Author Archives: Jason Chamberlain

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!


The (Math) Things Kids Say

Children’s thoughts about mathematics are reflections of their experiences (Steffe, von Glasersfeld, Richards, & Cobb, 1983). Let’s take a look: In the AIMS blog, there has been a good bit of talk about the “mathematics of children” (Steffe, 1991).  This covers a lot of ground.  When we talk about the way that children construct math… Continue Reading

Meet the Children Where They Are: Episode 4

Narrator:  “Although this series makes this interaction seem like a long period of time, it was actually only about 3 minutes…well, maybe 5!  Time flies when you’re having fun, okay?  Anyway, can Bob count two hidden piles of rocks?  Let’s get back to the action.” (Math research associate narrates…) RA:  So it was the bottom… Continue Reading

Meet the Children Where They Are: Episode 3

Narrator: “Last time on Meet the Children Where They Are: Has Mr. Unnamed nerdy-looking stereotypical math Research Associate (RA) found 5-year-old Bob’s zone of potential construction (ZPC)? Bob has counted 12 rocks and he came up with three different solutions.” (from title screen — video of construction site with concrete truck backing up) (fade to… Continue Reading

Meet the Children Where They Are: Episode 2

Narrator: “Last time on Meet the Children Where They Are: Mr. Unnamed nerdy-looking stereotypical math Research Associate (RA) is down in the mud getting ready to do some math with an unsuspecting 5-year-old child named Bob. Can he find Bob’s ZPC without damaging the child for life? Let’s head back to the construction site where… Continue Reading

Using Fingers to Build Number Sense

It has been a lot of fun to get back into the classroom during the last few weeks!  The Early Mathematics research associates have been interviewing students (3- and 4-year-olds) to see how they say the number word sequence, how they count objects, and how they subitize.  These were our initial interviews from which we… Continue Reading


Why? Why do you go to work daily? Why are you good at what you do? Why are you reading this mathematics blog? Why are students engaged in some math lessons and not others? Why do some students do well in school while some fall behind? I’m going to leave you, the reader, to answer… Continue Reading