In our work with early education teachers this fall, our AIMS center team, and our teacher colleagues have had a singular focus: counting. Even though we challenged the teachers this summer with the charge to “count everything” because we knew it was important, we see now that we underestimated the power of counting.
Since the beginning of the school year, we have chronicled fantastic growth in the number sense among the students of these teachers. In this blog post, I will revisit pieces of a blog I wrote last year to help examine why counting is such a powerful piece of building number sense.
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 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, young students come to associate these representations with the spoken word “five” as shown in the diagram below.
One common trait we notice initially in students with very little number sense was that their connections to the number words (such as “five”) were minimal. Some students 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.” Generally, counting was not associated with numbers in these students’ minds.
The counting experiences that we work with teachers to design offer students fun opportunities to make connections among MANY of the different representations of “five” in an authentic, playful environment. Our criteria for game/activity design were that they should be:
- authentically playful
- potentially perturbing (for self-motivated learning)
- able to be easily modified
- able to make student mathematical thinking visible.
To build number sense, students MUST come to know number words (in this example, “five”) as rich concepts with as many representations as possible and with repetition. The charge of “count everything” helps accomplish exactly this. As students 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?
Students with a flexible understanding of “five” create replacements when they can’t perceptually touch, see, hear, or feel items to be counted. Because they have counted to five so many times, they can imagine five 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” counting items. Students 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 and learning “addition.”
It is amazing that such simple experiences with counting in different contexts open up a complex world of opportunity for students to become flexible mathematical thinkers. After seeing what we have seen this fall, I can confidently say to early educators, “Have your students count, count, count! Count everything!” As educators, we can’t afford to miss this opportunity!
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