Density Curriculum

Introduction

Challenges in Understanding Density

Emphasizing the Relational Aspects of Density

What causes objects to sink or float? Why are some clouds tall while others are flat? What causes currents in the ocean? What makes hot and cold weather fronts? From most students' perspectives, these are unrelated questions. From a scientific perspective, there is a common causal pattern underlying each of them—relative density. The relationship between two or more densities causes the outcome.

Understanding relative density begins with understanding a more basic relationship—the relationship of mass to volume that defines density. This module seeks to help students to understand density as a relationship, and to grasp the generative concept of relative density. Many density units emphasize that density is a property of materials, related to different "material kinds." While this unit does introduce the idea that density is a property, it also alerts students and teachers to ways that this concept can be problematic for learners as they try to extend their understandings to more complex phenomena. As elaborated below, it emphasizes the relational aspects of density and density concepts, and introduces conditions in which density is dynamic. Our research suggests that these concepts offer an effective intermediate model and a conceptual bridge to reasoning about density in complex phenomena.

Difficulties Understanding Density

A wealth of research demonstrates how difficult it is for students to come to a scientific understanding of density. Most students think of weight and density as the same thing1. They tend to focus on one feature of an object (either weight, size, or shape) or another kind of material (for instance, a liquid is often described as thin, thick, or loose)2. This limited focus is also found when students explain sinking and floating. Typically, they focus only on the object that they are testing to see if it sinks or floats3 ignoring the liquid that the object is in.

Without a clear concept of density, students often explain differences in objects of the same volume but different mass as due to one object being hollow or "filled with air." While this is one possible explanation (and is a case of mixed density—the density of the material surrounding the hollow space plus the density of the air inside), students often do not realize the possibility that the object is not hollow, but is made of a substance of lesser density.

There are a number of reasons, from a cognitive and perceptual sense, that students would be inclined to focus on the weight of an object (and miss its density), or focus on the sinking or floating object (missing the role of the liquid), locating the cause of sinking or floating in the entity itself. We explain each in turn below.

Non-Obvious Causality

Density is an intensive quantity. You can't directly see or measure it. It must be inferred by holding either volume or mass constant and assessing the implications for the other variable4. This typically creates huge difficulties for students5. Everyday experience does not necessarily provide opportunities for us to hold the volume or mass of an object constant in order to make the existence of density obvious. Weight, on the other hand can immediately be perceived or felt as one lifts an object. The obviousness of the object's surface features (felt weight and/or size) attracts students' attention, making it unlikely that they will look beyond these features to infer the existence of density.

In order to develop separate notions of weight and density, students need to reason about non-obvious causes. The relationship between density and weight is perceptually non-obvious. It only becomes obvious in outcomes where what one would predict based on weight is discrepant with what one would predict based on density, and some experience reveals the discrepancy. Children do have enough of these experiences to lead them to develop some intuitive sense of density6. For example, solid objects can be the same size but have different weights; very large objects can weigh less than much smaller ones. As children notice that these objects are made of different materials, they develop an intuitive sense that there are heavier and lighter kinds of materials. These intuitive notions, while helpful in some respects, can be limiting in others, as they further an object-based focus that is problematic when extended to more complex density problems, such as sinking and floating.

Relational Causality

Understanding density involves understanding Relational Causality. Scientists define density as the mass of a substance per unit volume. It is the relationship between the mass (or weight, on Earth) of one unit of a material and the volume of that one unit. Neither feature (mass or volume) is sufficient to define density. Students need to reason about the relationship between mass and volume and understand that if the relationship between them changes, the density changes.

Similarly, in understanding the role of density in sinking and floating, students need to reason about the relationship between the densities involved either between object and fluid or fluid and fluid. This relational type of causality involves recognizing that an effect is caused by the relationship, often one of balance or imbalance, between elements of a system. Neither element is the cause by itself. Thinking about Relational Causality requires a departure from linear, unidirectional forms of causality where one object or entity acts as a causal agent on another, affecting an outcome in one direction only like one domino hitting another domino7.

Research shows that students typically assume simple linear, unidirectional cause and effect models when analyzing scientific phenomena. These assumptions are evident from infancy8. Causes are often perceived of as embedded in entities. These default assumptions about causality can lead to static linear, entity-based models of density that generate a wealth of perceptual problems and misconceptions. For example, students think that the density within a closed system does not change (e.g. the alcohol in a glass tube that acts as a thermometer) or don't realize that the density of materials in different phases of matter changes.

Students' problematic tendencies are compounded by certain teaching practices. For instance, common practices include teaching specific densities for various materials without letting students know that density can change or referring to certain objects as 'sinkers' and others as 'floaters' without reference to the liquid. A common activity in the primary grades is to make a list of objects that sink and objects that float. This disregards the fact that most objects will sink in some liquids and float in others, and supports a linear static model that contributes to a range of difficulties for students later. It makes it difficult for students to understand weather patterns, ocean currents, the make-up of our atmosphere, and so on.

One might argue that from a developmental perspective, Relational Causality belongs in the middle school. It is certainly true that middle school students are in a good developmental position to learn Relational Causality. However, this does not mean that younger children cannot begin building these concepts, or that it is a good idea to stress causal models that are simpler but are a poor fit for the scientific concept when teaching younger children.

The illustrations below show preschoolers playing with the concept of Relational Causality in a concrete manner. With a clothesline pulley mounted to the ceiling and another on the rice table, they are figuring out that they can balance the bottles on either side if they put the same amount of rice in each, and that any imbalance causes one bottle to go up and the other to go down.

Clothesline Pulley 1 Clothesline Pulley 2

Attempting to reason about sinking and floating with an entity-based, linear causal model leads students to view the surrounding fluid as playing a passive role. This reinforces a linear conception. Only in dramatic contexts, such as dropping an object into a very dense liquid, does the liquid's role in the relationship as part of the causal agent become obvious enough to challenge the notion that equates the entity with the cause.

Further, an over-emphasis on material type without a sense of the ways that density is dynamic can create an apparent contradiction. Students need to reconcile the notion of density as a property of material type characterized by a steady state model with the notion of density as a potentially dynamic feature of that same material when certain conditions such as temperature and/or pressure change.

The Microscopic and Macroscopic Causes of Differences in Density

In order to support the idea that density is defined by the amount of matter in a given space, this module offers students the underlying atomic theory for why there are differences in density. Teachers of younger students may choose to skip over these lessons and use boxes in its place. Crowdedness models use boxes that show different amounts of dots in a given amount of space. Therefore, they capture certain aspects of the underlying atomic theory (amount of particles in a given space due to bond strength and structure). However, for older students, the underlying atomic theory can play an important role in supporting the bigger picture of how density is dynamic and why it is defined relationally.

The unit introduces three causes that contribute to density: 1) atomic mass; 2) the strength and structure of atomic and molecular bonds; and 3) mixed density. In any given instance, these causes are possible contributors to density. Therefore, some lessons in this part of the module focus on how to consider what it means to have multiple contributing causes.

  1. Atomic Mass: The first cause involves zooming in to the micro-level to consider the atomic structure of the substance. Some types of atoms have more protons and neutrons than others. This contributes to the mass of the material, because protons and neutrons have a significant amount of mass compared to electrons. It also results in more stuff in the same amount of space. The weight of an atom depends upon the number of protons and neutrons it has. This information can be found on the periodic table. However, you can't directly compare the density of different elements based upon the number of their protons and neutrons alone, because density can have multiple contributing causes. (For example, the strength of the atomic bonds and the subsequent crowdedness of the atoms may compensate for the mass of individual atoms.)
  2. The Strength and Structure of Atomic and Molecular Bonds: The second cause of differences in density also involves zooming in to the micro-level. It has to do with how the atoms are bonded to other atoms (either the same type or different types) to create molecules of pure substances, or compounds, or how the molecules are bonded to other molecules. In some cases, they are bonded very closely (such as in a metal). In other cases they are bonded loosely and there is more space between them, resulting in fewer atoms packed into a certain amount of space. With stronger (tighter) bonds, there are more atoms per unit of space. It is the strength of the bonds that counts; the bonds themselves do not contribute mass or matter because they are not things; they are electrical attraction.

    It also makes a difference how the bonds are structured. In a solid, how the atoms or molecules are bonded (the bonds of the crystalline structure) contributes significantly to density. In liquids, scientists don't understand the bonds very well and they are studying them to try to understand them better. However, there are different amounts of space between the bonds of different liquids. In gases, the most important variable in terms of density is how spread out the atoms or molecules are. The impact on density due to atomic mass and to the strength and structure of atom and molecular bonds is outweighed by how spread out the atoms and molecules are due to pressure and temperature.
  3. Mixed Density: The third cause is most easily talked about as mixed density. The clearest and easiest example of this is with gases (such as water molecules in the form of steam) when they spread out in a room and there are lots of "air molecules" in between. The density of the air in the room, therefore, is actually a mixture. Other examples include a sponge with holes in it. The state of the molecules affects how spread out they are and whether or not other types of atoms or molecules fit between them. For instance, Styrene is a dense liquid. However, it can be blown into Styrofoam so that it increases in volume and has air in the spaces (that has a mixed density).

    It is important to note that there are cases where molecules or atoms are spread out, but it is not due to mixed density. Instead of air in between, there is simply space. The structure of the molecules also affects how spread out they are. The molecules in many plastics (polymers) are long and curly so when they fit next to each other there can be spaces (with vacuum, gases or liquids in them). However, in our testing of the module, most middle students found these details to be confusing.

A Word About Culture

There are many forms of reasoning in science that may interact with students' cultural tendencies. Cultural experience acts as a filter for how we interpret events in our world and interact with how we learn about the world. Contrast a western view of the natural world as something to be dominated and controlled versus a Native American view emphasizing the need to live harmoniously and as part of nature. A science lesson where students are taught to isolate and control variables in order to determine their effects would look quite different through each filter, as would a lesson on the food web and the connectedness of the components of the web. These filters pose slightly different challenges to the learning and thinking in each lesson. The same is true for teaching about causal models. Some cultures may encourage a more relational view of the world, whereas others may encourage a more linear one. Being alert and sensitive to such differences is important in a culturally diverse classroom.

Using the Term "Mass" Versus "Weight"

Middle school students often find it difficult to achieve a clear understanding of the difference between mass and weight. This makes sense because our experience of the two concepts is completely confounded here on Earth. We would probably recognize their difference more easily if we spent time on different planets where the gravitational attraction is different due to the differences in mass between planets. This module uses "mass" throughout or "felt weight" in instances where observations rely on students' perceptions rather than measurements. However, for younger students, teachers might want to substitute the term "weight on Earth" for mass if it helps students grasp the concepts.

"Air Molecules" Versus Molecules That Make up the Air

The module uses the term "air molecules" in quotations since there really is no such thing as an air molecule. Rather, there are a number of different kinds of molecules that make up the air. If you use the term "air molecule," we suggest telling the students that it is just a shortened way of referring to the molecules in the air (things like Oxygen, Nitrogen, Hydrogen, etc.) but that there's actually no such thing as a molecule of air. There are molecules of particular gases that make up the air, so another way to handle this issue is to just talk about gases, and not substitute air molecules for gases.