How many dots do you see?
Did you have to count them, or did you just see the quantity at once? If you just saw the quantity without needing to count the dots one at a time, you subitized! Subitizing is the ability to see a quantity without needing to count.
Subitizing builds number sense such as unitizing, conservation, composing and decomposing numbers. Students first learn to perceptually subitize, which is what you most likely did when you first looked at the picture of the die. When we perceptually subitize, we instantly see the pattern and know the quantity.
But what about this? How many dots do you see?
Your brain had to do something different from what you did when you looked at the first die. You most likely saw three groups of four dots and then put that together to know that there are 12 dots. This is called conceptual subitizing, which is when you see smaller groups instantly and then compose them to make the total quantity.
Students need a bridge to be able to move from perceptual to conceptual subitizing by strategic arrangement of dots with known numbers. This bridge is a crucial piece that often gets overlooked because we assume that students will just know how to conceptually subitize once they can perceptually subitize. My colleague, Jamie, had her preschoolers figure out the total number of dots on a slide.
Her students came up with the answer of four dots and determined that they just saw the dots or subitized (yes, let’s teach children the correct mathematical language!). Then Jamie showed them this slide and asked them to figure out the number of dots and defend how they knew.
Her students still saw four dots, but this time, the discussion involved the understanding of numbers being composed of other numbers. The colors of the dots helped their brains see 1 + 3= 4. This is the bridge! For the final slide that day, Jamie showed her preschoolers this slide, and once again asked them to figure out the number of dots and defend how they knew.
Even though the arrangement of the dots remained the same, the discussion changed because of the colors of the dots. Students saw two dots plus two dots or two groups of two dots. In one brief number talk, Jamie intentionally grew her 4-year-olds from counting one by one, to subitizing a group, to the “counting on” strategy and seeing how numbers are composed of other numbers!
Once students demonstrate that they can perceptually and conceptually subitize, then they can move to the discussion of computation strategies. So how do you get students to move to more efficient computation strategies? Number talk discussions are key to having students share their computation strategies because we know that students come to us with connections and a sense about numbers already. But sometimes students are stuck on a computation strategy, like counting on, that will not prove to be efficient when working with larger numbers.
If we want students to build number sense, computation strategies and fact fluency, we need to design lessons and tasks where students explore and discover the strategies. Remember, the person doing the thinking and talking is the person doing the learning. Number strings are an effective way to get students to explore and discover a computation strategy. The purpose of a number string talk is to guide students down the path so they discover a specific computation strategy.
I worked with a group of students who had fluency with their doubles facts. So, I wanted to get them to see how they could use their doubles facts to derive answers to other fact problems. I showed students this slide:
The students said the answer was 10 dots because they saw 5 + 5 = 10. Then I showed students this slide:
The students saw nine dots this time because 5 + 4 = 9. Then I told them that good mathematicians are always noticing and wondering, so what did they notice and wonder about the pattern of dots on the first slide compared to the pattern of dots on the second slide? They noticed that one dot was missing from the second slide. The discussion led them to the conjecture that they could use doubles and then subtract one, so that if they saw the fact 5 + 4, then they could think, 5 + 5 = 10 and 10 – 1= 9, so 5 + 4= 9. We wondered when this strategy would work or when it would be best to use. We discussed that we would have to prove when this strategy works and doesn’t work because good mathematicians always prove their conjectures. The students were hooked to prove their new strategy and were eager to use concrete math tools to explore and investigate this strategy!
We can use subitizing to teach addition and subtraction strategies as well as multiplication and division strategies. I am excited to use number talk strings to teach my third-grade son multiplication and division strategies this year. I use the rekenrek tool a lot with students for subitizing multiplication and division facts. Here is a free digital rekenrek (starts at 20, and you can add rows to go up to 100)
Do you need number talk string slides that have been purposefully created to teach specific fact fluency computation strategies? You are in luck, because we have spent hours creating these for you!
We have created some amazing resources for you that will help you teach fact fluency strategies through subitizing! For our email subscribers, you will receive a code to download a FREE sample set of Perceptual to Conceptual and Fact Fluency Subitizing Slides (58 slides!). To download the resources in their entirety, visit our shop (these are on sale this week)!
Subitize 1-5: Perceptual to Conceptual (39 slides)
Subitize 6-10: Perceptual to Conceptual (39 slides)
Subitize Addition & Subtraction: Visualizing the Strategies (Doubles, Doubles +1, Doubles -1, Make a Ten, Friendly Numbers, Subtract all but 1, Place Value) (82 slides)
Subitize Multiplication & Division: Visualizing the Strategies (Doubles, Double & Double, Double & 1 More Set, Identity Property, Tens, Using Tens, Square Numbers, Decompose a Factor, Decompose Both Factors, Half & Double) (67 slides)
