The Child's Ideas for 2. Heavy for Size

Why is Density So Hard for Students to Learn and for Teachers to Teach?

if you were holding small and much larger pieces of steel, nothing would tell you that the two objects have exactly the same density

Density has a reputation for being an extraordinarily hard concept for students and teachers alike. Indeed, it is common for many adults to have difficulty with the concept and for teachers to report that it is frustratingly hard to teach. What makes it such a thorny idea both for students to learn and teachers to teach?

One source of difficulty is that, unlike an object's weight, we don't have direct perceptual access to an object's density. If you hold two objects in your hand, you can judge (albeit somewhat crudely) whether one object is heavier than the other or whether they weigh about the same. However, comparing the weights of two objects only gives you some indication of their relative densities primarily when they are roughly the same size. If you compare a small (light) piece of steel with a much larger (and heavier) piece of aluminum, your senses tell you the aluminum piece is larger and heavier, not that it is less dense than the steel. Further, if you were holding small and much larger pieces of steel, nothing would tell you that the two objects have exactly the same density. Our senses simply tell us that one piece is larger and heavier.

Density is fundamentally a relational concept. It can't be defined ostensively by simply pointing to an object and saying “This one is dense.” Rather, density refers to an inferred relation between weight (or mass) and volume that uniformly characterizes objects made of a given material under standard conditions of temperature and pressure. Thus, building a concept of density depends upon relating many other concepts—weight, mass, volume, material, measurement, proportionality—that themselves are all under development by the young child.

A concept of density emerges as children puzzle about and seek deeper explanations for why objects weigh what they do or why some things might sink or float in a given liquid. An intermediate step is for them to form generalizations that some objects are made of heavier kind of materials than others (e.g., steel is a heavier kind of material than aluminum; aluminum is a heavier kind of material than wood). However, there are several challenges in helping them take this important step.

First, children initially think that small objects weigh nothing at all because they regard weight as felt weight, rather than as a measurable property related to the amount of material in the object. Helping them see that the weight of an object is equal to the sum of the weight of its parts—engages them in building an initial explanation of why objects weigh what they do. Understanding that tiny pieces of any material have weight, no matter how small, is necessary to prepare them to form generalizations about the weight differences of materials.

Second, children rarely have experience holding solid chunks of different materials in everyday life. They are usually handling specific objects that may be fashioned from multiple materials—such as pencils or toy cars—or objects that may be empty inside or filled with different materials—such as steel, aluminum or plastic cans that may be empty or filled with liquids like soup. Further, children have limited knowledge of what materials comprise these everyday objects. Consequently, some of the first ideas children develop about why things are heavy or light relates to how big they are or whether they are empty or full rather than generalizations relating weight to kind of material. Another step is to give them opportunity to explore different kinds of materials, as we do in the 3rd grade Inquiry curriculum, when children work with eight equal volume cubes made from a range of different materials (woods, plastics, and metals.)

Third, children initially use their pre-existing ideas to understand this new experience with the material cubes. They form generalizations such as “steel is heavy,” “aluminum and wood are light,” rather than the more nuanced “objects made of steel are heavier for their size than objects made of aluminum” or “steel is a heavier kind of material than aluminum.” Thus, students do not fully realize that these generalizations call for a fundamentally different sense of weight—heavy for size rather than total weight—and are about a fundamental property of materials that is the same across all sample sizes for that material. For example, children who have seen that a large chunk of steel is heavier than a same size chunk of aluminum may not infer that a sliver of steel must also be heavier than a sliver of aluminum that is the same size. Instead, they simply say the slivers are both light, or both weigh nothing because they are making direct perceptual comparisons of their felt weight, rather than reasoning using knowledge of constant weight/size relationships of different materials.

Fourth, in their experiences comparing the material cubes, the object made of the denser material is also absolutely heavier. In order to get them to more clearly differentiate heavy and heavy for size, they need experiences where the object made of the heavier kind of material is not also absolutely heavier: for example, experiences where two objects made of different materials are the same weight (on a balance scale), but quite different in size. In this case, the object made of the heavier kind of material is much smaller than the object made of the lighter kind of material, reinforcing it is heavier for its size. We provide this further experience in the 4th grade curriculum.

All of these factors help explain why density is so difficult to teach. It is tempting for teachers to introduce students to the concept of density relatively late (in middle school) in the context of giving them a formal mathematical definition (density = mass/volume). But this “final form” teaching of density gives students too little too late. It ignores the fact that the beginning of the teaching of density must occur implicitly well before any distinct words or formulas are introduced by helping students understand that: even tiny things have weight and take up space; weight and space are measurable quantities; objects made of some materials “weigh” more than others; and there are important nonverbal precursors (for example attending to heaviness for size) to the concept of density that don't look like its final form version. This is what the Inquiry Project, with its cross-grade learning progressions approach, seeks to do.

—Carol L. Smith