Pariksha SaathiLesson 8 of 8 · Mathematics Pedagogy
Evaluation in Mathematics: CCE, Open-Ended Questions, Rubrics and Maths Anxiety
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Formative and summative
Michael Scriven introduced the terms formative and summative evaluation in 1967. Formative assessment, or assessment for learning, happens during teaching through questions, observation, class work, quizzes and conversation; its purpose is to improve learning by giving feedback to both child and teacher. Summative assessment, or assessment of learning, comes at the end of a unit or term and judges how much has been achieved. Assessment as learning means children assessing their own and their peers' work, which builds reflection. Good feedback in mathematics is descriptive ('your tens are lined up correctly; check the ones column again') rather than only a mark or a tick.
Continuous and comprehensive evaluation
Section 29(2)(h) of the RTE Act 2009 requires comprehensive and continuous evaluation of the child's understanding of knowledge and ability to apply it. Continuous means assessment is spread through the teaching-learning process rather than left to the end; comprehensive means it covers both scholastic areas and co-scholastic ones such as attitudes, interests and working with others. In mathematics, CCE uses many tools: oral questions, observation of children using materials, worksheets, projects, maths journals in which children write how they solved a problem, checklists, anecdotal records, portfolios, and self and peer assessment. NEP 2020 continues this shift from rote-based tests to regular, formative and competency-based assessment, proposes a 360-degree holistic progress card, and proposes a national assessment centre, PARAKH.
Understanding versus procedure
Richard Skemp (1976) distinguished instrumental understanding, knowing a rule and how to use it (rules without reasons), from relational understanding, knowing both what to do and why. Procedural knowledge is the ability to carry out algorithms; conceptual knowledge is knowing the relationships behind them. A test that only asks 'Add 3/4 + 1/8' checks procedure, while 'Which is bigger, 3/4 or 4/5? Show it with a drawing', 'Explain why 0.5 and 1/2 are equal' or 'Write a story for 12 ÷ 3' checks understanding. The Position Paper criticises assessment that rewards only mechanical computation and recommends assessing mathematisation: whether children can formulate a problem, choose a method, reason and explain.
Open-ended questions
A closed question has one correct answer, such as 6 × 4 = ?. An open-ended question has many correct answers or many routes: 'The answer is 24. What could the question be?', 'Find two numbers whose sum is 50 and whose difference is less than 10', or 'Make three different shapes of area 6 square units on a geoboard'. Such questions have a low floor and a high ceiling, so every child can begin and able children can go further; they reveal how children think, encourage several strategies and reduce the fear of the single right answer. They are best scored with a rubric rather than simply marked right or wrong.
Portfolios and rubrics
A portfolio is a purposeful collection of a child's work over time: worksheets, drawings, project reports, problems solved in different ways and the child's own reflections. It shows growth rather than a single performance, and when children choose the pieces and say why, it also builds self-assessment. A rubric is a scoring guide that lists criteria and describes levels of performance for each. For a word problem the criteria might be understanding the problem, choice of strategy, accuracy of calculation and explanation, each described at 4 levels from beginning to excellent. An analytic rubric scores each criterion separately; a holistic rubric gives one overall judgement. Rubrics make marking fair and transparent, and when shared in advance they show children what good work looks like.
Mathematics anxiety
Mathematics anxiety is a feeling of tension and fear that interferes with handling numbers and solving problems; Richardson and Suinn developed the Mathematics Anxiety Rating Scale (MARS) in 1972. It is not built into the subject; it grows largely out of experiences: timed tests and speed drills, public correction and punishment, insistence on one right method, rote learning without understanding, comparison with other children, and adults who say mathematics is not for everyone or that girls are weaker at it. Anxiety uses up working memory, so an anxious child performs below her ability, avoids mathematics and falls further behind.
Reducing anxiety through assessment
The teacher reduces anxiety by using frequent low-stakes formative assessment instead of a few high-stakes tests, valuing the process and different methods, allowing time and oral explanation, treating mistakes as part of learning, using games, group work and real contexts, giving descriptive feedback, praising effort and strategies rather than speed, and avoiding labels and comparison. In CTET, the correct option is usually the one that assesses reasoning through varied tools and makes mathematics less threatening; options that rely only on timed paper-pencil tests of speed and memory are usually wrong.
Key facts
- Formative and summative evaluation: terms introduced by Michael Scriven (1967); formative is assessment for learning during teaching, summative is assessment of learning at the end.
- RTE Act 2009, Section 29(2)(h): comprehensive and continuous evaluation of the child's understanding of knowledge and ability to apply it.
- CCE tools in mathematics: oral questions, observation, worksheets, projects, maths journals, checklists, anecdotal records, portfolios, self and peer assessment.
- Skemp (1976): instrumental understanding is rules without reasons; relational understanding is knowing what to do and why.
- Procedure item: 'Add 3/4 + 1/8'. Understanding items: 'Which is bigger, 3/4 or 4/5? Show it with a drawing' or 'Write a story for 12 ÷ 3'.
- Open-ended question: 'The answer is 24. What could the question be?'; many answers, low floor and high ceiling, best scored with a rubric.
- Portfolio: a purposeful collection of work over time that shows growth. Rubric: criteria with described levels; analytic scores each criterion, holistic gives one overall judgement.
- Maths anxiety: Richardson and Suinn's MARS (1972); caused by timed tests, public correction, one right method, comparison and stereotypes; it uses up working memory.
- Reducing anxiety: low-stakes formative assessment, valuing process and many methods, mistakes as learning, games and group work, descriptive feedback, praise for effort.
- NEP 2020: regular, formative, competency-based assessment; a 360-degree holistic progress card; a national assessment centre, PARAKH.
Quick revision
Scriven 1967: formative (for learning) vs summative (of learning); assessment as learning; RTE Act Section 29(2)(h) CCE; CCE tools; Skemp 1976 instrumental vs relational understanding; assess understanding, not only procedure; Position Paper: assess mathematisation; open-ended questions, low floor and high ceiling; portfolios show growth; rubrics analytic and holistic; maths anxiety, MARS 1972, causes and remedies; NEP 2020 360-degree progress card, PARAKH.