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PRAXISCode: 7813🎓 ~80% Specialized CKT QuestionsSubtest of 7811 · 85 min⚠ Retiring August 2028

Praxis® Elementary Education:
Mathematics CKT (7813)
Practice Test & Study Guide

A specialist-level mathematics exam — 52 questions in 85 minutes across four content categories. Approximately 80% of questions assess Content Knowledge for Teaching (CKT): how teachers apply mathematics knowledge to the work of instruction. This is a subtest of the Elementary Education: Content Knowledge for Teaching (7811) assessment. Four-function calculator provided. Retiring August 2028.

45
Questions
85 min
Time limit
Varies
Passing score*
4
Content categories
~80%
Specialized CKT
4-func
Calculator provided
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Approximately 80% of all questions test specialized Content Knowledge for Teaching (CKT) — not just whether you know elementary mathematics. CKT is the specialized professional knowledge teachers use specifically in the work of teaching. Per ETS, it is “not knowledge that students are expected to learn, nor is it general knowledge of classroom management strategies or learning theory; it is content knowledge specialized to the work of teaching.” CKT questions on the 7813 require candidates to: interpret a student's mathematical error to identify the specific misconception; evaluate a student's explanation for validity and generalizability; select the most appropriate representation or manipulative for a specific concept; identify which student work samples demonstrate the same underlying reasoning. The CKT framework was developed through a partnership between ETS and TeachingWorks at the University of Michigan.

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An on-screen four-function calculator is provided — but since ~80% of questions test teaching knowledge, most don't benefit from calculator use. The four-function calculator is available throughout the 85-minute exam. Use it for the minority of questions requiring precise numerical computation. The vast majority of 7813 questions test whether you can identify what student errors reveal, select appropriate representations, evaluate explanations, and understand why mathematical procedures work — tasks where the calculator adds no advantage. Deep conceptual mathematical understanding is what the exam primarily tests. No personal calculators are permitted.

Elementary Education: Mathematics CKT (7813) — Test at a Glance

Key facts confirmed from the official ETS 7811 Study Companion. The 7813 is one of four separately timed subtests of the 7811.

Test code
7813
Subtest of 7811
Total questions
52
47 one-pt + 5 two-pt
Time limit
85 min
Independently timed
CKT emphasis
~80% of all questions
Specialized teaching knowledge
Category I
Counting & Whole Num Ops
30% · ~16 questions
Category II
Place Value & Decimals
25% · ~13 questions
Category III
Fractions, Ops & Ratios
25% · ~13 questions
Category IV
Equations, Measurement & Geometry
20% · ~10 questions

About the Praxis Elementary Education: Mathematics CKT (7813)

Understanding what CKT means — and why this exam is fundamentally more rigorous than a standard mathematics content test.

The Elementary Education: Mathematics CKT (7813) is a subtest of the Elementary Education: Content Knowledge for Teaching (7811) assessment. All four subtests of the 7811 — Reading and Language Arts (7812), Mathematics (7813), Science (7814), and Social Studies (7815) — measure Content Knowledge for Teaching (CKT): the specialized professional knowledge used specifically in the work of teaching.

The 7813 contains 52 questions (47 one-point and 5 two-point questions) administered in 85 minutes. Approximately 80% of questions measure specialized CKT — applying mathematics content to specific teaching tasks. The remaining ~20% measure standard content knowledge of elementary mathematics itself. An on-screen four-function calculator is provided.

The CKT framework was developed through a partnership between ETS and TeachingWorks at the University of Michigan, grounded in over 25 years of research on the mathematical knowledge needed to teach effectively. The 7813 is part of the broader 7811 series that retires in August 2028, being replaced by the new Elementary Education Fundamentals series. Always verify your state's current requirement at ets.org/praxis/states.

Content Knowledge for Teaching (CKT) — 10 Teaching Tasks Tested

~80% of all questions require applying mathematics content to these teaching tasks. The 7813 also includes 6 additional mathematics-specific CKT tasks beyond the standard 10.

CKT Task 1
Evaluating Texts and Materials
Evaluating mathematical texts, examples, representations, and tasks for how well they support a specific instructional goal
CKT Task 2
Creating and Modifying Materials
Creating and modifying examples, representations, and tasks to support particular instructional goals — including differentiation
CKT Task 3
Analyzing Language Systems
Analyzing mathematical language systems — understanding how mathematical notation and vocabulary affect student learning
CKT Task 4
Explaining and Demonstrating
Selecting the most accurate and mathematically accessible explanation or demonstration for a specific concept
CKT Task 5
Evaluating Explanations
Evaluating student mathematical explanations for validity, generalizability, mathematical precision, and completeness
CKT Task 6
Evaluating Strategies and Activities
Evaluating instructional strategies and activities for their effectiveness in eliciting, developing, or assessing student mathematical thinking
CKT Task 7
Evaluating Student Work — Strengths
Evaluating student mathematical work to identify specific strengths as well as areas for growth
CKT Task 8
Classifying Student Development
Evaluating student work to classify students' level of mathematical development on a developmental continuum
CKT Task 9
Identifying Misconceptions
Analyzing student work to identify patterns of thinking, misconceptions, and partial conceptions — the most frequently tested CKT task
CKT Task 10
Responding to Student Errors
Responding to student mathematical work to target the specific content issue in need of instructional attention
Mathematics-specific CKT tasks (unique to 7813):The 7813 also includes six additional tasks specific to mathematical reasoning — evaluating student conjectures for validity and generalizability; generating counterexamples to student conjectures; selecting and creating representations or manipulatives for a mathematical purpose; interpreting how a student's error would replicate across similar problems; and identifying tasks where student work that seems correct might actually mask incorrect mathematical thinking. These tasks make the 7813 particularly demanding.

Four Content Categories — Weighted Distribution

All four categories include both standard content knowledge (~20% of total) and specialized CKT questions (~80% of total). Counting and Whole Number Operations is the largest at 30%.

I. Counting & Whole Number Operations ~16 questions
30%
II. Place Value & Decimals ~13 questions
25%
III. Fractions, Operations with Fractions & Ratios ~13 questions
25%
IV. Early Equations & Expressions, Measurement & Geometry ~10 questions
20%

Official Exam Blueprint: 4 Content Categories

All content categories and question distributions confirmed from the official ETS 7811 Study Companion. ~80% of questions in every category test CKT — applying mathematics to teaching tasks.

Category I — 30%
Counting and Operations with Whole Numbers
Counting and skip counting 0–1,000; counting on from any whole number; connecting counting to cardinality (the last number counted represents the total quantity — a critical conceptual milestone, not just a rote skill); one-to-one correspondence between numbers and objects; subitizing — perceptual (recognizing 1–3 at a glance) and conceptual (recognizing 4–5 by parts); understanding that sets with higher counts are larger. All four operations with whole numbers: representations using manipulatives, drawings, and diagrams; connecting representations to expressions and equations; real-world problem situations for operations; properties of operations (commutative, associative, distributive); basic number theory (prime/composite, factors, multiples). CKT emphasis: diagnosing why a student miscounts or makes systematic errors in multi-digit addition/subtraction; evaluating whether a representation accurately models a multiplication situation; identifying how an error in a student's counting strategy would replicate; selecting the appropriate manipulative for teaching the cardinality principle.
30%
~16 questions
Category II — 25%
Place Value and Decimals
Conceptual understanding of digit value — not just "the 3 is in the tens place" but why the positional system works as it does; ten-times relationships between adjacent places (understanding powers of 10); comparing multidigit and decimal numbers; rounding multidigit and decimal numbers to any place; composing and decomposing multidigit numbers — understanding why grouping (regrouping) and ungrouping are mathematically helpful in performing operations, not just that they work. Using drawings, base-10 block representations, and number line representations to model place value; connecting representations to numerical equations and written descriptions. CKT emphasis: identifying what a student's written work reveals about their place value understanding vs. a procedural error only; selecting the best manipulative (base-10 blocks, place value charts, hundred charts) for a specific conceptual goal; evaluating whether a student's explanation of regrouping demonstrates conceptual vs. procedural understanding; diagnosing errors in decimal comparison or ordering.
25%
~13 questions
Category III — 25%
Fractions, Operations with Fractions, and Ratios
Fractions as part-whole relationships, as multiples of unit fractions, as numbers on a number line, as the result of division (a÷b = a/b), and as ratios — moving flexibly among all five conceptualizations; equipartitioning as the foundational precursor to fraction understanding; characteristics of fractions less than, equal to, and greater than 1; fraction equivalence (why equivalent fractions represent the same quantity); strategies for comparing fractions (benchmark fractions, common denominators, unit fraction reasoning). All four operations with fractions and mixed numbers — with conceptual understanding of why each works: why adding fractions requires a common denominator (same-size parts); why multiplying by a fraction less than 1 gives a smaller product; why dividing by a fraction less than 1 gives a larger quotient; applications of operations on fractions (scaling). Ratios as a relationship between two quantities; ratio tables; unit rates. CKT emphasis (most conceptually demanding area): evaluating whether a student's explanation of fraction division is valid; identifying what a student's area model for fraction multiplication reveals about their conceptual understanding; selecting a representation that correctly models a specific fraction comparison; identifying the specific misconception in a student who always cross-multiplies when comparing fractions.
25%
~13 questions
Category IV — 20%
Early Equations and Expressions, Measurement, and Geometry
Equivalence — what it means for algebraic terms, expressions, and equations to be equivalent; the equal sign as relational equivalence (not as "get the answer" — one of the most important and frequently misunderstood elementary algebra concepts); using equations to represent real-world and mathematical situations; solving equations using all four operations; order of operations including parentheses and the distributive property; two interpretations of variable — as an unknown quantity (solving) and as a varying quantity (understanding functions); independent and dependent variables. Measurement: recognizing which attributes are measurable and selecting appropriate units; solving problems involving length, area, perimeter, volume, surface area, time, money, and mass; unit conversions. Geometric figures and their properties: classifying 1D, 2D, and 3D figures; angles (types and measures); polygons and solids classified by attributes; properties of specific quadrilateral types and their hierarchy. CKT emphasis: identifying what students understand about the equal sign as relational equality vs. procedural "do the operation"; evaluating whether a student's conjecture about geometric shapes is valid; selecting an appropriate real-world measurement context for teaching a specific measurement concept.
20%
~10 questions

Key Topics and CKT Applications by Category

What the CKT framework looks like in practice for each of the four mathematics content categories — the specific mathematical tasks of teaching the exam tests.

Counting & Whole Numbers

Category I — Most Heavily Tested CKT Topics

30% · ~16 questions
Subitizing — perceptual vs. conceptual: perceptual subitizing (recognizing 1–3 items immediately without counting) vs. conceptual subitizing (perceiving a larger set by recognizing smaller subsets — e.g., seeing 7 as 4+3 by arrangement). CKT questions test whether you can identify which type of subitizing a task develops, recognize it in student responses, and select appropriate activities to build it
Cardinality — the most commonly tested counting CKT concept: understanding that the last number said when counting a set represents the total quantity (cardinality principle) is a conceptual milestone — not all children who can count to 10 understand this. CKT questions ask you to identify whether a student's response shows understanding of cardinality, how to assess it, and which tasks best reveal it
Identifying and interpreting student counting errors: one-to-one correspondence errors (saying more or fewer number words than objects); sequence errors (saying numbers out of order); cardinality errors (not understanding the last-number-said represents the total); recounting errors (counting already-counted objects). CKT questions present student counting behavior and ask you to identify the specific error type
Evaluating multi-digit computation representations: given a student's algorithm or drawn model for multi-digit addition, subtraction, multiplication, or division — evaluate whether it is mathematically valid; determine whether it would generalize to all numbers or only to specific cases; identify what misconception it reveals; select a more effective representation for the target understanding
Properties of operations — conceptual application: knowing the commutative, associative, and distributive properties well enough to evaluate whether a student's use of them is correct; identify which property justifies a specific equivalence; select a real-world context that naturally illustrates a specific property; evaluate whether a student's conjecture about properties is valid or identifies a counterexample
Place Value & Decimals

Category II — Most Heavily Tested CKT Topics

25% · ~13 questions
Conceptual vs. procedural place value understanding: a student can correctly regroup in subtraction without understanding WHY it works conceptually. CKT questions present student work and ask you to distinguish procedural fluency from conceptual understanding; select representations that develop conceptual understanding specifically; identify what a student's verbal explanation reveals about their understanding level
Regrouping — why it works conceptually: "borrowing" in subtraction and "carrying" in addition are procedural labels that can mask the underlying mathematics. CKT questions test whether you understand that regrouping works because of the base-10 structure — you can always exchange ten of one unit for one of the next-larger unit. You need to be able to evaluate whether a student's explanation of regrouping is mathematically accurate at this conceptual level
Decimal comparison misconceptions: common student errors include comparing decimals as if they were whole numbers (believing 0.25 > 0.5 because 25 > 5) or believing longer decimals are always larger. CKT questions ask you to identify the specific misconception, predict how it would appear in other decimal comparison problems, find multiple student work samples demonstrating the same error, and select a remediation approach
Base-10 representations — affordances and limitations: base-10 blocks naturally model place value up to hundreds; place value charts extend to larger numbers; number lines show relative magnitude; expanded form reveals additive structure. CKT questions ask you to evaluate which representation best supports a specific learning goal and identify the mathematical limitations of each representation (e.g., base-10 blocks don't naturally show the relationship between thousands and millions)
Fractions & Ratios

Category III — The Most CKT-Demanding Category

25% · ~13 questions
Five conceptualizations of fractions — and why they matter for teaching: part-whole (3/4 of a pizza), number on a number line (3/4 is between 0 and 1), multiple of a unit fraction (3/4 = 3 × 1/4), result of division (3 ÷ 4 = 3/4), and ratio (3:4 or 3 to 4). CKT questions ask you to identify which conceptualization a problem or student response reflects and select representations appropriate for each conceptualization
Equipartitioning — the foundation of fraction understanding: the ability to partition a whole into equal parts (equipartition) is the prerequisite for fraction understanding. CKT questions test whether you can identify when a student's drawing shows unequal partitioning, evaluate whether equipartitioning activities are appropriate for a developmental stage, and select activities that develop this foundational concept before formal fraction instruction
Fraction operation CKT — why procedures work: the most demanding area. CKT questions test your ability to evaluate whether a student's explanation of why the invert-and-multiply rule works is conceptually valid (does it appeal to the meaning of division, or is it just procedural?); evaluate why adding numerators AND denominators is incorrect (same-size parts required); evaluate whether a student's area model for fraction multiplication accurately represents the operation
Identifying fraction misconceptions and their replication: common fraction misconceptions include: not using the same whole when comparing fractions; treating fractions as two separate whole numbers; incorrectly adding denominators; assuming multiplication always gives a larger result; assuming division always gives a smaller result. CKT questions present a student error and ask you to identify other problems where the same error would appear or would NOT appear
Evaluating student conjectures about fractions: a student says "when you multiply two fractions, the product is always smaller than both fractions." Is this always true? (Only when both fractions are less than 1.) A student says "the bigger the denominator, the smaller the fraction." Is this always true? (Only when comparing unit fractions with the same numerator.) CKT questions ask you to evaluate the validity, identify the domain of truth, and find appropriate counterexamples
Equations & Geometry

Category IV — Early Equations, Measurement, and Geometry CKT Topics

20% · ~10 questions
The equal sign as relational equivalence — the most critical algebra concept: research shows that most elementary students interpret the equal sign as "get the answer" rather than as a symbol of relational equivalence. A student who writes 4 + 3 = 7 + 5 = 12 has an "answer" misconception. CKT questions test whether you can identify this misconception from student work, evaluate whether a student's response demonstrates relational understanding, and select instructional tasks that specifically develop relational understanding of the equal sign
Variable — two interpretations and why they matter: variable as unknown (solve for x in 3x = 12) vs. variable as quantity that varies (understand y = 2x as a relationship). Most elementary students are only exposed to the first interpretation. CKT questions test whether you understand both interpretations, can identify which a student is using, and can select tasks that introduce variable as a varying quantity appropriate for elementary learners
Evaluating student conjectures about geometric shapes: a student says "all squares are rectangles but no rectangles are squares." Evaluate this. A student says "you can tell a shape is a parallelogram because it leans." Evaluate this. CKT questions present student statements about geometric properties and ask you to evaluate whether they are mathematically valid, identify the specific misconception, and select a counterexample or corrective activity
Quadrilateral hierarchy — a frequently tested CKT area: the hierarchy of quadrilaterals (square → rectangle → parallelogram → quadrilateral) is one of the most commonly misunderstood geometric classification topics. CKT questions test whether you understand that all squares are rectangles (because they have all rectangle properties plus equal sides), identify student misconceptions about these relationships, and evaluate whether a definition is sufficiently precise to correctly classify shapes

Registration, Test Day & Scoring

Everything you need to know before and on exam day for this independently timed subtest.

Registration

Where to registerpraxis.ets.org
Subtest of7811 (can test independently)
Testing formatsIn-person or remote
Arrive (in-person)30 min early

Scoring

Score typeScaled score
Wrong answer penaltyNone
Passing scoreVaries by state
Two-point questions5 questions worth 2 pts each
Results available~5 weeks post-test

Test Day

Time limit85 minutes (independently timed)
CalculatorFour-function (on-screen) ✓
Personal calculatorNot permitted
~80% of questionsRequire CKT application

Retirement Timeline

7813 retiresAugust 2028
Parent series (7811)Also retiring August 2028
New Fundamentals series8002–8006 (live March 2026)
Verify your stateets.org/praxis/states

Passing Score Requirements by State

Passing scores are set individually by each state or licensing agency for each subtest.

Important: Before registering for the 7813, verify your state still requires it at ets.org/praxis/states. The 7813 is being replaced by the new Fundamentals series (8002–8006). Some states have already transitioned.

If you pass three subtests of the 7811 but fail one, you may retake only the failed subtest independently. There is no penalty for incorrect answers — always answer every question. The 5 two-point questions are worth more — prioritize them accordingly.

How to Prepare for the Praxis Elementary Education: Mathematics CKT (7813)

Strategies for the most rigorous elementary mathematics assessment — where knowing the math is necessary but not sufficient. You must also know how to apply mathematics knowledge to professional teaching tasks.

  • ~80% of questions test specialized CKT — studying mathematics content alone is not enough preparation. This is the defining insight about the 7813. For every major topic you study, you must also ask: What are the common student misconceptions? How would I identify a misconception from student work? What would a valid vs. invalid student explanation look like? What representation or manipulative best supports this specific mathematical concept at this developmental level? What task would elicit a particular type of student thinking? Practicing content in the context of these teaching questions is the only way to genuinely prepare for the 7813.
  • Category III (Fractions, Operations with Fractions, and Ratios) is the most conceptually demanding — prepare it with the greatest depth. Per ETS, the fraction CKT questions are the hardest on the 7813 because they require knowing not just how to work with fractions but why each operation works conceptually. Know all five conceptualizations of fractions and which representations support each one. Know why the invert-and-multiply rule for fraction division works conceptually. Know the most common fraction misconceptions and be able to identify them in student work and predict how they would replicate.
  • Know the two mathematics-specific CKT tasks that are unique to the 7813 — they are the hardest question types on the exam. Task 15: given a student's mathematical error, anticipate how it would replicate across similar problems and identify multiple student work samples demonstrating the same underlying incorrect reasoning. Task 16: identify tasks where student work that seems mathematically correct might actually mask incorrect thinking. Both require deep mathematical understanding AND the ability to think from a student's perspective — the most demanding combination on any Praxis exam.
  • Category I (Counting and Whole Number Operations) is the largest at 30% — go deeper than you think you need to on subitizing, cardinality, and counting development. Most candidates over-prepare multi-digit computation and under-prepare the foundational concepts of counting development (subitizing, one-to-one correspondence, cardinality, counting principles). Know the difference between perceptual and conceptual subitizing. Know the five counting principles. Know how to identify specific counting errors from student behavior. These developmental concepts are exactly what the CKT framework targets most heavily in Category I.
  • The equal sign as relational equivalence (Category IV) appears on nearly every 7813 — it's the most commonly tested elementary algebra CKT concept. Research shows that most elementary students interpret the equal sign as "the answer comes next." The 7813 tests whether you can identify this misconception from student work (e.g., 4 + 3 = 7 + 5 = 12), evaluate whether a student's response demonstrates relational understanding, and select instructional tasks that specifically develop relational understanding of the equal sign. Know this concept in depth — it bridges arithmetic and algebra and is one of the most important foundational mathematical concepts for elementary teachers to understand.
  • Download the official ETS 7811 Study Companion and work through all sample questions and discussion topics specifically for the 7813 subtest. The Study Companion is the most realistic CKT practice available. For the mathematics subtest specifically, work through all discussion questions that ask you to evaluate student explanations, identify what errors reveal, generate counterexamples, and select appropriate representations. These are the exact cognitive tasks the exam requires. Also note: the five two-point questions on the 7813 are worth more than the one-point questions — practice under timed conditions (85 minutes for 52 questions = approximately 98 seconds per question) so you can allocate more time to the two-point items.

Frequently Asked Questions

Answers sourced from the official ETS Praxis Elementary Education: Content Knowledge for Teaching (7811) Study Companion.

How many questions are on the Praxis Elementary Education: Mathematics CKT (7813)?
The exam contains 52 questions (47 one-point and 5 two-point questions) with an 85-minute time limit. Questions span four categories: Counting and Operations with Whole Numbers (30%, ~16q), Place Value and Decimals (25%, ~13q), Fractions, Operations with Fractions, and Ratios (25%, ~13q), and Early Equations and Expressions, Measurement, and Geometry (20%, ~10q). An on-screen four-function calculator is provided.
What is Content Knowledge for Teaching (CKT) on the Praxis 7813?
CKT is specialized professional content knowledge used in the work of teaching — approximately 80% of questions require applying mathematics to teaching tasks: identifying student misconceptions; evaluating explanations for validity; selecting representations and manipulatives; generating counterexamples; identifying how errors replicate across problems. The CKT framework was developed through a partnership between ETS and TeachingWorks at the University of Michigan, grounded in over 25 years of research.
What is the difference between the Praxis 7813 and 7003?
The 7003 (Elementary Education: Mathematics) is a content knowledge test with 40 questions — it tests whether candidates know elementary mathematics. The 7813 (Mathematics CKT) is a specialist-level assessment with 52 questions — approximately 80% of questions require applying mathematics to professional teaching tasks (identifying misconceptions, evaluating explanations, selecting representations). The 7813 is considered the more rigorous of the two.
Is a calculator provided on the Praxis 7813?
Yes. An on-screen four-function calculator is provided for the Mathematics CKT (7813) subtest. However, since ~80% of questions test specialized teaching knowledge, most questions don't require or benefit from calculator use. Deep conceptual mathematical understanding is what the exam primarily tests. No personal calculators are permitted.
Is the Praxis 7813 being retired?
Yes. The 7813 is a subtest of the Elementary Education: Content Knowledge for Teaching (7811) series, which retires in August 2028. It is being replaced by the new Elementary Education Fundamentals series (8002–8006), launched March 9, 2026. Both are currently live. Always verify your state's current requirement at ets.org/praxis/states before registering.
Can I retake only the 7813 if I fail it but passed the other 7811 subtests?
Yes. The four subtests of the 7811 are independently scored and can be retaken independently. If you pass 7812, 7814, and 7815 but fail the 7813, you can register for and retake only the 7813. There is no penalty for wrong answers — always answer every question. The five two-point questions count more than the one-point questions.
What is the passing score for the Praxis Elementary Education: Mathematics CKT (7813)?
Passing scores vary by state or licensing agency. Always verify the specific requirement for your state at ets.org/praxis/states before registering. Verify also that your state still requires the 7813 rather than a test from the new Fundamentals series.

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Sources: ETS Praxis Elementary Education: Content Knowledge for Teaching (7811) Study Companion (official PDF, praxis.ets.org/on/demandware.static/-/Library-Sites-ets-praxisLibrary/default/pdfs/7811.pdf); ETS official 7811 test page; TeachingWorks at the University of Michigan — High-Leverage Teaching Practices; Ball, D. L., Thames, M. H., & Phelps, G. (2008). "Content Knowledge for Teaching: What Makes It Special?" Journal of Teacher Education 59(5), 389–407. Praxis® is a registered trademark of ETS. This site is not affiliated with or endorsed by ETS or TeachingWorks. Passing score requirements vary by state — always verify at ets.org/praxis/states.
Last Updated: May 22, 2026