In this guide
Two children both write 42 − 17 = 35. A test that marks only the answer records two identical wrong answers. But ask them how they got it and you may find that one subtracted the smaller digit from the larger in each column, while the other correctly worked out 12 − 7 = 5 but forgot to reduce the 4 tens to 3. The first has not understood regrouping and needs it re-taught with materials; the second understands the idea and needs a procedural reminder, and perhaps one look at the bundles to see why the tens go down. Evaluation in mathematics is about finding that difference.
The CTET syllabus heading for both papers is "Evaluation through formal and informal methods", with "diagnostic and remedial teaching" close behind. Expect questions that describe what a teacher does and ask what kind of assessment it is, which question best tests understanding, or what the teacher should do with the result.
Why assess in mathematics?
| Purpose | Also called | Question it answers | Example in maths |
|---|---|---|---|
| Assessment for learning | Formative | What should I teach next, and how? | Watching children regroup with bundles and sticks |
| Assessment of learning | Summative | What has the child learnt by the end? | End-of-term test on fractions |
| Assessment as learning | Self-assessment | How am I doing, and what should I do next? | A child checks their own answers against an estimate |
| Diagnostic assessment | — | Exactly where is the difficulty? | A test that separates subtraction with and without regrouping |
The Right of Children to Free and Compulsory Education Act, 2009, in Section 29(2)(h), asks for comprehensive and continuous evaluation of the child's understanding of knowledge and ability to apply it. The NCF 2005 position paper on teaching mathematics criticised crude assessment that makes maths look like mechanical computation, and asked for assessment of mathematisation: reasoning, problem solving and communication, not only procedures. For the general theory, see assessment for, of and as learning.
Formal and informal methods
| Formal | Informal |
|---|---|
| Tests, exams, worksheets, quizzes | Observation, questioning, discussion, games |
| Planned, with a fixed time and format | Part of everyday teaching |
| Mainly summative, but can be diagnostic | Mainly formative |
| Produces marks or grades | Produces notes, impressions and next steps |
| Can cause anxiety if overused | Low stress; children may not notice they are being assessed |
Neither is better in every case. A good teacher uses informal methods daily and formal ones sparingly, and uses both to plan teaching.
What to assess in mathematics
- Concepts: Does the child understand what 3/4 or place value means?
- Procedures: Can the child carry out the steps accurately?
- Reasoning: Can the child explain why a method works, or spot a wrong claim?
- Problem solving: Can the child use maths in an unfamiliar situation?
- Communication: Can the child explain a solution in words, pictures or symbols?
- Disposition: Is the child confident, curious and willing to try?
A test made only of "compute this" items assesses the second point and nothing else.
Tools for evaluation in maths
- Open-ended questions with more than one correct answer.
- Observation with a checklist or anecdotal notes, while children work with materials.
- Oral questioning and short interviews: "How did you get that? Can you do it another way?"
- Maths journals, where children write or draw how they solved a problem.
- Portfolios, collections of a child's work over time that show growth.
- Projects, such as surveying the class and presenting the data.
- Rubrics that describe levels of performance, used for problem solving and projects.
- Self-assessment and peer assessment, with simple criteria children understand.
- Well-designed multiple-choice items, where each wrong option matches a known error.
Closed and open questions
| Closed | Open-ended |
|---|---|
| What is 6 × 4? | Find two numbers whose product is 24. How many pairs can you find? |
| Find the area of a 6 cm × 4 cm rectangle. | Draw two different rectangles with an area of 24 cm². |
| Find the mean of 3, 5 and 7. | Write five numbers whose mean is 5. |
| Is 0.4 greater than 0.35? | Write three decimals between 0.3 and 0.4 and explain how you know. |
Whole-number answers to the first open question include 1 × 24, 2 × 12, 3 × 8 and 4 × 6, and a child who also offers ½ × 48 is showing something a closed question would never reveal.
A diagnostic multiple-choice item
"Which number is the greatest? (a) 0.5 (b) 0.45 (c) 0.405 (d) 0.54"
The answer is (d), since 0.54 > 0.5 > 0.45 > 0.405. But the wrong options are informative too. A child who picks (c) probably thinks a longer decimal is larger. A child who picks (a) may think fewer decimal places mean a larger number. The item assesses understanding of decimal place value, and the choice of distractor points to the misconception. See fractions and decimals.
A simple problem-solving rubric
| Criterion | 1 (beginning) | 2 (developing) | 3 (secure) |
|---|---|---|---|
| Understanding the problem | Misreads what is asked | Understands part of it | Identifies what is known and what is asked |
| Strategy | No clear plan | A workable plan with gaps | An efficient plan, such as a table, drawing or equation |
| Answer and explanation | Wrong or unexplained | Correct but poorly explained | Correct, checked and clearly explained |
Using the results
- Give descriptive feedback ("You regrouped correctly in the tens; check the hundreds") rather than only a mark.
- Regroup children for re-teaching based on the error, not on overall marks.
- Plan remedial teaching for specific gaps. See errors and remedial teaching.
- Change your own teaching if most of the class shows the same error.
- Avoid labelling or comparing children publicly; it feeds the fear of maths that NCF 2005 warns about.
Scenario questions with reasoning
- Which question best assesses a child's understanding of addition? (a) 25 + 17 = ? (b) Write the table of 5 (c) Show two different ways to find 25 + 17 (d) Copy the sum neatly
- A teacher notes how each child uses bundles of sticks during a place-value activity. This is: (a) informal, formative assessment (b) formal, summative assessment (c) a standardised test (d) a placement test
- The main purpose of a diagnostic test in mathematics is to: (a) rank children (b) select children for competitions (c) award grades (d) identify the specific difficulties of each child
- "Write five numbers whose mean is 5." This is an example of: (a) a closed question (b) an open-ended question (c) a recall question (d) a true-false item
- In a multiple-choice question on comparing decimals, the wrong options are each based on a common error. This helps the teacher to: (a) identify the misconception behind a wrong answer (b) make the test harder (c) confuse children (d) save time
- A child keeps a maths journal describing how they solved problems each week. The journal mainly helps the teacher assess: (a) handwriting (b) speed (c) the child's thinking and communication (d) attendance
- Most of a class makes the same error in dividing fractions. The best action is to: (a) give more tests (b) move to the next chapter (c) blame the children (d) re-examine the teaching approach and re-teach with a different model
Answers and reasoning
- (c). Showing two methods needs understanding of the operation, not just a remembered procedure.
- (a). It happens during teaching and is used to guide the next steps.
- (d). Diagnosis pinpoints the gap so that remedial teaching can target it.
- (b). Many sets of numbers work (5, 5, 5, 5, 5 or 1, 3, 5, 7, 9), so children's choices reveal their understanding of mean.
- (a). Diagnostic distractors turn a wrong answer into information.
- (c). Journals show reasoning and mathematical communication.
- (d). A class-wide error is feedback on the teaching, not only on the children.
What to do next
- Rewrite five closed textbook questions as open-ended ones.
- Write one multiple-choice item on fractions whose three wrong options each match a misconception.
- Make a formal-informal table from memory with three tools in each column.
- Continue with errors and remedial teaching.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Central Board of Secondary Education website .
Get the next CTET guide by email
New guides every week. No spam, unsubscribe any time.