A really awesome teacher I have the privilege of working with and learning from shared this really nice opening conversation she had one day last week with her kindergarteners. Keri used a visual routine* called “look quick” to start off their math block.
CARD #1: She set out this card, and asked the students, “How many dots do you see and how do you see them?”
She gave the kiddos a bit of time to look at the ten-frame card and think about it. When it seemed that they had enough time to think about her questions, she asked them to share their ideas. Many of her students told her that there were 4 dots, and the reason that was given was, “I just know it.”
At this time of the year, they just look at the pattern and know that there are four. And that is what we want–we want them to just know. “Young children are able to perceptually subitize, or visualize and recognize amounts (usually five or fewer), at an early age. (page 34 in Jessica Shumway’s book, Number Sense Routines)
CARD #2: Then, Keri set out the second card she had selected for the day’s look quick sequence, and asked the students, “How many dots do you see and how do you see them?” They acknowledged that there were 4 dots, but their explanations were a bit different than I just know it.
Lots of the kiddos said that the card had 4 dots because they could see two and two and that makes 4. (Two dots on the top row and two dots on the bottom row). There were a few who said that there are 4 dots “because I counted them and got 4.”
CARD #3: This is the last card Keri had selected for the look quick routine. She, again, asked the students, “How many dots do you see and how do you see them?”
The students had different ideas as to how they found the 4 dots. Some kids started in the top left corner and counted by ones. Some saw three on the top and one more on the bottom and that made 4. Others started on the bottom and counted by ones. A few made two groups of 2. However, they all agreed that there were 4 dots.
The students who used a strategy other than counting by ones are “conceptually subitizing; which means that they recognize small amounts and combine them to see them as a unit. Douglas Clements explains that ‘subtilizing is foundational to children’s number sense. He states that children use counting and patterning abilities to develop conceptual subitizing.’ This more advanced ability to group and quantify sets quickly in turn supports their development of number sense and arithmetic abilities.” (Shumway, page 34) Those students who saw two groups of two (2 + 2) and those who saw a group of 3 and a group of 1 combined the two groups to get one group of 4 are conceptually subitizing.
This is all happening because the kids have had lots of opportunities every day, all year to think about numbers, explore numbers, and talk about what the see and what they know and what they think. Keri has made it an essential and integral part of the learning in her class.
LAST QUESTION: Keri set out all three of ten-frame cards she used in the look quick portion of her warm-up and asked her students to talk about what they noticed. She used a slightly amended version of the I Notice, I Wonder™ strategy from The Math Forum at Drexel University found in the book Powerful Problem Solving by Max Rey.
This is what they said:
“They’re the same. They make 4.”
“They’re the same because it’s 4 but they are different because that card is 2 and 2, and that card is 3 and 1, and that card is 4 and 0.”
AWESOME kids. AMESOME teacher.
*FOOTNOTE: Jessica Shumway has a really nice piece about Visual Routines: Seeing and Conceptualizing Quantities in her book, Number Sense Routines. She talks about the use of routines in life and in math class, and how we can easily expand what we are already doing with math routines to include a number sense component. On page 22, Shumway talks about the different routines, what they each help with, how they work, and ways to use the routine and questioning strategies.
Keri took the idea of using routines to build number sense that we had explored during some grade level work sessions, and adjusted it to fit her students and their needs, and has had great results. She has lots and lots of engaged confident math kids.
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