Subitizing and Decomposing Numbers for Early Math

By Dr. Deni Basaraba and Cassandra Hatfield, 
RME Assessment Coordinators

In 2013, the National Center for Education Evaluation and Regional Assistance (NCEE), in partnership with the Institute of Education Science (IES) released an educator’s practice guide focused on Teaching Math to Young Children. The intent of this Guide (and all similar Practice Guides) is to provide educators with evidence-based practices they can incorporate into their own instruction to support students in their classrooms. In this ongoing blog series we will focus on specific recommendations put forth in the Practice Guide Teaching Math to Young Children and provide practical suggestions for incorporating these recommendations into your classroom instruction.

Recommendation 1 in this Practice Guide is to teach number and operations using a developmental progression. Using and understanding a developmental progression for number serves as the foundation for later mathematics skill development. As noted in the work documenting the development of learning trajectories for mathematics (Clements & Sarama, 2004; Daro, Mosher, & Corcoran, 2011) as well as in our own work in the development of diagnostic assessments using learning progressions, developmental progressions can provide teachers with valuable information regarding students’ knowledge and skill development by providing a “road map for developmentally appropriate instruction for learning different skills” (Frye et al., 2013).

Specifically, the research recommends that teachers first provide students with multiple opportunities to practice subitizing, or recognizing the total number of objects in a small set and labeling them with a number name without needing to count them. According to Clements (1999), two types of subitizing exist:

• Perceptual subitizing: The ability to recognize a number without using other mathematical processes (e.g., counting).
• Conceptual subitizing: The ability to recognize numbers and number patterns as units of units (e.g., viewing the number eight as “two groups of four”).

The role of subitizing as it relates to numeracy (Kroesbergen et al., 2009) and procedural calculation (Fuchs et al., 2010) has been documented in the literature. Kroesbergen et al., (2009), for example, not only found that subitzing was moderately correlated to the early numeracy skills of kindergarten students, but that it also explained 22% of the overall variance observed n counting skills and 4% of the variance in early numeracy skills after controlling for language and intelligence. Moreover, research also indicates that instruction designed using a developmental progression can support students’ ability to subitize (Clements & Sarama, 2007), as evidenced by relatively large gains in the pretest to posttest gain scores observed for students receiving this type of instruction compared to a “business as usual” comparison condition.

To support students with subitizing and decomposing numbers, flash images of arrangements of dots visually for students for about 3 seconds. Then give students an opportunity to share what they saw. Over time, student’s verbal descriptions can transition to writing equations. For younger children, subitizing may be fast and efficient only when the number of objects is less than four (Sarama & Clements, 2009); numbers larger than this may require decomposition into smaller parts.. For students learning multiplication arrangements of multiple groups of dots can be shown to support visualizing equal groups.

How do you see this image?

5 and 5, minus 1 

4 and 4, plus 1 

2 and 2, doubled, plus 1 

2 groups of 4, plus 1

Print these dot cards or 10 frame cards on cardstock and put them on a ring. They can be used in various ways:

1. Hang them in places throughout the hallway of your school. Working on subitizing is a great way to keep students engaged during transition times.
2. Place them as a center for partners to flash the images and ask “How many?”
3. Independent think time: Students can be given an arrangement and write all the different ways they see the arrangement.
4. Warm-up activity to get students thinking prior to small group instruction

Hyperlink for dot cards: http://www.k-5mathteachingresources.com/support-files/dotcards1-12.pdf

Link for 10 frame cards: http://www.k-5mathteachingresources.com/support-files/large10frames1-10.pdf

References

Clements, D. H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5, 400-405.

Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical thinking and learning, 6, 81-89.

Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. Journal for Research in Mathematics Education, 38, 136-173.

Daro, P., Mosher, F. A., & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. (CPRE RR-68). New York, NY: Center on Continuous Instructional Improvement.

Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching mathematics to young children: A practice guide (NCEE 2014-4005). Washington, DC: National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education Sciences. Retrieved from the NCEE website: http://whatworks.ed.gov.

Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D., Hamlett, C. L., Seethaler, P. M., Bryant, J. D., & Schatschneider, C. (2010). Do different types of school mathematics development depend on different constellations of numerical versus general cognitive abilities? Developmental Psychology, 46, 1731-1746.

Krosebergen, E. H., Van Luit, J. E. H., Van Lieshout, E. C. D. M, Van Loosbroek, E., & Van de Rijt, B. A. M. (2009). Individual differences in early numeracy: The role of executive functions and subitizing. Journal of Psychoeducational Assessment, 27, 226-236.

Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge.

Developing Numerical Magnitude Understanding in Young Children

By Dawn Woods, RME Elementary Mathematics Coordinator

Research shows general math achievement is closely related to children’s understanding of numerical magnitudes, or the amount of a quantity (Gersten, Jordan, & Flojo, 2005; Siegler & Booth, 2004). Children develop the ability to quantify and order numbers through subitizing and counting (Clements and Sarama, 2009) and many children can answer the question “which is more, 5 or 3?” by five years of age. However, some children may be unable to tell which of two numbers is bigger or which number is closer to another number and may not have developed the “mental number line” representation of numbers (Gersten, Jordan, & Flojo, 2005; Griffin, Case, & Sigler, 1994; Clements and Sarama, 2009).

This concept of numerical magnitude is a core component of number sense, which is widely viewed as crucial to success in mathematics (National Council of Teachers of Mathematics, 2006). Furthermore, existing data on the relationship between mathematical proficiency and understanding of magnitudes are consistent with the view that helping young children develop a better understanding of numerical magnitudes may lead to improved performance on mathematics tasks (Laski & Siegler, 2007).

So with this research in mind, how can teachers and parents help young children develop a better understanding of numerical magnitudes? One way is to use a clothesline as number line in order to build understanding of numerical relationships (Suh, 2014). The list of activities below can help young children develop flexible thinking with numbers.

• Encourage young children to equally space and hang number cards on the number line, using benchmark numbers such as 0, 5, and 10 as a guide. As the child masters this range of numbers, expand or change the range. Encourage the child, as he/she hangs the number cards, to reason and talk about mathematical ideas such as: 
•  Is your number card closer to 0 or 5? How do you know? 
•  Is your number card closer to 5 or 10? How do you know? 
•  How far is 4 from 10? How do you know? 
•  Support children’s reasoning about comparing and ordering numbers by having them justify solutions. For example, 
•  Which number is bigger, 4 or 5? Why? 
•  Why is 225 smaller than 250? 
• Discussing placement of fractions and decimals highlights equivalency concepts. As children work with these number cards ask questions such as, 
•  Which fraction is equivalent to 1/2? 
•  Are 0.09 and 0.90 the same or different number? How do you know?

References
Clements, D. & Sarama, J. (2009). Learning and teaching early math: the learning trajectories approach. New York: Routlege. 

Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293 – 304. 

Griffin, S., Case, R., & Siegler, R. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 24–49). Cambridge, MA: MIT Press. 

Laski, E., & Siegler, R. (2007). Is 27 a big number? Correlational and casual connections among numerical categorization, number line estimation, and numerical magnitude comparison. Child Development 78(6), 1723-1743. 

National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics. Washington, DC: National Council of Teachers of Mathematics. 

Siegler, R. S., & Booth, J. (2004). Development of numerical estimation in young children. Child Development, 75, 428 – 444. Suh, J. (2014). 

Line ‘em up! Teaching Children Mathematics, 20(5), 336.