Precision Issues in STACK-Based Assessment: Maple and STACK/Maxima

Precision Issues in STACK-Based Assessment: Maple and STACK/Maxima Francesco Cosentino 1 Abstract: The Mathematics Department at the University of York broadly employed STACK based on Maxima, via a locally hosted Moodle, to assess modules across all stages of its mathematics programmes. As lecturer and Computing Officer managing the IT systems, I observed precision issues and potential impact on assessment, notably in the Year 3 Numerical Analysis module where students coded algorithms in Maple and Java. For example, a summative STACK quiz question on calculating by bisection the root 𝑟of 𝑓(𝑥)=𝑒𝑥−3𝑥in [0,1]to 9 significant digits revealed that students using Maple’s default 10-digit precision obtained final digits that differed from Maxima’s 16-digit result. These discrepancies arose from the interaction between precision settings, termination criteria, and rounding behaviour. Although such differences reflect genuine issues in numerical computing, they also highlight the need for clear expectations and appropriate grading tolerances when STACK quizzes rely on external software. This suggests tailoring STACK assessment design to both module stage and software context to maintain validity and fairness. Keywords: STACK, Maxima, Maple, Floating-Point Precision, Moodle, Computer-Aided Assessment, Mathematics Education, Assessment Fairness 1 Introduction Computer-aided assessment systems such as STACK have become integral to mathematics teaching, enabling scalable and automated assessment aligned with module learning outcomes. STACK’s integration with Moodle and its use of the Maxima computer algebra system make it a powerful tool for algebraic, symbolic, and certain numerical topics [ 1 , 3 ]. When STACK is used to assess floating-point numerical methods, however, practical issues can arise: differences in precision, rounding behaviour, and termination criteria across software environments may produce discrepancies that influence grading. These issues do not arise from STACK itself, but from how numerical tasks interact with its strict answer-matching mechanisms. This paper examines one such case in a Year 3 Numerical Analysis module at the University of York. 2 Pedagogical Context and Background STACK quizzes underpin modules across the undergraduate mathematics curriculum at York. In earlier years, assessments focus on symbolic algebraic manipulation where Maxima’s 1 University of Leeds, School of Computer Science, Woodhouse, Leeds LS2 9JT, United Kingrdom. F[email protected] exact symbolic capabilities excel. In contrast, Year 3 Numerical Analysis emphasises floating-point approximations, algorithmic stability, and implementation in programming languages such as Maple, MATLAB, and Java. Here, numerical precision is not incidental but a central learning outcome [4]. The challenge discussed in this paper did not arise from incorrect student algorithms alone, but from the interaction between student implementations, software defaults, and STACK’s strict numerical answer checks. Understanding these interactions is essential for quiz authors creating numerical content. 3 The Bisection Assessment Task Students were asked to implement the bisection method to find the root 𝑓(𝑥)=𝑒𝑥−3𝑥 in [0,1], to 9 significant figures, using the tolerances: • | 𝑓(𝑐)| ≤ 10−6 • (𝑏−𝑎) ≤ 10−10 3.1 Educational Rationale The bisection method is robust and deterministic, and therefore pedagogically suitable as a first root-finding method. The quiz asked for the final numerical value only, not the code. 3.2 Maple Implementation (student version) The student code shown was representative: Digits := 10; f := x -> exp(x) - 3*x; a := 0.; b := 1.0; tol_f := 10^(-6); tol_x := 10^(-10); c := (a + b)/2; while tol_x < b - a do if f(a)*f(c) < 0 then b := c; else a := c; end if; c := (a + b)/2; end do; Limitations include: Digits := 10 is insufficient for 9 significant figures; tol_f was unused; and there was no check for f(c) = 0. These issues were addressed in feedback. 3.3 Maxima Implementation (STACK) STACK evaluated the reference answer using Maxima, which defaults to 16-digit precision. This alone can yield small final digit discrepancies. 4 Results and Observations Students using Maple with low precision sometimes obtained answers that differed from Maxima’s output by one digit. This is not incorrect mathematically but failed STACK’s grading unless a tolerance was configured. While deterministic, the behaviour near convergence may appear inconsistent to students unfamiliar with rounding. For example, f(c) may evaluate as zero earlier in Maple, leading to surprising continued iteration or unexpected termination behaviour. 5 Discussion 5.1 Assessment Fairness Strict grading against a single numerical reference, without configured tolerance, can unintentionally penalise students using a different but valid environment. The issue lies not in the use of Maple, but in grading configuration. 5.2 Pedagogical Implications The situation prompted productive discussion on: •floating-point precision and software defaults [2, 3] •termination conditions and rounding •algorithm correctness versus software-specific behaviour 5.3 Summative vs Formative Use Numerical tasks are suitable for summative use, provided tolerances and expected behaviour are made explicit. Students should be marked on algorithmic understanding, not on alignment with a specific CAS default. 5.4 Implications for the STACK Community Quiz authors should: •test questions in both Maxima and student-used environments •configure answer tolerances (e.g., via answer_tolerance) •provide instructions on precision settings •pre-test numerical skills or constraints explicitly 6 Recommendations for Practice 1. Define explicit grading tolerances for floating-point answers. 2. Specify acceptable software or provide precision guidance. 3. Teach how rounding and termination affect algorithms. 4. Test questions with both STACK and external tools. 5. Use summative tasks when aligned with learning outcomes. 7 Conclusions Precision differences across symbolic and numerical tools can lead to small, but gradingrelevant discrepancies. Assessment design should address these differences to ensure fairness and support student learning. Acknowledgements The author thanks colleagues at the University of York and University of Leeds who contributed insights into these precision effects during STACK quiz reviews and computeraided assessment discussions. Bibliography [1] C. Sangwin. Computer Aided Assessment of Mathematics. Oxford University Press, 2013. [2] Maplesoft. Maple 2025 User Manual, Waterloo Maple Inc., 2025. [3] Maxima Project. Maxima Manual, version 5.47.0, 2025. [4] N. J. Higham. Accuracy and Stability of Numerical Algorithms. SIAM, 2002. Information on the author Francesco Cosentino is a Lecturer in Mathematical Data Science at the University of Leeds. He previously lectured at the University of York, teaching Computational Finance, Vector Calculus, and Object-Oriented Programming. Earlier in his career, he worked as a radiotherapy physicist (2010–2018), developing automation, quality control, and augmented reality tools for healthcare. He was also a Research Fellow in the Immersive Cognition Research Group at Leeds. His scholarly interests include data science, machine learning, the foundations of quantum mechanics, mathematical logic, and computational visualisation. He holds a PhD in Computer Science (Chester), MSc degrees in Medical Physics, Control Engineering, Mathematical Logic, and Optics, and an MPhys in Theoretical Physics.