Using concrete models to represent abstract concepts

Third graders in Minnesota need to understand how time, money and temperature can be used to solve real-world and mathematical problems (Math Academic Standard 3.3.3.1). This includes being able:

• to tell time to the minute, using digital and analog clocks and 
• to determine elapsed time to the minute.

Time, and especially the passage of it, are particularly abstract concepts that many third graders have difficulty understanding. Third grade educators, and other teachers of young children, must set up instruction that allows students to move from concrete to abstract (Making the PYP Happen, Subject Area Annex, Mathematics in the Primary Years Programme).

Recently, a G3 teacher introduced a model to her students that they could use to concretely represent the abstract concept of elapsed time.

The model allows students to show time passing on a timeline. Students represent the passage of time with different geological features:

• A mountain = 1 hour
• A hill = 30, 20, 15, 10, or 5 minutes
• A rock = 1 minute

The teacher explained to her students that these different features represent different amounts of time relative to the amount of time it would take to climb over one (i.e. it takes longer to climb over a mountain that it does to climb over a hill, whereas jumping over a rock takes relatively little time.)

The class used this anchor chart as they learned how to use the model.

The following are some examples of students using the concrete model to represent the abstract concept of the passage of time.

  

In this example, the child jumps over a small hill (10 minutes) and a rock (1 minute) before jumping over a larger hill (30 minutes) in order to get to an even time (5:00). 

When students are required to show their thinking in this way, they're able to justify their thinking (abstract) more easily because they can refer to the model (concrete). Thus, young learners can initiate, explore, discuss, document, and manage their thinking about abstract concepts, if given the appropriate structure (Defining Thinking Routines, Ron Ritchhart).

This same model could also be used to make sense of other abstract mathematical concepts, such as

• the passage of time on a greater scale (decade, century, millennia).
• representing multiplication as jumps on a number line.
• making change to a dollar could be concretely shown using this timeline model.

How could you use this concrete model (or other models) with your students for them to make sense of an abstract concept?

Click here to learn more about the Elapsed Time Mountain Strategy.