Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

I’m currently working with my Kinders and Firsties on subitizing. If you’re scratching your head about the word “subitizing”, you might not be alone. Consider this–how many fingers are displayed in the picture above. I’m pretty sure you answered eight. I’m also reasonably sure that you did not count the fingers one by one. That, my friends, is subitizing. Subitizing is a standard in both Kindergarten and 1st grade in Texas, although the TEKS don’t specifically mention the word:

K.2.D Recognize instantly the quantity of a small group of objects in organized and random arrangements

1.2.A Recognize instantly the quantity of structured arrangements

Subitizing allows students to move away from counting by ones to seeing numbers as chunks. For example, in the picture above, eight is seen as a chunk of five (the fingers on one hand) and three more. That’s much quicker than counting each finger. It also leads to a student understanding that 5 + 3 = 8.

So what are organized or structured arrangements? The pips on dice are a great example of an organized or structured arrangement. We instantly recognize the way the numbers look as represented by the pips on a die.

Another great structural tool is the ten-frame. First, we want students to recognize that if a row is full, it’s five and can be counted as five without counting each dot. That’s a huge step! I call it a “fast five”. Once they recognize that the full row is five, students can begin to count on. We run our finger across the top row and say “5” and then point to each dot as we count on (6, 7, 8). In time, the students will just recognize that the arrangement shows 8. Some might look at the empty spaces and know that two empty spaces means 8. Asking students how they know it’s eight is a terrific formative assessment, and one you can’t get with a paper and pencil activity. Think about what each of the following responses means in terms of understanding and number sense:

I counted all the dots.

I said 5…6, 7, 8

I know that 5 and 3 is 8

If all the spaces were full it would be 10, but 2 are empty so it’s 8

I can’t overstate how much practice Kindergarten and 1st graders need with this skill. It should be a part of your daily math instruction all year long. I keep my ten-frame flashcards on an O-ring, so I can easily grab them and spend a couple of minutes subitizing. In addition to ten-frames, students also need experience with different types of arrangements. I made a little I Have/Who Has game for practicing arrangements up to the number twelve. There are only twelve cards in the set, so either use it for small group instruction, or use partners for whole group.

Ready to try it? You can grab the I Have/Who Has game and ten-frame flashcards for free here. If you’re looking for more subitizing resources, check out Subitizing Cards: Perceptual and Conceptual.

Are you talking about multiplication facts? Jennifer Bay-Williams writes about using dot cards for teaching multiplication facts in her new book Math Fact Fluency. It’s a phenomenal book! Research and an explanation of using a strategy-based approach, but also tons of games.

Donna- I love your subitizing resources, but I am struggling with the perceptual to conceptual with one particular student and I’m wondering if you have some insight/advice. She has mastered all of the perceptual, but when it comes to the conceptual (like 2 and 3 on dice) she falls apart. I am wondering if I should be grouping the sets of cards that have 2 and 3 and present them in sequence until mastered and then shuffle in a few cards with 1 and 4 or something like that. I’m really stuck on how to move her beyond the groupings of 2 and 3.

I am looking for the link from the SDE conference in Las Vegas in July. I am getting ready to revamp my guided math centers and would like to use the information that you provided during this conference. Thanks

I am attempting to write a grant in my district for math manipulative that would be beneficial for students in developing their number sense. I am looking for manipulative ideas for that would be appropriate for 3-5 students that have gaps within their number sense and in turn struggle with the higher level concepts of the grade level.

Any suggestions for manipulatives would be greatly appreciated!!!! Thank you!

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Thank you! You continue to create fantastic resources for our young children!

Another outstanding, developmentally appropriate math resource! Thank you so much!

I teach 4th grade and multiplication is always difficult. Is there a way to do this with multiplication as well?

Are you talking about multiplication facts? Jennifer Bay-Williams writes about using dot cards for teaching multiplication facts in her new book Math Fact Fluency. It’s a phenomenal book! Research and an explanation of using a strategy-based approach, but also tons of games.

Donna- I love your subitizing resources, but I am struggling with the perceptual to conceptual with one particular student and I’m wondering if you have some insight/advice. She has mastered all of the perceptual, but when it comes to the conceptual (like 2 and 3 on dice) she falls apart. I am wondering if I should be grouping the sets of cards that have 2 and 3 and present them in sequence until mastered and then shuffle in a few cards with 1 and 4 or something like that. I’m really stuck on how to move her beyond the groupings of 2 and 3.

Hey, Jennifer! Yes, definitely focus on the 2 and 3 and then combine with others. It will just take time!

I am looking for the link from the SDE conference in Las Vegas in July. I am getting ready to revamp my guided math centers and would like to use the information that you provided during this conference. Thanks

Jeanne, I just emailed you.

I am attempting to write a grant in my district for math manipulative that would be beneficial for students in developing their number sense. I am looking for manipulative ideas for that would be appropriate for 3-5 students that have gaps within their number sense and in turn struggle with the higher level concepts of the grade level.

Any suggestions for manipulatives would be greatly appreciated!!!! Thank you!

Here is a great blog post from Blair Turner on essential manipulatives. It’s a good read!