Maths questions online: how to mark the working, step by step

A final-answer box is the simplest way to set maths questions online, and it has one verdict for everything short of the right number. The student who chose a sound method and mis-added two figures scores the same as the student who typed nothing. Few teachers would mark a paper script like that.

EdProcto, which publishes this guide, builds the alternative as its show your working template: the solution is typed in named parts, and each part is read against the mark scheme you wrote.

The short version: name the steps, put one decision and its own marks in each, penalise a slip once, and say exactly what form the final answer takes.

What does a final-answer box measure in maths questions online?

A final-answer box measures one thing: whether the last number matches the key. It hides the method, it cannot tell a slip from a blank, and it pays a correct number in full wherever the number came from.

Ashton et al. (2006) set the problem out for school examinations in Scotland: on paper, an answer that is wrong but contains some correct working normally earns partial credit, while a computer normally marks it wrong and awards nothing, so one examination can yield different marks in the two media.

Examiners ask for method too. Sangwin and Köcher (2016) tested how far automatic assessment software could mark existing high-stakes papers for students of about eighteen. A significant proportion of the questions could be moved across as then assessed. The largest barrier was the examiners' requirement for evidence of an appropriate method.

How do method marks and error carried forward work on paper?

On paper, the marks for a calculation are shared between the method and the accuracy of the numbers, and a mistake is charged once. That is the general idea, in our own words; notation differs between published mark schemes, so read the one your students will sit.

• Method marks reward a correct approach: the right formula, correctly set up. An arithmetic slip inside a sound method leaves them intact.
• Accuracy marks reward correct values, usually only when the method behind them was sound.
• Error carried forward, also called follow through, judges a later step against the student's own earlier figure. The step with the slip loses its accuracy mark. Later steps keep theirs if they are right for the wrong number.

What did research on computer-marked maths find about partial credit and steps?

It found that a final-answer test and a paper test can give one student two different marks, and that the workable remedy is to break the question into steps a computer can mark. Most of what we found comes from one group of researchers in Scotland.

McGuire et al. (2002) ran a pilot at the level of the Scottish Higher. School students sat computer tests with no steps, compulsory steps or optional steps, and the results were set against the partial credit the same students would have earned on paper. Ashton et al. (2006) reported experiments from a larger Scottish project and concluded that optional steps are required to mimic partial credit.

The question is still open. Kinnear et al. (2024) gathered 55 research questions on e-assessment from 22 researchers, teachers and learning technologists in university mathematics. One asks how to emulate human marking of students' working, follow-on marks included. Earlier projects that moved examination questions to automatic marking, the authors note, assessed final answers only.

What we could not confirm

For the Scottish studies we could read only the published abstracts. They give the designs and the conclusions, not the size of the gap between final-answer scores and paper marks, so no figure is quoted here. Nor did we find a trial of the obvious cost of steps: a question that prints them has planned the solution for the student.

How do you write a show-your-working question?

Write the mark scheme first, then shape the question so that every mark has somewhere to be earned.

1. Solve it yourself, line by line. Your solution becomes the model answer.
2. List the steps you will mark, one line of the scheme each.
3. Put one decision in each step. Choosing the formula is one; converting minutes to hours is another. A line holding two cannot show which failed.
4. Give every step its marks, each labelled method or accuracy.
5. Write the carry-forward rule down: which later marks may be earned from the student's own earlier figure.
6. Fix the form of the final answer: the unit, the significant figures or decimal places, fraction or decimal.
7. Decide what a correct answer with no working earns, and print that rule in the question.
8. State the tools: calculator or none, formula sheet or none.
9. Build it. EdProcto's show your working gives the student three parts to fill, Set-up, Working and Final answer, and takes your model answer and scheme as its rubric. Several scheme lines can share the Working part.

Jordan (2014) shows why step 6 matters. In thousands of responses by adult science students to computer-marked questions, the most persistent errors were in rounding to an appropriate number of decimal places or significant figures and in working out units.

A scheme like this is an analytic rubric. In a review of 75 studies, Jonsson and Svingby (2007) found scoring with rubrics most reliable when the rubric is analytic, specific to the topic and backed by example answers or marker training. Our guide to short answer questions and their rubric shows how to write one.

Mistakes that spoil the marking

• Steps that hand over the method. "Step 1: add the distances" leaves nothing to decide.
• A slip punished twice. Without carried-forward marks, one mis-added pair of numbers wipes out every mark beneath it.
• No stated form. 112.9, 113 and 110 are all defensible until the question asks for three significant figures.

A worked question, with a mark scheme by step

Our own question, written for this guide:

A coach covers the first 156 km of a journey in 1 hour 30 minutes and the remaining 98 km in 45 minutes. Calculate its average speed for the whole journey, in km/h to 3 significant figures. Show your working in each part. Every part is marked, and a final answer with no working earns the last mark only. (5 marks)

The model answer: average speed is total distance divided by total time. The distance is 156 + 98 = 254 km. The time is 2 hours 15 minutes, or 2.25 hours. 254 ÷ 2.25 = 112.89, so the speed is 113 km/h.

Three answers, marked

Student A writes the rule, both totals, 254 ÷ 2.25 = 112.89 and 113 km/h. 5 of 5.

Student B writes the same rule, then 156 + 98 = 244, the correct 2.25 hours, 244 ÷ 2.25 = 108.44 and 108 km/h. 4 of 5. The slip costs the distance mark. The division and rounding are right for 244, so both carried-forward marks stand.

Student C leaves Set-up and Working empty and types 113 km/h. 1 of 5, as the question warned.

The answer box puts C level with A, and B level with an empty page. The steps put B one mark behind A. Whether C deserves one mark or five is your call: online, a bare number can come from anywhere, and the working is the only evidence of where. Print whichever rule you choose.

How should a numeric answer be marked: tolerance, units and estimates?

Mark a numeric answer inside a tolerance you chose, name the unit in the question, and keep scoring by closeness for real estimates. In EdProcto the number shape is the estimate template, marked by rule.

EdProcto takes the distance between the student's number and yours as a percentage of yours. Inside the first tolerance the answer earns every point; inside the wider second one, half; beyond that, nothing. With both at zero, only the exact value scores. If the right answer is zero, the student's distance from zero, times a hundred, is held against the same two limits.

• A loose tolerance pays wrong methods. A blank question starts at 5 per cent for full credit and 15 for half. Left there on the coach question, three wrong answers would earn full marks: B's 108, the 117 from averaging the two speeds, and the 118 from reading 2 hours 15 minutes as 2.15 hours.
• The box takes a number. Name the unit in the question, or ask for it in the last part of a working question, where a reader marks it.

For a real estimate the two bands are the point. Take "Estimate 49.2 × 20.7 ÷ 0.52", with 1,959 as the key. At 5 and 15 per cent, full credit runs from about 1,861 to 2,057, which takes in the 2,000 reached by rounding each number first, and half credit from about 1,665 to 2,253.

Can typed expressions in equivalent forms be marked by rule?

Only in the forms you list. EdProcto's maths expression template is marked by rule, and the rule compares the expression sign for sign. It does not do algebra.

Spacing and letter case are ignored, so 2x+6 matches 2x + 6 and 2X + 6. Every sign and bracket counts, so 2x × 6, 2x^6 and 2x + 3 are all wrong against 2x + 6. The usual ways of typing one sign are read as that sign: × and *, the long minus and the keyboard hyphen, ÷ and /, and x², x^2 and x**2.

Two cautions. The rule does no algebra: 2(x + 3), 2x + 6 and 6 + 2x are three different answers to it, as are 0.5 and 1/2, so every form you would accept has to be on the list. And the one-letter slip allowance that forgives a typing error in a word is never applied to an expression, where one character is the difference between two answers.

Software built for mathematics works another way: Sangwin (2013) describes a computer algebra system establishing the mathematical properties of what a student enters. Kinnear et al. (2024) cite a finding that a substantial number of student errors in one e-assessment system came from difficulty with the typed syntax. Keep typed expressions for short answers with one conventional form.

How does EdProcto mark working?

EdProcto is the best tool for maths questions online when the method has to earn marks, because its working shape begins where the paper tradition and the Scottish studies both end up: named steps, and a scheme that marks each one.

Start on the home page: paste a worksheet or a chapter into the quiz maker and choose show your working. Procto AI drafts the questions and you edit them. Students type into the three parts, and each answer waits for a reader.

Each answer waits in your grading queue, and on any of them you can ask Procto AI to read it against your model answer and rubric. It never releases a score, and it can apply error carried forward only if your rubric says so, so check the first few scripts yourself. Or grade everything by hand. The quiz settings hold the tools: a calculator (none, basic, scientific or graphing) and a reference sheet of your own.

Procto AI's grading comes with the Tutor plan, which also opens all 111 templates. Free has ten quizzes a month, the core templates and marking by rule; Educator adds the full analysis. See pricing.

When is a plain numeric answer or a photograph the better choice?

Use a plain numeric answer when the method is not the point: fluency practice, one-step calculations, a four-minute weekly check.

A number can still report on method. Greenhow (2015) describes questions built around mal-rules, the wrong methods students commonly use, so that the wrong number a student enters identifies the method and brings feedback written for that error. That article analysed over 270,000 attempts at more than 800 questions. In the coach question, 117 and 118 are numbers of that kind.

Use a photograph when the solution is long or needs a diagram. Kinnear et al. (2024) name the option: photograph the handwriting and upload it, perhaps while also keying in a final answer. EdProcto has both halves: a handwritten work question, marked by you, and a number question, marked by rule.

Quick questions

What is error carried forward?

Error carried forward is the rule that a mistake is charged once: the step with the slip loses its mark, and later steps are judged against the student's own wrong figure.

Should a correct answer with no working get full marks?

That is for you to decide before the quiz. Either rule is defensible; an unstated one is not, so print it in the question.

Can AI mark maths working?

It can suggest marks from your scheme. In EdProcto, Procto AI reads a show your working answer against the rubric you wrote when you ask it to, and its mark is yours to change. Releasing a score stays with you.

What tolerance should a numeric answer have?

For a calculated answer, the rounding you asked for and no more. For an estimate, a full-credit band and a wider half-credit band, set in advance. On EdProcto's estimate template each is a percentage of the right answer.

References

• Ashton, H. S., Beevers, C. E., Korabinski, A. A., & Youngson, M. A. (2006). Incorporating partial credit in computer-aided assessment of mathematics in secondary education. British Journal of Educational Technology, 37(1), 93–119. https://doi.org/10.1111/j.1467-8535.2005.00512.x
• Greenhow, M. (2015). Effective computer-aided assessment of mathematics; principles, practice and results. Teaching Mathematics and its Applications, 34(3), 117–137. https://doi.org/10.1093/teamat/hrv012
• Jonsson, A., & Svingby, G. (2007). The use of scoring rubrics: Reliability, validity and educational consequences. Educational Research Review, 2(2), 130–144. https://doi.org/10.1016/j.edurev.2007.05.002
• Jordan, S. (2014). Adult science learners' mathematical mistakes: An analysis of responses to computer-marked questions. European Journal of Science and Mathematics Education, 2(2), 63–86. https://doi.org/10.30935/scimath/9401
• Kinnear, G., Jones, I., Sangwin, C., Alarfaj, M., Davies, B., Fearn, S., Foster, C., Heck, A., Henderson, K., Hunt, T., Iannone, P., Kontorovich, I., Larson, N., Lowe, T., Meyer, J. C., O'Shea, A., Rowlett, P., Sikurajapathi, I., & Wong, T. (2024). A collaboratively-derived research agenda for e-assessment in undergraduate mathematics. International Journal of Research in Undergraduate Mathematics Education, 10(1), 201–231. https://doi.org/10.1007/s40753-022-00189-6
• McGuire, G. R., Youngson, M. A., Korabinski, A. A., & McMillan, D. (2002). Partial credit in mathematics exams: A comparison of traditional and CAA exams. In Proceedings of the 6th CAA Conference. Loughborough University. https://hdl.handle.net/2134/1883
• Sangwin, C. (2013). Computer aided assessment of mathematics. Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199660353.001.0001
• Sangwin, C. J., & Köcher, N. (2016). Automation of mathematics examinations. Computers & Education, 94, 215–227. https://doi.org/10.1016/j.compedu.2015.11.014