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!
As a teacher, I want to know what I can do to support my students’ mathematical development. One of the new “teacher moves” I have learned from our research into Student Adaptive Pedagogy is giving a child a “recent experience of counting.” Today I want you to watch a video of me interviewing Eden. She… Continue Reading
I remember getting a Rubik’s cube when I was younger. I sat down immediately scrambled it up, put in the Def Leppard cassette on my Walkman and set out to solve the puzzle. It soon became apparent to me that it was not going to easy or quick. I would play with it for a… Continue Reading
Tiger got to hunt, Bird got to fly; Man got to sit and wonder, ‘Why, why, why?’ Tiger got to sleep, Bird got to land, Man got to tell himself he understand. -Kurt Vonnegut This poem nicely sums up part of our human nature in that we are at rest when we understand our experiences.… Continue Reading
As educators, we have all heard of the benefits of wait time after you ask students a question. In this video, you will see proof of it. At the beginning of our pilot with K-2 teachers we shared many student videos and, although they noticed a lot of student’s math behavior, they also noticed how… Continue Reading
Today we are going to watch a video of Sarina. This video was taken August 28, 2018. She is a great example of what productive struggle looks like in a student. She is going to solve 8 + 5, 9 + 4, and 8 + 6. Some questions to think about are: Did she solve… Continue Reading
This blog is related to a multi-part series titled “Creating Centers in the Classroom.” If you have missed that series, you can read part 1 HERE, part 2 HERE, part 3 HERE, part 4 HERE, and part 5 HERE. You are ready!! You’ve planned your centers and are excited about differentiating them based on student… Continue Reading
Currently, I am in the midst of raising a 16-year-old who has been learning to drive. Recently, I was reminded of what it feels like to do something new while I was doing a math demo lesson in a classroom and how circumstances can bring us back to those early behaviors. Fog. Here in the… Continue Reading
Counting. Nothing in mathematics gives us more of a return on our investment. In my first blog post in this series, we briefly looked at beginning counters, who start with rote counting and then move on to count concrete items. These types of counting are huge investments in their mathematical futures. In the second post,… Continue Reading
Have you ever had one of those “aha!” moments when the light bulb goes off, and you come to understand something that makes your whole life easier? Or better yet as a teacher, you witness your students have one of those “aha!” moments. For us, at the AIMS Center for Math and Science, we have… Continue Reading