Interpreting data often entails finding and confirming relationships in the data, and these relationships can have varying levels of complexity. Simple linear relationships are easier to spot than inverse relationships or interactions. Students may often fail to consider that more than one type of relationship may be present. For example, children investigating the health of a population of finches may wish to examine the weight of birds in the population. The weight of adult finches is likely to be a nonlinear relationship. That is, as both low weight and high weight are disadvantageous to survival, one would expect to find a number of weights in the middle, with fewer on both ends of the distribution.
The desire to interpret data may lead to the use of various statistical measures. These measures are a further step of abstraction beyond the objects and events originally observed. For example, understanding the mean requires an understanding of ratio. If students are merely taught to “average” data in a procedural way, without having a well-developed sense of ratio, their performance often degrades, mistakenly, into procedures for adding and dividing that make no sense. However, with good instruction, middle and upper elementary students can learn to simultaneously consider the center and the spread of the data.
Students also can generate various mathematical descriptions of error. This is particularly true in the case of measurement: they can readily grasp the relationships between their own participation in the act of measuring and the resulting variation in measures.
Scale models, diagrams, and maps are additional examples of modeling. Scale models, such as a model of the solar system, are widely used in science education so that students can visualize objects or processes that they cannot perceive or handle directly.
The ease with which students understand these models depends on the complexity of the relationship being communicated. Even preschoolers can understand scale models used to depict location in a room. Elementary school students can look beyond the appearance of a model to investigate the way it functions. However, extremely large and small-scale models often pose serious challenges for students. For example, middle school students may struggle to work out the positional relationships of the earth, the sun, and the moon, which involves not only reconciling different perspectives (what one sees standing on the earth, what one