Start Here: What are constrained and unconstrained skills?
A powerful insight on what drives reading and math achievement.
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. Despite important differences, all reading and math skills can be boiled down to two types–constrained and unconstrained. This post provides an overview of both types of skills and their distinct developmental pathways. Like everything on this Substack, this post is a work-in-progress. I will make updates as needed. Footnotes and citations are included at the end. Questions, comments, and suggestions are welcome.
Last updated: July 19, 2025
Three Big Ideas
An under-appreciated dynamic is at play below the picture of stagnant K-12 reading and math achievement. There are two types of skills—constrained and unconstrained. Most children reach relatively high levels of proficiency in constrained skills. It’s the unconstrained skills that limit further progress and are largely responsible for the achievement gap.
This academic insight about skill type is known among education researchers. They have written about and conducted research explicitly focused on constrained and unconstrained skills over the past twenty years. Yet, this key insight has yet to make its way into mainstream K-12 education circles.
Children and youth need to develop both types of skills together. A key challenge is that these skills follow distinct developmental pathways. Constrained skills are largely developed in formal classroom settings. Unconstrained skills are developed both inside and outside of classrooms. Improving K-12 reading and math achievement and closing achievement gaps will require new approaches across the classroom, community, and home.
The challenge of improving K-12 reading and math
Parents, educators, policymakers, and employers want to ensure children and youth develop skills needed to succeed in school and in life. The ability to read deeply and think mathematically are keys to doors of opportunity in a rapidly changing workplace and world.1
Despite waves of reform and billions of public and philanthropic dollars, we’ve made too little progress in raising academic achievement and closing gaps. These first two charts show reading and math scores for American nine-year-olds over nearly twenty years.2
Figure 1
Figure 2
A first-level conclusion is the obvious lack of progress in reading or math achievement prior to the impact of the COVID-19 pandemic. A closer look, however, reveals an important dynamic below the surface. There are two types of skills—constrained and unconstrained. Each has distinct developmental pathways. Children generally master constrained skills. It’s the unconstrained skills that quietly drive progress for K-12 reading and math.3
Constrained skill theory
In 2005, Scott Paris, a former University of Michigan professor, published the first of a series of academic papers describing differences in relationships between various reading skills over time.4 He divided reading skills into two categories based largely upon features related to their mastery: constrained and unconstrained.
Over the next 20 years, other researchers built upon this Constrained Skill Theory. Initial contributors continued to focus on early literacy, adding implications for instruction and policy. Later researchers added math skills and nonacademic5 (executive) skills.
Today, researchers suggest skills differ in four fundamental ways:
The amount of information associated with learning a particular skill.
The definition of mastery of a skill and whether it’s the same for everyone.
The amount of time typically needed to reach mastery of a skill.
The manner in which a skill is learned or acquired.
Constrained skills
Constrained skills involve a relatively limited amount of information.6 Everyone works with the same information.7 Skills like the alphabet, word reading, counting, addition, and subtraction. Mastery is clearly defined for constrained skills. The yardstick for mastery is the same for everyone.8
Constrained skills are relatively straightforward to teach and assess. Constrained skills are mostly learned in formal classroom settings. Most typically developing children master constrained skills within relatively limited amounts of time.9
We can see this dynamic at work in the math and reading scores of American nine-year-olds. Let’s revisit the reading and math scores we saw in Figures 1 and 2. This time, instead of scale scores, let’s look at proficiency levels of constrained skills.10
Figure 3 shows the percentage of nine-year-olds who were proficient in a constrained math skill: simple arithmetic.11 Between 2008 and 2020 virtually all American nine-year-olds were highly proficient in this constrained math skill.12
Figure 3
Now, let’s turn to reading. Figure 4 shows the percent of children who were proficient in simple reading tasks.13 Proficiency is not as universally high as math skills. However, the gap between student groups was limited to 6 to 8 points prior to the pandemic. (Mouse over the chart to see the percentages.) And all student groups were generally 90% proficient or higher.14
Figure 4
An even better way to appreciate the differences in children’s proficiency in constrained and unconstrained skills is to see their growth longitudinally. I’ve written a separate post on the growth of constrained skills from kindergarten through fifth grade.
Unconstrained skills
Unconstrained skills involve much broader amounts of information. Everyone doesn’t have access to the same information. Skills like vocabulary, comprehension, relational thinking, word problem solving, inhibition control, perspective taking, and knowledge.15 There is no universal finish line for mastery of these skills. There is always the opportunity to learn more.
It is not as straightforward to teach and assess unconstrained skills. Unconstrained skills are developed inside and outside of formal classroom settings.16 These “taught and caught” skills generally take more time to build than constrained skills.
Figure 5 looks at math proficiency again, this time for an unconstrained math skill: math operations and reasoning.17 Unlike the constrained math skill in Figure 3, we see significant and persistent gaps across student racial groups over nearly two decades.
Figure 5
Figure 6 shows the same information for an unconstrained reading skill: simple inferential comprehension.18 Once again, we see significant and persistent gaps in skills across student groups for nearly 20 years.
Figure 6
An even better way to appreciate the differences in children’s proficiency in constrained and unconstrained skills is to see their growth longitudinally. I’ve written a separate post on the growth of unconstrained reading and math skills from kindergarten through eighth grade.19
Two types of skills; two developmental pathways
From an early age, constrained and unconstrained skills interact with and reinforce each other’s development. I’ve identified a range of constrained and unconstrained skills that research describes as key components of reading comprehension and mathematical cognition. The development of both types of skills at the same time is critical for children and youth’s success in reading and math.
One might be tempted to think that constrained and unconstrained skills is an overly complicated way of describing “simple” versus “complex” skills. To some degree, this is true for some skills. For example, letter knowledge (constrained) is clearly much simpler than reading comprehension (unconstrained).
Many unconstrained skills, however, defy this simple categorization. For example, unconstrained skills such as vocabulary and declarative knowledge (facts) — critical for reading and math — can’t be categorized as merely more “complex” than constrained skills such as reading fluency or addition.
This table offers a summary of the characteristics of constrained and unconstrained skills.20 It is derived from descriptions of constrained and unconstrained skills in research papers about constrained skill theory. Both skill types follow distinct developmental pathways. Children and youth are likely to have more common opportunities and experiences to develop constrained skills than unconstrained skills.
Kids need to develop constrained and unconstrained skills together
Both constrained and unconstrained skills are important to success in reading and math achievement. Constrained and unconstrained skills interact and reinforce each other. For example, vocabulary (unconstrained) and word-reading (constrained) reinforce each other. Other unconstrained skills — like attention control, attention shifting, and inhibition control — also support word-reading skills. So, improving unconstrained skills also helps to improve constrained skills.
Despite skill gaps early in school, most kids master constrained reading and math skills by late elementary or middle school. The major problem for constrained skills is that it takes too long for some children to become proficient. We need to close early gaps in these constrained skills.
The greater challenge is unconstrained skills. These skills grow more slowly. Unlike constrained skills, early gaps in unconstrained skills don’t appear to close by the end of middle school. However, these skills do continue to grow through high school. They develop from opportunities and experiences inside and outside of the classroom.
Although schools and families are important, they can’t do this on their own. I believe that informal learning is a critical third part of the solution. Progress and improvement in K-12 reading and math achievement will require an explicit focus on growing constrained and unconstrained skills at the same time.
But wait, there’s more
If you’d like to learn more about constrained and unconstrained skills, check out these other posts:
A working list of constrained and unconstrained skills that support reading and math
Growth patterns of constrained and unconstrained skills from four nationally representative datasets
Works cited
Cantor, P., Osher, D., Berg, J., Steyer, L., & Rose, T. (2019). Malleability, plasticity, and individuality: How children learn and develop in context. Applied Developmental Science, 23(4), 307–337.
Immordino-Yang, M. H., & Fischer, K. W. (2010). Neuroscience bases of learning. In V. G. Aukrust (Ed.), International encyclopedia of education (3rd Edition, pp. 310–316). Elsevier.
Kuhn, M. R., & Levy, L. (2015). Developing fluent readers: Teaching fluency as a foundational skill. New York: The Guilford Press
Paris, S. G. (2005). Reinterpreting the development of reading skills. Reading Research Quarterly, 40(2), 184-202.
Stafford-Brizard, K. B. (2016). Nonacademic skills are the necessary foundation for learning. Education Week. Retrieved from http://www.edweek.org/ew/articles/2016/07/21/nonacademic-skills-are-the-necessary-foundation-for.html.
Stahl, K. A. D., & García, G. E. (2022). Expanding reading comprehension in grades 3–6: Effective instruction for all students. New York: The Guilford Press.
Although the data in this post focus on children in the United States, these concerns are global in nature.
These data come from the long-term trend version of the National Assessment of Educational Progress. These first two charts show reading and math performance for a nationally representative sample of U.S. nine-year-olds. The data show reading and math scale scores from five separate assessments administered from 2004 through 2022. For reading (Figure 1), a scale score of 200 indicates the ability to understand specific or sequentially related information. A scale score of 250 indicates the ability to search for specific information, interrelate ideas, and make generalizations. For math (Figure 2), a scale score of 250 indicates initial understanding of the four basic operations, an ability to compare information from graphs and charts, and a developing ability to analyze simple logical relations. (The NAEP website provides detail about these performance levels.)
I follow Immordino-Yang & Fischer (2010) to define skill as “the capacity to think and act in an organized way in a specific context.” A deep dive into what skills are and how they develop are beyond the scope of this post. I’ve written separate posts that offer a basic primer on skills and a summary of 14 principles of skill-building.
This term comes from Stafford-Brizard (2016), which discusses how skills such as attention, working memory, self-regulation, problem-solving, social awareness, and growth mindset are critical to how children learn. These abilities and competencies support more than engagement in academic subjects such as reading and math, hence the term “nonacademic skills.”
Although I describe constrained and unconstrained as two categories, in reality they exist along a continuum. Some constrained skills (e.g. letter knowledge) are more constrained than other constrained skills (e.g. reading fluency). The same is true for unconstrained skills; some are less constrained than others.
For example, English speakers all work with the same 26 letters of the alphabet and 44 sounds (phonemes).
For English speakers, you either know all 26 letters of the alphabet or not. You know all 44 phonemes or you do not. You know all the letter-sound combinations (roughly 240) or you do not. Proficiency is evaluated the same way for everyone.
The major problem for constrained skills is that it takes too long for some children to become proficient. We need to close early gaps in these constrained skills. Explicit and systematic instruction is needed in order for children to successfully learn these skills. Some children of course have learning challenges with constrained reading skills (dyslexia) and constrained math skills (dyscalculia). Good systematic instruction can help children develop these skills, particularly if challenges are diagnosed and addressed early. It’s important to remember, however, that constrained skills and unconstrained skills interact and reinforce each other. For example, vocabulary (unconstrained) and word-reading (constrained) reinforce each other. Other unconstrained skills — like attention control, attention shifting, and inhibition control — also support word-reading skills. So, improving unconstrained skills also helps to improve constrained skills.
The NAEP Long-Term Trend assessment measures knowledge and skills of 9-, 13-, and 17-year-olds. The NAEP LTT reports four performance levels: 150, 200, 250, and 300. The Digest of Education Statistics of the U.S. Department of Education has long reported proficiency in terms of the percentage of students at or above these performance levels. Level 150 for both reading and math is representative of constrained skills. Levels 200, 250, and 300 are representative of unconstrained skills. The NAEP LTT website offers descriptions of skills associated with each performance level.
From the NAEP website: “Students at this level know some basic addition and subtraction facts, and most can add two-digit numbers without regrouping. They recognize simple situations in which addition and subtraction apply. They also are developing rudimentary classification skills.”
We can see the effect of the pandemic in 2022. I don’t cover this topic in this post. My review of NAEP long-term trend data between 2020 (pre-pandemic) and 2022 shows that the overall drop in reading and math scores was heavily driven by declines in unconstrained skills.
From the NAEP website: “Readers at this level can follow brief written directions. They can also select words, phrases, or sentences to describe a simple picture and can interpret simple written clues to identify a common object. Performance at this level suggests the ability to carry out simple, discrete reading tasks.”
Recent efforts focused on the ‘science of reading’ promise to close this gap even further in the years ahead.
I follow Cantor et al. (2019) in defining knowledge as the ability to use facts, principles, and ideas to decide and do complex tasks. Knowledge is essential to skill building and sometimes is referred to as a skill.
There is a wide range of skills that could be classified as “unconstrained.” For example, “soft skills”, “durable skills”, and “deeper learning skills” are all examples of complex unconstrained skills. As my focus is improving K-12 math and reading achievement, I focus on the set of unconstrained skills identified in research as component skills for reading comprehension and mathematical cognition.
From the NAEP website: “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.”
From the NAEP website: “Readers at this level can locate and identify facts from simple informational paragraphs, stories, and news articles. In addition, they can combine ideas and make inferences based on short, uncomplicated passages. Performance at this level suggests the ability to understand specific or sequentially related information.”
This specific post is just focused on K-8 unconstrained reading and math skills. I have three additional posts on K-5 nonacademic (executive) skills, K-1 general knowledge, and high school math.
This table is inspired by similar summaries in Kuhn and Levy (2015) and Stahl and Garcia (2022).