Show me the data: Unconstrained K-8 reading and math skills
Exploring growth patterns of unconstrained reading and math skills.
Note: Unconstrained Kids unpacks, translates, and integrates academic research and data about constrained and unconstrained skills for people that run, fund, and assist organizations that teach and serve kids. This post presents a series of charts I created from among four nationally representative datasets to illustrate proficiency patterns in representative groups of constrained and unconstrained reading and math skills, unconstrained nonacademic skills, and unconstrained general knowledge. For reference, see this working list of constrained and unconstrained skills. Like everything on this Substack, this post is a work-in-progress. I will make updates as needed. A list of the skills included in these charts is included at the end. Questions, comments, and suggestions are welcome.
Last updated: March 15, 2025
Three big ideas
There are significant and meaningful proficiency gaps in unconstrained reading and math skills across socioeconomic groups early in elementary school. Unlike constrained skills, these gaps do not appear to narrow by the end of middle school.
One exception is the proficiency in reading words in context of other text. There are significant proficiency gaps in this skill across socioeconomic groups through third grade. This gap finally appears to narrow by the end of middle school.
A review of more recent data shows that achievement gaps in unconstrained reading and math skills between socioeconomic groups appear similar to the late 2000s.
About these data
The data in these charts come from two sources: the Early Childhood Longitudinal Study, Kindergarten Class of 1998-99 (ECLS-K) and the National Assessment of Educational Progress Long-Term Trend Assessment (NAEP LTT). You can read an overview of these and other datasets. The ECLS-K data were collected between Fall 1998 and Spring 2007. It’s the best dataset I’ve found to dig deep into differentiable K-8 reading and math skills. I used the EdSurvey R package to analyze the data. The NAEP LTT data were collected in 2008 and 2020. I used the NAEP Data Explorer to analyze the data. The data are organized by either household socioeconomic status or eligibility for free and reduced-price meals (a proxy for household income). In the ECLS-K dataset, socioeconomic status is defined using a combination of household income, parental education level, and parental occupation. The charts in this post focus on constrained reading and math skills in elementary and middle school.1
A continuum of unconstrained skills
Although researchers often broadly refer to skills as “constrained” or “unconstrained,” in reality skills fall along a “continuum of constraint” (Paris, 2005; Stahl, 2011; McCormick & Mattera, 2022). Constrained skills are often divided into “highly constrained” and “moderately constrained.” But unconstrained skills have degrees of constraint, too. This is is implicitly recognized in the design of the reading and math assessments used in the ECLS-K. The skills were explicitly ordered in a hierarchical manner. The expectation was that kids had to master a skill at a lower level in the sequence before they could master skills at the next higher level.2
As skills become less constrained, they require the acquisition, consolidation, integration, and generalization of more information. This “information” sometimes takes the form of proficiency in subordinate reading or math skills. Mastery of unconstrained skills requires more time than constrained skills. Moreover, unconstrained skills are more likely to be “caught and taught,” benefitting from experiences and practice outside of school along with classroom instruction. But kids don’t have equal opportunities to develop unconstrained skills.
Some gaps in unconstrained skills take longer to close
We’ve seen previously that gaps in constrained reading and math skills largely close by the end of elementary school. This is supported by explicit and systematic instruction and extra supports for kids with learning differences (dyslexia and dyscalculia.) Some take longer but close by middle school. By contrast, a general pattern emerges from the data that gaps in unconstrained skills take longer to close.
Example 1: Comprehending words in context
This chart shows proficiency in comprehending words in context of other text from kindergarten through eighth grade. Kids were asked to select the best word to complete a sentence (a cloze test). This unconstrained skill requires vocabulary knowledge and comprehension ability. There is a ten point proficiency gap in spring kindergarten between the highest and lowest socioeconomic groups. This gap widens to 36 points a year later at the end of first grade. (The gaps also widen between each socioeconomic group.) The gap between the highest and lowest socioeconomic groups is cut in half in third grade (17 points). It is all but closed by eighth grade (4 points). (Mouse over the chart.) This chart shows it is a mistake to assume that gaps in unconstrained skills cannot be closed.
Figure 1. Ability to read and understand words in the context of other text.
Most gaps in unconstrained skills do not close through middle school
Despite the example in Figure 1, most examples of unconstrained skills persist through middle school.
Example 2: Literal inference (comprehension)
This chart shows proficiency in literal inference (comprehension) from kindergarten through eighth grade. This involves using explicit cues to determine the meaning of written text. Very few kids at the end of kindergarten have proficiency in this reading skill. By a year later, a large 23 point proficiency gap emerges between the lowest and highest socioeconomic groups. This gap increases to almost 40 percentage points in third grade. The proficiency gap in this unconstrained reading skill is cut in half between third and eighth grade. Yet, but the end of middle school nearly a fifth of kids in the lowest socioeconomic group had not achieved proficiency in literal inference. This is the minimum level of reading comprehension generally assessed on state and national assessments of reading beginning in third grade.
Figure 2. Ability to make inference using cues explicitly stated in the text.
Example 3: Interpreting narrative text (comprehension)
This next chart shows growth in proficiency in interpretation of narrative text from first grade through eighth grade. This requires understanding of how an author conveys information and clues to the reader. The task in the ECLS-K reading assessment also asked kids to make connections between problems in a story and similar real-life problems. In first grade a six point proficiency gap separates the lowest and highest socioeconomic groups. This mushrooms to a nearly 30 point gap two years later in third grade. By the end of middle school, it balloons further to a forty point gap.
Figure 3. Ability to interpret narrative text using implicit cues and making connections with problems in the text.
Example 4: Understanding place value
Let’s turn to mathematics. This next chart shows proficiency in understanding place value of integers to the hundreds place. This is a form of relational thinking, which is an unconstrained math skill. The chart covers first grade through eighth grade. Once again, we find a small 7 point proficiency gap between the lowest and highest socioeconomic groups at the end of first grade. Two years later, this gap explodes to nearly 50 percentage points. Moreover, there are significant proficiency gaps between each socioeconomic group. This gap is cut in more than half by the end of middle school. Yet, more than 20 points separates the lowest and highest socioeconomic groups in eighth grade. Larger skill gaps persist for kids in lower socioeconomic groups.
Figure 4. Ability to understand place value of integers to the hundreds position.
Example 5: Word problem solving
The next chart shows proficiency in a prototypical unconstrained math skill: word problem solving. Solving math word problems involves engaging and applying math facts and procedures (constrained) as well as math concepts (unconstrained). This entails mathematical thinking: understanding a problem, developing a plan to solve the problem, executing the plan, and checking your work to ensure you got the correct answer. This is an iterative process that engages both procedural and conceptual math knowledge (Rittle-Johnson et al., 2001; Rittle-Johnson et al., 2015). Solving word problems also draws upon unconstrained reading skills—vocabulary and comprehension.
This chart shows proficiency in solving word problems that involved knowledge of measurement and rate from third through eighth grade. By the end of third grade, there is almost a 25 point proficiency gap between the lowest and highest socioeconomic groups. This gap virtually doubles by the end of fifth grade. It only slightly narrows by the end of middle school. There are clear proficiency gaps at each point between socioeconomic groups.
Figure 5. Ability to solve math word problems involving rate and measurement.
Some more recent data about unconstrained skills
These data are from 15-25 years in the past. Much has occurred in American education in reading and mathematics. Do we see similar trends in more recent data? For a more recent perspective, we turn to the NAEP Long-Term Trend Assessment. We review the proficiency levels of U.S. nine- and thirteen-year-olds in 2008 and 2020. The 2008 NAEP LTT data was collected one year after eighth grade data from the ECLS-K in the earlier charts. The 2020 NAEP LTT data was collected right before school closures due to the pandemic. We’ll first look at data for reading and then turn to mathematics.
Example 6: Intermediate-level reading comprehension
The next chart shows proficiency on the same set of unconstrained reading skills—interrelate ideas and make generalizations. (See description at the end.) The chart compares proficiency for kids who were eligible and not eligible for free and reduced-price meals. In both years and for both age levels, we see significant gaps in unconstrained reading skills by socioeconomic groups. There are some small improvements over this 12-year period for both groups, but the overall patterns for this unconstrained skill are largely unchanged.
Figure 6. Ability to comprehend intermediate-level informational and narrative text.
Example 7: Beginning math problem solving
We’ve come at last to our last chart in this post. This final chart shows proficiency on basic numerical operations and beginning problem solving. This is a mix of constrained and unconstrained math skills. (See description at the end.) This chart compares proficiency levels for kids who were eligible and non-eligible for free and reduced-price meals in 2008 and 2020. Just as we saw in Figure 6, there is a significant proficiency gap between socioeconomic groups in 2008 and 2020 for both age levels.
Figure 7. Ability to understand the four basic numerical operations and do simple problem solving.
Unlike the convergence in proficiency that we saw for constrained skills, the gaps in unconstrained reading and math skills are more persistent through elementary and middle school. Although unconstrained skills are more complex than constrained, there is more at play than just the complexity of the skills. Gaps in nonacademic skills and general knowledge strongly suggest the importance of opportunity to build these slower growing skills inside and outside of school.
But wait, there’s more
If you’d like to see more data about constrained and unconstrained skills, check out these other posts on Unconstrained Kids:
Skill definitions
ECLS-K Reading and Math Skills
The ECLS-K assessed elementary and middle school students on a range of constrained and unconstrained reading and math skills. The skills were considered hierarchical, i.e. one has to master the lower levels in the sequence before one could learn the material at the next higher level. The skills included in the charts in this post:
Comprehension of words in context. Selecting the best word to complete a sentence. This is an unconstrained reading skill.
Literal inference. Making inferences using cues that are directly stated with key words in text (for example, recognizing the comparison being made in a simile). This is an unconstrained reading skill.
Interpreting beyond text. Evaluation – demonstrating understanding of author’s craft (how does the author let you know...), and making connections between a problem in the narrative and similar life problems. This is an unconstrained reading skill.
Place value. Demonstrating understanding of place value in integers to the hundreds place. This is a constrained math skill.
Rate and measurement. Using knowledge of measurement and rate to solve word problems. This is an unconstrained math skill.
NAEP Long-Term Assessment Reading and Math Skills
The NAEP LTT has five performance levels each for reading and math that involve a mix of constrained and unconstrained skills. Each skill level involves successively greater degrees of unconstrained skills. The skill included in the charts in this post:
Interrelate ideas and make generalizations. Readers at this level use intermediate skills and strategies to search for, locate, and organize the information they find in relatively lengthy passages and can recognize paraphrases of what they have read. They can also make inferences and reach generalizations about main ideas and the author's purpose from passages dealing with literature, science, and social studies. Performance at this level suggests the ability to search for specific information, interrelate ideas, and make generalizations. This is a set of unconstrained reading skills.
Numerical operations and beginning problem solving. Students at this level have an initial understanding of the four basic operations. They are able to apply whole number addition and subtraction skills to one-step word problems and money situations. In multiplication, they can find the product of a two-digit and a one-digit number. They can also compare information from graphs and charts, and are developing an ability to analyze simple logical relations. This is a mix of constrained and unconstrained math skills.
Works cited
McCormick, M., & Mattera, S. (2022). Learning more by measuring more: Building better evidence on pre-k programs by assessing the full range of children's skills. MDRC: New York, NY.
Paris, S. G. (2005). Reinterpreting the development of reading skills. Reading Research Quarterly, 40(2), 184-202.
Stahl, K. A. D. (2011). Applying new visions of reading development in today's classrooms. The Reading Teacher, 65(1), 52-56.
ECLS-K researchers used a statistical method called “differential item functioning” to determine there was no bias in the K-8 reading and math assessments by socioeconomic status.
Reading Skills
Levels 1 to 4: Letter recognition, beginning sounds, ending sounds, sight words (constrained skills)
Level 5: Words in context (lowest level unconstrained skill in this dataset)
Level 6: Literal inference—making inferences using cues that were directly stated with key words in a text
Level 7: Extrapolation—identifying clues used to make inferences
Level 8: Evaluating fiction—demonstrating understanding of author’s craft and making connections between a problem in the narrative and similar life problems
Level 9: Evaluating nonfiction - comprehension of biographical and expository text
Level 10: Evaluating complex syntax—evaluating complex syntax and understanding high-level vocabulary (highest level unconstrained skill in this dataset)
Note that oral reading fluency is generally considered the “least constrained” constrained reading skill. Fluency was not assessed in the ECLS-K study.
Math Skills
Levels 1-4: Number and shape, relative size, ordinality and sequence, addition and subtraction, multiplication and division (constrained skills)
Level 6: Place value—demonstrating understanding of place value in integers to hundreds’ place (lowest level unconstrained math skill in this dataset)
Level 7: Rate and measurement—using knowledge of measurement and rate to solve word problems
Level 8: Fractions—solving problems using fractions (demonstrating understanding of the concept of fractional parts)
Level 9: Area and volume—solving word problems involving area and volume (highest level unconstrained skill in this dataset)