Techniques Grading

Intent

Techniques Grading (TG) is a mastery-based grading approach designed for proof-intensive courses, where course objectives are framed as the techniques covered in the course. TG draws on principles from specifications and standards-based grading to create a structured system that supports students in achieving mastery in writing correct mathematical proofs. TG intends to provide a transparent grading system where grades are assigned based on the product of the students’ work and which enables students to demonstrate proficiency in proof techniques through multiple attempts and clear, criteria-driven evaluations.

Problem

In most proof-based courses, the core challenge is not merely to convey facts about mathematical objects but to develop students’ “habits of mind” and mastery of essential techniques in combinatorics, algebra, or analysis. The goal is to help students think critically, adopting the specific problem-solving mindset of experts in the field. In this context, theorems, definitions, and complex proofs are valuable primarily as tools for cultivating analytical thinking. The TG approach focuses on fostering a deeper, discipline-specific way of thinking by encouraging students to “think like an analyst” or “think like a topologist.”

Solution

TG is structured around several key elements to ensure students achieve proficiency in proof techniques, with an emphasis on binarity (objectives are binary or pass/fail), product focused (objectives are stated in terms of the student’s work product), multiple chances (students are given multiple chances to achieve each objective), and transparency (the system is clear to students).

• Binarity and product-focus are inherent in the system’s structure, where each proof is graded based on whether it meets a set standard, making the evaluation straightforward and focused on mastering specific skills.
• At the per-proof level, the S/P/U (Satisfactory/Partial/Unsatisfactory) marking scheme provides multiple chances, allowing students to try again to demonstrate mastery on each proof. Additionally, because the technique objectives are common and fundamental, students have numerous opportunities to showcase their skills across the many proofs assigned throughout the semester.
• Transparency is emphasized by the TG syllabus, which clearly outlines not only the expectations for students but also the course’s overall goals. It provides a detailed list of skills students need to acquire, directly linking the grade to the student’s mastery of these skills, and clarifying the standards for assessment from the start of the term. This transparent and structured approach ensures that students understand what they need to do to succeed, with multiple opportunities to demonstrate their competence.

Applicability

The TG system works well in proofs-based courses, particularly when the focus is on developing and mastering specific techniques. In these courses, non-content objectives like participation and exam performance are typically not challenging, as most students meet the required standards. The real challenge lies in defining the technique objectives—the key skills students must master. These are identified by analyzing patterns in past assignments and textbooks, as well as considering well-established learning objectives. Techniques related to fundamental theorems or equivalent definitions, like Taylor’s Theorem or the Heine-Borel Theorem, are natural candidates. To refine these objectives, the source suggests focusing on techniques that appear frequently in course materials. While the list of potential techniques can be long, only the most frequently used ones are selected, and less common techniques are still included through direct instruction or supplemental homework. This approach ensures that students receive repeated opportunities to practice and master essential techniques, making the TG system effective in fostering deep learning in proofs-based courses.

Here are the challenges that could be faced while implementing this play.

First, student buy-in could be a significant hurdle. The complexity of the TG syllabus compared to traditional grading can cause initial confusion and reluctance, as students are unfamiliar with it. To address this, the source mentioned to be transparent about the differences, emphasizing the system’s goals and the detailed skills it aims to develop. Despite initial resistance, the source mentions that most students eventually appreciate the play and report no difference in learning outcomes compared to traditional grading.

Second, the problem cases involve situations where the subjective grading assessment can differ from the TG-based grade. These discrepancies occasionally highlight the need for adjustments in the grading criteria or reveal that the TG system is, in fact, correct. Although this issue is common across all grading systems, the TG system allows for refinement and improvements over time, making it adaptable.

How to Implement

The TG objectives are divided into three main categories: technique objectives, which make up the majority; exam objectives, which assess performance on quizzes and exams; and participation objectives. To earn a specific letter grade, students need to meet the requirements for that grade within each category. A student’s overall grade is determined by the lowest letter grade achieved across the categories. If a student does not meet the minimum “C” level in any one category, they will receive a “D” or an “F.” In cases where there is a discrepancy between categories, a plus or minus grade may be assigned, with the primary influence on these distinctions coming from the technique objectives.

• Technique Objectives: These form the core of TG, defining the specific proof techniques that students must master. Each objective is binary (pass/fail) and linked to writing a clear, correct proof demonstrating the targeted technique. Technique objectives are chosen based on their broad applicability, allowing students multiple opportunities to demonstrate mastery across various assignments.
• Grading of Proofs (S/P/U System): Proofs are graded using a three-tier system: Satisfactory (S): A proof meets the high standards required and counts towards the objective. Progressing (P): The proof has correctable flaws, allowing the student to revise and resubmit. Unsatisfactory (U): The proof has fundamental errors that require a new approach. Students are allowed unlimited resubmissions for “P” proofs, supporting iterative learning.
• Exam Objectives: In addition to technique objectives, exams assess foundational knowledge in a low-stakes format, focusing on recall, definitions, and analysis rather than on-the-spot proof writing. Performance on exams contributes to the final grade but is not the primary focus.
• Participation Objectives: These objectives incentivize student engagement through ungraded activities, such as journal reflections, creative assignments, and pre-class preparations. Participation objectives aim to encourage consistent effort and self-reflection.
• Final Portfolio: At the end of the term, students compile a portfolio of their S-marked proofs. Each proof is accompanied by an explanation of the technique it demonstrates, requiring students to reflect on and articulate their understanding. This portfolio serves as a summative assessment of their mastery over the course objectives.
• Multiple Chances and Transparency: TG is designed to be transparent, with clear criteria for each objective and multiple opportunities to demonstrate mastery. The syllabus communicates these standards at the outset, providing students with a roadmap of what is expected and how they will be assessed throughout the course.

TG’s structure incorporates elements from specifications and standards-based grading systems but is adapted to address the unique demands of teaching proof-writing skills, emphasizing skill mastery over time, clarity in expectations, and the iterative improvement of students’ work.

See Also

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Source

Source: Andrew A. Cooper (2020) Techniques Grading: Mastery Grading for Proofs Courses, PRIMUS, 30:8-10, 1071-1086, DOI: 10.1080/10511970.2020.1733151

Described by: Debarati Basu, (basud@erau.edu)

References

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