What is Five?: Foster a Flexible Understanding of Number

What an opportunity I have here at the AIMS Center!  We have been afforded the time and resources to read mountains of research about how children come to understand number, replicate that research, and then begin designing mathematical learning opportunities that help children along while eliciting their mathematical thinking.  After a period of reflection upon this year’s work, I would like to share some key takeaways.

As the collage above shows, there are many different ways to represent the number five.  In the middle (placed strategically), you will see the spoken word “five”.  You also see other visual or physical representations of the number five: taking five steps (kinesthetic movements), numerals (5), dice patterns, recognizing finger patterns, COUNTING finger patterns one finger at a time, sequentially raising/counting fingers one at a time, counting other perceptual items, the written word “five”, etc.  Because language is the fabric that binds human communication, and with many experiences, children come to associate these representations with the spoken word “five” as shown in the diagram below.

One common trait we noticed in preschool children with very little number sense was that their connections to the number word “five” were extremely limited.  Some children assigned a number word to a person (i.e., “I am 4; Jose is 2).  Some thought “five” meant for them to put up one hand with all fingers extended.  To others, “five” meant a verbal exercise, saying, “1, 2, 3, 4, 5.”

The research-driven learning experiences that we are designing offer children fun opportunities to make connections among MANY of the different representations of “five” — in an authentic play environment.  Our criteria for game/activity design were that they should be: authentically playful, child-initiated, child-directed, socially interactive, potentially perturbing (for self-motivated learning), able to be easily modified, and able to make student mathematical thinking visible.

To build number sense, children MUST come to know number words (in this example, “five”) as rich concepts with as many representations as possible.  As they construct, they have the opportunity to become FLEXIBLE in their understanding (see below).  This flexibility allows them to become competent problem solvers at a young age.  So what do they do?

Children with a flexible understanding of “five” create replacements when they can’t perceptually touch, see, hear, or feel items to be counted.  If they can imagine 5 blocks, they can represent those by: counting a “visualized” (in the mind) dot pattern they know, by putting up five fingers and counting them, or by counting fingers as they raise them.  These replacements are what the research calls “figurative material”.  Children being able to use this figurative material to solve problems is the BRIDGE between counting things they can see and understanding number words as abstract concepts.

So looking back on this incredible year here at the AIMS Center, I am amazed to see that seemingly simple experiences around number words open up a complex world of opportunity for very young children to become flexible thinkers.  As educators, we can’t afford to miss this!


2 Responses to What is Five?: Foster a Flexible Understanding of Number

  1. Mike,

    We have spent some time discussing those differences around the idea of imagination. There certainly are varying degrees of imagination at work for each of those mediums.
    What are problem-solving, creativity, and innovation? They seem to me to be one’s imagination putting figurative material together in a new or different way.
    Here’s a link to an interesting video I found yesterday about imagination:


    Thanks for your input.

  2. High five to you on this fine article!

    Or perhaps high 5?

    High 1,2,3,4,5?

    Very interesting take on the nature of figurative material. Wondering if there is a connection to the ways that people also make literal and figurative sense of the world in every aspect. It makes me wonder about the differences between reading a book, listening to an audio version, or watching the movie version. What might be the difference in the use of (for lack of a better phrase) brain power?

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