Perhaps the interaction above sounds familiar?
Indulge me for a moment as I ramble on like an 'Ole Timer who is reminiscing about days gone by....
Yes, sometimes 1/8 is greater than 1/4, but that is a discussion for another day when we chat about the importance of identifying the whole. For today, let's talk about why my students could not even answer the simple warm-up question --- and let me begin by acknowledging that I had NO ONE TO BLAME BUT MYSELF!
Do you remember that I mentioned that this was my second year of teaching? My first year was in a 1st grade classroom filled with eager six year olds and shelves full of manipulatives (or Tinker-Tools as my colleague fondly called them!). Every day, we took out the bins full of blocks and counters and soft pom poms and we used these mathematical tools to build our math understanding. Adding 7 + 4? Well, let's take out 7 blue pom poms then we'll add 4 red pom poms. If we push them all together, how many pom poms do we have now? This is what math looked like every day in 1st grade – we counted things (yes, actual things), we moved things, we stacked things, we put things side by side to compare their sizes, we weighed things, we explored math concepts.
The next year when I changed grades and began teaching 5th grade, the blocks and counters (and most definitely the pom poms) went away and we replaced those concrete math tools with #2 pencils and eraser marks. That is why I had no one to blame but myself.
My 5th grade students, all of 10 years old, were not quite ready to be abstract thinkers – Piaget has been telling us that for nearly 50 years! During the elementary years, children rely on their concrete experiences to develop abstract thinking because children under the age of 12 are typically still functioning in the Concrete-Operational phase of learning. Rather than dive too deeply in the Piaget pool, let's talk about this topic in more practical terms
– let's talk about CRA.
CRA stands for Concrete-Representational-Abstract.
It is a three-part instructional strategy, with each part building on the previous instruction to promote student learning and retention and to address conceptual knowledge (American Institute for Research, 2016).
- During the CONCRETE phase of instruction, the students use concrete materials to explore the mathematical concepts. Counters to show a value of six. Actual containers of water to understand the concept of liquid volumes. Geoboards and rubber bands to create trapezoids and parallelograms. Attribute blocks to build an understanding of equivalent fractions.
- Those concrete experiences can then be REPRESENTED by creating drawings, using tallies, making little circles to represent the apples in the story, or other graphical representations of the concrete model.
- Finally, we come to ABSTRACT representations that use standard mathematical notations, such as written numbers, equal signs, and operation symbols (+ - x ÷) to represent the problem. This phase can be used in conjunction with the Concrete and Representational phases but can only be used alone once students have a conceptual understanding of the mathematical topic.
If I had allowed my 5th grade students to explore with these Tinker-Tools before asking, "Which is greater?", do you think any of them would have responded that 1/8 is greater?
(Nope, me neither)
Do you think they would have conceptually understood which was greater?
(Yep, me too!)
The learning progression is not a mystery....
In order to develop an abstract understanding of any topic, it must be rooted in a concrete experience. That is true for all of us, not just children.
Consider this personal example: My children have seen me drive my car hundreds, no, thousands of times. But when they were old enough to drive, just seeing a representation of driving was not enough – they had to have a concrete experience of what it felt like to press the brake to truly understand just how much pressure must be applied to smoothly stop the car (and I can attest that they needed that concrete experience many, MANY times before they truly owned that knowledge!).
In order to develop an abstract understanding of any topic, it must be rooted in a concrete experience. That is true for all of us, not just children.
Consider this personal example: My children have seen me drive my car hundreds, no, thousands of times. But when they were old enough to drive, just seeing a representation of driving was not enough – they had to have a concrete experience of what it felt like to press the brake to truly understand just how much pressure must be applied to smoothly stop the car (and I can attest that they needed that concrete experience many, MANY times before they truly owned that knowledge!).
So.... look around your classroom.... Do you have manipulatives that students can access? Are they hidden away in a closet? Are they only used when you decide they are needed and then under specific guidance from you? Tinker-Tools should be plentiful and accessible at all times in your classroom. A bit of warning, however, from the authors of the widely acclaimed book Teaching Student-Centered Mathematics about correctly using manipulatives in your classroom:
The most widespread error that teachers make with manipulative materials is to structure lessons in such a manner that students are being directed in exactly how to use a model, usually as a means of getting answers. There is a natural temptation to get out the materials and show children exactly how to use them. Children will blindly follow the teacher's directions, and it may even look as if they understand. A rote procedure with a model is still just that, a rote procedure.
– Teaching Student-Centered Mathematics, Van de Walle and Lovin (2006)
