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.

Don't teach concepts to children

Yep! You read it right. Don't teach concepts TO children.

Instead, give your students opportunities to EXPLORE concepts.

Let them DISCOVER the complexities of concepts by INVESTIGATING concrete examples from a variety of times, places, and situations.

In doing this, children will CONSTRUCT UNDERSTANDING of these timeless, abstract, universal, and transferrable ideas that we call concepts.

Letting students construct their own understanding of life's BIG IDEAS is known as inquiry, which is more than question asking. Teaching in this way doesn't come naturally to me, so until it does, I've been trying to follow these five steps.

1. Start with a concept.
2. Pick a specific, concrete example of a person, place, situation, or thing that illustrates that concept.
3. Create an opportunity for students to explore that concrete example. 
4. Check for understanding by having them write a concept statement.
5. Reflect on their thinking and decide next steps.

If you think using these steps could help your students construct their own understanding of life's BIG IDEAS, try them out and let me know how it goes. That is exactly what one second grade teacher did recently. Here is her story:


1. Start with a concept.

In grade 2, teachers collaboratively decided to have their students explore how people influence the world. During this unit of inquiry, students also explore the genre of biography, as they read about various heroes.

2. Pick a specific, concrete example of a person, place, situation, or thing that illustrates that concept & 3. Create an opportunity for students to explore that concrete example.

During the last several weeks, students in second grade have read, listened to, and watched biographies and have learned a lot about specific heroes.

After all this learning about individual heroes, the teacher posed two questions to the students: "Who is a hero?" and "What is a hero?"

Students responded using the Chalk Talk thinking routine (for more on Chalk Talk, see p. 78 of Making Thinking Visible by Ritchhart, Church, and Morrison). In the last step of the thinking routine, when students were sharing the thinking, they identified common themes and ideas that emerged. Those were notated by the teacher and are written in black marker and boxed in the pictures below.

4. Check for understanding by having them write a concept statement. (This step is very similar to the thinking routine Headlines found in Making Thinking Visible by Ritchhart, Church, and Morrison, p. 111)

The next day, the teacher simply gave students a blank piece of paper and prompted them to write all they understood about heroes. The following were the student responses:

They stand up for people.

(they) doing awesome

Then, the students worked collaboratively to create one, concise concept statement.

When each individual statement was read, the teacher confirmed with the class if it was true or not. Then, students had to decide if the statement had any new information to be added; if it said anything different.

If it did have some new and different information, students had to work together with their teacher to revise the existing concept statement. The images below show the progression the concept statement went through as the class collectively constructed their understanding of the concept of hero.

First, the students started with ...

Next, they added a couple of attributes and took out the pronouns.

2nd graders then added how heroes did this.

To fit better with the vocabulary of the IB Learner Profile, they decided to replace kind.

2nd graders realized that heroes don't just help people.

The students struggled with what is the "right" thing. That value-laden statement is based on one's perspective and is not an absolute truth, so they knew they had to get rid of it.

But the students felt something was missing when they took out that phrase, so they put the idea back in, while acknowledging that "right" is subjective, based on the perspective of the "hero".

At this point, the teacher invited me into the conversation to see what a fresh eye could add. To make sure that "hero" was front and center, I suggested moving the dependent clause to the end of the independent clause.

Then, I suggested to the students that they change some of the wording so that it sounded more like a second grader. Often times when crafting concept statements (sometimes called generalizations, central ideas, or enduring understandings) we create hard-to-understand statements that don't sound like how we normally talk. Concept statements should be written in language that is easy to understand to all who read it.

Finally, I had the students test the statement to see if it was true in any time and in any place. To do this, we briefly brought up "heroes" from different times and from different places that they had already learned about. We decided their definition of a hero held up when applied to all those heroes, so it was indeed timeless and universal.

The students were comfortable that their concept statement encapsulated all the learning they had done up until that point, so the teacher wrote it down on their whiteboard.

5. Reflect on their thinking and decide next steps.

The second graders' understanding of hero is certainly deep and complex. The next step now is to continue to explore the concept of hero: read more biographies, watch age-appropriate biopics, read age-appropriate news articles, interview heroes local to their community, reflect on how their own actions could lead others to believe they themselves are heroes, etc.

As they continue to learn about other "heroes", second graders should be encouraged to continually revisit and revise their definition of hero as needed.

After reading about how one second grade teacher set up opportunities for her students to EXPLORE the concept of hero, instead of directly teaching this big idea to her students, how could you or have you had your students construct understanding of life's BIG IDEAS, known as concepts?

Teaching Conceptually using Tug-of-War

Currently, our G4 students, with the help of their teachers, are trying to understand that maintaining health requires knowledge of body systems and the diseases that affect them. They are trying to understand this big idea through the conceptual lenses of causation and responsibility as they inquire into one’s personal responsibility for health.

To guide students through this inquiry, a G4 teacher recently used the Tug-of-War thinking routine from Making Thinking Visible by Ritchhart, Church, & Morrison (p. 199). After working through this routine with her students, the teacher invited others on her team to analyze, discuss, and reflect on her students’ thinking. Here’s her story:

1. Start with a concept.

For this particular learning engagement, the teacher wanted the students to understand that choices individuals make not only affect the individual, but others around them too.

2. Pick a specific, concrete example of a person, place, situation, or thing that illustrates that concept.

This teacher decided to guide her students through an exploration of the recent measles outbreak in the United States. Investigating the facts around this concrete situation would allow her students to construct understanding of the big idea that choices individuals make (particularly about their health) not only affect the individual, but others around them too.

3. Create an opportunity for students to explore that concrete example.

First, students read and viewed a diverse variety of complex texts.

• LA Times: Disneyland measles outbreak: Infected woman took 2 flights before diagnosis
• The Guardian: Measles outbreak spreads in US after unvaccinated woman visits Disneyland
• Public Radio: Disneyland measles tally rises, unvaccinated OC high school students kept home
• National Vaccine Information Center (NVIC): Vaccine Laws

The students went through the texts as a whole group, a format appropriate for reading such complex texts. To understand more about how to support students as they read complex texts, learn from the experts: Fisher & Frey’s article “Close Reading in Elementary Schools”.

After learning more about the facts around this recent outbreak, the teacher drew a tug-of-war rope across the middle of the whiteboard. The class identified and framed the two opposing sides of the dilemma that they were exploring. They labeled one end of the tug-of-war rope “Yes, all people should have to get a vaccine” and on the other end of the rope they wrote “No, people should be allowed to file an exemption.”

Students then generated “tugs” or reasons that “pull you toward,” that is, support each side of the dilemma. The students wrote their ideas on individual sticky notes. Because this was the first time the teacher had led the students through this thinking routine, she required each of them to write a tug for each of the opposing sides.

After, as a class, the students determined the strength of each tug and placed it on the tug-of-war rope, placing the strongest tugs at the farthest end of the rope and the weaker tugs more toward the center.

4. Check for understanding by having them write a concept statement.

The teacher did not have her students write out a concept statement at the end to check for conceptual understanding. She did say that having the students make their thinking visible on the sticky notes and listening to the ensuing discussion was one way to check for conceptual understanding. However, to be assured that she knew what all of her students understood about the concepts of decisions, their consequences, responsibility, and causation, the teacher decided that next time as a follow-up, she would want the students to draft some statement at the end to check for their understanding of these timeless, abstract, universal, and transferable concepts.

5. Reflect on their thinking and decide next steps.

During the reflection meeting, other important points came up:

• The students were very engaged during this thinking routine.
• The thinking routine “I Used to Think …, Now I Think …” could have also been used in combination with Tug-of-War to be able to show students how their own thinking about this topic had changed.
• Right now, students are learning about determining what is fact and what is opinion during their focused literacy lessons. As students were determining the strength of each tug, they naturally started classifying tugs with facts as stronger and those that had opinions as weaker. The teacher noticed that students were applying their understanding of fact and opinion in a real-life context, thus solidifying their understanding of this literacy concept.

After reading about how this teacher used the thinking routine “Tug-of-War” with her students to examine perspectives, reason, identify complexities of a current event and construct a deep understanding of concepts, how could you or have you used this thinking routine with your own students?