If we consider education standards as specifying the “content of the intended curriculum” (Porter et al., 2011, p. 103) and primary informers of EEEs’ written curricula (i.e., the materials, resources, and guides teachers use for teaching; Remillard, 2005), we can identify teachers’ opportunities to teach spatial reasoning. Using this argument, it logically follows that written curricula would provide the content of the learning targets and suggested instructional practices on which EEEs can layer pedagogical practices to teach toward the standards. This layering of intended and written curricula plus pedagogy should increase students’ opportunities to learn specified skills. However, there is variation in intended and written mathematics curricula in the U.S. based on state policy and local resources.

While the National Council of Teachers of Mathematics (2000, 2006) outlined standards for what mathematics content and processes should be taught at each grade level in U.S. schools, they are not the same as mandated education standards. Instead, education standards adopted at the state level are what teachers are held accountable for teaching, and no national intended or written curriculum exists. Further, there is debate on how well-aligned standards, written curricula, and instructional practices are in the classroom (Rivera, 2011; Polikoff, 2012; Pak et al., 2020). While the provision of written curricula has been proposed as a support for teaching practice (Whitehurst, 2009; Chingos and Whitehurst, 2012), written curricula in the U.S. are locally selected and, therefore, have less transferability to alternative contexts (Polikoff, 2018). This means that the intended curricula should be studied to identify ways for EEEs to broadly incorporate spatial reasoning into their teaching practices. Further, due to the predictive nature of early mathematics experiences on later academic achievement (Duncan et al., 2007), it is critical to better identify how EEEs can incorporate spatial reasoning teaching practices into their intended curricula through pedagogical practice without detracting from children’s other mathematics learning, and how this can be done for all children, not just a few.

Numeric relational reasoning and spatial reasoning are two primary constructs of mathematical thinking and development that warrant instructional emphasis in K-2 (National Council of Teachers of Mathematics, 2006; National Research Council, 2009; Perry, 2016). However, spatial reasoning skills are seldom referenced explicitly within intended curricula, such as the CCSS-M (Gilligan-Lee et al., 2022). The current layering of the intended and written curricula in the U.S. generally underrepresents spatial reasoning (Gilligan-Lee et al., 2022); where (i.e., in what content standards) and how (i.e., implicitly, optionally, or explicitly) the skills are included varies, meaning many students may not be receiving opportunities to learn to reason spatially. Boda et al. (2022) found implicit opportunities to teach spatial reasoning within the K-2 CCSS-M, but those connections were mainly within the Measurement and Data or Geometry domains (Common Core State Standards (CCSS) Initiative, 2010). Given the small number of standards in those domains compared to the overall number of mathematics standards at each grade (e.g., three out of 26 second-grade standards are in the Geometry domain), spatial reasoning’s representation is considered relatively sparse.

Despite few standards related to spatial reasoning, Fowler et al. (2019) asserted that spatial reasoning can be taught through overarching standards. Davis et al. (2015) supported that idea, claiming that most spatial reasoning skills in their emergent conceptualization (see also Table 1; Pinilla, 2024) could be taught using standard mathematics curricula. This means that by intentionally planning instruction using mathematics education standards (e.g., the CCSS-M), EEEs could provide children with opportunities to learn spatial reasoning skills without interrupting the scope and sequence of their existing instruction. However, few EEEs are prepared to supplement their given mathematics instructional tools (Ginsburg et al., 2006), and adopted, commercially produced curricular packages do not typically offer spatially relevant connections (Kalyankar, 2019). Given the minimal explicit representation of spatial reasoning in common education standards (i.e., CCSS-M) and knowledge that recommended curricular reforms (Gilligan-Lee et al., 2022) may prove difficult, this work seeks to locate existing connections between the standards and spatial reasoning skills.

I developed an a priori coding scheme using Krippendorff’s (2019) content analysis components and followed Saldaña’s (2016) protocol coding methods. Whereas Davis et al.’s (2015) emergent conceptualization of spatial reasoning included 40 skills, I used 38 as a priori categories to deductively code all K-2 CCSS-M standards. Two skills named in Davis et al.'s (2015) conceptualization were considered overarching skills (i.e., transforming and understanding); they were too broad to make meaningful connections with standards when initially coding. I abductively inferred (Krippendorff, 2019) the presence of the 38 skills in each standard using constant comparison (Glaser and Strauss, 1967) to decide whether each spatial reasoning skill was implicit in a standard. I assigned the skills dichotomous values of 1 for present or 0 for not present. Coding in this phase was utilitarian; if there was any relation, the skill was coded as present.

I next collaborated with an experienced educator to establish intercoder reliability, which is fundamental in enhancing the transparency of methods and quality of content analyses (O’Connor and Joffe, 2020). I hired the educator as a project consultant and trained them on the coding scheme by providing a project overview and time to independently review and give written feedback on the definitions of the 38 spatial reasoning skills (see Appendix A for definitions). In their feedback, the consultant documented their understandings and sought clarity between some terms that were close in meaning. For example, they questioned the difference between some inner-element subelement skills, like packing and fitting, which are subelements of Constructing. Due to the close meanings and anticipated challenges in differentiating them meaningfully within standards, we condensed three pairs of inner-element subelements into groups. Within the Altering element, scaling was grouped with dilating/contracting. Within the Constructing element, packing and fitting were made into one group, while sectioning was grouped with de/re/composing. While the structure of the conceptual framework (see Figure 1) remained, we coded from thereon with the grouped codes, reducing the number of codes to 35.

I assigned the consultant a proportional stratified random sample of 20% of the K-2 CCSS-M to double-code for the presence of spatial reasoning skills. Our initial agreement was 83.44% across spatial reasoning skills. While there is no definitive guideline, Miles et al. (2014) recommended that coders reach 85 to 90% overall agreement; more granularly, Miles and Huberman (1994) suggest that coders establish 80% agreement across 95% of the codes for 20% of the data to establish intercoder reliability. To reconcile differences, we engaged in consensus conversations and discussed the nine spatial reasoning skills on which our agreement was less than 80%. Through these conversations, we came to negotiated understandings of how those skills appeared in standards and reached an overall agreement of 95.25%. I applied the updated meanings for those spatial reasoning skills about which we had the consensus conversations to all independently coded standards. I calculated descriptive statistics to narrate how spatial reasoning is implicitly represented across the K-2 CCSS-M.

I extended the analysis from dichotomously identifying the implicit presence of spatial reasoning in the standards to applying a four-point scale to determine the extent to which spatial reasoning was represented explicitly, optionally, implicitly, or absent in the standards. While initial results indicated that spatial reasoning is somewhat frequently represented within the K-2 CCSS-M, the degree of inference required to see those connections required tacit knowledge of the skills and pedagogy. I engaged in second-cycle coding to answer the second research question and determine how explicitly the skills were represented in the standards. I recoded all spatial reasoning skills marked as present in a standard as explicit, optional, or implicit.

Standards explicitly connected to spatial reasoning name one or more skills (e.g., composing) in the standards themselves. For example, composing is present in standard K.G.6 (Common Core State Standards (CCSS) Initiative, 2010, p. 12), which explicitly calls for children to compose shapes. Skills coded as optional were those given as optional representations in a standard; diagramming is named in standard 2.OA.1 (Common Core State Standards (CCSS) Initiative, 2010, p. 19), wherein a child could draw a diagram to solve a problem, but it is not required to demonstrate that skill to master the standard. Implicitly represented skills are those within pedagogical reach by an experienced teacher but not named explicitly or as an optional representation in the standard. For instance, K.CC.4.b requires children to understand that the last number said when counting objects is the total regardless of the arrangement, which implicitly relates to the rearranging skill. Finally, spatial reasoning skills with no connection to a standard are those that could not be incorporated when teaching toward a standard without significant planning and alterations to the teaching practice; they were those coded as not present in the first cycle. All K-2 CCSS-M were coded using this more granular scheme, and findings were narrated using descriptive statistics and graphically.

On average, almost 10 standards per grade level (M = 9.67) implicitly related to Moving skills (i.e., changing the position of something by using spatial transformations; see Appendix A). Of Moving’s subelements, balancing (i.e., bringing into proportion or visual equilibrium by creating equivalence or sameness; see Appendix A) was represented most frequently (M = 8.33), with standards from the Operational and Algebraic Thinking domain of the CCSS-M across grades often found to relate with the skill implicitly. For example, standard 2.OA.3 reads, “Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2 s; write an equation to express an even number as a sum of two equal addends” (Common Core State Standards (CCSS) Initiative, 2010, p. 19). Given the example of pairing objects in the standard, I inferred that students would work with concrete manipulatives when pairing and counting by 2 s and use the equals sign to symbolize balance when writing the equation. This inference was supported when discussing balancing with the EEE consultant engaged in the coding; she referenced using a pan balance when teaching about the equals sign. While the example illustrates a high-inference representation, we agreed on the skill’s implicit presence in the standard, which gives credence to the idea that pedagogy and shared understandings of the tools used in mathematics instruction are within the knowledge and practices of experienced educators and researchers.

However, the representation of other Moving subelements tended to decrease as grade level increased. Whereas there were implicit representations of reflecting, rotating, and translating in three standards each at kindergarten, there was no representation of those skills by second grade (see details in Appendix B). This difference is associated with how the Geometry domain standards are written across the grades; kindergarten standards state that children will be composing shapes and modeling compositions (i.e., K.G.4, K.G.5, and K.G.6), yet second-grade standards focus on partitioning shapes into fractional pieces (i.e., 2.G.2 and 2.G.3), and the Moving subelements were not found to be represented therein.

Connections to Moving skills were generally limited to those implicit opportunities. No standards across grade levels provided optional opportunities to teach any Moving subelements, but one standard at kindergarten did explicitly connect to the translating skill. CCSS-M K.G.6 says, “Compose simple shapes to form larger shapes, For example, ‘Can you join these two triangles with full sides touching to make a rectangle?’” (Common Core State Standards (CCSS) Initiative, 2010, p. 12), which requires students to translate shapes, thus forming the single explicit connection between Moving skills and the K-2 CCSS-M.

Altering (i.e., modifying or changing something’s appearance; see Appendix A) was the least represented element from the spatial reasoning conceptual framework in the K-2 CCSS-M. Its comprising subelements were found to implicitly relate with fewer than three standards per grade level (M = 2.67). Of Altering’s subelements, dilating/contracting and scaling skills were found most frequently. In kindergarten standards, the Geometry domain contained references to these skills. However, at the first- and second-grade levels, any representation was within the Number and Operations in Base Ten domain. Specifically, those standards that call upon children to “bundle” ones into 10s or 10s into 100 s (e.g., 1.NBT.2.a and 2.NBT.1.a) implicitly relate to scaling, or understanding the correspondence between unit sizes (see Appendix A). However, other Altering subelements (i.e., folding and shearing) were found to have very little representation across grades, with one or fewer standards implicitly related to each skill per grade (see Appendix B). Further, no standards offered optional or explicit connections to Altering skills.

Situating subelements were implicitly represented in 10 standards per grade level, on average. Locating skills were identified most often in an average of eight standards per grade. Specifically, standards in the Measurement and Data domain (e.g., 2.MD.1 or 2.MD.2) tended to relate to locating. For example, when comparing an object to a measurement device (e.g., measuring the height of a shelf with a yardstick), students locate the corresponding number on that measurement device that tells the object’s size. Additionally, time-telling standards (e.g., 1.MD.3 or 2.MD.7) require students to locate the numbers on an analog clock and determine their meaning. However, these connections are all implicit.

Surprisingly few standards were related to dimension-shifting (i.e., moving between two and three dimensions; see Appendix A) across grades (M = 1.33). Some standards call upon students to recognize shapes as being two- or three-dimensional (i.e., K.G.3) or to compose two- or three-dimensional shapes (i.e., 1.G.2) but not work across dimensions. The intersecting, orienting, and pathfinding subelements were either infrequently represented or not at all. Orienting (i.e., understanding and operating on the relationships between the positions of objects in space with respect to one’s own position; see Appendix A) was best represented in kindergarten Geometry domain standards (i.e., K.G.1, K.G.2, K.G.3) in which students were to recognize objects regardless of their orientation. Intersecting (i.e., meeting or crossing at a point or in a plane; see Appendix A) was only represented in first- and second-grade standards that called upon students to represent data (i.e., 1.MD.4 and 2.MD.9). No standards contained a reference to pathfinding.

Similarly to skills in the Moving element, Situating skills were most often implicit within standards. One kindergarten standard offered an optional connection to dimension-shifting (i.e., moving between two and three dimensions; see Appendix A) but there were no other optional or explicit connections between this set of skills and the K-2 CCSS-M. Of note, optional connections were found to the Situating element in two kindergarten standards, but they did not connect similarly with any of its comprising subelements.

On average, the Constructing element was implicitly represented in 17 standards per grade level. Of those, de/re/composing and sectioning (i.e., putting together parts to make a whole, taking wholes apart or cutting into parts, and putting together again in a different way; see Appendix A) was the group of subelements most frequently represented. On average, 15.67 standards per grade level required students to compose, though most often, those compositions were of numbers; the implicit connections to spatial reasoning emerged through the examples of how students would demonstrate mastery of a standard. For example, 1.OA.1 reads,

This standard includes examples of concrete objects and drawings, which provide optional connections to the de/re/composing subelement; on average, two standards per grade level contained such optional connections, and two standards per grade level explicitly connected to them (see Table 2).

The subelement de/re/arranging is also relatively well represented implicitly (M = 9.33) through standards that specify the use of concrete representations in their examples. However, there was only one standard at kindergarten that offered an optional connection to that subelement (i.e., K.CC.4.B). Conversely, the un/re/packing and fitting subelements were infrequently represented in the standards (M = 1.67). Implicit relations were found within the Measurement and Data domain (e.g., 1.MD.2 asks for students to lay multiple copies of a comparator end to end when measuring an object, thus determining how many of those pieces fit into the same length) and the Geometry domain (i.e., 2.G.2 and 2.G.3 address partitioning shapes in different ways). No optional or explicit connections were found between standards and un/re/packing and fitting.

The Interpreting element was better represented in the standards than all others, with at least one of its subelements appearing implicitly in an average of 21 standards across grade levels (M = 21.33). For example, 13 of the 25 kindergarten standards called upon students to model their solutions, which implicitly supports the modeling subelement (i.e., constructing [scale or dimension-shifted] representations of real spaces to simplify problems when interpreting information; see Appendix A). That high frequency of representation was slightly lower in grades 1 and 2, wherein modeling was represented implicitly in nine standards each. When assessing optional representations, there were connections to two standards per grade level on average, and modeling was only explicitly described in a single standard in each kindergarten and grade 2 (see Figure 2 and Appendix B).

Conversely, the comparing subelement (i.e., judging sameness or difference by distinguishing between forms using appearance-based relational reasoning; mapping correspondences between two or more forms; see Appendix A) was implicitly represented in standards with greater frequency as grade levels increased. Whereas 14 kindergarten standards implicitly called upon students to use comparing skills, 19 of the 28 grade 2 standards did the same (see Table 2). However, the number of optional and explicit representations of comparing were found to be similar to that of modeling. I located optional practices to support comparing with increasing frequency across grades, which averaged almost three standards per grade (M = 2.67) and explicit connections to one standard per grade on average. For example, 1.MD.1 is “Order three objects by length; compare the lengths of two objects indirectly by using a third object” (Common Core State Standards (CCSS) Initiative, 2010, p. 16). The standard states that students should compare the length of physical objects, which is a spatial representation and, therefore, explicitly connects to the comparing subelement.

Relating skills [i.e., showing or establishing connections between two or more (things) to make sense of how (things) are spatially organized; see Appendix A] were often implicitly represented in the same standards as comparing, though less frequently because relating’s definition focuses on the sensemaking of spatial organization. Between one to two standards offered optional connections to relating per grade level (M = 1.67), but no explicit connections were found. Diagramming and designing were also often implicitly represented, with an average of almost seven standards per grade relating to diagramming (M = 6.67) and 6 relating to designing (M = 6). The degree of representation varied between the elements, however. I located optional connections to diagramming in 4 standards per grade on average and explicit connections to just over 2 (M = 2.33). For designing, I found an optional and explicit connection to less than one standard per grade level on average (M = 0.33 and M = 0.67, respectively).

The outlier Interpreting subelement was symmetrizing (i.e., interpreting and/or explaining balanced proportions through equivalent structures, often as a bilateral reflection; see Appendix A). I initially assessed standards for its presence similarly to how I examined them for representing balancing; however, the words “equivalent structure” in the symmetrizing definition led to me looking more closely for ideas related to an identical image rather than just balanced proportions. After clarifying its meaning, I found that only one kindergarten CCSS-M content standard (i.e., K.G.4) implicitly related to symmetrizing because it calls on students to compare spatial structures.

Sensating was implicitly represented in almost 16 standards per grade level on average (M = 15.67), with less overlap between subelements than in other elements. Tactilizing (i.e., making perceptible by touch, or tangible; see Appendix A) was the best-represented Sensating subelement with over 11 standards per grade level on average relating to the skill implicitly (M = 11.33). This prevalence was attributed to the numerous standards that call for children to use concrete representations (e.g., moving objects or counting fingers); if children use tangible materials, they are physically engaging and, thus, may be using tactilizing skills. However, that potential connection is implicit, at best, as merely using a physical representation does not necessitate engaging in reasoning spatially. The second best implicitly represented subelement was imagining (M = 6.33), followed by visualizing (M = 3.33). Some keywords in determining if a standard related to imagining or visualizing included stating the use of “mental images.” For example, K.OA.1 is “represent addition and subtraction with objects, fingers, mental images, drawings, sounds [e.g., claps], acting out situations, verbal explanations, expressions, or equations” (Common Core State Standards (CCSS) Initiative, 2010, p. 11). Given that students could use mental images, it is a reasonable conclusion that the Sensating subelements of imagining (i.e., forming mental images; see Appendix A) and visualizing (i.e., imagining and mentally transforming spatial representations in space; see Appendix A) are connected as not just implicit, but also optional representations. However, that standard was the only one across all three grade levels that offered an optional connection, and none explicitly connected to the skills.

Additionally, the final three Sensating subelements (i.e., perspective-taking, projecting, and propriocepting) were almost or entirely absent from the standards. Projecting was implicitly represented in one standard per grade level in the Measurement and Data or Operations and Algebraic Thinking domains (see Supplementary Figures B1–B3 in Appendix B for details).