Concrete to Pictorial to Symbolic/Abstract Thinking

The conversation of the development that students travel down when learning new concepts, that is the development from concrete to pictorial to symbolic/abstract thinking has popped up many times in discussions with colleagues. It is mentioned in almost every quality math resource. On page 80 in Marian Small’s resource Making Math Meaningful K to 8 she talks about the importance of making informed decisions about where students are on the continuum of  concrete to symbolic thinking and adapting practice to meet those needs. She also mentions that there is a place for all three levels of thinking throughout all the grades. I also recently got to ask her a question after her ignite talk at OAME about what her thoughts were on this development and she clarified something for me. She said she also wants students to know that they can use all three levels of thinking at different times from one task to another and one might work better than another for understanding the mathematics during a certain task. 

What was clear to me through listening to Marian and then reading about sequencing solutions using a concrete to symbolic approach from the 5 Practices, is that it is important to link the mathematical thinking between the different representations. Not sure if I have used the right wording there for what I mean by linking them but hopefully it is a little clearer by the end of the post.

Here is an example  of how a teacher I was working with and myself tried to do this. We used the classic caterpillar task with a grade 6 class.

Task: A grade six class needs 5 leaves  each day to feed two caterpillars. How many leaves would the students need each day to feed 12 caterpillars?

At first they may seem like it may not be too challenging for a grade 6 class but it is! I have used this task from grade 3 to 8 and even in the junior/intermediate grades we often see a solution of 60 leaves.


Here is a picture of the solutions we used:

File_002 (2)

We chose these three because we felt it showed the idea of solving this problem by using 6 groups of 5 to get an answer of 30 the clearest as we move from concrete on the right to symbolic on the left. We were able to link and show the students the 6 groups of 5 in all three representations. The tiles built in groups of two to five on the right, show how you can get an answer of 30. The pictorial representation in the middle shows the same thing but the students drew it and then labeled 5 beside each picture for a total of 30. The symbolic representation on the right shows the number sentence for 6 times 5 for an answer of 30.

The students also used some different strategies. One group found the unit rate of 2.5 leaves per caterpillar and then times it by 12 to get 30. We were going to use that one as the symbolic representation but then decided it didn’t link with the other solutions because they did not use unit rate and 2.5 times 12 is not evident in the other solutions.

Here are two pics of other solutions that came up:

All three are clearly solid solutions but we thought the other three fit better together to show the way you could answer this problem moving from a concrete to a symbolic way of thinking. Although we did think afterwards maybe the one on the right in this picture is a better pictorial choice because it somewhat abstracts the leaves by using circles to represent the leaves instead of actual pictures of the leaves. This is ideally what we want students doing instead of spending to much time drawing fancy pictures!

If we had chose to sequence our solutions on which ones show the multiplicative relationship in a rate problem the best it would have been a whole different order.How we ordered them and the solutions we chose seemed to fit best in what we wanted to show but looking back I think we would do it slightly different next time. I’m interested to hear some peoples thoughts on this or hear about ways you have sequenced solutions from concrete to symbolic. Thanks for reading!

About stamp36

Instructional Leader for Trillium Lakelands District School Board
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