Physics Quantitative Literacy: Concepts & Assessment

• PQL is defined as the integrated ability to use mathematics in physical contexts, emphasizing proportional, covariational, and sign reasoning.
• Research instruments like PIQL, GERQN, and PMQ assess PQL by analyzing measurement uncertainty, conceptual blending, and student reasoning patterns.
• Studies reveal modest pre/post instructional gains in PQL, prompting curricular redesigns that explicitly integrate math with physics for improved lab and analysis outcomes.

Searching arXiv for recent and foundational papers on Physics Quantitative Literacy, PIQL, GERQN, and related lab-uncertainty work. arXiv_search query: "Physics Quantitative Literacy PIQL GERQN measurement uncertainty introductory physics" max_results: 10 Physics Quantitative Literacy (PQL) denotes the ability to apply quantitative reasoning to physical situations and, more specifically, to blend conceptual and procedural mathematics with physical meaning-making. In the literature, it is described as the interconnected skills, attitudes, and habits of mind that support the sophisticated use of elementary mathematics in physics, rather than computation in isolation or conceptual physics in isolation. Across current research, PQL encompasses proportional reasoning, covariational reasoning, reasoning with signed quantities, unit interpretation, statistical and probabilistic reasoning, and modeling and sense-making. The central issue is whether learners can interpret equations, graphs, units, uncertainty, and functional relations as statements about physical systems, and can use those interpretations to analyze data, design measurements, and justify conclusions (Brahmia et al., 2020, Smith et al., 2020, Smith et al., 2019).

1. Definitions and scope

PQL has been defined in several closely related ways. One formulation describes it as the ability to apply quantitative reasoning to physical situations, with key components including numerical fluency, algebraic fluency, statistical and probabilistic reasoning, and modeling and sense-making. Another defines it as the ability to interpret equations, apply mathematics in context, and connect mathematical relations to physical meaning. A third emphasizes “the interconnected skills, attitudes, and habits of mind that together support the sophisticated use of elementary mathematics in the context of physics” (Lewandowski et al., 2017, Brahmia et al., 1 Aug 2025, Smith et al., 2020).

These definitions are consistent in treating PQL as domain-specific quantification. In physics, familiar mathematical operations such as forming a ratio, reading a slope, interpreting an area under a curve, or assigning a sign are not merely formal procedures. They are tied to physical interpretation: densities, rates of change, accumulated quantities, directions, decreases, or interaction strengths. The literature therefore distinguishes PQL from routine symbolic manipulation such as solving algebraic equations without regard to units, sign conventions, or physical plausibility (Brahmia et al., 2020, Brahmia et al., 1 Aug 2025).

Measurement uncertainty occupies a special place within this broader construct. In laboratory contexts, PQL underpins students’ capacity to design experiments, analyze data, and draw conclusions with appropriate attention to uncertainty. In that setting, statistical reasoning about spread, mean values, and uncertainty becomes part of the same quantitative-literacy framework that also includes algebraic and graphical reasoning (Lewandowski et al., 2017).

2. Cognitive and theoretical structure

The dominant assessment framework for introductory-course PQL organizes it around three core mathematical-reasoning constructs: proportional reasoning, covariational reasoning, and reasoning with signed quantities. In one formulation, unit interpretation is treated alongside sign reasoning, including dimensional analysis and the multiple “natures of negativity” in physics: sign as direction, sign as type, sign as indication of decrease, and facility with dimensional analysis. Proportional reasoning involves interpreting, applying, and constructing ratios and identifying linear scaling. Covariational reasoning concerns how one physical quantity changes as another varies, including interpretation of slopes and areas under curves (Brahmia et al., 2020).

This framework is interpreted through conceptual blending theory and Sherin’s symbolic forms. In that view, mathematical symbol structures such as “Ratio,” “Scaling,” “Opposition,” “Quantity-Measurement,” and “Covariation-Dependence” become inseparable from their physical meaning. PQL is therefore characterized not by separate mathematics and physics resources, but by their integration into a blended space in which learners can create, interpret, and apply physical quantities and equations (Brahmia et al., 2020, Brahmia et al., 2019).

Conceptual blending analyses of PIQL items make this integration explicit. In the Lead–Steel exchange problem, students must coordinate an invariant ratio with a symbolic variable. In the Slide Steepness problem, they must construct a new quantity from height and base and interpret it as steepness. The analyses identify stable resources, such as recognizing an invariant ratio, and unstable projections, such as failure to sustain symbolic reasoning with a general variable or selecting a non-optimal quantity despite an expert-like attempt to quantify. The proposed hierarchy runs from recognition of generic structures, to projection into the blend, to back-projection that checks physical meaning and algebraic correctness (Brahmia et al., 2019). This suggests that partially correct reasoning is structurally important rather than incidental.

3. Assessment instruments

Research on PQL has produced several instruments with different scopes. The Physics Measurement Questionnaire (PMQ) focuses on measurement uncertainty in laboratory reasoning. The Physics Inventory of Quantitative Literacy (PIQL) is a reasoning inventory for calculus-based introductory physics. The General Equation-based Reasoning inventory of QuaNtity (GERQN) is an algebra-based adaptation intended for students who have taken algebra but are not enrolled in, or have not completed, calculus (Lewandowski et al., 2017, Smith et al., 2020, Zimmerman et al., 20 Apr 2025).

Instrument	Domain emphasis	Format and scoring
PMQ	Measurement uncertainty; point and set paradigms	Free-response probes RD, UR, SMDS, DMSS; responses coded P, S, or N
PIQL v2.2	Proportional reasoning, covariation, signed quantities	20 items: 14 single-response multiple-choice and 6 MCMR; optional four-level MCMR rubric
GERQN v3.0	Algebra-based PQL with sign, covariation, proportional reasoning	16 items; MCSA and MCMR; approximately 30 minutes

The PMQ was designed to characterize student reasoning into point or set paradigms. Point-paradigm reasoning treats individual measurements as capable of yielding the true value and tends to neglect spread. Set-paradigm reasoning treats measurements as a distribution centered on the true value and emphasizes repeated measurements, averages, and characterization of spread. Four PMQ probes have been central in published analyses: Repeating Measurements (RD), Using Repeated measurements (UR), Same Mean, Different Spread (SMDS), and Different Mean, Same Spread (DMSS) (Lewandowski et al., 2017).

The PIQL was developed iteratively from a protoPIQL of 18 items to a final version 2.2 with 20 items. Its items are designed to probe proceptual blends of mathematics and physics rather than rote calculation, and it is explicitly presented as a reasoning inventory rather than a concept inventory. Multiple-choice/multiple-response items were included because they can capture the complexity of blended reasoning by allowing students to select all statements that “must be true.” For MCMR items, a four-level rubric distinguishes “Completely Correct,” “Some Correct,” “Both Correct & Incorrect,” and “Completely Incorrect” (Brahmia et al., 2020, Smith et al., 2020).

The GERQN retains the same three facets as the PIQL but adapts them for algebra-based courses by restricting to scalar quantities, limiting covariation to single-variable linear relationships, and emphasizing contexts accessible without calculus. Its representative items include sign interpretation for $v(t)=-2t+10$, symbolic manipulation of $F=ma=5x-20$, and graphical interpretation of water height as a function of volume in a bottle made of two cylinders (Zimmerman et al., 20 Apr 2025).

4. Psychometric results and the structure of the construct

PIQL development has been accompanied by extensive psychometric analysis. Early protoPIQL results showed mean difficulty approximately $0.75$, five items with $p_i>0.80$, three items with $D_i<0.30$, and Cronbach’s $\alpha=0.67$. In the final PIQL v2.2, only one item had $p>0.80$, all discrimination indices exceeded $0.30$, six items had $D_i>0.60$, item-total correlations were all above $0.25$, and Cronbach’s $F=ma=5x-20$0, meeting the stated criterion for individual-level reliability (Smith et al., 2020).

A central psychometric question has been whether the three expert-defined subdomains are orthogonal in student reasoning. Confirmatory factor analysis of the three-factor model consistently produced poor fit, with $F=ma=5x-20$1, well below the target $F=ma=5x-20$2. By contrast, exploratory factor analysis and a unidimensional CFA on the final version yielded good fit, with $F=ma=5x-20$3, $F=ma=5x-20$4, $F=ma=5x-20$5, and $F=ma=5x-20$6 or $F=ma=5x-20$7, indicating that PQL behaves as an approximately single coherent construct in the studied population (Brahmia et al., 2020, Smith et al., 2020).

Network-based module analysis reinforced that result. Factors and student-response modules did not map cleanly onto the expert categories of ratios, covariation, and negativity; items cross-loaded, modules intermingled subdomains, and the module structure changed across PreMech, PostMech, and PostEM. Items 11 and 12 clustered together reliably, and multiple-response items sometimes split across modules, showing that students could activate one intended element of reasoning without its companion (Smith et al., 2019). This suggests that the analytic separation of PQL into three facets is useful for item design and instructional diagnosis, but not necessarily a faithful description of how introductory students organize their reasoning.

The GERQN extends this psychometric program to algebra-based physics. For its filtered data, item difficulty ranged $F=ma=5x-20$8–$F=ma=5x-20$9 for most items, point-biserial discrimination exceeded $0.75$0 for all kept items, Cronbach’s $0.75$1 with 95% CI $0.75$2, Ferguson’s $0.75$3, and test–retest Pearson correlations were $0.75$4–$0.75$5. Factor analysis yielded $0.75$6, Bartlett’s $0.75$7, $0.75$8, and a one-factor CFA with $0.75$9 and $p_i>0.80$0, again supporting a single latent PQL construct (Zimmerman et al., 20 Apr 2025).

Performance data further sharpen the picture. At one institution, 2,758 PIQL administrations from 2,078 students produced mean scores of $p_i>0.80$1 out of 20 at PreMech, $p_i>0.80$2 at PostMech, and $p_i>0.80$3 at PostEM, with effect sizes $p_i>0.80$4 from PreMech to PostMech and $p_i>0.80$5 from PostMech to PostEM. In a matched GERQN cohort of $p_i>0.80$6, average score remained near 60% correct from PreMech to PostEM. Across both instruments, the recurring result is that PQL does not improve strongly under standard instruction, even when research-based curricula are in use (Brahmia et al., 2020, Zimmerman et al., 20 Apr 2025).

5. Measurement uncertainty and laboratory reasoning

In laboratory studies, PQL has been examined through students’ reasoning about measurement uncertainty. The PMQ operationalizes this domain by distinguishing point and set paradigms. Within the set paradigm, repeated measurements are treated as a distribution of values with intrinsic spread, and two standard metrics are foregrounded: the sample standard deviation,

$p_i>0.80$7

and the standard error of the mean,

$p_i>0.80$8

These metrics support the shift from isolated readings toward distribution-based inference (Lewandowski et al., 2017).

A large pre/post study in PHYS 1140 at the University of Colorado Boulder analyzed 525 matched students from a Fall 2016 enrollment of 588. Responses were coded using expanded codebooks derived from Volkwyn et al., with a primary coder coding all responses and a secondary coder coding a 10% subset; inter-rater agreement was 78% with $p_i>0.80$9. Pre/post distributions were compared using Mann–Whitney $D_i<0.30$0 tests at $D_i<0.30$1, and shifts in mean fractions of point or set scores were computed with 95% confidence intervals using multinomial variance (Lewandowski et al., 2017).

The results were probe-specific. On RD, point reasoning decreased significantly ($D_i<0.30$2), but set reasoning did not increase significantly ($D_i<0.30$3); many responses moved to the mixed or unclear category. On UR, point reasoning decreased slightly ($D_i<0.30$4), while set reasoning was already above 80% both pre and post, indicating a ceiling effect. On SMDS, the shift toward set reasoning was small and not statistically significant ($D_i<0.30$5 for set, $D_i<0.30$6 for point). On DMSS, by contrast, there was a large shift toward set reasoning ($D_i<0.30$7) and away from point reasoning ($D_i<0.30$8); this was the only shift judged both statistically and practically significant for the set paradigm (Lewandowski et al., 2017).

The PMQ findings identify an important asymmetry. Traditional labs appeared more successful in improving reasoning about data comparison when spread was implicit than in improving reasoning about data collection decisions. This supports the instructional recommendation that students be engaged explicitly in deciding how many measurements to take and why, rather than encountering uncertainty primarily at the reporting stage (Lewandowski et al., 2017).

6. Instructional designs for developing PQL

Several instructional programs treat PQL as a teachable practice rather than an indirect by-product of standard problem solving. In introductory labs, Holmes and Bonn’s framework centers on an iterative cycle of Compare → Reflect → Iterate. Students collect two independent data sets, compute a continuous comparison statistic,

$D_i<0.30$9

interpret the magnitude of $\alpha=0.67$0, and then design follow-up measurements. The decision criteria are explicit: $\alpha=0.67$1 indicates “Unlikely different” or overestimated uncertainties; $\alpha=0.67$2 indicates ambiguity; $\alpha=0.67$3 indicates “Likely different,” prompting a search for systematics or model limitations (Holmes et al., 2015).

In the pendulum example, repeated iterations moved one group from $\alpha=0.67$4 to $\alpha=0.67$5 to $\alpha=0.67$6, with the final result supporting a conclusion about breakdown of the small-angle approximation. At the class level, SQILab participation was associated with an increase from $\alpha=0.67$7 to $\alpha=0.67$8 trials per student, from $\alpha=0.67$9 to $p>0.80$0 swings per trial, and a reduction in mean uncertainty from $p>0.80$1 to $p>0.80$2 (Holmes et al., 2015). Within the language of PQL, this framework shifts attention from percent-error style point comparisons toward distribution-sensitive, model-aware inquiry.

For lecture and recitation contexts, PIQL-based work recommends explicit instruction on the multiple meanings of the negative sign, scaffolded ratio-and-proportion tasks contextualized in mechanics, covariational activities using dynamic graphing tools, and integrated unit-analysis “sanity checks” on all problem solutions. MCMR item analysis is used diagnostically, because it can distinguish partial understanding from mixtures of correct and incorrect reasoning (Brahmia et al., 2020).

At the curricular level, the Rutgers Extended Analytical Physics program embeds PQL directly into an extended, credit-bearing calculus-based sequence. The EAP pathway adds an extra credit hour each semester, yielding approximately 25% more contact time, and integrates algebra, precalculus, and the intuitive underpinnings of calculus—rates, accumulation, and variation—precisely when needed for physics topics. Its stated design features include flexible entry, representative instructors, a supportive environment, deep learning focus, and “Calculus Foundations Without Calculus” (Brahmia et al., 1 Aug 2025).

Upper-division work on integrated conceptual and quantitative tutorials points to an additional design constraint. In the Quantum Interactive Learning Tutorial study, the hybrid version that integrated conceptual and quantitative reasoning improved conceptual post-test performance for graduate students and for one undergraduate cohort, but a weaker-prepared undergraduate cohort performed less well than students using the conceptual-only version. One interpretation offered in the paper is that integrated quantitative work must be commensurate with prior knowledge and carefully scaffolded to avoid cognitive overload (Justice et al., 11 Dec 2025). A plausible implication is that PQL instruction is not merely a matter of adding more mathematics; calibration and sequencing are central.

7. Generalization, equity, and open directions

A consistent result across PIQL and GERQN studies is that PQL shows only small gains under ordinary course conditions. That pattern has motivated both instrument development and curricular redesign. In algebra-based physics, the GERQN was developed because the PIQL includes instantaneous rates of change and vector products not expected in courses for the larger population of students who have completed only Algebra I. The algebra-based instrument retains sign reasoning, covariational reasoning, and proportional reasoning, but simplifies the mathematical demands while preserving the physics contexts (Zimmerman et al., 20 Apr 2025).

The equity dimension of PQL has become more explicit in recent work. The critique developed in the Rutgers/TIPSSS paper is that rigid prerequisite structures and placement systems reward procedural fluency in algebra and trigonometry while overlooking the conceptual quantitative reasoning that physics actually demands. In that account, labeling students as “underprepared” obscures structural causes and underestimates the role of course design. The EAP model is presented as an alternative in which PQL development is embedded within the physics sequence rather than outsourced to prerequisite remediation (Brahmia et al., 1 Aug 2025).

Outcome data from Rutgers connect this design to persistence. Comparing the two years before EAP with the seven years after launch, first-year physics passing rates rose from roughly 60% to 80% for all engineering students, 50% to 70% for female-identifying students, and 40% to 60% for underrepresented minority students. Six-year STEM degree completion rates rose from approximately 30% to 50% for all students, 20% to 45% for female-identifying students, and 15% to 40% for underrepresented minority students, with a ten-year follow-up confirming that gains persisted into Analytical Physics II grades (Brahmia et al., 1 Aug 2025).

Several open directions recur across the literature. One is extension beyond current construct boundaries: measurement-uncertainty research explicitly calls for analysis of systematic uncertainty reasoning beyond the point–set framework and for finer gradations within set reasoning. Another is broader validation: PIQL data have largely come from one large research institution’s calculus-based courses, and GERQN authors likewise call for further multi-institutional work. A third is refined scoring and longitudinal modeling, including richer use of MCMR responses, network analyses, and possible item-response models (Lewandowski et al., 2017, Smith et al., 2020, Zimmerman et al., 20 Apr 2025). Taken together, these directions treat PQL not as a narrow skill deficit, but as a coherent and assessable dimension of physics learning whose development depends on both cognition and curriculum.