Commentary on “Alignment Between Children’s Numeracy Performance,the Kindergarten Common Core State Standards for Mathematics,and State-Level Early Learning Standards”

Abstract

Litkowski et al. compare preschoolers’ performance on three counting items to various standards. We clarify that the items Litkowski and colleagues found to be too easy for kindergarten were actually goals for 4s/PKs in the National Research Council’s report Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity but that they were included as kindergarten standards to ensure that all children had an opportunity to learn these crucial competencies. The helpful analysis in their article of the variability across present state early childhood standards indicates that the kindergarten Common Core State Standards–Mathematics need to remain unchanged for the same reason. We suggest that research funding in early childhood is better spent on research on high-quality instructional contexts for all children than on survey research. And we address the important question of what more-advanced children should learn in kindergarten by pairing standards those children already know with crucial standards that need a lot of time and attention.

Litkowski et al. (2020) raise several issues in early childhood mathematics education in their comparison of preschoolers’ performance on three counting items to the Common Core State Standards–Mathematics (CCSSM; National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) and to individual state standards. They raise the question of whether two cardinality standards are appropriate for kindergarten children and suggest the “need to conduct large, nationally representative studies measuring children’s abilities on items that more closely assess the specific mathematics skills included in the CCSSM and early learning guidelines.” We believe the study is a valuable contribution, but we wish to clarify and justify the inclusion of the cardinality tasks in the CCSSM kindergarten standards and present alternatives to Litkowski and colleagues’ implications.

The issue of the appropriateness of the CCSSM for kindergarten has been discussed by many, with some claiming the standards are too demanding (e.g., Clements et al., 2017, 2019) and others that they are too easy (Clements et al., 2017). Litkowski et al. (2020) report preschoolers’ performance on three counting items in the kindergarten CCSSM: one verbal counting task aligned with K.CC.A.1 (Count to 100 by ones and by tens; K.CC = kindergarten–counting and cardinality) and two cardinality tasks aligned with parts of K.CC.B.5: Count to find “how many” and “give n.” They found that 18.9% of 5.5-year-olds and about 9% of 5-year-olds could count to 100 by ones (counting by tens was not assessed), that 86.5% of 5.5-year-olds and somewhat over 65% of 5-year-olds counted and then correctly answered the “how many” question for 16 objects, and that 53.3% of 5.5-year-olds and about 33% of 5-year-olds could give 16 objects (the percentages for 5-year-olds are from the graphs in the article). On this basis, they claim that the cardinality standards in K.CC.B.5 are too easy and are not appropriate for kindergarten.

We would like to clarify why the K.CC.B.5 (Count to find “how many” and “give n”) cardinality competencies are in the CCSSM kindergarten standards. The kindergarten, Grade 1, and Grade 2 CCSSM standards drew heavily on the research-based foundational and achievable goals identified in the National Research Council’s (NRC; 2009) report Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. This NRC report identified foundational and achievable goals for four age-related steps: for 2s/3s, 4s/PKs, kindergarten, and Grade 1. The two cardinality competencies in K.CC.B.5 are goals for the 4s/PKs (see NRC, 2009, Box 5.7 on p. 141). These competencies were included as CCSSM kindergarten standards to ensure that all children would have opportunities to learn and be fluent in these tasks because they are fundamental to functioning well in other standards in kindergarten and beyond. At that time, some states did not have standards for prekindergarten, and the standards that did exist were quite varied. Therefore, the committee believed that the crucial knowledge in K.CC.B.5 must be included in the kindergarten standards. But we can see from the Litkowski et al.’s (2020) data that about one third of 5-year-olds still need to learn to count and answer the “how many?” question for 16 objects, and about two thirds of 5-year-olds and almost half of 5.5-year-olds need to learn to give 16 objects accurately. So these items do not seem to be too easy to be standards, although, of course, we do not want kindergarten children spending a lot of time doing what they already know how to do, an issue we address later.

Litkowski and colleagues also provide a useful summary of how present state standards address the three standards on which they focused (see Table 1 in Litkowski et al., 2020). Reviewing this summary reveals the inconsistency and imprecision across states in these standards (cf. Scott-Little et al., 2012). Furthermore, some states still do not address one or both of these learning goals. So the situation has not changed so much that it is safe to consider dropping the K.CC.B.5 standard. Furthermore, kindergarten has a special status as the first year of CCSSM standards. For equity purposes, we must include in this first-year crucial foundational and achievable goals as identified in the NRC’s report so that all children will have opportunities to learn these goals in kindergarten and be prepared for Grade 1.

Litkowski and colleagues also conclude that we “need to conduct large, nationally representative studies measuring children’s abilities on items that more closely assess the specific mathematics skills included in the CCSSM and early learning guidelines.” But it is difficult and expensive to gather such data for those populations for which we need it most, which is the knowledge of children in home-based care. Limitations of knowledge in this population was a concern expressed in Recommendation 8 of the NRC report: Early childhood education partnerships should be formed between family and community programs so that they are equipped to work together in promoting children’s mathematics (NRC, 2009). It is more direct for research funding to be oriented to what children are able to learn and do with good instruction—a distinctly different approach than a large-scale survey that does not directly affect learning. Another advantage of such research is that it would simultaneously inform teachers about features of such good instruction. Such increased knowledge also might create a better atmosphere for improving state standards for prekindergarten to include vital knowledge as identified in the NRC report. Meanwhile, the CCSSM kindergarten standards need to remain as they are to ensure that all students have an equitable opportunity to learn the crucial core competencies.

Litkowski and colleagues raise another important issue that needs to be addressed. Fairly high percentages of the center-based children could do the “how many” and to a lesser extent, the “give n” cardinality tasks. What should these children be doing in kindergarten? First, there are four different arrangements of items to be counted described in the first part of K.CC.B.5 (line, rectangular array, circle, and scattered). We do not know if the assessed children could do all of these, so experience with the rectangular array, circle, and scattered arrangements might be fruitful. Nor do we actually know whether the children counted 16 dots in a line accurately and then answered “how many?” The focus for K.CC.B.5 is on accurately counting; knowing the “how many?” answer is standard K.CC.B.4b. So additional practice in counting fluently might be warranted. Furthermore, such counting can be embedded in activities simultaneously building other standards, such as the crucial K.A.NBT.1:

Work with numbers 11–19 to gain foundations for place value: Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 12)

This is one of the most important kindergarten standards and can prepare children for learning and applying place value ideas well in Grade 1. Children who need experience with the count to find “how many” and “give n” cardinality tasks for ten and below can develop this knowledge while working on kindergarten standards K.CC.C.6 and 7 (compare numbers) and K.OA.A.1 through 4 (understand addition and subtraction). So no children have to spend time doing what they already know but can learn new kindergarten standards. And children who need to learn K.CC.B.5 can do this while also learning other kindergarten standards.

The variability in the prekindergarten state standards so usefully summarized by Litkowski and colleagues also points out the importance of another important task for the field: Helping everyone become aware of the research-based foundational and achievable goals identified in the NRC’s report for 2s/3s and for 4s/PKs so that all children can have appropriate experiences at these ages. The National Council of Teachers of Mathematics and the National Association for the Education of Young Children do have a small book for teachers that summarizes these goals and describes related learning activities (National Council of Teachers of Mathematics, 2010). This book is not outdated, but perhaps it could be retitled and reworked. One or more journal articles could also summarize the NRC research-based foundational and achievable goals to make them more widely known. These could provide a basis for reworking state prekindergarten standards, as Litkowski and colleagues have so clearly indicated is needed.

Finally, increasing opportunities to learn for all 4s/PKs is crucial to kindergarten success for all children. So it is important to make widely available what is already known about young children’s thinking and potential for learning in high-quality instructional contexts (e.g., Clements & Sarama, 2013; Ginsburg et al., 2003; Griffin, 2004). Funding could be directed both to additional research about high-quality instructional contexts and to making available what is known at this date about such contexts.

In summary, Litkowski and colleagues’ discussion of whether two cardinality standards are appropriate for kindergarten children in the context of CCSSM and state standards raises several important issues that we agree need continuing attention in policy, research, and practice. We provided the reasons for the inclusion of the cardinality tasks in the CCSSM kindergarten standards and presented alternatives to Litkowski and colleagues’ implications, including the need for research and development on what children are able to learn and do with good instruction, embedding counting experiences in tasks also developing other standards, and raising awareness of very young children’s learning potential with effective teaching.