Welcome back to our deep dive into mathematical representations! Today, we are taking a look at symbolic representations and how we can translate between symbolic, concrete, and visual representations. First, let’s do a two-sentence recap of this series so far:
We have already focused on concrete representations and the immense value of manipulatives, the range of visual representations we want to encourage with our students, and how we can use numerals and operations to represent thinking symbolically. We are centering our conversation around Lesh’s Translation Model, which encompasses the range of ways we represent our thinking, and stresses the importance of making connections between representations.
Today we are talking about verbal representations. While it’s an essential form of representation for our students, it is ignored by the very popular Concrete-Pictorial Abstract model. This is one of the key reasons I have moved away from that model and gravitated more towards Lesh’s Translation Model (which can be seen above).
The language we use to communicate our thoughts and ideas is another equally important representation. This can be oral, written, signed, or any way that a student would look to communicate language. James Heddens writes that students “need to be given opportunities to verbalize their thought processes: verbal interaction with peers will help learners clarify their own thinking.”
If we go back to our previous examples from concrete, visual, and abstract thinking, we have a student with five yellow counters and four red counters. The student then sketched their counters and wrote the number sentence 5+4=9 on their paper.
So how does verbal representation come into play? Perhaps after the activity, a student shows you their sketch of the counters. When you ask them about their drawing, they may share “I had nine counters. Four of them were red and five of them were yellow, and that makes nine.” That statement is a verbal representation of the concept. They have also just translated their visual representation to a verbal representation.
Connecting Two Verbal Representations
If wanted, we could take it a step further by asking the student to write their thoughts down. This will require the student to revisit their thoughts communicated orally and condense them into a written description, like “Four counters and five counters make nine counters.” This extra step of condensing their language into a second form, allowed students to connect two verbal representations. WOW!
Verbal representation is essential to our work, especially in the early grades. Our students who may not have the ability to write words or numbers will often communicate their understanding orally. This NEEDS to be a part of the discussion when we talk about deepening student understanding, and it’s a huge reason why I far prefer Lesh’s Translation Model to the Concrete-Pictorial-Abstract model.
What’s Up Next?
This series is going to dive deep into each of the representations discussed in Lesh’s Translation Model, and then we are going to put it all together so we can make a big impact on your math teaching this year.
If you missed Part One about Concrete Representations, Part Two about Visual Representations, or Part Three about Symbolic Representations, check them out so you have all of the info you need before we move on!