People use models every day without even realizing it. Models create a structure for predictions (inferences) that can be used or adapted as situations change. For example, students may predict dinnertime from past experience, but their informal model of this event lets them adapt their prediction if conditions change (e.g., if they must leave early for a school band concert). Learning that occurs by developing a model can be applied in a range of contexts.
A model is a relational system that highlights aspects of a phenomenon that the modeler deems important and diminishes the rest. Statistical models capture variability of data, enabling students to experience the power of statistics by making inferences about a context.
In this article, we apply a framework for model development in an experiment with paper helicopters conducted by fifth graders. The model development sequence focused on comparing two distributions, an important statistical model for a broad class of problems. We emphasize the important role of the teacher in this process and the learning opportunities available when students engage with statistical ideas before being taught underlying skills and concepts.
THE MODEL DEVELOPMENT SEQUENCE
A model development sequence (see figure 1) is a set of related tasks, beginning with a model eliciting activity (MEA) and followed by model exploration activities (MXAs) and model application activities (MAAs) (Lesh and Doerr 2003). MEAs reveal students' initial ideas about a problem situation. Although their initial ideas are often not well-formed or sophisticated, students evaluate and revise their ideas to be increasingly useful through discussion. In MXAs, students explore the underlying structures and representations of their initial models, and MAAs engage students in applying and adapting their models to new problem situations.
Model Eliciting Activity (MEA)
In the MEA, students worked in small groups to investigate the question, “How long does a paper helicopter stay in the air?” Our aim was for students to answer this question using data from the paper helicopters they had made (see figure 2; video 1 shows how to make a paper helicopter, and the appendix is a paper template). Students had tools available, including metersticks and stopwatches, but no initial instructions on how to launch the helicopters or measure flight durations. We wanted students' initial ideas about helicopter flights to motivate the need for a systematic process to generate, collect, and represent data to respond to the inquiry question.
Groups threw or dropped their helicopters from different heights. Some groups timed flights with stopwatches; others counted seconds. Few measured or recorded their drop height.
The teacher introduced a number line on which students collated the class data. The initial representation (see figure 3) provided a common artifact for students to interpret and critique. The teacher drew attention to the variability in flight times and asked whether the class could answer the inquiry question, “How long does a paper helicopter stay in the air?”
After five minutes of discussion in pairs, the teacher directed students to each write a conclusion and an explanation. The initial play with helicopters revealed important insights to students: the need to measure and record drop times, the need to consider possible sources of variability, and noticing (conjecturing) that drop height affected flight time. Students' were instructed to make their initial conclusions on the basis of the first model.
Model Exploration Activity (MXA)
Collect enough data from two different heights to show evidence with dot plots that height matters.
Students were not given specific heights to use, how much data to collect, nor how to represent their data on dot plots. Again, our purpose was to build from students' own ideas and focus the discussions on improving their models.
Groups created a variety of dot plot models for flight times in rank order, with different scales, and with the same scale for each dot plot (see figure 4), most of which could be used as evidence to argue whether flight time was related to drop height. However, some representations made the comparison easier than others, and this became the focus for the discussion.
The models generated by the children as evidence that “drop height matters” led to a discussion on useful characteristics of representations created by some groups. The teacher asked such questions as, “What are some of the properties of some of these [representations] that make that [height matters] easy to see?” The teacher guided students' attention to characteristics of graphs that make them easier to compare (e.g., stacked dot plots with drop heights labeled and on the same scale, as in figure 4c).
By the end of the model exploration activity (MXA), the students had learned a useful model for comparing two groups using labeled dot plots with equal scales.
Model Application Activity (MAA)
In the MAA, students applied what they learned from the MEA and MXA to design a way to collect evidence of the difference in flight durations between two types of helicopters. Short-wing helicopters were constructed by trimming the wings at the vertical line marked on figure 2. They were told the manufacturer wanted to answer the question, “How big of a difference is there between the flight duration of short-wing and long-wing paper helicopters?” Students worked in pairs to design a method to collect and analyze data, draw conclusions, and support their conclusions with evidence using pictures or graphs. The MAA furthered the development of the students' models for comparing groups of data because they now had to account for the extent of the overlap in data for the long-wing and short-wing helicopters.
Collectively, the model development sequence (Lesh and Doerr 2003) provided a framework to elicit (MEA), explore (MXA), and apply (MAA) a sequence of models grounded in students' experiences. The MEA elicited a need for a process to compare two distributions. In the MXA, students explored useful representations for comparing distributions. And in the MAA, they extended their statistical models for comparison in a new situation.
The teacher played a key role in guiding the development of students' models by—
- encouraging students to play and make sense of the flight of paper helicopters;
- introducing representations (e.g., a common number line for class data);
- asking students to interpret and critique representations (e.g., “Drop height seems to matter. How can we tell this from the class data?”);
- selecting student representations for the class to compare (e.g., figure 4);
- asking questions about characteristics of useful representations (e.g., “What are some of the properties of some of these [representations] that make that [height matters] easy to see?”); and
- privileging aspects of each discussion that promoted the overall goals of the sequence of activities (e.g., “Can you see anything that makes that [graph] easier to interpret?”).
These aspects of the teacher's role highlight two important instructional implications for using a modeling approach to learning mathematics. First, as seen in the unstructured play during the MEA and in the openness of the initial question for investigation, the children were free to approach the problem in ways that made sense for them. This generated a multiplicity of approaches as students decided how and what data to collect, interpreted their data, and critiqued the usefulness of various representations. The multiplicity of approaches generated opportunities to compare the usefulness of representations to convey information to others (“How is that one easier to interpret?”) and provide evidence for conclusions (“How can we tell?”). Thus, the sequence engaged students in the critical activity of the evaluation and improvement of models.
Second, rather than isolate concepts and skills such as constructing graphs or calculating averages, a statistical modeling approach emphasizes the power and relevance of statistics. The instructional goal was to build on students' own ideas and experiences so they could learn to develop a model for comparing two groups that could be used and reused in structurally similar situations.
Data from this article come from research funded by the Australian Research Council (DP120100690) and the University of Queensland, Australia. An expanded version of this article was published in the Statistics Education Research Journal (Doerr, delMas, and Makar 2017).
Doerr, Helen M., Robert delMas, and Katie Makar. 2017. “A Modeling Approach to the Development of Students' Informal Inferential Reasoning.” Statistics Education Research Journal 16, no. 2 (November): 86–115.
Lesh, Richard A., and Helen M. Doerr, eds. 2003. Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching. Mahwah, NJ: Lawrence Erlbaum Associates.
Katie Makar, email@example.com, an associate professor at the University of Queensland, Australia, was previously a classroom teacher for 15 years. She is interested in inquiry-based mathematics teaching and children's informal statistical reasoning.
Helen M. Doerr, firstname.lastname@example.org, professor emerita at Syracuse University in Syracuse, New York, is interested in modeling approaches to teaching and learning statistical reasoning.
Robert delMas, email@example.com, an associate professor in the Department of Educational Psychology at the University of Minnesota, is interested in developing activity-based approaches to advance students' statistical understanding.