Show me the data: Unconstrained skills in high school
Unconstrained skills keep growing in secondary school.
Note: Unconstrained Kids unpacks, translates, and integrates academic research and data about constrained and unconstrained skills for people that run, fund, and assist organizations that teach and serve kids. This post presents a series of charts I created from among four nationally representative datasets to illustrate proficiency patterns in representative groups of constrained and unconstrained reading and math skills, unconstrained nonacademic skills, and unconstrained general knowledge. For reference, see this working list of constrained and unconstrained skills. Like everything on this Substack, this post is a work-in-progress. I will make updates as needed. A list of the skills included in these charts is included at the end. Questions, comments, and suggestions are welcome.
Last updated: March 15, 2025
Three big ideas
Unconstrained math skills continue to grow through high school. The greatest gains appear for kids in lower socioeconomic households.
We have less information about growth of unconstrained reading skills in high school.
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:
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.
The seven levels were: algebraic expressions; multiplicative and proportional thinking; algebraic equivalents; systems of equations; linear functions; quadratic functions; and, log and exponential functions (geometric sequences).