Back when I was pursuing my teaching degree, I learned that students learn by doing. As we studied Bruner and Piaget and other great thinkers who influenced mathematics teaching and learning, we learned about three stages in which students represent their mathematical thinking:
Since then, I’ve learned that “students learn by doing” is only a partial truth. I’ve come to discover that students learn by thinking about what they’re doing (Kamii, 2007). It’s in the reflection that meaningful learning takes place, not simply in the “doing” itself. Furthermore, the three stages of representation aren’t really stages at all. They are classifications, or categories, of representation. They do not necessarily occur sequentially, nor is one category of representation necessarily more sophisticated than another. Instead, the thinking behind the representation is what determines the level of sophistication.
To that end, I’ve adapted an existing framework to explain how students represent their thinking. The Lesh Model, pictured below, includes the three original representations — concrete (physical), pictorial (visual), and abstract (symbolic). And it also includes two additional categories: verbal and contextual. The addition of these two categories introduces more inclusive ways in which we use math to describe the world around us.
WHAT’S THE POINT HERE?
We see from the diagram that each representation is connected to every other representation. Since each category is connected to every other category with a two-way arrow, one can see that the categories interact with one another rather than occurring in a set sequence. It’s when students make these connections between and among representations that true learning occurs.
Let’s take a closer look at the five representation classifications:
- Physical representations use concrete objects to facilitate thinking about thinking about, representing, and manipulating math ideas. Concrete objects may include tiles, counters, paper strips, etc. Note that just because an object is physical does not mean that it concretely represents a mathematical idea. For example, coins are physical objects, but they do not concretely represent their mathematical values.
- Symbolic representations convey ideas through formal math representations such as numerals, equations, variables, and other symbols.
- Visual representations such as pictures, sketches, diagrams, charts, number lines, graphs, etc. use visual means to promote thinking about math ideas.
- Contextual representations situate math ideas in real-life, every day, imaginary, or math contexts.
- Verbal representations use oral or written language to describe, discuss, interpret math ideas.
WHY DOES THIS MATTER?
When looking at the Lesh Model, you probably focused your attention on the five representational displays: physical, symbolic, visual, contextual, verbal. The real power of the diagram, however, lies with the arrows that connect those representations. With the added arrows, the Lesh Model, originally called the Lesh Translation Model, emphasizes the ways in which the different representations are connected. The arrows call our attention to the notion that meaningful learning takes place during the thinking, processing, and reflection upon what one has done. By connecting representations, students develop and display their thinking in new ways.
TIPS FOR TEACHERS:
- Take inventory of the supplies available to students that encourage engagement in all five types of representations. What manipulatives are available? Have you provided students with a variety of writing materials? Are there different types of paper available to them? Is there opportunity to contextualize math — both in your discussions and on their paperwork? Is there a list of various symbols and their meanings posted somewhere in the room?
- Create anchor charts for each of the five categories of representation. Be sure to list or add pictures of many examples for each category.
- Frequently ask students to verbally connect multiple representations. Put a Lesh Translation Model up in the room and have students trace the arrow that shows the connection they are making (e.g., trace the arrow between physical and visual when explaining the connection between a fraction tile representation and a number line sketch).
TIPS FOR LEADERS:
- Lead a conversation with your teachers about the different categories of representation. Ask them to solve a math problem together and then lead a discussion about how to connect the different representations to one another.
- When observing a math class, notice if students are connecting representations in addition to simply following a process of using manipulatives the way the teacher demonstrated.
- Ask your teachers if their inventory of materials for each of the five categories of representation is lacking. Are there any tools that need to be purchased or located to support mathematical representation?
More on this topic of representation in the coming weeks…stay tuned!
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.