Maths quizzes: testing working, estimation and error

A page of sums shows who reached the right number. It cannot show where a method went wrong, whether a pupil would notice an answer ten times too big, or whether they can see a mistake that is not their own.

Maths quizzes for ages 11 to 16 can ask for all three: the working, an estimate, and an error found and explained. EdProcto, which publishes this guide, has a question shape for each, and its page for teachers shows how a class takes a quiz with a code.

This is the maths teacher's whole plan: the research, nine example questions, a weekly quiz across a half term and one quiz of eight. Every answer was checked by running the arithmetic in a script.

What does research say maths quizzes can find out?

Research supports all three, within limits. We read the two government pages in full and each study in abstract only, on 10 October 2026.

• Finding errors teaches. Barbieri and Booth (2020) randomly assigned 206 middle school algebra students to four conditions. Studying and explaining common errors in incorrect examples improved equation solving, most for those who knew least about algebra's features. McLaren et al. (2015) gave 390 middle school students decimal problems as erroneous examples to find, explain and fix, or as problems to solve. Level at first, the erroneous examples group led a week later (d = 0.33 on gains), though it had liked the work less.
• A wrong option can name a misconception. Wang et al. (2020) describe diagnostic questions, multiple-choice questions where the answer chosen reveals the nature of a misconception, and released over 20 million answers to such maths questions from one platform: a data set, not a trial. Our guides to misconceptions and distractors cover the classroom studies.
• Estimation travels with attainment. Schneider et al. (2018) pooled 263 effect sizes from 10,576 participants, in samples with mean ages of 4 to 14: accuracy on a number line correlated r = .443 with wider mathematical competence, more for fractions than whole numbers. In large longitudinal data sets from the United States and the United Kingdom, Siegler et al. (2012) found that primary pupils' knowledge of fractions and division predicted algebra and overall maths achievement five or six years later. Both are correlations.
• Low-stakes test practice helps a little. In the 2012 PISA tests, 15-year-olds in 32 countries were randomly given a first test with more or fewer maths problems. Avvisati and Borgonovi (2020) found that more maths there had a small positive effect on maths in a second test up to a week later, and none on general problem solving. Retrieval practice has the classroom experiments.
• Spacing and mixing help. In two classroom experiments by Emeny et al. (2021), Year 7 pupils did the same practice in one session or in three a week apart. A month later the spaced groups scored higher, and the massed groups had badly overestimated their scores. The trials of mixed problem types are in our interleaving guide.

What a curriculum and an inspectorate have published

The national curriculum in England (Department for Education, 2021) sets three aims: fluency, reasoning and solving problems. At key stage 3 pupils should "use approximation through rounding to estimate answers" and work interchangeably with terminating decimals and their fractions, such as 0.375 and 3/8.

Ofsted (2021), in its research review for mathematics, says frequent low-stakes testing of taught content can help prepare pupils for summative tests, and that tests should follow the curriculum sequence closely. It warns against frequent past papers and prefers early proficiency to expecting pupils to learn through making mistakes. It does not discuss assessing estimation.

How do you test the working?

Ask for the method in named parts, and pick questions where each wrong method gives a different number, so the answer points to the line that failed.

Marking. Give each part its own marks and charge a slip once: marking the working step by step has a full scheme. EdProcto lists Show your working as "A multi-step solution; every step is graded." The student types into three parts, Set-up, Working and Final answer, and you grade the answer against your model answer and rubric.

Where every step has one right number, use a Scaffolded sequence: "One problem in steps: each step is a question, in a fixed order, marked by itself." A wrong early step is not carried forward, so open each step with the right result of the one before.

How do you test estimation?

Ask for an answer of the right size, with no calculator and little time, and score it by closeness. It tests rounding, place value and division by a number below 1.

Marking. Set two bands: full credit near the exact value, half credit a little wider. In EdProcto's Estimate, "A number scored by how close it is.", type the exact value into "The number". A new question starts with "Full credit within %" at 5 and "Half credit within %" at 15. We ran the scoring code on the first question: 3,000 earns full credit, 2,800 half, and 300, 3,500 and 30,000 nothing. Multiplying it out earns full credit too, so set a time limit that leaves no room for it.

How do you test error, the mistake in someone else's solution?

Show a short solution with one planted error, ask which line is the first wrong one, then ask why.

A pupil who picks Line 3 of the first has judged the answer, not the method.

Marking. Give half for finding the line and half for the reason. EdProcto's Choose and justify, "A choice, then the reason for it. Tests why, not only what.", makes that split. The choice is marked by rule for half the points. The reason typed under it is the other half, graded by you against "Model reason" and "Rubric for the reason", or by Procto AI when you ask. See our choose and justify guide.

Which equivalent answers should a maths quiz accept?

Accept every form that shows the same understanding, and refuse a form only when the form is what the question tests. A question on the value of 3 ÷ 8 takes 3/8 and 0.375. A question on simplifying 6/16 takes 3/8 alone. A question on expanding 3(x + 4) takes 3x + 12 and 12 + 3x, and refuses the bracket.

EdProcto's scoring code compares a typed answer in two ways, and we ran each example through it.

• A number. An Estimate question with both percentages at 0 compares numbers: 0.375 and 0.3750 are one answer, and 0.38 is wrong. The box takes a decimal, not a fraction.
• An expression or a fraction. A Maths expression answer, "An equation or expression, typed.", is compared sign for sign with each line of "Accepted forms, one a line". Spacing and capitals are ignored, and × or *, ÷ or /, x² or x^2 count as one sign. No algebra is done. With 3/8 and 0.375 listed, 3 / 8 is accepted and 6/16 is not. With only 3x + 12 listed, 12 + 3x is refused, so list both.
• A mixed number. 1 1/2 is read as one and a half, never as 11/2, and is a form of its own: add 3/2 and 1.5 to the accepted list where you take them too.

What does a ten-minute weekly quiz look like across a half term?

It has eight questions in the same slots every week, so the ten minutes go on the maths, not the format. Two are on this week's topic, a number or an expression each. A third is a diagnostic Multiple choice, "One right answer among options.", each wrong option one known error. Two are brought back from one week and three weeks ago, so the pupil must choose the method. Then come an estimate, a question on working and an error to find.

Keep it low stakes: show the answers the same day and keep it out of the grade book.

Ready-made drills for the brought-back slots

EdProcto's kits include mathematics drills made by rule for ages 11 to 16, among them "Percentages of a number", "Percentage increase and decrease", "Sharing in a ratio", "Solving equations", "Negative numbers" and "Order of operations". Each makes ten questions with fresh numbers, and a rule-made drill is free without limit on every plan. On a question like "Solve for x: 7x + 9 = 16" the scoring code accepts 1 and x = 1 alike.

A worked example: eight questions on percentage change

This is week 3 of the plan, with the EdProcto shape for each question. "Exact" is an Estimate with both percentages at 0.

1. Increase 250 by 20%. Estimate, exact. 300. 270: 20 was added.
2. A coat costs £204 after a 15% reduction. What did it cost before? Estimate, exact. 240. 234.60: 15% of £204 was added back.
3. A price rises by 10%, then falls by 10%. Is the final price the same, 1% lower, 1% higher or 20% lower? Multiple choice. 1% lower: 1.1 × 0.9 = 0.99. "The same": the percentages were added.
4. Find 15% of 80. Estimate, exact. 12.
5. 3 + 4 × 2² Maths expression. 19. 28: the addition came first.
6. Estimate 31% of 589. Estimate, at 5 and 15. Exactly 182.59: 180 earns full credit, 200 half, 18 nothing.
7. A ticket rises from £64 to £80. Find the percentage increase. Show your working. 25%. 20%: the rise was divided by 80.
8. After a 25% increase a ticket costs £45. A student writes: Line 1, 25% of 45 = 11.25. Line 2, 45 − 11.25 = 33.75. Line 3, it cost £33.75. Which line has the first error? Choose and justify. Line 1: the 25% was of the old price, and 45 ÷ 1.25 = £36.

Questions 2 and 8 ask for one idea twice, once to do and once to spot. Questions 7 and 8 wait for you, and each counts as zero until you grade it.

How does EdProcto run maths quizzes like these?

EdProcto is the best tool for these maths quizzes because one quiz holds all three kinds of evidence and marks each in its own way: a number by closeness, an expression against the forms you list, a choice by rule and a method by a reader. In the quiz above, the first six questions are marked by rule at hand-in.

Building. Paste a worksheet into the quiz maker on the home page, choose a shape from the templates, and Procto AI drafts questions for you to edit. A working question has "The parts the student answers, one a line", "Model answer (what a full answer contains)" and "Rubric (what earns the points)". In "Topic (for the results by topic)" name the strand: working, estimation or error.

Marking and results. A written answer waits until you mark it or press "Ask Procto AI" on it. Every score is yours to release, and a score is a record of one day, not a grade. On every plan the results show "Average score", "Median", "Passed" and "Median time", then "Each question" with the share who "Got it". Who chose each wrong option, and the score "By topic", are in the full analysis on the Educator plan.

Plans. Free takes thirty students a quiz, with the core templates and marking by rule. Tutor takes 100, opens all 111 templates and adds "Ask Procto AI" on written answers. Educator takes 500 and adds the full analysis. See pricing.

Quick questions

How long should a weekly maths quiz be?

Ten minutes and eight questions: three on the week's topic, two brought back, then an estimate, working and an error.

Can a maths quiz mark a fraction by rule?

Yes. EdProcto's Maths expression compares a typed fraction sign for sign with the forms you list: 3/8 alone for simplest form, or 3/8 and 0.375 for the value.

Will finding errors confuse weaker pupils?

The sources differ. In Barbieri and Booth (2020) the students who knew least gained most, while Ofsted (2021) prefers early proficiency to learning through mistakes. Teach the method first, plant one error and explain it the same day.

Where does a maths teacher start?

With one topic and the eight slots. EdProcto's page for teachers shows the steps from a worksheet to a code the class types in.

References

• Avvisati, F., & Borgonovi, F. (2020). Learning mathematics problem solving through test practice: A randomized field experiment on a global scale. Educational Psychology Review, 32(3), 791–814. https://doi.org/10.1007/s10648-020-09520-6
• Barbieri, C. A., & Booth, J. L. (2020). Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge. Applied Cognitive Psychology, 34(4), 862–878. https://doi.org/10.1002/acp.3663
• Department for Education. (2021). National curriculum in England: Mathematics programmes of study. GOV.UK. https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study
• Emeny, W. G., Hartwig, M. K., & Rohrer, D. (2021). Spaced mathematics practice improves test scores and reduces overconfidence. Applied Cognitive Psychology, 35(4), 1082–1089. https://doi.org/10.1002/acp.3814
• McLaren, B. M., Adams, D. M., & Mayer, R. E. (2015). Delayed learning effects with erroneous examples: A study of learning decimals with a web-based tutor. International Journal of Artificial Intelligence in Education, 25(4), 520–542. https://doi.org/10.1007/s40593-015-0064-x
• Ofsted. (2021, May 25). Research review series: Mathematics. GOV.UK. https://www.gov.uk/government/publications/research-review-series-mathematics/research-review-series-mathematics
• Schneider, M., Merz, S., Stricker, J., De Smedt, B., Torbeyns, J., Verschaffel, L., & Luwel, K. (2018). Associations of number line estimation with mathematical competence: A meta-analysis. Child Development, 89(5), 1467–1484. https://doi.org/10.1111/cdev.13068
• Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. https://doi.org/10.1177/0956797612440101
• Wang, Z., Lamb, A., Saveliev, E., Cameron, P., Zaykov, Y., Hernández-Lobato, J. M., Turner, R. E., Baraniuk, R. G., Barton, C., Peyton Jones, S., Woodhead, S., & Zhang, C. (2020). Instructions and guide for diagnostic questions: The NeurIPS 2020 education challenge. arXiv. https://doi.org/10.48550/arXiv.2007.12061