25.3 Assessing Mathematics
Key Takeaways
- PECT 0012.3 applies SAS assessment types — screening, diagnostic, formative, summative, and benchmark — to mathematics, then uses results to plan, modify, differentiate, and accommodate.
- Math evidence should include interviews, observation, and work samples, not only multiple-choice items; 'how did you get that?' is often the diagnostic question.
- Error analysis separates a wrong fact (6×7=43) from a wrong strategy (subtracting the smaller digit from the larger in each column) so reteaching matches the break.
- The PECT Module 3 on-screen calculator is a test tool; in the classroom, calculators and other tools are appropriate when they match the goal, not when mental number sense is the learning target.
- Accommodations (extra time, read-aloud of a word problem, manipulatives, scribe) keep the math intent; sending a child to the hall is not an accommodation.
25.3 Assessing Mathematics
PECT lens: Objective 0012.3 tests assessment in mathematics: selecting appropriate assessments for different purposes (screening, diagnosis, benchmark, formative, summative); interpreting results; and using those results to plan, modify, and differentiate instruction and to make accommodations. Module 3 remains 45 selected-response items, 90 minutes, with a standard on-screen calculator. That calculator is a test accommodation of the exam format, not a claim that every kindergarten task should be calculator-first.
SAS — Pennsylvania's Standards Aligned System — still names the types you learned in 0002.1. 0012.3 is those types aimed at mathematics. A color-coded spreadsheet that never changes a small group, a manipulative, or a reteach is not yet assessment in the PECT sense.
Match the math decision to the type
| Type | Math question it can answer | PreK–4 snapshot | What it is not |
|---|---|---|---|
| Screening | Who might need a closer look at number, counting, or operations? | Brief universal fall measure of counting, quantity, or early operations | A disability diagnosis or a report-card grade |
| Diagnostic | Which counting principle, fact, or strategy is missing, and why is this hard? | Interview: Count these. How many? How did you see 8? | A two-minute whole-class thumbs-up |
| Formative | What should I teach in the next number talk or small group? | Whiteboard 8+5; anecdotal note that three children counted all | |
| Summative | Did they meet the taught PA Core / ELS math standard after instruction? | Unit rubric on composing 10; Grade 3–8 PSSA mathematics | Daily checks that still change this week's teaching |
| Benchmark | Is the grade on track toward year-end math standards at this checkpoint? | Fall/winter/spring standards-aligned math pulse | A sticky-note during centers |
| Authentic (method) | Can they use the math in a real task? | Fair-share snack; measure the garden bed | A packet that never asks children to quantify |
Screening is a gate. Good math screeners are brief, given to everyone, and sensitive enough to flag children who may need more. A child below a concern range is not thereby labeled with a math disability. Next comes diagnosis: a counting-principles probe, a place-value interview, a look at work samples, a talk with families about home-language number words. Universal screening includes children who already have IEPs and children who look typically developing.
Diagnostic math assessment is narrow and deep. It should name a skill, not a vibe: one-to-one not yet coordinated on sets of eight; counts on for addition to 10; subtracts smaller-from-larger by place. Classroom Diagnostic Tools on SAS exist in eligible grades; teachers also diagnose with interviews, running records of problem solving, and targeted probes. Diagnostics are not universal screeners and not daily formative checks.
Formative math assessment happens during instruction and changes teaching. The same interview is formative if you pull a composing-8 group tomorrow. It is wasted if you only file it. Summative math assessment judges a period of teaching against the standard. PSSA mathematics is a statewide summative in grades 3–8; PreK, K, 1, and 2 do not take the PSSA. Young children still have summative evidence: a rubric, a portfolio of dated work, an end-of-unit performance task. Benchmark math is a periodic standards checkpoint, often three times a year — not tomorrow's exit ticket.
Interviews, observation, and work samples — not only multiple choice
Selected-response items can sample facts and some procedures. They hide strategy. PECT 0012.3 expects you to know that mathematics assessment in PreK–4 is multi-method.
| Method | What it uniquely shows | Limit |
|---|---|---|
| Interview (How did you get that? Show me.) | Strategy, counting principles, language, whether a right answer was luck | Time; requires a focus question |
| Observation (centers, games, snack) | One-to-one, fair shares, whether the child uses math without a worksheet | Not systematic if undated and unfocused |
| Work samples / photos | Representations over time; CPA movement | Products without talk can hide counting-all |
| Performance task | Authentic use (measure, share, graph a class question) | Rubric must name the math, not neatness |
| Probe / CBM-style | Repeated brief samples (facts, oral counting) to graph growth | A flat line means change instruction, not repeat the same probe forever |
| Multiple choice | Efficient sample of some skills | Cannot see why; poor as the only PreK–2 measure |
Construct an interview from the standard: a small set of objects, count them, how many?, what if I hide two?, how did you know? Record what the child does, not a character label. Observation in the block area ("Maya gave each bear one plate, then said three") is math evidence when it is dated and tied to a principle.
Error analysis: wrong fact versus wrong strategy
A mark of wrong is not an instructional plan. Error analysis asks what kind of miss it is so reteaching matches.
| Error type | Example | What to teach next |
|---|---|---|
| Wrong fact | 6 × 7 = 43; 8 + 5 = 12 | Targeted fact work after meaning (arrays, derived facts) — not only more speed sheets |
| Wrong strategy / procedure | 51 − 27 = 36 because the child subtracts the smaller digit from the larger in each column (1 from 7, 5 from 2) | Place-value subtraction, regrouping with concrete then pictorial models |
| Conceptual | Writes 6 + 4 = 10 + 5 = 15, treating equal as "put the next answer" | Meaning of equal as same amount; balance contexts |
| Counting-principle miss | Recites past the set; cannot cardinalize | One-to-one and cardinality on small sets |
| Language / misread | English learner computes the wrong operation because fewer was unknown | Preview language; keep the math task with objects |
| Careless / recording | Correct oral answer, reversed digits | Recording supports; do not reteach the entire operation |
Exam gold: 51 − 27 = 36 with an interview that reveals smaller-from-larger is a strategy error. Flashcards of 51 − 27 will not fix it. Base-ten blocks or drawings that force regrouping might. Conversely, 7 × 8 = 54 with a correct array setup is more likely a fact miss. Matching the reteach to the error is using assessment to modify instruction.
Interpret, then plan, modify, differentiate, accommodate
A useful math-assessment loop:
- Clarify the standard (PA Core or ELS Mathematical Thinking).
- Choose the type that answers the decision you actually have.
- Collect more than one source when the decision is high-stakes (screener + interview + work sample).
- Interpret with error analysis, language, and access in mind.
- Act — regroup, reteach the missing principle, change the representation, extend.
- Recheck (formative or a later probe) to see whether the action worked.
Accommodations change how the child shows the math without changing the target. Typical PreK–4 math accommodations (must match the IEP or 504 when those exist): extra time; separate setting; read-aloud of directions and word problems (you are not reading the numeral the child must identify if reading numerals is the goal); scribe; enlarged print or fewer items per page; manipulatives; a multiplication table or calculator when computation is not the measured skill. Modifications change the target (fewer standards, different criterion) and are IEP-level decisions — do not quietly modify by sending the child out.
Sending a student with a disability to the hallway during the math assessment is not an accommodation. Neither is scoring only neatness, nor using a norm-referenced battery to claim mastery of this week's taught standard (that remains a criterion-referenced job from 0002.2).
Tools: Module 3 calculator versus number-sense goals
PECT Module 3 provides a standard on-screen calculator. Candidates may use it on the exam. That fact is logistics. It is not a teaching rule that PreK–4 fluency should be outsourced to a device.
Classroom decision rule:
| Goal of the lesson or assessment | Tool use |
|---|---|
| Counting principles, subitizing, composing small numbers | Fingers, objects, ten-frames — not a calculator for 3+2 |
| Mental strategies, number talk | No calculator; the thinking is the work |
| Multi-digit computation after sense is built, or a rich problem where the target is modeling, not arithmetic | Calculator or other tool can be appropriate |
| IEP accommodation when calculation is not the skill being measured | Calculator, table, or scribe as written |
| PECT Module 3 itself | On-screen calculator is available; it does not change what 0012.1–0012.3 ask you about children |
If a stem says the learning target is number sense or counting on, the child who is handed a calculator for 4+3 is in the wrong condition. If the target is a Grade 4 multi-step application and calculation is not the measured skill, a calculator can be a legitimate tool or accommodation. Teach children when a tool helps and when the mind is the tool.
Scenario: Grade 1 fall screener
Ms. Lee gives every child a brief counting-and-quantity screener. Four children flag. She does not call families to announce a disability. She runs a diagnostic interview (one-to-one, cardinality, compare two sets), watches snack, and learns one child was exhausted, one is an English learner who counts accurately in Spanish, and two need a small-group cardinality cycle. Screening found them; diagnosis explained them; instruction changed.
Scenario: interview beats the bubble (Grade 2)
On a multiple-choice item, both children mark 15 for 8+7. An interview shows one child made 10 and added 5; the other counted from 1 on fingers and miscounted yesterday but guessed today. Formative next steps differ. The bubble alone could not tell.
Scenario: error analysis (Grade 3)
Devon writes 51 − 27 = 36. Mr. Alvarez asks show me. Devon always subtracts the smaller digit from the larger. That is a strategy error rooted in place value, not a missing 51 − 27 fact card. Reteach with base-ten blocks and a drawing of regrouping. A calculator would hide the strategy if the goal is to learn regrouping; later, a calculator might be fine when the goal is a story problem about leftover seats.
Scenario: PSSA versus PreK (summative grain)
A principal asks for "PSSA-style bubbles" in kindergarten so the school will be ready. 0012.3: K does not take the PSSA. Kindergarten summative math is a performance task and dated work samples against ELS/PA Core K counting and cardinality. Grade 3 does take PSSA mathematics; that statewide summative still does not replace unit evidence or daily formative checks.
Scenario: accommodation that keeps the target (Grade 4)
An IEP lists extra time, a scribe, and read-aloud of word problems. During a fractions concept check (equal shares of the same whole), the teacher reads the story, allows fraction strips, and scribes the explanation. The child still must show equivalent shares. A hallway coloring sheet would have dropped the target. A calculator is irrelevant to this particular concept check.
Exam traps for 0012.3
- Screener = diagnosis = IEP.
- Only multiple choice counts, especially in PreK–2.
- Treating every miss as a missing fact (strategy errors need models, not only flashcards).
- PSSA as the only math assessment that matters; or giving PSSA-style batteries in kindergarten because Grade 3 is tested.
- Calculator for 3+2 when the goal is counting or composing.
- Exclusion as accommodation.
- NRT percentile as mastery of this week's PA Core standard.
- Collecting data that never change grouping, representation, or reteaching.
If you can say, "This tool answers this math decision, the error is this kind, and tomorrow I will teach this, with this access support," you are inside 0012.3. If you only have a score, you are not finished assessing.
Every Grade 1 student completes a brief fall math measure. Several score below the concern range. What is the most appropriate 0012.3 next step?
A Grade 3 student writes 51−27=36. An interview shows the student subtracted the smaller digit from the larger in each column. What is the best 0012.3 interpretation?
Which statement about tools and accommodations best matches 0012.3 and PECT Module 3?