TECHNICAL NOTE / FRAMEWORK & EVIDENCE
Methodology & Standards
An eight-observation, un-timed instrument for grades 7–10. It samples how physical relationships are interpreted and expressed through arithmetic and algebra. It is not a standardized test or a comprehensive measure of science or mathematics attainment.
01 / DIAGNOSTIC FRAMEWORK
Physical interpretation and mathematical representation
Each domain links a qualitative physical model to a quantitative relationship and a symbolic representation. These are connected tasks, not independent measures of intelligence or two separate percentage scores. The adaptive pair distinguishes an arithmetic connection from either a qualitative comparison or an algebraic extension.
- L1 / QUALITATIVEPhysical intuition
- Compare equal-volume masses, position spacing, balanced forces, or frictionless ramps without calculation.
- L2 / QUANTITATIVEArithmetic relationship
- Calculate a unit ratio, speed, signed force difference, or force–distance product using concrete values.
- L3 / SYMBOLICAlgebraic representation
- Rearrange a density or work formula, interpret slope with unit conversion, or solve a signed equilibrium equation.
Every domain starts at Level 2. An answer matching the fixed key selects Level 3 next; any other answer selects Level 1. Four domains × two observations = eight responses. Scratch paper and calculators are welcome; elapsed time is not scored.
02 / CURRICULUM CONNECTIONS
Explicit standards mapping
The table distinguishes sampled skills from broader standards. NGSS performance expectations include scientific practices and evidence that these brief selected-response tasks do not fully assess. A related standard is not a claim of complete alignment or mastery.
| Domain / sampled skill | NGSS / scope of connection | Common Core / mathematical skill |
|---|---|---|
| DensityCompare mass at equal volume; calculate mass per volume; isolate volume. | MS-PS1-2 (opens in a new tab) Analyze substance properties before and after interaction to determine whether a chemical reaction occurred. Density is a relevant property; reaction evidence is not sampled here. | 6.RP (opens in a new tab) · 7.RP (opens in a new tab) Ratios, unit rates, and proportional relationships (including 7.RP.A.2). Formula isolation extends to HSA-CED.A.4. |
| MotionCompare position spacing; calculate speed; interpret slope and convert time units. | MS-PS2-2 (opens in a new tab) Plan an investigation of changes in motion in relation to force and mass. Describing motion is a prerequisite; investigation design and acceleration are not sampled. | 6.RP (opens in a new tab) · 8.EE (opens in a new tab) Unit rates and unit conversion (6.RP.A.3d); interpret unit rate as slope (8.EE.B.5). |
| EquilibriumIdentify support and gravity; combine opposing forces; solve for a balancing force. | MS-PS2-2 (opens in a new tab) Balanced and unbalanced forces connect to changes in motion. These observations do not require planning an investigation. MS-PS2-1, cited in the field guide, concerns third-law interactions between objects; balancing forces on one object is not a third-law test. | 8.EE (opens in a new tab) Solve a linear equation in one variable (8.EE.C.7). Signed force addition also draws on 7.NS. |
| WorkCompare frictionless ramps; calculate work; isolate force in W = Fd. | MS-PS3-1 (opens in a new tab) · MS-PS3-2 (opens in a new tab) MS-PS3-1 concerns kinetic-energy graphs relating mass and speed; these are not sampled. MS-PS3-2 concerns potential-energy models as object arrangements change; equal-height ramps provide a related gravitational-energy context, not a full assessment. | 6.RP (opens in a new tab) · 7.RP (opens in a new tab) Multiplicative reasoning and proportional force–distance relationships. Rearranging W = Fd connects directly to HSA-CED.A.4 (opens in a new tab), beyond the middle-school domains. |
03 / RESEARCH BASIS
Sources and their role
These sources inform the instructional design. They do not validate this particular eight-response instrument. University-level findings are background evidence, not proof of effectiveness for grades 7–10.
- Arnold B. Arons — Teaching Introductory Physics (opens in a new tab)
Wiley, ISBN 978-0-471-13707-8. A teaching reference supporting conceptual development and reasoning about physical quantities before routine formula use. The publisher lists August 1996; the field guide uses the commonly cited 1997 edition year.
- Philip Southey — The Ratio Table: a tool for understanding ratios (opens in a new tab)
2019 Physics Education Research Conference proceedings, published 2020. Connects algebraic ratio operations to physical meaning and “per” units. This supports the ratio/units bridge; it is not a general validation study of math-to-physics transfer.
- Nakakoji & Wilson (2018) — First-Year Mathematics and Its Application to Science: Evidence of Transfer of Learning to Physics and Engineering (opens in a new tab)
Education Sciences, 8(1), 8. Direct research on transfer between university mathematics and science contexts; included to distinguish transfer evidence from the narrower ratio instruction studied by Southey.
- Next Generation Science Standards — official standards (opens in a new tab)
Primary source for performance expectations, clarification statements, assessment boundaries, scientific practices, and disciplinary core ideas.
- Common Core State Standards — Mathematics (opens in a new tab)
Primary source for ratios and proportional relationships, expressions and equations, and rearranging literal equations.
Domain-specific physics education research
- DENSITYSmith, Snir & Grosslight. Using conceptual models to facilitate conceptual change: The case of weight-density differentiation. Cognition and Instruction, 9(3), 221–283. (opens in a new tab)
- MOTIONInvestigation of student understanding of the concept of velocity in one dimension. American Journal of Physics, 48, 1020–1028. (opens in a new tab)
- EQUILIBRIUMExplaining the ‘at rest’ condition of an object. The Physics Teacher, 20, 10–14. (opens in a new tab) · Students’ preconceptions in introductory mechanics. American Journal of Physics, 50, 66–71. (opens in a new tab)
- WORK & ENERGYChi, Feltovich & Glaser. Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121–152. (opens in a new tab)
04 / VERIFICATION & SCORING DISCLOSURE
Deterministic evaluation. No generative AI.
Evaluation is completely rule-based and unassisted by generative AI. A selected answer identifier is compared with a fixed answer key. Fixed rules determine the next observation, evidence level, response explanation, and workout focus. Identical responses produce identical results; there are no hidden weights, random branches, or model-generated judgments.
- Level 3 · Algebraic
- Level 2 and the subsequent Level 3 response both match their keys.
- Level 2 · Arithmetic
- Level 2 matches its key; the subsequent Level 3 response does not.
- Level 1 · Intuitive
- Level 2 does not match its key; the subsequent Level 1 response does.
- Intuition not yet confirmed
- Neither Level 2 nor the subsequent Level 1 response matches its key. No demonstrated level is assigned.
Level 1 is not separately observed on the Level 2 → Level 3 branch. Any lower-level ability implied by a higher-level result is an inference, not an additional observation. The map reports evidence from this session, not permanent ability.
The workout targets the lowest demonstrated domain, treating unconfirmed intuition as the starting point. Ties follow the fixed order: Density, Motion, Equilibrium, Work. Practice responses do not change the diagnostic map.
Automated rule checks cover adaptive branching and evidence-level assignment. Such software checks are not psychometric validation. No reliability coefficients, diagnostic accuracy estimates, or norm-referenced cutoffs are reported for this instrument.
Distractor rationale: candidate explanations, not mind-reading
Each alternative answer has a prewritten explanation of the operation or physical model it represents. Choosing an alternative suggests a possible friction point; it cannot establish the student’s reasoning. Guessing, misreading, and other methods can produce the same answer.
Density L2 · Mass in each unit of space
A uniform block has a mass of 24 g and a volume of 8 cm³. How much mass is there in each 1 cm³?
Fixed model: Share 24 g across 8 equal cubic centimeters: 24 ÷ 8 = 3 g/cm³.
- 192 g/cm³
- Multiplication combines the two quantities; it does not find mass per unit volume. Divide 24 by 8.
- 16 g/cm³
- Subtracting mass and volume combines incompatible units. ‘Per cubic centimeter’ asks for division.
- ⅓ g/cm³
- This reverses the ratio: volume ÷ mass. Mass per unit volume is 24 ÷ 8, not 8 ÷ 24.
Motion L3 · Slope is a physical rate
On a position–time graph, a straight line passes through (1 min, 60 m) and (3 min, 300 m). What is its speed in m/s?
Fixed model: v = Δx/Δt = (300 − 60)/[(3 − 1) × 60] = 240/120 = 2 m/s.
- 120 m/s
- 120 is the rate in meters per minute. Convert the 2-minute interval to 120 seconds.
- 1.67 m/s
- 300 ÷ 180 uses one point relative to the origin. This line starts elsewhere; use differences between the two points.
- 4 m/s
- This uses a one-minute interval. The time change from 1 min to 3 min is 2 min = 120 s.
Equilibrium L2 · Combine opposing pushes
Two horizontal forces act on a box: 18 N to the right and 12 N to the left. What is their combined force?
Fixed model: Choose right as positive: 18 − 12 = 6 N to the right.
- 30 N to the right
- Adding magnitudes ignores direction. These forces oppose one another, so subtract their magnitudes.
- 6 N to the left
- The difference is 6 N, but the larger force points right. The resultant points with the larger force.
- 0 N — opposing forces cancel
- Opposing forces cancel only when equal in magnitude. Here one exceeds the other by 6 N.
Work & Energy L3 · Rearrange the work relationship
A steady force aligned with motion does 360 J of work over 6 m. Using W = Fd, which expression isolates F and gives its value?
Fixed model: Divide both sides of W = Fd by d: F = W/d = 360/6 = 60 N.
- F = Wd = 2,160 N
- Multiplying by d again does not undo F × d. Divide both sides by d to isolate F.
- F = d/W ≈ 0.017 N
- This inverts the ratio. From W = Fd, force is work divided by distance.
- F = W − d = 354 N
- Subtraction does not undo multiplication. Work and distance also cannot be directly subtracted.
Diagnostic responses remain in memory during navigation and are not saved as student records. Optional product feedback is stored separately and does not contain assessment responses. This exploration supports a next learning step; it does not replace classroom evidence or a formal assessment.