This is an important topic, so this post is longer than usual. Citations and additional commentary is in the footnotes. (Mouse over the footnotes to read them.)

Thanks to Malai Amfahr, Madison Brown, Kristin Droege, Siobhan O’Loughlin-Reardon, and Kara Richardson for providing feedback on an earlier (and lengthier) draft of this post. Any errors or omissions are entirely my own.

Last updated: July 15, 2025

Three Big Ideas

1. Unconstrained skills, which develop from experiences inside and outside of school, quietly drive reading and math success. They’re largely responsible for the achievement gap.

2. Informal learning organizations—such as museums, libraries, zoos, aquariums, afterschool programs, and extracurricular programs—are uniquely positioned to provide out-of-school experiences that help children and youth build and strengthen unconstrained skills. There’s evidence they’ve already been doing this.

3. Strategically aligned efforts between schools, families, and informal learning organizations offers a promising path to raise reading and math achievement overall and close stubborn achievement gaps.

Families and schools are critically important to children’s learning and development.1 They are the foundation of children and youth’s academic growth and achievement. But a third institution also plays an important role–informal learning organizations.2 A third of school children’s weekly lives is spent in activities other than school, meals, and sleep. Families routinely rely upon out-of-school programs and experiences to fill some of this time.

“Children learn best in active, engaged, constructive, and interactive environments, when the material they are learning is meaningful to them, and when they receive consequential feedback and probing questions.”3

This post focuses on the important role that informal learning plays in children and youth’s learning and skill development. At their best, informal learning organizations excel in creating and providing active, engaged, constructive, interactive, and meaningful experiences. These experiences support children and youth’s learning and development across a range of important outcomes.

In this post, I summarize findings from rigorous empirical research studies about the impact of informal learning organizations on academic achievement. I focus specifically on their ability to build unconstrained skills. This particular type of skill depends upon opportunities to develop inside and outside of school. Unconstrained skills are largely responsible for the achievement gap between student groups. I believe a focus on unconstrained skills is a pressing opportunity to improve K-12 reading and math achievement. Informal learning organizations are uniquely positioned to play an important role to improve these outcomes.

Informal learning is a big tent

The time children and youth spend in the classroom is unquestionably important for their learning and development. It’s also relatively limited. On average, school age children in the U.S. spend 31 of 168 hours per week in school. Add to this the average time for meals (about 8 hours) and sleep (about 67 hours), and this leaves about 60 hours a week for other activities.4

Informal learning is a big tent that includes a variety of organizations, program types, and experiences:

• Afterschool programs with a mix of academic (e.g. reading, math) and enrichment activities (e.g. science, arts, games)

• Botanical gardens and arboretums with opportunities to learn about plants and trees and their habitats

• Extracurricular activities with a specific content focus (e.g. sports, music, art lessons, Scouts)

• Libraries with with opportunities to engage in arts and crafts, STEM projects, reading clubs, and special interest groups

• Museums, art galleries, and historical sites with opportunities to learn about science, social studies, and the humanities

• Zoos and aquariums with opportunities to learn about animals and their habitats

Some experiences have aspects of formal learning – a defined curriculum and activities that feel like an extension of the school day. Engagement is structured and managed by adults. Some informal learning experiences are completely unstructured. Engagement is completely driven by children and youth interests and learning is largely incidental. Other experiences fall somewhere in between.5

How informal learning matters for reading and math

A child learns a good number of facts, procedures, and concepts in the process of becoming proficient in reading and math. Despite important differences, all reading and math skills can be boiled down to two types–constrained and unconstrained.8 Each skill type has a distinct developmental pathway.

Constrained skills involve a relatively limited amount of information.9 Everyone works with the same information.10 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.11 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.12

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, perspective taking, and knowledge. 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.13 These skills generally take more time to build than constrained skills.

From an early age, constrained and unconstrained skills interact with and reinforce each other’s development. I’ve described elsewhere the constrained and unconstrained skills that research describes as key components for reading and math development. The development of both types of skills at the same time is critical.14 My review of data over the past 25 years shows that the primary challenge with reading and math achievement overwhelmingly lies with unconstrained skills.

Children need repeated and varied experiences to build proficient skills

Up close, the process of skill-building can look messy. It’s a repeated cycle of building, collapse, and rebuilding. Children build specific skills for specific contexts. “Context” means more than the physical environment. It also includes a child’s emotional state, the external support they receive, the domain or subject area, and the specific task involved.15

In the initial stages of skill-building, a small change in the context can rapidly cause one’s skill level to quickly collapse.16 A child might be able to solve a math problem one day or in one situation but not the next day or a different but otherwise similar situation.17 They might be able to exhibit a skill with one person, but not another. Repetition and exposure to multiple opportunities for skill-building across multiple contexts helps children build and rebuild skills towards mastery.

This is particularly important for unconstrained skills. Skills like vocabulary, inference, perspective taking, inhibition control, and knowledge (facts, procedures, concepts) are both “taught and caught.” They are learned through direct instruction in formal learning settings. They are also learned indirectly through less formal learning opportunities.

Unconstrained skills involve more information to acquire, integrate, and master. These slower-building skills don’t develop as readily on school time (weeks, quarters, semesters, years).18 They are built and reinforced over longer periods through experiences inside and outside of school. Children who participate in out-of-school programs have more varied opportunities to build and rebuild these skills outside of the classroom and home.

Research shows informal learning can build unconstrained skills

My review of rigorous empirical studies find that informal learning programs can build unconstrained skills:

• Afterschool programs: self-perception (self-esteem, self-concept, self-efficacy), problem behaviors (inhibition control), social understanding (perspective taking)19

• Extracurricular activities: math problem solving, reading comprehension, and expressive vocabulary20

• Museums: factual knowledge, conceptual knowledge, critical thinking (e.g. interpretation, evaluation, comparison)21

• Zoos and aquariums: factual knowledge, conceptual knowledge, inhibition control22

Impact on skills is not automatic. It depends upon program quality and the design of the experiences. Too many programs and experiences fail to demonstrate impact under rigorous scrutiny. But it is possible.

I don’t mean to suggest that out-of-school organizations shouldn’t help children and youth develop constrained skills like word-reading or arithmetic. But the comparative advantage of informal learning organizations is their opportunity to help children and youth build different skills in different ways from what they experience in formal classroom settings. Different.

Access to informal learning opportunities is a key challenge

Not all of the roughly 50 million school age children and youth in the United States have the same opportunity for these experiences. Most afterschool and extracurricular activities are paid programs. The majority of families who send their children to these programs are middle income families.23 Families in the top-fifth of household incomes spend about $2,200 annually on out-of-school activities. This is just over three times the $700 spent by families in the bottom-fifth of household incomes.24

Between 1998 and 2020, school-age children in high-income households were much more likely than their peers in low-income households to participate in a range of potential skill-building activities: sports (59% vs 24%), music, dance, and language lessons (41% vs 21%), and organized clubs (24% vs 11%). Over this period, 43% of children and youth from low-income households did not participate in any organized extracurricular activities compared to just 14% of children and youth from high-income households.25

These differences also extend to other informal learning settings. Roughly half of school-aged youth in the United States visit a museum, art gallery, or historical site once or more a year. Children in lower socioeconomic households and those who live in rural areas, however, are less likely to visit museums over their entire childhood.26

There are many organizations working to increase access to quality afterschool programs, summer programs, and expanded learning opportunities.27 Like other issues, progress depends upon policy, funding, and willpower. While we advocate for more support, we can also demonstrate what’s possible with the resources currently in hand.

How we can leverage the power of informal learning to improve reading and math

Informal learning plays an important role in learning and skill-building alongside the classroom and home. Each informal learning experience contains the seed of opportunity to build and strengthen a number of unconstrained skills. Here are seven ideas to leverage the superpower of informal learning to improve reading and math achievement.

1: Take skills seriously. We should use research insights from developmental psychologists about how skills work to inform the design of programs and experiences for children and youth. Elsewhere I’ve summarized a set of key principles about skills and skill-building.

2: Take skill-building seriously. Researchers find programs that involve supervised practice – “practicing new skills in the presence of someone who can observe and provide feedback to ensure its optimal adoption and application” – are more effective than programs with unsupervised practice.28 Impactful afterschool programs are active, focused, and explicit regarding skill-building.29 In other words, they use active forms of learning to support skill-building, have at least one component of their program focused on developing skills, and have clear and explicit learning objectives related to skill growth.30 Programs with these components have shown impact on a range of academic and non-academic outcomes.

3: Focus on unconstrained skills. Research studies provide examples of how a single well-designed informal learning experience can boost unconstrained skills:

• Conversation cards that structured a museum visit increased dialogue between preschool children and parents, transfer of knowledge across exhibits, and increased knowledge acquisition up to two weeks afterwards.31

• Guided visits during trips to the zoo supported increased learning and knowledge acquisition.32

• The combination of activities prior to and after field trips to zoos, aquariums, and museums positively impacted student interest and knowledge acquisition.33

• Hands-on activities during visits to zoos and aquariums increased students’ cognitive engagement and knowledge-building.34

• The combination of strategically-designed worksheets, social interaction, and free choice increased knowledge acquisition during a museum visit.35

• Guided discussion about art works during museum visits increased critical thinking skills up to three weeks later.36

And this impact is not necessarily short-lived. For example, researchers find that under the right conditions, the impact of visits to zoos and aquariums on children’s knowledge can last from months to years.37

4: Build systems to support unconstrained skills. Supporting individual organizations is good. Building systems of informal learning is even better. The FrameWorks Institute offers the metaphor of “charging stations” as a way to think about this:

“We want to make sure that people in our community are a part of high wattage, densely networked charging systems that support early childhood development. That’s why we need to build a network of connected charging stations that includes daycares, pre-kindergarten (pre-K), libraries, museums, science centers, and afterschool programs so that young children have lots of opportunities to charge up their development.”38

Some children and youth live in neighborhoods full of charging stations: libraries, sports fields, organized clubs, community centers. Others have less ready access to “charge up.” This means fewer opportunities to build and rebuild skills across multiple settings. The Forum for Youth Investment offers a roadmap for designing community-level systems of support.

5: Let informal learning be informal. Informal learning organizations don’t have to look like school to make an impact. They don’t have to follow a curriculum (though they can if they want). Informal learning organizations don’t have to cover a set of grade-level state standards each year. They can directly engage children and youth’s interests and curiosity. Children and youth can practice hobbies and favorite activities or discover new ones. They can move at their own pace. Let informal learning do what it’s good at. Awe and wonder. Fun and joy. Love of learning. Let the goose lay the golden eggs.

6: Rethink alignment between formal and informal learning. Alignment does not mean that informal learning organizations should replicate the structures and work of school systems. It means combining the strengths of formal education with those of informal learning. In “all-day schools” in European countries such as Germany, Finland, Greece, Spain, and Croatia, learning seamlessly shifts from formal classroom instruction at the end of the academic day to informal extracurricular activities after school.39 The Learn Everywhere initiative in New Hampshire gives academic credit for student experiences outside of school, such as music lessons, karate classes, robotics clubs, and foreign language classes. Creating stronger linkages between formal and informal learning gives more children and youth more opportunities to build constrained and unconstrained skills at the same time.

7: Rethink impact and outcome measurement. Greater focus on unconstrained skills and system-building requires different approaches to impact and outcome measurement. We need to make sure assessments are appropriate for efforts focused on unconstrained skills. We should also include as outcomes key drivers of skill-building, like wonder, awe, interests, motivation, and self-beliefs. We also will need to rethink time; it may take longer to see impact than a semester or academic year.40 Our theories of change should explicitly mention skill-building. For example, the QuEST (Quality, Engagement, Skills, and Transfer) program logic model from the Forum for Youth Investment helps organizations identify which skills they are trying to grow and how likely the qualities of the program experience they provide will grow those skills. There are also multiple observation tools for out-of-school programs that include a review of skill-building program features among other program components.

Unconstrained skill is a powerful lever to improve reading and math achievement

“To move the boulder, you need to be smart and strategic. Because of the complexity you face, you can’t change everything. You can’t change most things. You can’t even change a respectable fraction of things! But with a bit of prodding and catalyzing, you can change something. A well-chosen something. We’ll call that “well-chosen something” a Leverage Point (a term popularized by the systems theorist Donella Meadows).” Dan Heath, Reset: How to change what’s not working

Envisioning K-12 reading and math achievement as a boulder might conjure up the image of poor Sisyphus. Each morning he rolled a boulder up a hill only to see it roll back to the bottom at the end of the day. When it comes to metaphors about moving boulders, I’m more of an Archimedes guy. I am looking for a lever and a place to stand.

The research and data I’ve seen screams out that a focus on building children and youth’s unconstrained skills is a powerful leverage point. In Dan Heath’s words, it’s well-chosen. Families and schools are obvious places to focus this lever.41 But informal learning organizations are important places to focus efforts, too.

Families and schools routinely rely upon informal learning organizations. Families sign up their children for extracurricular activities, after school programs, and summer camps. Families take their children to the zoo, children’s museum, and science center. Schools regularly send classes on field trips to zoos, aquariums, science centers, and museums.

Working together with families and schools, informal learning organizations offer a promising path toward improved reading and math achievement. Each visit to an afterschool program, extracurricular activity, zoo, aquarium, library, or museum has the power to be a transformative, skill-building experience.

But wait, there’s more

• An overview of constrained and unconstrained skills.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

• Insights about the 14 principles of skill building.

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Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). The process of skill-building occurs at two levels: short-term, small-scale changes in specific contexts and longer-term, large-scale changes across multiple contexts. Skill building depends upon a variety of experiences and opportunities for developing, overlearning, and generalizing skills across contexts and emotional and mental states. This can be supported with the intentional design and development of learning opportunities and experiences intended to optimize skill development with high, personalized support (Cantor et al., 2019).

“Learning is a class of short-term changes leading to new knowledge and skill. Development is a class of long-term changes leading to new knowledge and skill. Learning and development occur at the same time.” (Fischer & Granott, 1995)

Skill building happens at two levels: a) shorter-term, small-scale changes within particular domains or contexts and b) longer-term, large-scale changes independent of domain (Fischer & Bidell, 2006). Psychologists refer to these small-scale changes as microdevelopment. The large-scale changes are referred to as macrodevelopment.

There are multiple pathways for skill building. (See Principle 12.) Developmental psychologists describe this as a “constructive web.” Shorter-term changes in skill building create the individual strands; longer-term changes comprise the entire web (Fischer & Bidell, 2006).

These shorter-term small scale changes (microdevelopment) involve the acquisition of skills for specific contexts, tasks, or problems. Skill acquisition can occur within relatively limited amounts of time – as short as minutes, hours, days, or weeks (Fischer & Granott, 1995; Fischer & Bidell, 2006).

Microdevelopment is involved when children are expected to learn, grow, and master skills in “school time” – minutes, hours, days, weeks, semesters (Fischer, 2008). This includes combining, differentiating, and reorganizing specific skills in particular tasks and contexts, as well as generalizing them (Fischer & Yan, 2002).

By contrast, macrodevelopment involves broader-scale change over months and years. This larger scale process incorporates the many varied activities and domains involved in microdevelopment. These skills are gradually consolidated, generalized, and related across contexts over long periods.

Developmentally, these changes appear as periodic clusters of jumps, drops, and reorganizations of skills across multiple strands (domains). (Fischer & Bidell, 2006). In this process of reorganization and rebuilding of skill hierarchies, macrodevelopment adds broad changes in mental capacity and frameworks (Fischer & Yan, 2002).

These shorter-term, small-scale changes in skill level and longer-term, large-scale changes in skill ability converge and integrate. Both levels of skill building affect and place limits on the other. Longer-term, large-scale development is limited (constrained) by the amount and degree of skill development from shorter-term, small-scale skill building.

In turn, broader scale, cross-domain skills (macrodevelopment) create limits on the developmental range of short-term, small scale skill development. The developmental range is the difference in a person’s ability when they work independently (the functional level) and with high support (the optimal level) (Fischer & Bidell, 2006).

Skills are gradually constructed and reconstructed in both the short-term (in seconds, minutes, hours, or days) and long-term (in months or years) (Fischer & Yan, 2002). Both levels of skill building work together to “form complex skills that stabilize and generalize to new contexts over time” (Cantor et al., 2019).

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

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.

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

Fischer, K. W. (2008). Dynamic cycles of cognitive and brain development: Measuring growth in mind, brain, and education. In A. M. Battro, K. W. Fischer, & P. Léna (Eds.), The educated brain (pp. 127–150). Cambridge University Press.

Fischer, K. W., & Granott, N. (1995). Beyond one-dimensional change: Parallel, concurrent, socially distributed processes in learning and development. Human Development, 38(6), 302–314.

Fischer, K. W., & Yan, Z. (2002). Darwin’s construction of the theory of evolution: Microdevelopment of explanations of variation and change in species. In N. Granott & J. Parziale (Eds.), Microdevelopment: Transition processes in development and learning. Cambridge University Press.

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.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). The transfer of skills learned in one context to other contexts is a major challenge in education. In most cases, skill transfer (generalization) is time-intensive and requires intentional effort. Skills learned in one setting can be applied in another setting (Osher et al., 2020). But it is highly unlikely that mere exposure will do the trick. This is because skills are context sensitive. Change the context, change the skill. (See Principle 3.) Shortcuts to transfer, however, might be possible through the intentional leverage of existing skills, frameworks, and concepts (prior knowledge).

“When a person borrows someone else’s bicycle and rides it or rides his familiar bicycle on an unfamiliar type of terrain (e.g. across a grassy field instead of on a road or sidewalk), he must adapt his skill to the context of the new bicycle or terrain. He cannot immediately ride skillfully by using the skill he possesses from before. When a woman attempts to perform analysis of variance with a different computer program or when she tries to analyze the data in a study with an unfamiliar design, she has to work to adapt her skill. It can take days or weeks of hard work to generalize her skill to the new context.” (Fischer et al., 1993)

The transfer of knowledge and learning from one setting or domain to another is a well-established challenge in education (Barnett & Ceci, 2002). This process is referred to as the generalization of skills.

Near transfer involves the application of skills or knowledge learned in one context (task, activity, domain) to a similar context. Far transfer involves the use of knowledge or skills very different from the original context, task, or activity (Fischer & Bidell, 2006).

Education researchers and psychologists readily find evidence of near transfer from programs and interventions. But evidence of far transfer is much harder to find:

“Contrary to the common expectation that knowledge generalizes readily, researchers have had great difficulty finding generalization beyond very similar tasks. What occurs readily is near generalization, in which a student performs a task that differs modestly from the one where she learned the skill…Educational assessments indicate that students who perform well in a class typically generalize their knowledge only to material similar to what was explicitly taught in the class. They do not show far generalization to distinctly different material. The generalization about generalization is that near generalization is to be expected, but far generalization to very different tasks and situations does not usually occur.” (Fischer & Immordino-Yang, 2002)

Researchers have struggled to find evidence of far transfer of skills from interventions or activities developed in one domain to another. Take, for example, the popular belief that playing chess can improve math skills. While not specifically focusing on math facts, procedures, or concepts, the idea is that chess helps critical thinking, problem solving, and pattern recognition. Each of these skills are involved in math.

Unfortunately, experimental studies repeatedly fail to find a relationship between playing chess and math outcomes (Sala & Gobet, 2017; Blanch, 2022). Researchers similarly fail to find far transfer of skills from playing video games (Sala et al., 2018), playing brain games (Simons et al., 2016), participating in music training (Sala & Gobet, 2020), and cognitive training (e.g. working memory) (Gobet & Sala, 2023).

Skill transfer obviously happens all the time. The math skills we learn in specific tasks in early elementary school become generalized and used in many settings throughout our adult lives – such as modifying a recipe, calculating the savings on a discounted purchase item, or deciding how much paint to buy for a home improvement project.

So, how does skill transfer occur? Better yet, how can we help it happen for children to support their success in school subjects like math and reading? Fischer and Immordino-Yang (2002) say there are two fundamental approaches:

“For the structures to become generalized, either students must 1) learn them extensively over long periods involving diverse tasks, or 2) instruction must be grounded in common frameworks and concepts that are part of students’ prior knowledge base.” (Fischer & Immordino-Yang, 2002)

Approach One: Build skills over long periods with diverse tasks

Fischer and Bidell (2006) tell us that “learning is not a simple transmission of information through a conduit from one person to another or from one task to another.” It takes time and effort to build generalized skills. As we see in Principle 10, skill building is a repeated process of construction, collapse, and reconstruction.

Skills are repeatedly rebuilt with variations in tasks, context, and support. Even minor changes can cause a skill to collapse and require reconstruction. The skill needs to be built and rebuilt with variations so it can be sustained in different contexts and under different states (emotion, energy, etc.) (Fischer, 2008). “Through this slow process, people gradually build a more general skill” (Fischer & Bidell, 2006).

Building new, generalizable skills in subjects such as reading and math usually takes a long time. Developing expertise in a broad area often requires 5 to 10 years of learning. Skill proficiency in a specific task or smaller domain can be achieved in weeks or months – but it still takes time (Fischer & Yan, 2002; Fischer & Bidell, 2006).

Skills are built for participation in specific mental and physical tasks and contexts. (See Principle 3.) Over time they can and will gradually extend from specific contexts to new contexts through real practice in real contexts (Fischer & Bidell, 2006; Cantor et al., 2019).

There is recent research that supports this approach. Researchers used a “spiral” supplemental literacy curriculum for three academic years (Grades 1 through 3) to build vocabulary, science content knowledge, and comprehension skills. The curriculum progressed from simple to complex concepts. The concepts were cyclically reintroduced with each encounter building upon previous lessons.

The researchers found evidence of far transfer of skills taught in the curriculum on state reading and math tests in Grade 3 (Kim et al., 2024). 1This impact was not a one-time occurrence. The researchers found a continued positive impact (far transfer) on state reading and math tests a year later in Grade 4.

Time is an important factor. In this experimental study, there was no evidence of far transfer of taught vocabulary and science concepts on reading comprehension after two years of intervention. It took a full three years of intervention for this impact to appear (Kim et al., 2024).

Researchers similarly found evidence for far transfer in dialogic argumentation skills (debate) after three years. Middle school students who participated in a twice weekly dialogic argumentation intervention showed evidence of enhanced argumentation skills on topics outside of what was covered in the program (Crowell and Kuhn, 2014). The intervention also showed positive far transfer on argumentative writing skills (Kuhn and Crowell, 2011).2

Approach Two: Leverage common or familiar frameworks and concepts (prior knowledge)

“Fortunately, a broader view gives a less bleak portrait. The key is to examine frameworks and concepts that students commonly use instead of those they learn in [school]. Typically, students who cannot generalize the frameworks and concepts that they learn in school can easily generalize their own favored frameworks and concepts.” (Fischer & Immordino-Yang, 2002)

The notion of leveraging children’s prior knowledge in their learning is not a new idea. What is novel is using prior knowledge – children’s “own favored frameworks and concepts” – to create a “shortcut” in the ordinary time-intensive mechanism of skill construction, collapse, and reconstruction to generalize skills. (See Principle 10.)

Children have already gone through this process of skill building with their own “favored skills” and knowledge. Providing experiences, opportunities, and instruction based upon familiar ways of thinking and/or existing knowledge can accelerate the process of generalization of new skills. The alternative – extensive, long-term practice in diverse tasks – takes more time for generalized skill development than the typical intervention, program, curriculum module, or school course allows (Fischer & Immordino-Yang, 2002).

A positive example of this approach is the work of Robbie Case and colleagues on the mental number line (Case et al., 1996). The mental number line represents numbers in a sequential order. Each number has a specific place and relationship to other numbers. The mental number line helps children develop “number sense,” the understanding that numbers are both discrete units and part of a continuous system. This fundamental understanding supports later math achievement.

Case and colleagues developed an approach to teaching the mental number line to Pre-K and early elementary children that explicitly builds upon skills they already developed for understanding quantity and for counting. The intervention delivered positive results after as few as ten weeks. Children showed evidence of far transfer in unrelated tasks such as musical scales (which are based in numbers) and distributing gifts at birthday parties.

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

Works Cited

Barnett, S. M., & Ceci, S. J. (2002). When and where do we apply what we learn?: A taxonomy for far transfer. Psychological Bulletin, 128(4), 612.

Blanch, A. (2022). Chess Instruction Improves Cognitive Abilities and Academic Performance: Real Effects or Wishful Thinking? Educational Psychology Review, 34(3), 1371–1398.

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.

Case, R., Okamoto, Y., Griffin, S., McKeough, A., Bleiker, C., Henderson, B., Stephenson, K. M., Siegler, R. S., & Keating, D. P. (1996). The role of central conceptual structures in the development of children’s thought. Monographs of the Society for Research in Child Development, 61(1/2), i–295.

Crowell, A., & Kuhn, D. (2014). Developing dialogic argumentation skills: A 3-year intervention study. Journal of Cognition and Development, 15(2), 363–381.

Fischer, K. W. (2008). Dynamic cycles of cognitive and brain development: Measuring growth in mind, brain, and education. In A. M. Battro, K. W. Fischer, & P. Léna (Eds.), The educated brain (pp. 127–150). Cambridge University Press.

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

Fischer, K. W., Bullock, D. H., Rotenberg, E. J., & Raya, P. (1993). The dynamics of competence: How context contributes directly to skill. In R. Wozniak & K. Fischer (Eds.), Development in context: Acting and thinking in specific environments. Lawrence Erlbaum Associates.

Fischer, K. W., & Immordino-Yang, M. H. (2002). Cognitive development and education: From dynamic general structure to specific learning and teaching. In E. Lagemann (Ed.), Traditions of scholarship in education. Spencer Foundation.

Fischer, K. W., & Yan, Z. (2002). Darwin’s construction of the theory of evolution: Microdevelopment of explanations of variation and change in species. In N. Granott & J. Parziale (Eds.), Microdevelopment: Transition processes in development and learning. Cambridge University Press.

Gobet, F., & Sala, G. (2023). Cognitive Training: A Field in Search of a Phenomenon. Perspectives on Psychological Science, 18(1), 125–141.

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.

Kim, J. S., Gilbert, J. B., Relyea, J. E., Rich, P., Scherer, E., Burkhauser, M. A., & Tvedt, J. N. (2024). Time to transfer: Long-term effects of a sustained and spiraled content literacy intervention in the elementary grades. Developmental Psychology.

Kuhn, D., & Crowell, A. (2011). Dialogic argumentation as a vehicle for developing young adolescents’ thinking. Psychological Science, 22(4), 545–552.

Osher, D., Cantor, P., Berg, J., Steyer, L., & Rose, T. (2020). Drivers of human development: How relationships and context shape learning and development. Applied Developmental Science, 6–36.

Sala, G., & Gobet, F. (2017). Does chess instruction improve mathematical problem-solving

ability? Two experimental studies with an active control group. Learning Behavior, 45(4), 414–421.

Sala, G., & Gobet, F. (2020). Cognitive and academic benefits of music training with children: A multilevel meta-analysis. Memory & Cognition, 48(8), 1429–1441.

Sala, G., Tatlidil, K. S., & Gobet, F. (2018). Video game training does not enhance cognitive ability: a comprehensive meta-analytic investigation. Psychological Bulletin, 144(2), 111–139.

Simons, D. J., Boot, W. R., Charness, N., Gathercole, S. E., Chabris, C. F., Hambrick, D. Z., & Stine-Morrow, E. A. L. (2016). Do “brain-training” programs work? Psychological Science in the Public Interest, 17(3), 103–186.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). Children develop skills along multiple pathways. The “constructive web” is a powerful metaphor for understanding the dynamic relationships that shape the pathways for children’s skill development. The strands in the web are jointly built by the learner (the “web builder”) and the support conditions that they receive for skill building. These support conditions are akin to the tree branches or corner of a wall that a spider might use to build a web (Fischer & Bidell, 2006).

The constructive web “highlights the opportunity for instructional and curricular designs that address these alternative pathways and, in so doing, channel student effort toward the pathway where individual progress will be greatest, therein optimizing developmental range and literacy [and math] development for all students” (Cantor et al., 2019).

“There are various routes to effective skill development for reading and math. The job of education is to provide support for children with different neuropsychological profiles to develop effective yet flexible skills. Children use whatever capacities they have to learn the most important skills they need. Although there is often a modal way of learning a specific skill, people can adapt their capacities to learn skills in diverse ways.” (Immordino-Yang and Fischer, 2010)

We’re accustomed to marking growth and development milestones in terms of month, years, and grade levels. Physical, social, and language milestones. Reading and math skill benchmarks. These markers are based upon the idea of a typical (or average) sequence of growth and development.

The messy reality is there is no single pathway for building skills. Multiple pathways are possible (Cantor et al., 2019).

Successful classroom performance often requires following a prescribed set of procedures. “Examples of such procedures include “borrowing” as an algorithm when subtracting, remembering rules of thumb like “when two vowels go walking the first one does the talking,” or learning to apply strategies such as visualization, prediction, and self-questioning as the solution for comprehension difficulties” (Snow, 2008).

Learning these procedures might be tremendously helpful for some children. But for others they could be a hindrance. Some children who are successful in math or reading use “tricks, reminders, and strategies” that differ from what is formally taught in the classroom (Snow, 2008). Fischer and colleagues found three developmental pathways for developing word-reading skills (Fischer & Bidell, 2006).

In mathematics, “derived strategies” can greatly differ from the prescribed procedures or algorithms in the classroom.1 These different approaches appear to represent less “traditional” skill pathways that are nonetheless effective. Moreover, such approaches in math appear to be associated with greater conceptual understanding of math principles than rote memorization of algorithms (Dowker, 2019).

Given that children's skill development is both uneven and stable at the same time (See Principle 11), it naturally follows that the sequence of skill development – the developmental pathways of individual skills and groups of skills – can and does vary by child as well.

“[V]ery few developmental challenges can be solved in only one way. For many challenges, there is an “easy way,” the way that the majority of learners adopt and that teachers might usefully offer to their students, but also alternative routes to the same end. Most children start to talk by babbling, then producing single words, and then gradually increasing the length and complexity of their utterances. But some never babble, some start with longer, only partially segmented utterances, and some rely on gesture for a long period before moving to speech. Most children crawl, then “cruise” by walking upright with support, and then walk independently. But some scoot instead of crawling; others prefer to be upright from a very early age and never either scoot or crawl before walking. As far as we can tell, the linguistic and locomotive skills of the “easy route” groups end up indistinguishable from those of the ones who have taken alternative paths.” (Snow, 2008)

Developmental psychologists describe these varied pathways as a “constructive web.” The strands in the web are jointly built by the learner (the “web builder”) and the support conditions that they receive for skill building. These support conditions are akin to the tree branches or corner of a wall that a spider might use to build a web (Fischer & Bidell, 2006).

The web strands are fragile and tentative at first. They need support from surrounding strands. With practice and experience, the strands can become a stable part of the web. Unlike spiders, however, humans build their webs of skills by interacting with other people. (See Principle 8.)

“Within the web, strands represent pathways along which a child develops simultaneously, with pathways demonstrating responsiveness to emotion and support, the capacity for resilience, and variability in sequence, synchrony, and developmental range.” (Cantor et al., 2019)

The existence of multiple pathways for learning and development suggests powerful opportunities to discover and support children’s journeys along their unique pathways. In formal learning settings, a key challenge will be to create individualized support in the context of standardized instruction and assessment (Snow, 2008; Immordino-Yang, 2008; Christoff, 2008).

This could be one place where organizations outside of the classroom (out-of-school, family support, informal learning) can leverage their greater flexibility to meet individual student needs. This could also help support children’s ability to develop broader skill sets for critical thinking and problem solving (Christoff, 2008). Technology such as artificial intelligence might be helpful if it could help children discover and follow the most productive developmental pathways that work for them.

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

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.

Christoff, K. (2008). Applying neuroscientific findings to education: The good, the tough, and the hopeful. Mind, Brain, and Education, 2(2), 55–58.

Dowker, A. (2019). Individual differences in arithmetic: Implications for psychology, neuroscience and education. Routledge.

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

Immordino-Yang, M. H. (2008). The smoke around mirror neurons: Goals as sociocultural and emotional organizers of perception and action in learning. Mind, Brain, and Education, 2(2), 67–73.

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.

Snow, C. E. (2008). Varied developmental trajectories: Lessons for educators. Mind, Brain, and Education, 2(2), 59–61.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). Up close, skill building can look highly variable. This is especially so for skills that children are able to do independently. Under conditions of high support, however, skill development looks more predictable and stable. Moreover, children are capable of engaging in higher, more complex skills sooner when they receive strong social support to do so.

“When children's problem solving is analyzed according to support and its related changes in performance, the stage debate disappears. Development has strongly stage-like properties under conditions of high support, but not under conditions of low support.” (Fischer & Immordino-Yang, 2002)

Viewed up close, skill level can vary greatly from one moment to the next. A child’s ability can change for many reasons: how they feel (emotion), level of support, the task, goal, etc. (Cantor et al., 2019). “Even from moment to moment, a person performs a task differently as she or he adapts to variations in the situation, social context, or emotions of self and others” (Fischer & Bidell, 2006).

Although skill levels can vary for multiple reasons, there are predictable patterns in this unevenness. A very powerful source of this variation is the level of social support a child receives. In conditions of high support, a child is able to more consistently produce skills closer to the optimal level of their developmental range. (See Principle 9.)

Such support, however, does not always have to come from interaction with another person. Based upon the situation, well-designed instructional resources and materials (e.g., texts, diagrams, etc.) can be used to support a child’s optimal skill level (Fischer & Immordino-Yang, 2002).

Modeling or providing examples is another way to move a child’s ability towards their optimal level. For example, a beginning learner might more readily sound out words and suggest other words that rhyme with them when given examples or words to choose from by her teacher. When this support is removed, the child’s skill will likely collapse (Fischer & Immordino-Yang, 2002).

Under conditions of low support, skill building can appear quite uneven–even chaotic. (See Principle 10.) Under conditions of high support, however, skill building has a more stable, “stage-like” growth pattern (Fischer & Immordino-Yang, 2002). Together and within the context of the developmental range (See Principle 9), this explains the variation and stability of skill development.

The developmental psychologist Kurt Fischer and his colleagues identified multiple stages of skill development that correspond with child and brain development (Fischer & Rose, 1998; Fischer & Bidell, 2006; Fischer, 2009; Mascolo and Fischer, 2010). Each stage represents the ability for greater skill complexity.

We have two examples from Mascolo and Fischer (2010) for math and narrative understanding. The age ranges are based on research with middle-class American and European children and could vary across social groups. The skill examples should be considered illustrative rather than definitive.

The first example is for very young children. The math skills involves counting objects. The narrative skill involves simple description of an event the child experienced. Under conditions of high support (optimal level), a child can demonstrate these skill levels as early as 18-24 months. Under conditions of low support (independently or functional level), a child can demonstrate these skill levels between 2-5 years.

Skill stage: Single representations

Age range of optimal skill level: 18-24 months

Age range of functional skill level: 2-5 years

Math skill: The child begins to count objects. They slowly learn to associate a number with an object. They learn cardinality.

Narrative skill: The child can provide simple descriptions of individual events. (“We went to the park.”) Adults move the narrative forward (shift focus) using questions. (“What did we do? Who did we see? What did we do next?”)

The difference in skill level between the two levels (function and optimal) is the child’s developmental range. (See Principle 9.) Under conditions of high support, the child can consistently demonstrate the optimal level as early as 18-24 months. With practice, they are likely to demonstrate this skill level on their own (functional level) on the earlier part of the 2-5 year age range. Without lots of practice under high support, they are likely to demonstrate this skill on their own near the latter part of the 2-5 year age range.

Our second example occurs about a decade later. The math skill this time is simple algebraic expressions. The narrative skill is the ability to understand complex stories (likely both oral and written text). Once again, the child potentially can reach these levels of math and narrative skills sooner with high support. The timing of independent ability (functional level) will depend upon multiple factors, including the opportunity to reconstruct and reconstruct these skills under different conditions of support. (See Principle 10.)

Skill stage: Single abstractions

Age range of optimal skill level: 10-12 years

Age range of functional skill level: 13-20 years

Math skill: The child has initial understanding of simple algebraic expressions with abstract variables representing quantity (2x=4).

Narrative skill: The child can understand complex stories. These stories involve characters with distinct mental states and motives. The narrative has organized plots and subplots. These are driven by conflicts between characters and attempts to resolve these conflicts.

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

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.

Fischer, K. W. (2009). Mind, brain, and education: Building a scientific groundwork for learning and teaching. Mind, Brain, and Education, 3(1), 3–16.

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

Fischer, K. W., & Immordino-Yang, M. H. (2002). Cognitive development and education: From dynamic general structure to specific learning and teaching. In E. Lagemann (Ed.), Traditions of scholarship in education. Spencer Foundation.

Fischer, K. W., & Rose, S. P. (1998). Growth Cycles of Brain and Mind. Educational Leadership, 56(3), 56–60.

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.

Mascolo, M. F., & Fischer, K. W. (2010). The dynamic development of thinking, feeling, and acting over the lifespan. In R. M. Lerner (Ed.), Handbook of life-span development (Vol. 1, pp. 149–194). Wiley.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). Building new skills is a journey. The terrain is full of growth spurts, drops, and plateaus. Each person’s journey depends upon where they start. While new learners have the most ups and downs, even experts will experience the occasional drops in skill levels as they acquire new knowledge and skills.

“Analysis of growth curves shows a prototypic pattern for building and generalizing a new skill: People build a skill and then repeatedly rebuild it in a wavelike pattern of construction and reconstruction, not in a straight line or monotonic upward progression. Encountering a new task or situation, people first move down to a low level of complexity…They then gradually build a more complex skill for coping with the task by repeatedly rebuilding it with variations.” (Fischer & Bidell, 2006)

When we develop new skills, we combine new knowledge and skills with what we already know and are able to do. We rebuild our hierarchy of skills. (See Principle 2.)

This process involves repeated patterns of skill construction, skill collapse, and skill reconstruction. Minor changes in context (task, support, motivation, emotions, etc.) can cause a skill to collapse and require reconstruction (Fischer & Bidell 2006).

Skill acquisition for beginners (novices) is experienced through significant growth spurts and drops. Through repeated practice, beginners reach intermediate levels of skill ability.

At this point, skill building becomes less chaotic. Growth patterns shift to take on more of a wavelike pattern of construction and reconstruction. At this intermediate stage, learners are able to maintain a higher skill level for a longer time. But they still can experience abrupt drops in ability (Fischer, 2008).

It typically takes months or even years to become an expert capable of sustaining a high level of skill performance independently (Fischer & Bidell, 2006).

Even experts occasionally experience drops in skill levels when learning a new task, concept or activity. The difference is that these drops tend to be smaller and short-lived (Fischer & Bidell, 2006; Fischer, 2008). Regardless of age, domain, or expertise, everyone experiences this pattern of construction and reconstruction when learning new skills.

Growth curves for learning a new task. Source: Fischer (2008).

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

Works Cited

Fischer, K. W. (2008). Dynamic cycles of cognitive and brain development: Measuring growth in mind, brain, and education. In A. M. Battro, K. W. Fischer, & P. Léna (Eds.), The educated brain (pp. 127–150). Cambridge University Press.

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

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.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). The developmental range is the difference in a person’s skill level with social support (optimal level) and without support (functional level). With the right amount of challenge, rigor, feedback, and support, we can intentionally help children drive accelerate progress across their developmental range (Cantor et al., 2019).

“Skills are not all or none: A person does not suddenly move from not being able to perform a task to performing it well. Instead, a person’s skills vary frequently between two upper limits – a functional level, the most complex performance that a child or adult performs without support, and an optimal level, the most complex performance that she or he performs with explicit support, such as through modeling or priming. As a person builds a more competent, automated skill, she no longer requires extensive support, and can routinely perform well at a functional level.” (Fischer and Immordino-Yang, 2002)

The developmental range is the difference in a person’s ability with and without social support (Fischer & Bidell, 2006). At the low end is the functional level. This is the highest skill level for a particular task or activity a person is capable of independently. When a person receives high support, their skill level for the same task typically increases (Mascolo, 2020). This is called the optimal level.

We routinely operate at different skill levels for different tasks. “In optimal contexts–with high support, familiar tasks, and motivation to perform–children show a true upper limit on performance” (Fischer et al., 1993). Everyday we use skills across our developmental range.

As adults, we operate at our functional level when we prepare a familiar meal, read a newspaper, or drive a familiar route in low to moderate traffic. We move closer to our optimal level when we follow a recipe to prepare a meal for the first time; read a complex text after hearing a relevant lecture; or use GPS to drive an unfamiliar route (Fischer & Yan, 2002; Mascolo, 2020).

This concept of the developmental range builds upon Lev Vygotsky’s Zone of Proximal Development. The ZPD emphasizes the level of support a person receives to complete a task or engage in an activity. The developmental range goes further to incorporate the conditions under which a task or activity is engaged: skill area, features of the task, context, emotions, motivation, time of day, etc. (See Principles 3, 6, and 7.)

A person’s skill level for the same task can vary moment to moment based upon a combination of these factors. People show one level of ability when they act independently and a much higher ability in a highly supported context (Fischer et al., 1993; Fischer, 2008; Mascolo & Fischer, 2010).

We are most aware of our developmental range when we learn something new. When prompted by a more experienced teacher or mentor, we can understand a new idea or practice a new skill at a relatively high level. We are operating closer to our optimal level.

But when we leave and try to explain the new idea or demonstrate the skill to a friend we find our ability drops precipitously. We are operating closer to our functional level. This idea of the developmental range applies across tasks, ages, and cultures. The range grows larger with age at least through the late twenties (Fischer & Bidell, 2006).

Instructional and program design can intentionally support the acceleration of students’ progress across their developmental range. Examples include: 1) intentional leverage of prior knowledge; 2) activities that support retention of knowledge and information in long-term memory; 3) presentation of material in multiple modalities and contexts; and, 4) use of strategies such as well-designed project-based learning, well-designed service learning, and well-designed collaborative learning (Cantor et al., 2019).

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

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.

Fischer, K. W. (2008). Dynamic cycles of cognitive and brain development: Measuring growth in mind, brain, and education. In A. M. Battro, K. W. Fischer, & P. Léna (Eds.), The educated brain (pp. 127–150). Cambridge University Press.

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

Fischer, K. W., Bullock, D. H., Rotenberg, E. J., & Raya, P. (1993). The dynamics of competence: How context contributes directly to skill. In R. Wozniak & K. Fischer (Eds.), Development in context: Acting and thinking in specific environments. Lawrence Erlbaum Associates.

Fischer, K. W., & Immordino-Yang, M. H. (2002). Cognitive development and education: From dynamic general structure to specific learning and teaching. In E. Lagemann (Ed.), Traditions of scholarship in education. Spencer Foundation.

Fischer, K. W., & Yan, Z. (2002). Darwin’s construction of the theory of evolution: Microdevelopment of explanations of variation and change in species. In N. Granott & J. Parziale (Eds.), Microdevelopment: Transition processes in development and learning. Cambridge University Press.

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.

Mascolo, M. F. (2020). Dynamic skill theory: An integrative model of psychological development. In M. F. Mascolo & T. R. Bidell (Eds.), Handbook of integrative developmental science: Essays in honor of Kurt W. Fischer (pp. 91–135). Routledge.

Mascolo, M. F., & Fischer, K. W. (2010). The dynamic development of thinking, feeling, and acting over the lifespan. In R. M. Lerner (Ed.), Handbook of life-span development (Vol. 1, pp. 149–194). Wiley.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). Children’s interaction with others–family, friends, teachers, mentors–play an important role in skill building. Good design of experiences and opportunities for skill building intentionally takes into account the social dimension of learning.

“Individuals learn what they do, and particularly what they do with others. Social interaction—especially exchanges that are scaffolded by more accomplished others—has the effect of raising an individual’s performance to levels beyond that which can be sustained while working alone.” (Mascolo, 2020)

We don’t acquire skills and knowledge entirely on our own. Skills are learned partly from other people–whether directly or vicariously by watching them (Immordino-Yang, 2008). We imitate other people’s actions–including their thoughts and beliefs. In this way, social learning is a major driver of skill building.

Children rely upon trusted adults–parents, caregivers, teachers, mentors–and peers for emotional clues about how they should behave. “They imagine how other people feel and think, and those thoughts in turn influence how they feel and think” (Immordino‐Yang, 2011).

Social support is an important part of the context in which skills are built. (See Principle 3.) Interaction with others is critical to social emotions. (See Principle 4.). “We are influenced by not only our own emotions, but also by others’ emotions, behaviors, and mental states” (Immordino-Yang, 2007).

Social interaction is important for both intrinsic and extrinsic motivation. (See Principle 5.) Skills are developed and practiced with others. In situations of high support, a mentor, coach, or teacher can help a child achieve levels of competence they couldn’t reach on their own.

This type of support “awakens and directs the process of development” (Mascolo & Fisher, 2010). The presence or absence of social support can have dramatic effects on the skill level a child is able to produce (Fischer et al., 1993). (See Principles 9, 10, and 11.)

Social interaction is an important part of the relational dimension of “conditions for learning,” such as trust, attachment, and attunement (Cantor et al., 2019). The desire to overcome obstacles and undertake goal-oriented behaviors is often aroused by social emotions triggered by others (e.g. admiration, compassion, etc.) (Hardway, 2020).

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

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.

Fischer, K. W., Bullock, D. H., Rotenberg, E. J., & Raya, P. (1993). The dynamics of competence: How context contributes directly to skill. In R. Wozniak & K. Fischer (Eds.), Development in context: Acting and thinking in specific environments. Lawrence Erlbaum Associates.

Hardway, C. (2020). Of interest and engagement: The emotional force of learning and development. In M. F. Mascolo & T. R. Bidell (Eds.), Handbook of integrative developmental science: Essays in honor of Kurt W. Fischer (pp. 232–261). Routledge.

Immordino-Yang, M. H. (2007). A tale of two cases: Lessons for education from the study of two boys living with half their brains. Mind, Brain, and Education, 1(2), 66–83.

Immordino-Yang, M. H. (2008). The smoke around mirror neurons: Goals as sociocultural and emotional organizers of perception and action in learning. Mind, Brain, and Education, 2(2), 67–73.

Immordino‐Yang, M. H. (2011). Implications of affective and social neuroscience for educational theory. Educational Philosophy and Theory, 43(1), 98–103.

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.

Mascolo, M. F. (2020). Dynamic skill theory: An integrative model of psychological development. In M. F. Mascolo & T. R. Bidell (Eds.), Handbook of integrative developmental science: Essays in honor of Kurt W. Fischer (pp. 91–135). Routledge.

Mascolo, M. F., & Fischer, K. W. (2010). The dynamic development of thinking, feeling, and acting over the lifespan. In R. M. Lerner (Ed.), Handbook of life-span development (Vol. 1, pp. 149–194). Wiley.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). It is impossible to learn anything if we are not motivated. Motivation, emotion, and cognition are deeply intertwined. Our beliefs about ourselves, the degree of control we feel, and our expectations about the likely value of our efforts significantly shape our motivation. Extrinsic rewards are likely to be less energizing for children than intrinsic motivation. Relatively small things can make a big difference to support skill building and learning (Dweck et al., 2011).

“Motivation nudges us to convert intention into action. It pushes us to start doing something new or to restart something that we have stopped. It controls our decision to persist at a specific work goal in the face of distractions and competing priorities. It leads us to invest more or less mental and physical effort to enhance both the quality and quantity of our work.” (Clark, 2006)

Motivation is a psychological process that shapes whether we begin and persist at a task. It also influences whether we invest sufficient mental resources to be successful (Cantor et al., 2019).

Motivation is described as “perhaps the indispensable element needed for school success” (Sternberg, 2017). It is said that motivation might matter more than cognition for students’ academic success (Dweck et al., 2011). “Without motivation, even the most capable expert will fail” (Clark, 2006).

Emotion, cognition, and motivation are deeply intertwined (Meyer & Turner, 2006: Hardway, 2020). Negative emotion is called “one of the biggest killers of motivation” (Clark, 2003). (See Principle 6.) Although multiple factors shape motivation, researchers heavily focus on student beliefs about themselves and their learning goals, how much control they think they have over the outcomes, and their expectations for success (Clark and Saxberg, 2018; Cantor et al., 2019).

Academic motivation is conservatively estimated to account for about 30% of learning, transfer, and application in adolescents and adults. This is roughly the same level of impact as learning strategies (Clark and Saxberg 2018; Cantor et al., 2019).

Intrinsic motivation involves pursuing knowledge and activities for their own sake. Extrinsic motivation involves engaging in an activity (including learning) for external rewards (Hardway, 2020). Intrinsically motivated behaviors are guided by ends in themselves. Intrinsic motivation is based on curiosity; it is inherently interesting or enjoyable.

The positive energy that comes with intrinsic motivation helps us to expand knowledge, skills, and competencies. Our reward is the interest or enjoyment that we experience (Harackiewicz and Knogler 2017) Children can show different levels of intrinsic motivation for learning particular topics (Hardway, 2020).

Intrinsic motivation can be thought of as a heightened state of interest. Interest in a topic or task stems from a combination of the situation (context) and the individual. Novelty can help spark interest. While the context or situation can jump start motivation, individual or personal interest is more enduring (Hardway, 2020; Wigfield and Cambria, 2010).

When engaged in interesting activities there is little distinction between what someone thinks is important and what they like to do. Positive emotional reactions and cognitive functioning are intertwined. Cognition and attention control feel relatively effortless (Harackiewicz and Knogler 2017).

“Intrinsic motivation is associated with deeper focus, creativity, confidence, and achievement” (Cantor et al., 2019). If we want to help children build skills, we have to awaken their internal motivation.

Motivation doesn’t always start from the inside. Extrinsic motivation is based upon the expectation for some gain or reward. These rewards could be financial, good grades, or praise.

Rewards could also be avoidance of negative consequences such as criticism, shame, or stress (Harackiewicz and Knogler 2017). How others think or feel about us (social emotions) can also be a source of extrinsic motivation. (See Principle 6.)

Adults are greatly motivated by financial and other tangible incentives (Clark, 2003). But children appear to operate differently. A major experimental study in three large urban districts paid students for reading books, performance on interim exams, or for classroom grades. All three approaches failed to make an impact on student achievement (Fryer, 2011).

There is some evidence that financial incentives may decrease the motivation of school-age children (Clark, 2003). Yeager (2024) suggests that young people ages 10 to 25 are heavily motivated by non-financial incentives: status and respect.

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

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.

Clark, R. E. (2003). Fostering the work motivation of individuals and teams. Performance Improvement, 42(3), 21–29.

Clark, R. E. (2006). Motivating individuals, teams, and organizations. In J. A. Pershing (Ed.), Handbook of human performance technology: Principles, practices, and potential (Third, pp. 478–497). Pfeiffer.

Clark, R. E., & Saxberg, B. (2018). Engineering Motivation Using the Belief-Expectancy-Control Framework. Interdisciplinary Education and Psychology, 2(1).

Dweck, C. S., Walton, G. M., & Cohen, G. L. (2011). Academic tenacity: Mindsets and skills that promote long-term learning. Gates Foundation.

Fryer, R. G., Jr. (2011). Financial incentives and student achievement: Evidence from randomized trials. The Quarterly Journal of Economics, 126(4), 1755–1798.

Harackiewicz, J. M., & Knogler, M. (2017). Interest: Theory and application. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), Handbook of competence and motivation: Theory and application (Second). The Guilford Press.

Hardway, C. (2020). Of interest and engagement: The emotional force of learning and development. In M. F. Mascolo & T. R. Bidell (Eds.), Handbook of integrative developmental science: Essays in honor of Kurt W. Fischer (pp. 232-261). Routledge.

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.

Meyer, D. K., & Turner, J. C. (2006). Re-conceptualizing emotion and motivation to learn in classroom contexts. Educational Psychology Review, 18(4), 377–390.

Sternberg, R. J. (2017). Intelligence and competence in theory and practice. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), Handbook of competence and motivation: Theory and application (Second). The Guilford Press.

Wigfield, A., & Cambria, J. (2010). Students’ achievement values, goal orientations, and interest: Definitions, development, and relations to achievement outcomes. Developmental Review, 30(1), 1–35.

Yeager, D. S. (2024). 10 to 25: The science of motivating young people: A groundbreaking approach to leading the next generation-and making your own life easier. Avid Reader Press.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). “To foster learning, start with emotion” (Mascolo, 2020). Children’s feelings impact their thoughts and actions. We can help children build complex skills by connecting facts and procedures we want them to learn with “intrinsically emotional, subjective and meaningful experiences” (Immordino-Yang, 2016).

“It is the emotional dimensions of knowledge that allow people to call up memories and skills that are relevant to whatever task is at hand. Without the appropriate emotions, individuals may have knowledge but they likely won’t be able to use it effectively when the situation requires. Emotions are the rudder that steers thinking.” (Immordino-Yang and Damasio, 2007)

“It is neurobiologically impossible to build memories, engage complex thoughts, or make meaningful decisions without emotion. The brain is metabolically expensive. We don’t waste energy and oxygen thinking about things that don’t matter to us. We only think deeply about things we care about.” (Immordino-Yang, 2016)

Emotions are critical to human life; they literally help keep us alive. Fear keeps us from falling off cliffs. Disgust stops us from eating rotten food. Romantic love drives us to pair up and have children. Familial love compels us to create bonds with and care for our children (Immordino-Yang, 2016).

Human emotions such as anger, fear, happiness, and sadness have cognitive and physical dimensions. They engage brain systems for cognition (e.g. fight or flight, memories) and body regulation (e.g. blood pressure, heart rate, etc.) (Immordino-Yang & Sylvan, 2010).

Emotions are both internally-driven and externally triggered. Emotions can be positive or negative–joy, pride, relief, anxiety, anger, hopelessness (Pekrun, 2017). Emotions such as pride, joy, admiration, contempt, and envy can be externally triggered by social interactions with others (Hareli & Parkinson, 2008, Immordino-Yang, 2011; Pekrun, 2017).

Emotion helps build skills by serving as a “rudder” for thinking (Immordino-Yang and Damasio, 2007). It helps guide and direct our attention, thoughts, and behavior (Fischer & Bidell, 2006; Mascolo & Fischer, 2010; Mascolo, 2020; Hardway, 2020).

Emotion helps us recognize and call up relevant information – such as which procedure to use to solve a math problem. Success in subjects we tend to think of as purely rational – like science, technology, engineering, and mathematics – depends upon making emotional connections between concepts. Emotion helps engage the brain networks that manage attention, motivation, and evaluation of potential solutions (Immordino-Yang & Faeth, 2010; Immordino-Yang, 2011).

Emotion is not an “add-on” to cognitive skills (Immordino-Yang, 2016). Cognition is “emotion-dependent”; cognition and emotion work together (Pekrun, 2017). Emotion is the fuel for interests, motivation, and purpose.

Emotion helps us remember and apply what we learn. Much of educators’ focus in the classroom–student learning, focused attention, problem solving, comprehension, student motivation, social engagement–”are both profoundly affected by emotion and are subsumed within the process of emotion” (Immordino-Yang and Damasio, 2007).

“Emotions such as interest, anxiety, frustration, excitement, or a sense of awe in beholding beauty become a dimension of the skill itself” (Immordino-Yang, 2016). In this way, academic activities are never “emotionally neutral” (Immordino-Yang and Faeth, 2010). There is no aspect of skill building that is not dependent upon and affected by emotion.

Takeaway: “To foster learning, start with emotion” (Mascolo, 2020). Children’s feelings impact their thoughts and actions. We can help children build complex skills by connecting facts and procedures we want them to learn with “intrinsically emotional, subjective and meaningful experiences” (Immordino-Yang, 2016).

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

Works Cited

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

Hardway, C. (2020). Of interest and engagement: The emotional force of learning and development. In M. F. Mascolo & T. R. Bidell (Eds.), Handbook of integrative developmental science: Essays in honor of Kurt W. Fischer (pp. 232-261). Routledge.

Hareli, S., & Parkinson, B. (2008). What’s Social About Social Emotions? Journal for the Theory of Social Behaviour, 38(2), 131–156.

Immordino‐Yang, M. H. (2011). Implications of affective and social neuroscience for educational theory. Educational Philosophy and Theory, 43(1), 98–103.

Immordino-Yang, M. H. (2016). Emotions, learning, and the brain: Exploring the educational implications of affective neuroscience. W. W. Norton & Company.

Immordino-Yang, M. H., & Damasio, A. (2007). We feel, therefore we learn: The relevance of affective and social neuroscience to education. Mind, Brain, and Education, 1(1), 3–10.

Immordino-Yang, M. H., & Faeth, M. (2010). The role of emotion and skilled intuition in learning. In D. A. Sousa (Ed.), Mind, brain, and education: Neuroscience implications for the classroom (pp. 67–81). Solution Tree Press.

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.

Immordino-Yang, M. H., & Sylvan, L. (2010). Admiration for virtue: Neuroscientific perspectives on a motivating emotion. Contemporary Educational Psychology, 35(2), 110–115.

Mascolo, M. F. (2020). Dynamic skill theory: An integrative model of psychological development. In M. F. Mascolo & T. R. Bidell (Eds.), Handbook of integrative developmental science: Essays in honor of Kurt W. Fischer (pp. 91–135). Routledge.

Mascolo, M. F., & Fischer, K. W. (2010). The dynamic development of thinking, feeling, and acting over the lifespan. In R. M. Lerner (Ed.), Handbook of life-span development (Vol. 1, pp. 149–194). Wiley.

Pekrun, R. (2017). Achievement emotions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), Handbook of competence and motivation: Theory and application (Second). The Guilford Press.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). Children’s knowledge about the world around them (“knowledge that”) is an essential part of skill building (“knowledge to”). We need to seek and develop opportunities inside and outside of the classroom to give children the opportunity to build both types of knowledge.

“The development of complex dynamic skills does not occur in isolation; it requires the layering and integration of prerequisite skills and domain-specific knowledge, as well as the influence of contextual factors.” (Cantor et al., 2019)

Knowledge is the ability to use facts, principles, and ideas to decide and do complex tasks (Cantor et al., 2019). This definition implicitly incorporates the central concepts of “knowledge that” and “knowledge how.”

“Knowledge that” concerns conceptual or content knowledge: vocabulary, facts, ideas, theories, principles, concepts. “Knowledge how” involves competencies and skills. This type of knowledge concerns how to do a specific task or action (Rata, 2019).

“Knowing how informs knowing that, and vice versa” (Mascolo, 2009). Both types of knowledge are essential for learning and development. Knowledge is both “caught and taught.” “Knowledge that” is more readily learned through explicit instruction.

“Knowledge how” is usually acquired through practice, experience, and observation. Knowledge is acquired in both formal and informal learning experiences (Johnson & Majewska, 2022). There are opportunities for “knowledge how” experiences inside and outside of formal classroom settings.

All new knowledge and skills are built from existing knowledge and skills (Mascolo, 2009). Children’s prior knowledge and experiences have a strong influence on learning (Bruer, 2008; Osher et al., 2017). We acquire facts to the degree they can be integrated with what we already know and have experienced (Dehaene, 2011).

General knowledge represents children’s breadth and depth of understanding of their social and physical environment (i.e., the social, physical and natural world) and their ability to draw inferences and comprehend implications (West et al., 2000). It’s important to keep in mind that children’s prior knowledge could be correct or incorrect. It could also be incomplete or inconsistent with what is being taught (Cantor et al., 2019).

Knowledge is critical for reading comprehension (Hirsch, 2003; Pearson, et al., 2020; Catts, 2022). Content knowledge, background knowledge, text structure knowledge, and morphological knowledge1 are key inputs to successful reading comprehension.

Vocabulary, a critical component skill for both reading and math, is also a type of knowledge. “The knowledge of a word not only implies a definition, but also implies how that word fits into the world” (Stahl, 2005).

Factual (or declarative) knowledge, procedural knowledge, and conceptual knowledge are the three primary components of mathematics. Hattan and Lupo (2020) remind us there are multiple types of knowledge that children bring to bear for reading comprehension (and presumably mathematical thinking): cultural, linguistic, principled, strategic, multimodal, multiple text use, and conditional.

These different types of knowledge cut across “knowledge how” and “knowledge to. The breadth and depth of these different aspects of knowledge justify its description as a “large problem space” to be solved (Snow & Kim, 2007).

Knowledge is complex, has multiple dimensions, and requires significant effort to build strong, relevant interconnections (schema). Any assessment of knowledge can only provide a partial snapshot of a child’s ability to use facts, principles, and ideas to decide and do complex tasks.

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

Works Cited

Bruer, J. T. (2008). Building bridges in neuroeducation. In A. M. Battro, K. W. Fischer, & P. Léna (Eds.), The educated brain (pp. 43–58). Cambridge University Press.

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.

Catts, H. W. (2022). Rethinking how to promote reading comprehension. American Educator, 45(4), 26.

Dehaene, S. (2011). The number sense: How the mind creates mathematics. Oxford University Press.

Hattan, C., & Lupo, S. M. (2020). Rethinking the role of knowledge in the literacy classroom. Reading Research Quarterly, 55, 283–298.

Hirsch, E. D. (2003). Reading comprehension requires knowledge of words and the world. American Educator, 27(1), 10–13.

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.

Johnson, M., & Majewska, D. (2022). Formal, non-formal, and informal learning: What are they, and how can we research them? Cambridge University Press & Assessment Research Report.

Mascolo, M. F. (2009). Beyond student-centered and teacher-centered pedagogy: Teaching and learning as guided participation. Pedagogy and the Human Sciences, 1(1), 3–27.

Osher, D., Cantor, P., Berg, J., Steyer, L., & Rose, T. (2017). Science of learning and development: A synthesis. American Institutes for Research.

Pearson, P. D., Palincsar, A. S., Biancarosa, G., & Berman, A. I. (Eds.). (2020). Reaping the rewards of the reading for understanding initiative. National Academy of Education.

Rata, E. (2019). Knowledge‐rich teaching: A model of curriculum design coherence. British Educational Research Journal, 45(4), 681–697.

Snow, C. E., & Kim, Y.-S. (2007). Large problem spaces: The challenge of vocabulary for English language learners. In R. K. Wagner, A. Muse, & K. Tannenbaum (Eds.), Vocabulary acquisition and its implications for reading comprehension (pp. 123–139). Guilford.

Stahl, S. A. (2005). Four problems with teaching word meanings (and what to do to make vocabulary an integral part of instruction. In E. H. Hiebert & M. L. Kamil (Eds.), Teaching and learning vocabulary: Bringing research to practice. Erlbaum.

West, J., Denton, K., & Reaney, L. M. (2000). The kindergarten year: Findings from the early childhood longitudinal study, kindergarten class of 1998-99. (NCES 2001-023). National Center for Education Statistics.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). Children develop skills and knowledge from goal-oriented activities. This is likely to be more successful when children are able to connect these activities with prior knowledge and skills, engage with others (including experts in the skill), and attach positive meaning to the experience for themselves.

“Skills do not spring up fully grown from preformed rules or logical structures. They are built up gradually through the practice of real activities in real contexts, and they are gradually extended to new contexts through this same constructive process.” (Fischer & Bidell, 2006)

Knowledge and learning depend upon activity. When we do things in the world, we literally shape our brain structure – the neurons, synapses, and brain activity.1 Mere exposure is not enough to change our brain – or to build skills. Learning facts is important but not enough. We have to be actively involved (Fischer, 2009).

To support skill building, an action must have two basic components. First, the action has to be focused on a goal. For a baby learning to walk, the goal can be as simple as taking steps from one parent to another.

Second, the action has to be meaningful. The baby’s effort to learn to walk is shaped and influenced by the meaning she attaches to this experience. For example, the encouragement and approval she receives from her parents. (See Principles 6, 7, and 8.) As she improves her walking skills, she associates walking with feelings of independence and freedom for herself.

A variety of “goal-oriented” activities help children build skills. Some of these activities may be explicitly focused on subjects like reading, math, science, and social studies. But “goal oriented” doesn’t just mean academics. It also can include various forms of play, tinkering, sports, music, hobbies, outdoor challenges, etc.

“Action” is not limited to physical activity. Skills are developed and organized to support real mental activities, too. Thinking is a form of internalized action. Feeling involves the experience of activity (Mascolo, 2009). The documented impact of mindfulness interventions on executive and self-regulation skills is one positive example of “mental action.”

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

Works Cited

Barrett, L. F. (2020). Seven and a half lessons about the brain. Houghton Mifflin.

Fischer, K. W. (2009). Mind, brain, and education: Building a scientific groundwork for learning and teaching. Mind, Brain, and Education, 3(1), 3–16.

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

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.

Mascolo, M. F. (2009). Beyond student-centered and teacher-centered pedagogy: Teaching and learning as guided participation. Pedagogy and the Human Sciences, 1(1), 3–27.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). We can intentionally design of learning environments and experiences to help children grow new, more complex skills and domain-specific knowledge. This can be particularly beneficial for children affected by chronic stress and adverse childhood experiences (Cantor et al., 2019; Osher et al., 2020).

“In ordinary English usage, skill implies both person and context simultaneously. People have a skill for riding a bicycle, a skill for listening to their friends, a skill for repairing Toyota engines, a skill for doing analysis of variance. A person cannot have a skill independent of context. Skill requires a collaboration between person and context. This conception means that skills vary not only between people but also across contexts for a given person.” (Fischer et al, 1993)

We do not have abstract, general skills. We have skills for specific contexts. We have different abilities in different contexts (Fischer & Bidell, 2006). “Context” means many things at once: the physical environment, emotional state, social support, domain or subject area, and the specific task involved (Fischer & Immordino-Yang, 2002).

Context helps explain why a child who is able to solve a math problem one day or in a particular situation is unable to repeat this performance the next day or in a different or similar situation (Fischer & Bidell, 2006). Context also explains the challenge of transferring skill and ability developed in one situation or domain to another. (See Principle 13.)

Researchers found that children who were adept in complex mental math calculation in street markets in Kolkata and Delhi were not able to successfully apply the same math concepts in classroom settings. Moreover, children who were able to successfully complete these same calculations in classroom settings struggled to apply these concepts to real-world situations like the street markets (Banerjee et al., 2025).

Change the context, change the skill.

Chronic stress and adversity can have profoundly negative effects on skills. Insufficiently supportive physical and social contexts can hinder skill development. Developmentally rich relationships and experiences, however, can buffer the effects of stress and trauma (Cantor et al., 2019). Teachers, mentors, coaches, and other adults can create the conditions to better support skill building.

Change the context, change the skill.

Family obviously is the most important context for skill building. Early childcare and school are the next most important contexts for early development (Cantor et al., 2019). However, every encounter and interaction that a child has with other people – in houses of faith, youth programs, sports clubs, mentoring programs, cultural institutions, and more – is an opportunity for social learning and skill development (Bronfenbrenner & Morris, 2006; Osher et al., 2020). Each encounter is an opportunity to support and contribute to a child’s skill building.

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

Works Cited

Banerjee, A. V., Bhattacharjee, S., Chattopadhyay, R., Duflo, E., Ganimian, A. J., Rajah, K., & Spelke, E. S. (2025). Children’s arithmetic skills do not transfer between applied and academic mathematics. Nature, 639(8055), 673–681.

Bronfenbrenner, U., & Morris, P. A. (2006). The bioecological model of human development. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 793–828). John Wiley & Sons, Inc.

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.

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

Fischer, K. W., Bullock, D. H., Rotenberg, E. J., & Raya, P. (1993). The dynamics of competence: How context contributes directly to skill. In R. Wozniak & K. Fischer (Eds.), Development in context: Acting and thinking in specific environments. Lawrence Erlbaum Associates.

Fischer, K. W., & Immordino-Yang, M. H. (2002). Cognitive development and education: From dynamic general structure to specific learning and teaching. In E. Lagemann (Ed.), Traditions of scholarship in education. Spencer Foundation.

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.

Osher, D., Cantor, P., Berg, J., Steyer, L., & Rose, T. (2020). Drivers of human development: How relationships and context shape learning and development. Applied Developmental Science, 6–36.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). Skilled activity is the result of multiple skills collaborating with each other. Skills are organized hierarchically. Lower-level skills support higher, more complex skills. Skills collaborate within and across levels. Skills build together, though some take longer to fully develop than others. The relative contributions of individual skills and groups of skills are not static and change over time.

“As skills become integrated and differentiated at later levels, the component skills subordinate themselves to new forms of organization and mutual regulation. The very process of creating new skills through self-organizing coordination leads to a multileveled hierarchical structuring of living skills.” (Fischer & Bidell, 2006)

Skills organize themselves into multilevel hierarchies. New skills at higher, more complex levels are built from lower-level skills. Skills integrate with each other to create newer, more complex skills. Nonetheless, the component skills within this hierarchy of skills continue to operate as subsystems that can function as standalone skills (Fischer & Bidell, 2006).

For example, the early basic skill of recognizing letters combines with other skills to support more complex skills of phonics, decoding, spelling, word-reading, reading fluency, and reading comprehension. But when needed the basic skill of letter identification can be used on its own.

Skills are highly interactive. They operate in a sort of continuous feedback loop. Skills interrelate and collaborate with each other to support goal-oriented activity. Individual skills and networks of skills directly and indirectly support reading and math achievement (LeFevre et al., 2010; Rittle-Johnson et al., 2017; Hjetland et al., 2019; Kim, 2023).

Some skills contribute by supporting the development and integration of other skills. What we call “executive function” is a group of highly interactive component skills: attention control, attention shifting, inhibition control, and working memory (Dawson & Guare, 2018; Cantor et al., 2019). These skills make key contributions to math- and reading-specific skills such as problem solving, word-reading, and comprehension.

Skills are highly dynamic. Their individual contributions to skilled action change over time. For example, in the early phases of learning to read, word-reading subskills such as letter identification, phonics, and decoding are major drivers of reading achievement.

As children improve reading fluency, the contribution of word-reading skills to reading comprehension decreases relative to other skills (LAARC, 2015; Kim & Wagner, 2015). Researchers find that by fourth grade, other skills such as attention control, working memory, vocabulary, grammar, perspective taking, and listening comprehension support reading comprehension of average readers at a level equal to (or greater) than word-reading skills (Kim, 2020).

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

Works Cited

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

Hjetland, H. N., Lervåg, A., Lyster, S. A. H., Hagtvet, B. E., Hulme, C., & Melby-Lervåg, M. (2019). Pathways to reading comprehension: A longitudinal study from 4 to 9 years of age. Journal of Educational Psychology, 111(5), 751.

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.

Kim, Y. S. G. (2020). Hierarchical and dynamic relations of language and cognitive skills to reading comprehension: Testing the direct and indirect effects model of reading (DIER). Journal of Educational Psychology, 112(4), 667.

Kim, Y. S. G. (2023). Simplicity meets complexity: Expanding the simple view of reading with the direct and indirect effects model of reading. In S. Q. Cabell, S. B. Neuman, & N. Patton Terry (Eds.), Handbook on the science of early literacy (pp. 9–22). The Guilford Press.

Kim, Y.-S. G., & Wagner, R. K. (2015). Text (Oral) Reading Fluency as a Construct in Reading Development: An Investigation of Its Mediating Role for Children From Grades 1 to 4. Scientific Studies of Reading, 19(3), 224–242.

Language and Reading Research Consortium. (2015). Learning to Read: Should We Keep Things Simple? Reading Research Quarterly, 50(2), 151–169.

LeFevre, J. A., Fast, L., Skwarchuk, S. L., Smith‐Chant, B. L., Bisanz, J., Kamawar, D., & Penner‐Wilger, M. (2010). Pathways to mathematics: Longitudinal predictors of performance. Child Development, 81(6), 1753–1767.

Rittle‐Johnson, B., Fyfe, E. R., Hofer, K. G., & Farran, D. C. (2017). Early math trajectories: Low‐income children’s mathematics knowledge from ages 4 to 11. Child Development, 88(5), 1727–1742.

Last updated: June 4, 2025

Key Takeaway

Skill is the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010). Skills can be intentionally developed with positive social relationships, rich learning environments, and well-designed learning experiences. Children have different starting points and follow different developmental pathways. (See Principle 12.) Nonetheless, they have similar capacity to develop basic and complex skills under the right conditions (Cantor et al., 2019).

“Skills are self-organizing. Part of the natural functioning of skills is that they organize and reorganize themselves. These self-organizing properties go beyond maintenance to include growth of new, more complex skills.” (Fischer & Bidell, 2006)

One of the many remarkable features of the human brain is its capacity for self-organization. In response to stimuli and experiences, the brain’s internal processes organize its functioning and development. The brain reorganizes itself multiple times between when a child is born and when she graduates high school.1 This reorganization supports progressively complex skill development (Fischer 2008; Fischer 2009).

Skills mirror this ability. Skills developed for specific purposes combine with other skills developed for other purposes to create new, integrated skills. In this way, simple skills self-organize into more complex skills that can be used to support an increasingly wider range of activities.

For example, the ability to count objects is a skill we develop in early childhood. This basic skill, which starts with concrete objects, becomes more complex when it’s applied to abstract representations of numbers (numerals). This ability eventually supports even more complex skills such as adding numbers together.

Addition later supports multiplication (which is really multiple addition). Addition and multiplication in turn support even more complex (and abstract) skills such as word problem solving and algebraic thinking. These skills support even more advanced math skills such as probability and calculus.

Developmentally and interpersonally rich experiences are the breeding grounds for skills. For example, consider the skills that help us manage our thoughts, emotions, and actions: attention control, attention shifting, inhibition control, and working memory (Dawson & Guare 2018; Cantor et al., 2019). Stable, responsive relationships and supportive learning environments support the development and integration of these “executive skills,” which in turn support the development of other skills (Stafford-Brizard, 2016).

“By stimulating the brain’s self-organizing and reorganizing properties and integrating subsystems of skills, this feedback loop gives rise to the capacity to self-regulate and, ultimately, gives meaning to experiences, including stressful experiences.” (Cantor et al., 2019)

But wait, there’s more

• Insights about the other 13 principles of skill building.

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

Works Cited

Barrett, L. F. (2020). Seven and a half lessons about the brain. Houghton Mifflin.

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.

Dawson, P., & Guare, R. (2018). Executive skills in children and adolescents: A practical guide to assessment and intervention. The Guilford Press.

Fischer, K. W. (2008). Dynamic cycles of cognitive and brain development: Measuring growth in mind, brain, and education. In A. M. Battro, K. W. Fischer, & P. Léna (Eds.), The educated brain (pp. 127–150). Cambridge University Press.

Fischer, K. W. (2009). Mind, brain, and education: Building a scientific groundwork for learning and teaching. Mind, Brain, and Education, 3(1), 3–16.

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). John Wiley & Sons, Inc.

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.

Stafford-Brizard, K. B. (2016). Building blocks for learning: A framework for comprehensive student development. Turnaround for Children.

Last updated: August 30, 2025

I’m obsessed with skills. What they are. How they develop. I’m particularly interested in understanding the factors that support or inhibit the growth of unconstrained skills that quietly drive PK-12 math and reading achievement.

I’ve read dozens of academic articles, reports, and books about skills over the past 12 months. The development and refinement of skills to support goal-oriented action is an essential part of human life. In PK-12 education, “skill” is not always consistently defined (Afflerbach et al., 2008). I think about skills the way developmental psychologists describe them: the ability to think and act in an organized way in a specific context (Immordino-Yang and Fischer, 2010).

“Context” means many things at once: the physical environment, emotional state, social support, domain or subject area, and the specific task involved (Fischer & Immordino-Yang, 2002). Proficient skills are virtually automatic; their use requires little attention or conscious awareness (Mascolo, 2020).

Check out this post for a basic primer on skills:

This series of posts (14 Principles of Skill Building) summarizes key principles about skills I draw from an interdisciplinary body of knowledge known as the Science of Learning and Development.1 I lean heavily upon insights from a field known as Dynamic Skill Theory.

14 Principles of Skill Building

1. Skills self-organize and increase in complexity over time.

2. Skills are hierarchical, interactive, and dynamic.

3. Skills are built in specific contexts. Change the context, change the skill.

4. Skills are built through action.

5. Knowledge is essential to skill building.

6. Emotion is the fuel for skill building.

7. Skill building without motivation is impossible.

8. Skills are built through social interaction.

9. Skill level varies across a developmental range.

10. Skill building is construction, collapse, and reconstruction.

11. Skill building is varied and stable at the same time.

12. Skill building can occur along multiple pathways.

13. Skills built in specific contexts gradually transfer and generalize to new contexts.

14. Skill building involves small-scale changes in specific contexts and large-scale changes across contexts.

I’ve written a post with key insights for each principle. Although my overall focus is improving PK-12 math and reading skills, these principles apply to many types of skills. Each principle stands on its own. As you’ll see, the 14 principles are highly interconnected.

My read of the research suggests that instruction, interventions, and programs informed by these principles are better positioned for success.

But wait, there’s more

• The skills that quietly drive PK-12 math and reading achievement.

• Data that shows divergences in proficiency between two types of math and reading skills for more than 20 years.

Works Cited

Afflerbach, P., Pearson, P. D., & Paris, S. G. (2008). Clarifying differences between reading skills and reading strategies. The Reading Teacher, 61(5), 364–373.

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.

Fischer, K. W., & Immordino-Yang, M. H. (2002). Cognitive development and education: From dynamic general structure to specific learning and teaching. In E. Lagemann (Ed.), Traditions of scholarship in education. Spencer Foundation.

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.

Mascolo, M. F. (2020). Dynamic skill theory: An integrative model of psychological development. In M. F. Mascolo & T. R. Bidell (Eds.), Handbook of integrative developmental science (pp. 91–135). Routledge.

Osher, D., Cantor, P., Berg, J., Steyer, L., & Rose, T. (2020). Drivers of human development: How relationships and context shape learning and development. Applied Developmental Science, 6–36.

Last updated: March 15, 2025

What I’m doing now

This Substack is part of a larger project. I am working to bring a powerful 20-year-old insight from academia into the mainstream. The current focus here is WHAT constrained and unconstrained skills are and WHY they matter. I am in the throes of research and writing a book on HOW to improve them. My focus (the “dependent variable”) is improvement of PreK-12 reading and math achievement. I am particularly concerned about closing gaps in slow-growing unconstrained skills.

This requires thoughtful approaches to building and repairing skills. I am focused on the nature and nurture of skills, how kids learn to read and think mathematically, and how ecological systems support or inhibit skill development. This includes careful attention to the challenge of “skill transfer,” i.e. the generalization of skills from one context to another. We need to understand what works, for whom, and under what conditions. I intentionally use the term “kids” so that people and organizations outside of the schoolhouse (where “students” learn) know that I am talking to you, too. My goal is to help people who run, fund, and assist organizations that teach and serve kids.

I have a full-time day job; research and writing will be a slow process. But you don’t have to wait to start paying attention to unconstrained skills for the kids that you teach or serve. Academic researchers strongly encourage you to do so.

What you can do now

Constrained skills are largely developed in the classroom. Unconstrained skills are built in formal and informal learning environments: the schoolhouse, at home, and in the community. My advice is to look at how you can put more focus on unconstrained skills wherever you serve kids. But this does not mean sacrificing support for constrained skills. Kids need both.

One potential place to start is your data. Examine patterns of constrained reading and math skills. (You can’t do this with composite scores that combine constrained and unconstrained skills!) Point-in-time data can give you snapshots of relative levels of skill proficiency. Longitudinal data can show you trajectories of skill growth over time. Compare the patterns in your data to the data I’ve collected so far.

Ask questions about how your school or program is organized to build unconstrained reading, math, and nonacademic skills. And don’t forget knowledge building; it is the ultimate unconstrained skill. And don’t let anyone convince you that you can’t measure unconstrained skills. Anything can be measured.

TeamCraft

How to Measure Anything

A common scenario faced by a data middle manager is to field questions from execs and peers about something that, in their perception, is intangible (i.e. it cannot be measured…

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3 years ago · Gokhan Ciflikli

Schools

Schools have a crucial role to play. They have the resources, materials, and people to build kids’ constrained and unconstrained skills. Schools make critical decisions about how to spend time and money on curriculum, assessments, and interventions. Researchers have long advocated for educators to give more attention to unconstrained skills starting as early as Pre-K.

As important as schools are, they can’t build unconstrained skills alone. Snow and Matthews (2016) succinctly describe the challenge:

The research on effective schools by researchers and writers such as Tony Bryk and Karin Chenoweth show that schools never “beat the odds” on their own. They always have help. This is especially true for schools serving our most vulnerable children. We need to help schools build “exoskeletons” of additional support.

Home and community

Kids spend 80% or more of their waking hours outside of school. Children from higher socioeconomic families routinely receive active support at home. They participate in enrichment activities outside of school. They get extra academic tutoring when they need it. There are many families on the other end of the socioeconomic ladder who also make sure their kids get similar opportunities. This often happens with great sacrifice and determination. But too many kids from less advantaged households don’t receive a fair chance to fully develop their skills.

Although schools and families are critical, I believe that community organizations and institutions are especially well-suited to help kids build unconstrained skills. Think libraries, museums, zoos, aquaria, science centers, Scouts, 4-H, YMCAs, Boys and Girls Clubs, and the like. Mentoring programs could also play an important role in helping children build unconstrained skills.

Unconstrained skills are key leverage points for change

In his new book, Reset, Dan Heath suggests that the solution to solving tough problems is to realign (“restack”) existing resources behind strategically selected leverage points. Unconstrained skills is such a leverage point for education. This aligns with the recommendation of Bailey et al. (2017) to focus interventions on “trifecta skills”—skills that (1) are malleable, (2) fundamental for later success, and (3) would not have developed anyway in the absence of intervention. I believe many of the unconstrained skills identified by academic researchers meet these criteria. The challenge is finding strategies to repair unconstrained skills that are effective, pragmatic, and scalable for PreK-12 kids.

Studies dating to the 1980s repeatedly demonstrate that dialogic reading builds children’s vocabulary, language, and comprehension skills. The Home Literacy Model and the Home Numeracy Model document how activities at home contribute to classroom gains in reading and math (Sénéchal & LeFevre, 2002; Sénéchal, 2006; LeFevre et al., 2009; Skwarchuk et al. 2014). Research by McCormick et al. (2020) suggests home activities supporting unconstrained skills can positively support classroom reading and math achievement. Although the term “unconstrained skills” might be new, encouraging families to talk, read, sing, and play with their kids is not.

Share this Substack with others whom you think should read it. I would love to hear your thoughts about how to put more focus on unconstrained skills in the schoolhouse, at home, and in community organizations. Let me know if there are studies, programs, or interventions you think I should know about. I’m open to comments, suggestions, or alternative viewpoints. In the meantime, I’ll keep reading, thinking, and writing.

But wait, there’s more

If you’d like to learn more about constrained and unconstrained skills, check out these other topics:

• A basic primer on skills (they are essential to human life)

• Constrained and unconstrained skills drive achievement in reading and math

• What researchers have been saying to each other for the past 20 years about constrained and unconstrained skills

• A working list of constrained and unconstrained skills that support reading and math

• An annotated summary of the major research papers behind constrained skill theory

• Charts of growth patterns of constrained and unconstrained skills from four nationally representative datasets

Works cited

Bailey, D., Duncan, G. J., Odgers, C. L., & Yu, W. (2017). Persistence and fadeout in the impacts of child and adolescent interventions. Journal of Research on Educational Effectiveness, 10(1), 7-39.

LeFevre, J. A., Skwarchuk, S. L., Smith-Chant, B. L., Fast, L., Kamawar, D., & Bisanz, J. (2009). Home numeracy experiences and children’s math performance in the early school years. Canadian Journal of Behavioural Science, 41(2), 55.

McCormick, M. P., Weissman, A. K., Weiland, C., Hsueh, J., Sachs, J., & Snow, C. (2020). Time well spent: Home learning activities and gains in children’s academic skills in the prekindergarten year. Developmental Psychology, 56(4), 710.

Sénéchal, M. (2006). Testing the home literacy model: Parent involvement in kindergarten is differentially related to grade 4 reading comprehension, fluency, spelling, and reading for pleasure. Scientific Studies of Reading, 10(1), 59-87.

Sénéchal, M., & LeFevre, J. A. (2002). Parental involvement in the development of children’s reading skill: A five‐year longitudinal study. Child Development, 73(2), 445-460.

Skwarchuk, S. L., Sowinski, C., & LeFevre, J. A. (2014). Formal and informal home learning activities in relation to children’s early numeracy and literacy skills: The development of a home numeracy model. Journal of Experimental Child Psychology, 121, 63-84.

Snow, C. E., & Matthews, T. J. (2016). Reading and language in the early grades. The Future of Children, 57-74.

Last updated: March 15, 2025

Three big ideas

1. There are small early gaps in constrained reading and math skills between boys and girls. Girls have slightly higher proficiency in reading. Boys have slightly higher proficiency in math. These gaps close by the end of middle school if not sooner.

2. Girls have a small advantage over boys in unconstrained reading skills through middle school. Boys have a small(er) advantage over girls in unconstrained math skills, though it’s possible this may have narrowed in recent years.

3. This does not apply to all unconstrained skills. Girls appear to start school with higher levels of working memory and cognitive flexibility. But these differences in nonacademic (executive) skills appear to close in elementary school. And there are no significant differences between boys and girls in general knowledge at least in early elementary school.

About these data

The data in these charts come from three sources: the Early Childhood Longitudinal Study, Kindergarten Class of 1998-99 (ECLS-K), Early Childhood Longitudinal Study, Kindergarten Class of 2010-11 (ECLS-K:2011), and the National Assessment of Educational Progress Long-Term Trend Assessment (NAEP LTT). I've provided an overview of these and other datasets I used to analyze constrained and unconstrained skills. The ECLS-K data were collected between Fall 1998 and Spring 2007. It’s the best dataset I’ve found to dig deep into discrete K-8 reading and math skills. It’s also unique in its inclusion of data about general knowledge. The ECLS-K:2011 data were collected between Fall 2010 and Spring 2016. It’s the best dataset I’ve found to dig deep into executive function skills. I used the EdSurvey R package to analyze the ECLS-K and ECLS-K:2011 data. The NAEP LTT data were collected in 2020 prior to school closures due to the pandemic. I used the NAEP Data Explorer to analyze the NAEP LTT data.

Boys, girls, and skills

In recent years, some have raised concerns about the academic progress of boys relative to girls. Girls appear to be thriving. Boys on multiple measures—school readiness, GPA, advanced placement classes in high school—are reported to be behind girls. In this post, I examine constrained and unconstrained skills for boys and girls across four sets of skills: reading skills, math skills, nonacademic skills, and general knowledge. An important caveat is that my best information for skill proficiency of boys and girls is limited to elementary school and middle school. I’ve not yet found a source of differentiable data for both math and reading skills to provide a good picture for high school.

Skill is the capacity to act in an organized way in a specific context (Fischer & Bidell, 2006). Knowledge is the ability to use facts, principles, and ideas to decide and do complex tasks (Cantor et al., 2021). Knowledge is an essential part of skill building. The core difference between constrained and unconstrained skills is opportunity. Most kids appear to reach high levels of proficiency in constrained skills by the end of middle school (if not sooner). There are greater differences in mastery of unconstrained skills. Constrained skills involve limited amounts of information. Everyone has relatively equal access to this information. Constrained skills are relatively straightforward to teach and assess in classroom settings. Of course this requires explicit and systematic instruction and extra supports for kids with learning differences.

By contrast, unconstrained skills involve acquisition and mastery of broader amounts of information. Not everyone has equal access to the same information. Unconstrained skills are not as easily taught and assessed. But they can be taught. (And anything can be measured.) Unconstrained skills benefit from instruction, experiences, and practice inside and outside of the classroom. There are greater gaps between kids in unconstrained skills than constrained skills.

Let’s see what we find for boys and girls.

Boys, girls, and reading

Observers for years have noted that girls outperform boys on assessments of reading achievement. Some have noted that this gap also extends to writing. This concern is not limited to the United States. We first begin with constrained reading skills and then turn to unconstrained reading skills.

Constrained reading skill

The first chart shows the growth pattern for sight words. This is the highest level of constrained reading skill included in the ECLS-K study. Kids need to be able to automatically recognize in the neighborhood of 200 common words to be proficient readers (Snow & Matthews, 2016).1 In spring of kindergarten, girls have a slight 3 point advantage (17% vs 14%) over boys in sight word proficiency. By the end of third grade this gap had narrows to a single point (98% vs 99%) and completely closes by the end of elementary school.

Figure 1. Ability to recognize common words by sight.

Astute observers will note recognition of common sight words is not equivalent to decoding or reading fluency. The ECLS-K did not include a direct measure of these (constrained) reading skills. It did include, however, a measure of kids’ ability to complete the missing word in simple sentences (a cloze test). While word comprehension is an unconstrained skill, proficiency on this task provides some indication of word reading ability. Girls had a six point advantage (50% vs 44%) on reading words in context by end of first grade. However, by fifth grade this gap closed to a single percentage point (96% for boys and 97% for girls). It appears that boys and girls were on equal footing for constrained reading skills by the end of elementary school.

Unconstrained reading skill

We next shift to an unconstrained reading skill: interpreting narrative text. (Skill definition is at the end of this post.) Besides requiring proficiency in (constrained) word reading skills, a child who is proficient in interpreting narrative text engages multiple unconstrained skills at once: vocabulary, grammar, perspective taking, inference, reasoning, text structure knowledge, comprehension monitoring, and background knowledge. At the end of first grade only 3% of boys and girls are proficient in this unconstrained reading skill. But over the next 7 years, a five point advantage gradually develops for girls by the end of middle school. Girls appear to go into high school with a small advantage in unconstrained reading skills.2

Figure 2. Ability to interpret narrative text using implicit cues and making connections with problems in the text.

These data from the ECLS-K study are over 15 years old. A great deal has occurred in education and society since 2007. Do we continue to see similar patterns today? To answer this question, we turn to data from collected in 2020 just before school closures caused by the pandemic.

The NAEP Long-Term Trend Assessment assessed a nationally representative sample of about 8,700 13-year-olds in reading and math. This next chart shows proficiency for boys and girls on a set of unconstrained reading skills. The reading skills are ordered from top to bottom from most to least constrained. (Definitions for these skills are included at the end of the post.) Girls appear to have a 4 to 9 point advantage over boys for every level of unconstrained reading skill.

Figure 3. Ability to read and comprehend written text (different levels of complexity).

2020 was not an outlier. I compared average reading scale scores on the NAEP LTT for 13-year-old boys and girls over a fifteen year period: 2008, 2012, 2020, and 2023. Girls had a statistically significant advantage over boys in reading achievement at each period ranging between 8 to 10 points. A more nuanced picture suggests that boys and girls reach eventual parity in constrained skills, with the overall advantage in reading enjoyed by girls due to higher performance in less constrained reading skills.

Boys, girls, and math

Observers for years have observed that boys outperform girls on assessments of math achievement. Some have noted this outperformance is in specific areas of math. Researchers at Stanford University found outperformance in the U.S. is concentrated in specific geographies. This issue is not limited to the United States. Once again, we start with constrained skills and then move to unconstrained skills.3

Constrained math skill

This first chart compares proficiency in multiplication and division between boys and girls. Multiplication and division is the highest level constrained math skill in the ECLS-K dataset. The data cover the period from spring of kindergarten through eighth grade. As we’d expect, this skill is virtually non-existent at the end of kindergarten. The small disparity between boys and girls (2% vs 1%) while statistically significant, certainly is not substantive. From this point, proficiency grows in tandem for boys and girls through late elementary school. Boys enjoy a 1 to 3 point advantage through elementary school. By the end of middle school this small gap is erased as girls fully catch up.

Figure 4. Ability to solve simple multiplication and division problems.

Unconstrained math skill

Next, we have a prototypical unconstrained math skill: problem solving. Math problem solving 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).

This next chart includes data from the ECLS-K, starting in spring of first grade through eighth grade. Children are asked to use their knowledge of rate and measurement to solve word problems. At the end of first grade, boys and girls are on a level playing field: neither has any proficiency in this unconstrained math skill. A year later, however, boys have a five point advantage over girls (16% vs 11%). This small advantage persists through the end of eighth grade. Skill is context specific (Fischer & Biddell, 2006). The advantage girls have in unconstrained reading skills and constrained math skills does not add up enough to close the gap in math word problem solving.

Figure 5. Ability to solve math word problems involving rate and measurement.

These data are from 15-25 years in the past. Much has occurred in American education, including efforts to support girls in STEM fields. For a more recent perspective, we turn again to the NAEP Long-Term Trend Assessment. The data in this next chart are for 13-year-olds in 2020 just before the pandemic closed schools across the country. The chart shows proficiency for boys and girls on a set of unconstrained math skills. The skills are ordered top to bottom from most to least constrained.

Although we can’t directly compare these data to the ECLS-K assessment, girls in 2020 appear to have caught up to some degree in math skills. There is parity between boys and girls for the first two levels of math skills. Boys have a five point advantage for the least constrained math skill — moderately complex math procedures and reasoning. (Skill definitions are included at the end of this post.)

Figure 6. Understanding of numerical operations and solve math problems (different levels of complexity).

I compared average math scale scores on the NAEP LTT for 13-year-old boys and girls over a fifteen year period: 2008, 2012, 2020, and 2023. Boys had a statistically significant advantage over girls in math achievement at each period except for 2012 in which they were statistically tied. Statistically significant differences in scale scores for the three other periods ranged from 3 to 7 points. (This is smaller than the 8 to 10 point difference for girls in reading.) A more nuanced picture suggests that boys and girls reach eventual parity in constrained math skills, with the overall advantage in math enjoyed by boys due to higher performance in less constrained math skills.

Boys, girls, and nonacademic skills

So far, we’ve seen parity between boys and girls on constrained reading and math skills by late elementary and/or middle school. Girls have a small advantage in unconstrained reading skills. Boys have a smaller advantage in unconstrained math skills. However, unconstrained skills are not limited to reading and math. Nonacademic skills — also known as executive skills — play an important role in supporting goal-oriented behavior (Stafford-Brizard, 2016).

Executive skills — working memory, attention control, attention shifting, and inhibition control — support all goal-oriented human activity. Research studies demonstrate these skills also directly and indirectly support reading and math achievement (Cragg et al., 2017; Cartwright, 2023). To compare boys and girls on nonacademic skills, we turn to a third dataset: the Early Childhood Longitudinal Study, Kindergarten Class of 2011. The ECLS-K:2011 followed a nationally representative sample of roughly 18,000 kids from kindergarten through fifth grade (Fall 2010 through Spring 2016). In this post, we examine two nonacademic skills — working memory and cognitive flexibility (attention shifting).

Working memory

Working memory is the ability to hold and process information in our minds while performing complex tasks (Dawson and Guare, 2018). There are two types of working memory – visual and verbal. Working memory helps us to do a wide range of daily tasks. The environments in which children grow up – positive and negative – can significantly shape the development of their working memory. Working memory has been shown to have an indirect effect on reading and math achievement (Cragg et al., 2017; Kim, 2020).

This next chart shows the percentage of boys and girls with average or higher scores for working memory from kindergarten through fifth grade.4 There is a four point gap (54% to 58%) in working memory between boys and girls at the start of elementary school. By second grade this small gap is erased. Apparent differences at second and fifth grade are not statistically significant (they fall within the margin of error.)5

Figure 7. Verbal working memory capacity.

Cognitive flexibility (Attention shifting )

Cognitive flexibility is the ability to consider multiple pieces of information or ideas at one time and switch between them while engaging in a task (Cartwright, 2023). This ability helps us to adjust our plans or actions in the face of obstacles or changing conditions (Dawson and Guare, 2018). Cognitive flexibility is developed through consistent practice in everyday routines and experiences. Differences in children’s environments and experiences affect their ability to adjust their thought patterns, look at issues from alternative perspectives, and learn to adapt to change. In reading, cognitive flexibility is thought to help, for example, in actively shifting our attention back and forth between sounds and meanings of printed words (Duke and Cartwright, 2021; Cartwright, 2023). Cognitive flexibility can also help students evaluate different strategies and approaches for solving math problems.

This next chart shows the percentage of boys and girls with passing scores for cognitive flexibility from kindergarten through fifth grade.6 Similar to working memory, there is a small 3 point gap in the fall of kindergarten between boys and girls. This gap widens to six points in favor of the girls by second grade. However, this gap closes by the end of elementary school. The one point difference in fifth grade is not statistically significant (it falls within the margin of error.)7

Figure 8. Capacity to shift attention (cognitive flexibility).

Boys, girls, and general knowledge

Knowledge is the ability to use facts, principles, and ideas to decide and do complex tasks (Cantor et al., 2021). Sometimes knowledge itself is referred to as a skill in the context of reading or mathematics (e.g. background or content knowledge). Indeed, the ability to access and use facts, principles, and ideas to engage in goal-oriented behavior is an important part of skill development. Children’s knowledge about the world around them is an essential part of skill building.

The ECLS-K study included an assessment of K-1 children’s general knowledge of basic natural science and social studies concepts. These are topics that were not formally included in typical kindergarten and first grade curricula in the late 1990s. In fact, researchers expected children’s knowledge in these areas to stem from “his or her family background, home educational environment, and preschool experiences” (Rock & Pollack, 2002).8

Unlike reading and math skills, it isn’t possible to assign proficiency levels to children’s general knowledge (Rock & Pollack, 2002). ECLS-K researchers instead constructed a scale score based upon a two-stage assessment process. Children received a first-stage assessment that consisted of 12 questions. If they got 7 or more questions correct, they were routed to the higher-level assessment at the second stage. Otherwise, they received the lower-level form.

To make it easier to interpret the results, in this next chart I show the average number of questions answered correctly by boys and girls on the first-stage assessment. The chart shows results at three points in time: fall kindergarten, spring kindergarten, and spring first grade. There is rough parity between boys and girls in fall of kindergarten (4.9 for boys and 4.7 for girls). The general knowledge of boys and girls and boys grow in tandem through the end of first grade. Although the minute one-tenth difference in favor of boys is statistically significant, substantively boys and girls are equivalent.9

Figure 9. General knowledge about natural science and social studies concepts.

Summing up

We reviewed data for boys and girls across four sets of skills. “Gaps” and “advantages” should be understood to refer to average group differences in proficiency or ability. These terms do not refer to individual children:

• Reading skills. Despite early gaps in constrained reading skills in favor of girls, there is parity between boys and girls by middle school (if not sooner). For unconstrained skills, however, girls appear to hold a consistent small advantage. Depending upon the specific reading skill, this appears to be a 5 to 9 point advantage at least through middle school.

• Math skills. Despite early gaps in constrained math skills favor of boys, there is parity between boys and girls by middle school (if not sooner). For unconstrained math skills, data from roughly 15 years ago suggest boys have a small advantage. But more recent data suggests this gap might have narrowed.

• Nonacademic skills. Girls appear to start school with a small advantage in working memory at the start of school. But this gap between boys and girls appears to close quickly in early elementary school. (This is a distinctly different pattern than what we see for kids from different socioeconomic groups.) For cognitive flexibility (attention shifting), girls seem to have a small advantage at the start of school that widens temporarily. But by the end of elementary school this gap is completely closed as well. (Again, this is a different pattern than what we see across socioeconomic groups.)

• General knowledge. Substantively, there is no major difference between girls and boys in general knowledge across the first two years of school. This is important to know as general knowledge is moderately correlated with K-1 reading and math achievement. Unfortunately, the ECLS-K dataset did not continue to assess general knowledge after first grade.

Constrained and unconstrained skills are constantly interacting with each other as children develop reading and math skills. Nonacademic skills and general knowledge support the development of these skills — as well as other higher order and more abstract skills. There appear to be temporary differences between boys and girls in constrained skills and unconstrained nonacademic skills. It appears that longer-lived differences between boys and girls in unconstrained reading and math skills drive overall differences in reading and math achievement.

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:

• The four datasets used to analyze constrained and unconstrained skills

• Constrained reading and math skill proficiency in elementary school

• Unconstrained reading and math skill proficiency through middle school

• Nonacademic (executive) skills in elementary school

• World knowledge in kindergarten and first grade

• Unconstrained math skill proficiency in high school

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:

Sight words. Recognizing common words by sight. This is a constrained 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.

Multiplication and division. Solving simple multiplication and division problems and recognizing more complex number patterns. 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:

Partially developed skills and understanding. 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 is a set of unconstrained reading skills.

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.

Understand complicated information. Readers at this level can understand complicated literary and informational passages, including material about topics they study at school. They can also analyze and integrate less familiar material about topics they study at school as well as provide reactions to and explanations of the text as a whole. Performance at this level suggests the ability to find, understand, summarize, and explain relatively complicated information. This is a set of unconstrained reading skills.

Beginning skills and understandings. Students at this level have considerable understanding of two-digit numbers. They can add two-digit numbers but are still developing an ability to regroup in subtraction. They know some basic multiplication and division facts, recognize relations among coins, can read information from charts and graphs, and use simple measurement instruments. They are developing some reasoning skills. This is a mix of constrained and unconstrained math 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 constrained math skills.

Moderately complex procedures and reasoning. Students at this level are developing an understanding of number systems. They can compute with decimals, simple fractions, and commonly encountered percents. They can identify geometric figures, measure lengths and angles, and calculate areas of rectangles. These students are also able to interpret simple inequalities, evaluate formulas, and solve simple linear equations. They can find averages, make decisions based on information drawn from graphs, and use logical reasoning to solve problems. They are developing the skills to operate with signed numbers, exponents, and square roots. This is a set of unconstrained math skills.

Works Cited

Cantor, P., Osher, D., Berg, J., Steyer, L., & Rose, T. (2021). Malleability, plasticity, and individuality: How children learn and develop in context. In P. Cantor & D. Osher (Eds.), The science of learning and development: Enhancing the lives of all young people (pp. 3-54). New York: Routledge.

Cartwright, K. B. (2023). Executive skills and reading comprehension: A guide for educators. New York: The Guilford Press.

Cragg, L., Keeble, S., Richardson, S., Roome, H. E., & Gilmore, C. (2017). Direct and indirect influences of executive functions on mathematics achievement. Cognition, 162, 12-26.

Dawson, P., & Guare, R. (2018). Executive skills in children and adolescents: A practical guide to assessment and intervention. New York: The Guilford Press.

Duke, N. K., & Cartwright, K. B. (2021). The science of reading progresses: Communicating advances beyond the simple view of reading. Reading Research Quarterly, 56, S25-S44.

Fischer, K. W., & Bidell, T. R. (2006). Dynamic development of action and thought. In R. M. Lerner & W. Damon (Eds.), Handbook of child psychology: Theoretical models of human development (6th ed., pp. 313–399). New York: John Wiley & Sons, Inc.

Kim, Y. S. G. (2020). Hierarchical and dynamic relations of language and cognitive skills to reading comprehension: Testing the direct and indirect effects model of reading (DIER). Journal of Educational Psychology, 112(4), 667.

Najarian, M., Tourangeau, K., Nord, C., & Wallner-Allen, K. (2018). Early Childhood Longitudinal Study, Kindergarten Class of 2010–11 (ECLS-K: 2011), first-and second-grade psychometric report (NCES 2018-183). National Center for Education Statistics, Institute of Education Sciences, US Department of Education.

Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27, 587-597.

Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362.

Slotkin, J., Kallen, M., Griffith, J., Magasi, S., Salsman, J., & Nowinski, C. (2012). NIH toolbox. Technical Manual.

Snow, C. E., & Matthews, T. J. (2016). Reading and language in the early grades. The Future of Children, 57-74.

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.

Last updated: March 15, 2025

Three big ideas

1. Unconstrained math skills continue to grow through high school. The greatest gains appear for kids in lower socioeconomic households.

2. We have less information about growth of unconstrained reading skills in high school.

3. Reading and math proficiency are critical to prepare kids for post-secondary education, employment, and citizenry. We need more high quality data about kids’ development of constrained reading and math skills in high school.

About these data

The data in these charts come from the High School Longitudinal Study of 2009 (HSLS:09). I've provided an overview of this and other datasets I used to analyze constrained and unconstrained skills. The HSLS:09 study followed a nationally representative sample of about 23,000 kids from ninth to eleventh grade. The data were collected in the 2009-2010 and 2011-2012 school years. The data are organized by household socioeconomic status. The charts in this post focus on unconstrained math skills in high school. Differentiable data that allows us to examine growth in unconstrained reading skills are not available.

Unconstrained skills in high school

I’ve reviewed growth patterns of a variety of constrained and unconstrained skills in elementary and middle school: reading and math skills, nonacademic skills and general knowledge. A general pattern emerges of universal or close-to-universal mastery of constrained reading and math skills between late elementary and the end of middle school.

Significant proficiency gaps exist, however, for unconstrained skills across socioeconomic groups through middle school. Growth does occur but it’s not enough to close gaps before kids enter high school. What happens to unconstrained skills after middle school? Does growth continues or does it stagnate as kids progress through high school? Are there meaningful differences in growth for kids in different socioeconomic households? To answer these questions, we review growth of unconstrained math skills in a nationally representative sample of high school students.

Algebraic thinking

The High School Longitudinal Study of 2009 (HSLS:09) study followed kids from ninth to eleventh grade. This study had a special focus on algebraic reasoning; this is an unconstrained math skill. Algebraic reasoning involves pattern recognition, analysis of relationships, making generalizations, and use of symbols to represent math concepts. The HSLS:09 assessed high school kids across seven levels of algebraic reasoning.1

The lowest of the seven levels is algebraic expressions. (Skill definitions follow at the end of this post.) This is the most constrained algebra skill in the dataset. Higher levels of algebraic reasoning involve the acquisition, consolidation, integration, and generalization of progressively more information. They also depend upon mastery of lower level skills—like algebraic expressions.

This first chart shows growth in the percentage of students proficient in algebraic expressions between ninth and eleventh grade by household socioeconomic status. The HSLS:09 organizes socioeconomic status into three groups: bottom quintile, the middle three quintiles (the middle 60%), and the top quintile. There are significant proficiency gaps across all three socioeconomic groups in both ninth and eleventh grade.

Figure 1. Understanding the basics of algebra and algebraic expressions (numbers, variables, and mathematical operations).

The proficiency gap between the bottom and top quintiles shrinks from 22 to 12 percentage points from ninth grade to eleventh grade. Given that kids in the top quintile were already close to full proficiency, this improvement is the result of proficiency gains (12 points) among kids in the lowest socioeconomic group. Kids in the bottom four quintiles make substantive growth in this unconstrained math skill during the first three years of high school.

This next chart is for the same group of kids. The algebra skill this time is less constrained—multiplicative and proportional thinking. (Skill definitions are at the bottom of this post.) This is the next algebraic skill in the hierarchical sequence. The overall proficiency patterns for this unconstrained math skill looks similar to algebraic expressions in the previous chart.

Figure 2. Ability to solve word problems involving proportions and identify equivalent algebraic expressions involving multiplication.

There are clear differences in proficiency levels by socioeconomic status in both ninth and eleventh grade. However, all three groups experienced meaningful growth over the first three years of high school. The students who are in the bottom socioeconomic quintile experience the most growth — increasing proficiency by almost half (from 41% to 61%) from ninth to eleventh grade. The kids in the bottom quintile admittedly had more room to grow than the other groups. The overall gap between the bottom and top quintiles shrinks from 39 to 28 percentage points by eleventh grade.

We need more high school data

So far, I have found other data to examine growth in unconstrained skills in high school. I reviewed the Education Longitudinal Study of 2002, which followed students from tenth to twelfth grades. I believe too much has occurred over the past twenty years in math education to be certain of the relevance of these data today. The most recent NAEP Long-Term Trend Assessment for 17-year-olds was conducted in 2012. A review of these data finds similar differences in math skill mastery by eligibility for free and reduced-price meals as the HSLS:09. But point-in-time data from the NAEP LTT can’t be used to tell a story of growth. Only high-quality longitudinal data can provide insights about unconstrained reading and math skills in high 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:

• The four datasets used to analyze constrained and unconstrained skills

• Constrained reading and math skill proficiency in elementary school

• Unconstrained reading and math skill proficiency through middle school

• Nonacademic (executive) skills in elementary school

• World knowledge in kindergarten and first grade

• Constrained and unconstrained skills of boys and girls in elementary and middle school

Skill definitions

High School Longitudinal Survey of 2009

The Mathematics Assessment in Algebraic Reasoning was designed to provide a measure of student achievement in algebraic reasoning. The test assessed a cross-section of understandings of major domains of algebra and key processes. All of these are unconstrained skills. Skills from the HSLS:09 included in this post:

Algebraic expressions. Students able to answer questions such as these have an understanding of algebraic basics, including evaluating simple algebraic expressions and translating between verbal and symbolic representations of expressions.

Multiplicative and proportional thinking. Students able to answer questions such as these have an understanding of proportions and multiplicative situations and can solve proportional situation word problems, find the percent of a number, and identify equivalent algebraic expressions for multiplicative situations.

Last updated: March 15, 2025

Three big ideas

1. Knowledge is the ability to use facts, principles, and ideas to decide and do complex tasks. Knowledge is sometimes referred to as a “skill” in the context of reading and math (vocabulary knowledge, conceptual knowledge). This is akin to the way that cognitive scientists describe skill building as constructing and generalizing new knowledge to different tasks and contexts.

2. Early knowledge gaps as children start elementary school appear to be heavily shaped by their experiences outside of school. Twenty-five years ago, these gaps did not appear to narrow in the first years of school. It’s possible this might be changing with the adoption newer knowledge-building curricula.

3. Knowledge appears to be moderately correlated with overall reading and math achievement. Below the surface, however, is a dynamic relationship between knowledge and constrained and unconstrained reading and math skills.

About these data

The data in these charts come from the Early Childhood Longitudinal Study, Kindergarten Class of 1998-99 (ECLS-K). I've provided an overview of these and other datasets I used to analyze constrained and unconstrained skills. The ECLS-K data were collected between Fall 1998 and Spring 2007. It is unique in that it includes information about children’s knowledge in addition to their reading and math skills. I used the EdSurvey R package to analyze the data. The data are organized by household socioeconomic status. In the ECLS-K dataset, socioeconomic status is defined using a combination of household income, parental education level, and parental occupation.

What is knowledge?

Knowledge is the ability to use facts, principles, and ideas to decide and do complex tasks (Cantor et al., 2021). Sometimes knowledge itself is referred to as a skill in the context of reading or mathematics (Kim, 2023; Rittle-Johnson et al., 2001). Indeed, the ability to access and use facts, principles, and ideas to engage in goal-oriented behavior is an important part of skill development. Children’s knowledge about the world around them is an essential part of skill building.

Knowledge is critical for reading comprehension (Hirsch, 2003; Pearson, et al., 2020). General knowledge represents children’s breadth and depth of understanding of their social and physical environment (i.e., the social, physical and natural world) and their ability to draw inferences and comprehend implications (West et al., 2000). Content knowledge, background knowledge, text structure knowledge, and morphological knowledge are component reading skills.

Vocabulary, a critical component skill for both reading and math, is also a type of knowledge. “The knowledge of a word not only implies a definition, but also implies how that word fits into the world” (Stahl, 2005). Factual (or declarative) knowledge, procedural knowledge, and conceptual knowledge are the three primary components of mathematics. Hattan and Lupo (2020) remind us there are multiple types of knowledge that children bring to bear for reading comprehension (and presumably also mathematical thinking): cultural, linguistic, principled, strategic, multimodal, multiple text use, and conditional.

Knowledge is a “caught and taught” skill, developing from both informal and formal learning experiences. The breadth and depth of these different aspects of knowledge illustrates its nature as a “large problem space” (Snow & Kim, 2007). Unlike constrained skills, any assessment of knowledge will only give us a partial glimpse of a child’s ability to use facts, principles, and ideas to decide and do complex tasks.

Gaps in general knowledge are hard to close

The ECLS-K study includes an assessment of K-1 children’s general knowledge of basic natural science and social studies concepts. These are topics that were not formally included in typical kindergarten and first grade curricula in the late 1990s. In fact, researchers expected children’s knowledge in these areas to stem from “his or her family background, home educational environment, and preschool experiences” (Rock & Pollack, 2002).1

Unlike reading and math skills, it isn’t possible to assign proficiency levels to children’s general knowledge (Rock & Pollack, 2002). ECLS-K researchers instead constructed a scale score based upon a two-stage assessment process. Children received a first-stage assessment that consisted of 12 questions. If they got 7 or more questions correct, they were routed to the higher-level assessment at the second stage. Otherwise, they received the lower-level form.

To make it easier to interpret the results, Figure 1 below shows the average number of questions answered correctly by kids on the first-stage assessment. Kids are organized by household income status (quintiles). The chart shows results at three points in time: fall kindergarten, spring kindergarten, and spring first grade. Kids in the bottom group on average answer about 3 correct (out of 12) in fall kindergarten. By contrast, kids in the top group on average answer nearly 7 questions correctly. There are clear gaps in the average performance of kids in each socioeconomic group. (Mouse over the chart.)

Figure 1. General knowledge about natural science and social studies concepts.

By the end of kindergarten, both the top and bottom socioeconomic groups average one more correct question — maintaining the 4 question gap. This gap narrows slightly by the end of first grade. However, the average of 6.5 correct answers for kids in the lowest socioeconomic household in spring of first grade is roughly equivalent to that of the highest socioeconomic group in the fall of kindergarten. The overall pattern of gaps between socioeconomic groups remains unchanged.

To put these data in perspective, this next chart shows the growth pattern for a constrained math skill (relative size) for this same group of kids over the same time period. Early gaps in fall kindergarten are largely closed by the end of first grade. Kids across all socioeconomic groups on average are at or close to full proficiency.

Figure 2. Counting beyond ten and ability to compare size of different objects.

This next chart shows a similar growth pattern for a constrained reading skill—letter-sound correspondence at the end of words. Again, there were early and significant proficiency gaps at school entry. By the end of first grade, however, these gaps had significantly closed. All groups were over 90% proficiency, with all but one within a few percentage points of 100% average proficiency.

Figure 3. Ability to associate letters with sounds (letter-sound correspondence) at the end of words.

The relationship between knowledge, reading, and mathematics

Supporting children’s world knowledge is important in its own right. It’s also connected to their reading and math skills. The ECLS-K researchers found moderate correlations between scale scores for general knowledge and K-1 scale scores for reading and math (Rock & Pollack, 2002). The correlations between general knowledge and reading were fairly stable from fall kindergarten through spring first grade (between 57% and 59%). The same was true for general knowledge and mathematics (between 64% and 67%).

Kids need opportunities to develop general knowledge

A key difference between constrained and unconstrained skills is opportunity. After all, a constrained skill involves a relatively limited amount of information. All kids who attend school get access to the same information. Mastery is clearly defined and the same for everyone. But unconstrained skills depend more upon opportunities to develop these slower-growing skills inside and outside of school.

Rock and Pollack (2002) made this observation in a technical manual for the ECLS-K:

It is interesting to note that gains from a full year of schooling (fall to spring, in both kindergarten and first grade) in terms of standard deviation units [(i.e. the spread of individual scores from the average)] on general knowledge appear to be considerably less than those that were demonstrated in both reading and mathematics. Also, there is less differential in growth rates exhibited between adjacent [testing] rounds than in reading and mathematics. The rate of growth during the summer between kindergarten and first grade is closer to the growth during the school year intervals than was found in reading and mathematics. It would appear that the general knowledge test is measuring information that is not necessarily included in most kindergarten and first-grade curricula but is associated more with the child’s out-of-school experiences. [Emphasis added.]

These differences in opportunity are made more plain by examining differences in general knowledge across socioeconomic levels within a single racial group. This next chart shows correct responses to the first-stage general knowledge questions by Black students across five socioeconomic groups.

Figure 4. General knowledge about natural science and social studies concepts.

The initial pattern at kindergarten entry is a little different than for the full sample in Figure 1. Black kids from the three highest socioeconomic levels average roughly the same correct number of questions (3.6 to 3.8). By spring of kindergarten this bunching between the higher income groups is erased. We see the same stair-step pattern that exists in the full sample. This pattern repeats in spring of first grade.

There is more current use of knowledge-building curricula in schools compared to 25 years ago. It’s unclear if a similar study today would yield the same results as we see in these late 1990s data. The key point is that knowledge is caught and taught. Kids’ experiences inside and outside of school contribute to the growth of their world knowledge. It plays a complex and vital role in building constrained and unconstrained reading and math skills. Knowledge is the ultimate unconstrained skill.

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:

• The four datasets used to analyze constrained and unconstrained skills

• Constrained reading and math skill proficiency in elementary school

• Unconstrained reading and math skill proficiency through middle school

• Nonacademic (executive) skills in elementary school

• Unconstrained math skill proficiency in high school

• Constrained and unconstrained skills of boys and girls in elementary and middle school

Works Cited

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Hattan, C., & Lupo, S. M. (2020). Rethinking the role of knowledge in the literacy classroom. Reading Research Quarterly, 55, S283-S298.

Hirsch, E. D. (2003). Reading comprehension requires knowledge of words and the world. American Educator, 27(1), 10-13.

Kim, Y. S. G. (2023). Simplicity meets complexity: Expanding the simple view of reading with the direct and indirect effects model of reading. In Cabell, S. Q., Neuman, S. B. & Patton Terry, N. (Eds.), Handbook on the Science of Early Literacy, New York: The Guilford Press, pp. 9-22.

Paris, S. G., Carpenter, R. D., Paris, A. H., & Hamilton, E. E. (2005). Spurious and genuine correlates of children’s reading comprehension. In S. Paris & S. Stahl (Eds.), Children's Reading Comprehension and Assessment (pp. 149-178). New York: Routledge.

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Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362.

Rock, D.A. & Pollack, J.M. (2002). Early childhood longitudinal study-kindergarten class of 1998–99 (ECLS–K), psychometric report for kindergarten through first grade (NCES 2002–05). National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education. Washington, DC.

Snow, C. E., & Kim, Y.-S. (2007). Large problem spaces: The challenge of vocabulary for English language learners. In R. K. Wagner, A. Muse, & K. Tannenbaum (Eds.), Vocabulary acquisition and its implications for reading comprehension (pp. 123–139). New York, New York: Guilford.

Stahl, S. A. (2005). Four problems with teaching word meanings. In E. H. Hiebert & M. L. Kamil (Eds.), Teaching and learning vocabulary: Bringing research to practice, (pp. 95-116). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

West, J., Denton, K., & Reaney, L. M. (2000). The kindergarten year: Findings from the early childhood longitudinal study, kindergarten class of 1998-99. NCES 2001-023. National Center for Education Statistics.