 분석: 통계적 추리 지도를 위한 교육적 효용성 고찰

This study analyzes the characteristics of dot plots and box plots that have been identified to be effective in developing the understanding of distribution and variability. Moreover, the study examines the use of dot plots and box plots in statistical teaching and learning and discusses the utility in Korean mathematics curriculum. With respect to the results obtained, the dot plot is firstly expected to bridge the gap between the elementary and middle school curriculum since the dot plot has a very simple system that can be used to construct and express a distribution by dealing with numerical data even at the elementary school level. Secondly, box plots aid in comprehensive understanding of distribution concepts and summary concepts. These concepts are expected to help establish links of statistical content elements that are currently perceived segmentally within the middle school curriculum. Third, dot plots and box plots provide an experience of an informal inferential reasoning for data sets at the elementary and middle school levels, and are expected to be a springboard for learning formal statistical reasoning that will be addressed later in high school.

ASSESSMENT OF INFORMAL AND FORMAL INFERENTIAL REASONING: A CRITICAL RESEARCH REVIEW

As a contribution to the discussion on the assessment of informal inferential reasoning (IIR) and the transition from this to formal inferential reasoning (FIR), we present a review of research on how these two types of inferential reasoning have been conceptualized and assessed. Based on our review, we discuss the need to redefine the conceptions of IIR and FIR in order to create an integrated description of inferential reasoning that includes not only ideas of IIR and FIR, but also the whole activity of argumentation, which involves the production of both statistical and contextual reasons. Current descriptions of IIR and FIR list the facts that might be brought from data analysis to the process of inferential reasoning. The approach we propose considers how the facts, both statistical and contextual, can be used as arguments, leading to assessments of students’ inferential reasoning focusing on articulating the statistical and contextual reasons students present to support an inference. First published May 2019 at Statistics Education Research Journal Archives

통계적 추리에서 표집 분포와 신뢰구간에 대한 이해 조사

This study is a part of the design research project for the development of informal inferential reasoning. With informal inferential reasoning, students can better learn formal statistical inference, which have been shown in recent statistical education researches. The goal of this study is to investigate how the undergraduates who have not completed basic statistics course understand the sampling distribution and the confidence interval, and to find which misconceptions they have. For this purpose, we developed tasks related to sampling distribution and confidence interval, and conducted the surveys and follow-up interviews with 27 participants selected as the sample. The results were analyzed quantitatively and qualitatively. From the analysis, many participants did not understand the concept of sampling distribution and the confidence interval, and had various misconceptions. The participants showed a tendency to choose a shape similar to the population distribution instead of correct sampling distribution and did not recognize the effect of sample size. They also did not understand that the mean of the sample mean is the same as the mean of the population. In addition, the participants showed errors in interpreting the confidence interval such as a fixed interval without variability, or an interval containing a proportion of elements of the population as much as the confidence level, or an interval containing a proportion of the sample as much as the confidence level. With these findings, implications for the teaching of statistical inference were derived.

Exploring informal inferential reasoning: the case of comparing two data sets problem

Informal inferential reasoning as part of statistical reasoning is extremely important in learning. However, this reasoning has not been widely studied in Indonesia. This article explores informal inferential reasoning of students in solving comparing two data sets problem. This qualitative research was involved 118 junior high school students in Mataram West Nusa Tenggara. Instrument in this study is Assessment of Informal Inferential Reasoning (AIIR) which was modified and validated by mathematics and mathematics education experts. Student responses are classified based hierarchical cognitive framework. The findings showed that the existing levels was still relevant for the students of junior high school in Mataram. However, there were subjects which could not be classified in the levelling. The characteristic reasoning of the subjects are tends to ignore the variability and distribution aspects of the data in making predictions, but they can indicate significant value differences between two groups of data. They encountered an error in certain aspects, but they are able to produce an almost right conclusions. This study recommends the need for a new levelling to assess informal inferential reasoning of junior high school students by considering the ability of local and global view of data.

Modelling with authentic data in sixth grade

This article explores 6th-grade students’ modelling with data in generating models for selecting an Australian swimming team for the (then) forthcoming 2016 Olympics, using data on swimmers’ times at various previous events. We propose a modelling framework comprising four components: working in shared problem spaces between mathematics and statistics; interpreting and reinterpreting problem contexts and questions; interpreting, organising and operating on data in model construction; and drawing informal inferences. In studying students’ model generation, consideration is given to how they interpreted, organised, and operated on the problem data in constructing and documenting their models, and how they engaged in informal inferential reasoning. Students’ responses included applying mathematical and statistical operations and reasoning to selected variables, identifying how variation and trends in swimmers’ performances inform model construction, recognising limitations in using only one performance variable, and acknowledging uncertainty in model creation and model application due to chance variation.

THE EMERGENCE OF PROBABILISTIC LANGUAGE FROM STUDENTS’ UNCERTAINTY IN LEARNING INFORMAL INFERENTIAL REASONING

In our daily life, we often draw conclusion regarding some phenomenon to get new information from sampleof the population. By doing so, we unconsciously learn the inferential statistics informally or known as Informal Inferential Reasoning (IIR). Therefore, it can be said that IIR is close to our live. As a result, it is wise to use IIR in introducing inferential statistics to students. In introducing IIR, a set of learning design needs to be made. In this research, the design would use Realistic Mathematics Education (RME) as the approach of the learning and dot plot as a visual representation. This paper describes the probabilistic language students used in expressing some uncertainty when making generalization. The method used is design research. The involved subjects in this research were six students of VII-8 at SMPN 7 Tangerang Selatan. The results showed that the probabilistic language appears when students generalize the data. Moreover, probabilistic language also appears when students added samples in growing sample activity. Probablistic language which appeared in this research are “mungkin”, “hampir sama”, “kurang sama”, and so on. Keywords : design research; informal inferential reasoning; probabilistic language; uncertainty.

STUDENTS’ EXPRESSIONS OF UNCERTAINTY IN MAKING INFORMAL INFERENCE WHEN ENGAGED IN A STATISTICAL INVESTIGATION USING TINKERPLOTS

This paper reports on the results of a study investigating the potential to embed Informal Statistical Inference in statistical investigations, using TinkerPlots, for assisting 8th grade students’ informal inferential reasoning to emerge, particularly their articulations of uncertainty. Data collection included students’ written work on a statistical investigation as well as audio and screen records. Results show students’ ability to draw conclusions based on data, recognizing that these are constrained by uncertainty, and to use them to make inferences. However, few students used probabilistic language for describing their generalizations. These results highlight the need for working on probabilistic ideas within statistics, helping students to evolve from a deterministic perspective of inference to include uncertainty in their statements.
 First published November 2016 at Statistics Education Research Journal Archives

Developing students’ reasoning about samples and sampling variability as a path to expert statistical thinking

This paper describes the importance of developing students’ reasoning about samples and sampling variability as a foundation for statistical thinking. Research on expert–novice thinking as well as statistical thinking is reviewed and compared. A case is made that statistical thinking is a type of expert thinking, and as such, research comparing novice and expert thinking can inform the research on developing statistical thinking in students. It is also posited that developing students’ informal inferential reasoning, akin to novice thinking, can help build the foundations of experts’ statistical thinking.

Young children's explorations of average through informal inferential reasoning

This study situates children's early notions of average within an inquiry classroom to investigate the rich inferential reasoning that young children drew on to make sense of the questions: Is there a typical height for a student in year 3? If so, what is it? Based on their deliberations over several lessons, students' ideas about average and typicality evolved as meaning reasonable, contrary to atypical, most common (value or interval), middle, normative, and representative of the population. The case study reported here documents a new direction for the development of children's conceptions of average in a classroom designed to elicit their informal inferential reasoning about data.

Formative Use of Intuitive Analysis of Variance

Students’ informal inferential reasoning (IIR) is often inconsistent with the normative logic underlying formal statistical methods such as Analysis of Variance (ANOVA), even after instruction. In two experiments reported here, student's IIR was assessed using an intuitive ANOVA task at the beginning and end of a statistics course. In both experiments, students were provided feedback regarding the normative logic underlying ANOVA and how their reasoning compared with it. Additionally, students in Experiment 2 were given an assignment in which they analyzed and interpreted other students’ performance on the intuitive ANOVA task. Results indicate that the feedback combined with the assignment (which required active explanation of both normative and non-normative reasoning applied to the task) led to more normative inferential reasoning at the end of the course, whereas providing feedback alone did not. Implications are discussed for using the intuitive ANOVA task as a formative classroom tool to help students improve their conceptual understanding of ANOVA.

Informal Inferential Reasoning: Interval Estimates of Parameters

This research examined the informal inferential reasoning of senior secondary school students (age 17) when engaged in a computer-simulated sampling activity calling for the estimation of population parameters. The students undertook a task involving interval estimation of parameters within a computer-simulated environment. The research observed the students while they made and then explained their parameter estimates in order to better understand how the students formed the interval estimates, with particular attention to different strategies they adopted in forming these estimates. Activities involved sampling and estimating across three different sample size situations followed by a reflection stage to compare the estimates. Results of the analysis of the discussion between the students and the researcher during the students' activities are presented. A number of strategies for forming an interval estimate emerged. The students experimented with choosing different strategies for forming the interval estimate when a new sample (observed values) was drawn. The research findings are useful for informing the teaching of interval estimation to school-aged students.

Students’ emergent articulations of uncertainty while making informal statistical inferences

Research on informal statistical inference has so far paid little attention to the development of students’ expressions of uncertainty in reasoning from samples. This paper studies students’ articulations of uncertainty when engaged in informal inferential reasoning. Using data from a design experiment in Israeli Grade 5 (aged 10–11) inquiry-based classrooms, we focus on two groups of students working with TinkerPlots on investigations with growing sample size. From our analysis, it appears that this design, especially prediction tasks, helped in promoting the students’ probabilistic language. Initially, the students oscillated between certainty-only (deterministic) and uncertainty-only (relativistic) statements. As they engaged further in their inquiries, they came to talk in more sophisticated ways with increasing awareness of what is at stake, using what can be seen as buds of probabilistic language. Attending to students’ emerging articulations of uncertainty in making judgments about patterns and trends in data may provide an opportunity to develop more sophisticated understandings of statistical inference.