Proofs Literature Reading Group by debbie Borkovitz

 This is the first in a series of five blog posts from members of NE-COMMIT’s Summer 2026 Proofs Literature Reading Group, written in response to the call: For those of us who were in this summer’s group reading proof literature, write a bit about what you got from the group, such as articles you found interesting or plans for the fall. Our audience is NE-COMMIT and anyone else curious about improving teaching mathematics.

Debbie, who started our reading group, describes the classroom moments that drew her to the research on proof, and how the articles are reshaping her Discrete Math course this fall.

The series: The posts move from how the group started, to changes in proof courses, to ideas for non-proof courses, and finally to the theory behind it all.

1. Proofs Literature Reading Group, by Debbie Borkovitz (this post)
2. Teaching Proofs through Critical Reading, by Dana Rowland
3. Rethinking Proof Teaching: Reading Proofs as Much as Writing Them, by Maria Fung
4. Possible Uses of Comprehension Questions in Non-Proof Service Courses, by Michael Nathanson
5. What “I Don’t Understand This Proof” Means: A Cross-Walk of Four Theories, by Ileana Vasu

 

         In the Spring of 2019 on the last day of Discrete Math in my first year of teaching at Boston University, a student told me that before this class he had never realized you could solve a math problem by thinking about it. He’d just “followed orders.” He was a junior math major at the time. He was more blunt than most students, but he’s not alone, and in the years since, I’ve felt it most important to center problem solving when I teach Discrete Math. The first few times at BU I tried to do more of the classic “odd plus odd is even” introductory proofs as well, but I found that most students did not see the point of them and put them in the category of things to memorize, not to think about.

 

         In years since I’ve tried to integrate proof more into the kinds of problems we solve in class, which means focusing especially on induction and combinatorial proof. However, students are much better at problem solving than at writing formal proofs, so a lot of the proofs end up too hard for them, which right now, means they just go to an LLM. My main out-of-class assignments in the class had been what I first called “Challenge Problems” and later “Excursions,” where students solved an interesting problem and wrote up their solution and proof. These assignments don’t work now that students have access to AI, but the truth is that they didn’t work for many students well before AI, especially with regard to proof. Enough students did the assignments as I’d hoped and learned a lot from them, so I kept them year after year. These students worked on a problem, wrote about their thinking, and improved their work with feedback over another draft or two or three.  But others came to office hours a lot, and I ended up walking them through the solution, or relied on classmates to do the heavy thinking for them. Once, well before AI, a student used generating functions when we hadn’t studied them in class, so in my office I asked him what a generating function was, and he had no idea. It turned out his brother was a graduate student in math and had done the proof for him. So now everyone has a robot sibling who is a graduate student in math; I guess that’s equalizing in some way.

 

         It had been in the back of my mind for many years that I should read more of the Math Education literature about learning and teaching proofs, but I never seemed to get around to actually doing it. In another blog post (link) I wrote about the short in-class writing assignments I gave in Graph Theory last spring, which helped me realize that I was vastly misunderstanding where my students were starting from, and that coupled with the issues in Discrete Math last fall finally motivated me to ask NE-COMMIT colleagues if they were interested in reading with me. Several were, and the Proofs Literature Reading Group was born. Even though we only met three times this summer, the group has been incredibly helpful.

 

         For Discrete Math this fall I’m reworking some of the in-class activities and almost all of the out-of-class assignments, including making several changes in how I teach proof, many of which were directly inspired by articles we read in the group and our discussion of them. First, I think there is a much better way to integrate proof into my problem solving activities than straight out asking students to write the whole proof of what they solve. I think it makes a lot more sense to have students first work until they understand a problem well, and then give them proofs that I’ve written to analyze. Two of the articles we read were really helpful for thinking about how to organize such an activity.

 

         The 2022 article “Lessons Learned About Incorporating High‑Leverage Teaching Practices in the Undergraduate Proof Classroom to Promote Authentic and Equitable Participation” by Melhuish, Dawkins, Lew, and Strickland has many ideas about how to launch proof activities and to run them in ways that all students participate. In Discrete Math I often choose problems that can be solved with different approaches, so after having students discuss their approaches informally, I plan this fall to distribute several different proofs that I’ve written. Each student in a group will get one of these proofs to work through themselves, and then I’ll put them together to teach their proof to the others. In the first meeting of the Proofs Literature Group, we had a great discussion of this article, and talked about things like the pros and cons of doing activities like these as jigsaws, where students first meet with others with the same proof, and then meet in mixed groups.

 

         I learned a lot about how to structure questions about proofs from the 2012 article, “An Assessment Model for Proof Comprehension in Undergraduate Mathematics” by Mejia-Ramos, Fuller, Weber, Rhoads, and Samkoff. I think any mathematician who teaches proof, no matter the pedagogy, would benefit from reading this article. The article proposes seven types of questions for assessing proof comprehension. The first group focuses on local understanding of a proof, such as definitions, examples that illustrate a given term in the proof, identifying the purpose of a sentence in a proof and the proof technique, justifying how various claims are supported, and what warrants are implicit in the proof. We spent some time in the group trying to understand what warrant means in this context; it encompasses things like the rules of algebra or steps that the author considers obvious that are not explicitly stated in the proof. The second group of questions focus on a more holistic understanding of the proof, such as being able to summarize it, understanding the modular structure, being able to do a similar proof, and illustrating with examples. The authors provide sample questions, and in the second meeting of the group, we chose a standard proof about the irrationality of a square root and made up questions for all the categories, which was really fun and quite helpful. For my class, I will use the framework in this article as a guide for creating the questions that students discuss in class around the written proofs. I will also give more quizzes and assignments that focus on answering questions like these and fewer that ask students to write their own proofs.

 

         Another idea I’m excited about did not come from the Proofs group. At Mathfest, Oscar Levin gave a talk about using Sudoku variants to teach proof, as substitutes for the “odd plus odd is even” type of proof. Many students already like to solve puzzles, and they are often using standard proof techniques such as proof by contradiction or separating into cases. A theorem such as, “The top right cell in this puzzle must contain a 9” may actually be of similar complexity and use similar logic as some of the standard simple number theory proofs we often use to introduce proof, but it avoids the issue of why were are proving something the students think is obvious in such a convoluted way. So I’m interested to try some puzzles to introduce proof techniques and to have students write some proofs on their own that are at an appropriate level of complexity but are still interesting. From these proofs we can work on things like the grammar, conventions, and format of proofs. I think some of these will also lead to good in-class activities, as described in the article by Melhuish et. al. article. 

Teaching Proofs through Critical Reading by Dana Rowland

 This is the second in a series of five blog posts from members of NE-COMMIT’s Summer 2026 Proofs Literature Reading Group, written in response to the call: For those of us who were in this summer’s group reading proof literature, write a bit about what you got from the group, such as articles you found interesting or plans for the fall. Our audience is NE-COMMIT and anyone else curious about improving teaching mathematics.

Teaching both Discrete Math and Real Analysis this fall, Dana turns the readings into a concrete list of changes, with special attention to equity and to helping students read proofs critically, including ones generated by AI.

The series: The posts move from how the group started, to changes in proof courses, to ideas for non-proof courses, and finally to the theory behind it all.

1. Proofs Literature Reading Group, by Debbie Borkovitz
2. Teaching Proofs through Critical Reading, by Dana Rowland (this post)
3. Rethinking Proof Teaching: Reading Proofs as Much as Writing Them, by Maria Fung
4. Possible Uses of Comprehension Questions in Non-Proof Service Courses, by Michael Nathanson
5. What “I Don’t Understand This Proof” Means: A Cross-Walk of Four Theories, by Ileana Vasu

This fall I am teaching proof in Discrete Mathematics and Real Analysis.  At my institution, many students encounter mathematical proof initially as part of the Discrete Mathematics course (required for math and computer science majors among others) and reinforce their proof writing in Linear Algebra, but really dive in when they reach Abstract Algebra or Real Analysis.  Recently we moved to offering these upper-level courses once every two years, so students now have a wider variation and on average less mathematical maturity when they take Real Analysis than in the past.  Also, the availability of AI encouraged me to rethink my out-of-class assignments and overall approaches to assessment in both courses.

I approached the readings from this context.  Though I appreciated reading the theory behind how students’ engage with learning proof, I was particularly interested in concrete interventions to apply in my classroom. Below, I note my key take-aways based on three of the articles we read, and outline some specific changes I plan to make in my fall classes as a result.

Our first discussion was based on "Lessons Learned about Incorporating High-Leverage Teaching Practices in the Undergraduate Proof Classroom to Promote Authentic and Equitable Participation," by Melhuish, Dawkins, Lew, and Strickland.  This was my favorite of the articles we read due to its focus on creating activities for student-centered approach to teaching proof, with an added emphasis on equity.  While the lessons used in the paper were developed for an Abstract Algebra course, the underlying themes apply to any mathematical course that emphasizes proof.  While reading, I recognized that my approach to teaching proof thus far had focused primarily on supporting students in constructing proof but hadn’t given as much attention to tasks related to proof validation and proof comprehension (which arguably need more attention in the age of AI as a way of getting students to critically examine a generated proof).  Furthermore, my structure for in-class proof-writing tasks didn’t always provide equitable access points for students with varying backgrounds.  As noted in the conclusion (p. 312), “structuring group work to distribute responsibility and providing opportunity beyond just formal proof production” are essential for more equitable participation.

Some implementation action items that I plan to incorporate into my teaching of proofs include:

• Adjust “task launch” for proof writing (p. 297) to focus more attention on “unpacking key definitions and relationships in theorems” to be more explicit about to prior knowledge students need to engage with a particular task by creating a “public record” of necessary definitions and key ideas.

•  Give more attention to students’ appropriate use of quantification (there exists an x, or for all x) and identification of mathematical objects and their referents when verbalizing (for example, ask students to “explain what type of object is being referenced by certain symbols” (p. 309).

• Include activities that focus on analyzing, critiquing and comparing approaches to proof. Having students discuss proofs that are provided to them can provide a different access point (p. 303). The provided proofs might include different approaches from student work from prior semesters and/or AI generated results.

• Assign content-specific group roles so that each group member is responsible for leading the discussion on a different prompt.

The next paper we considered was “An assessment model for proof comprehension in undergraduate mathematics,” by Mejia-Ramos, Fuller, Weber, Rhoads, and Samkoff, which as the title suggests, focused on the type of questions to ask to assess student understanding.  The authors note that “If comprehension tests only ask students to reproduce a proof by rote, students are likely to develop a superficial understanding of that proof and emphasize form over substance” (p. 4), and propose alternative types of questions to use to assess different dimensions of understanding.  These questions focus on aspects such as the meaning of terms and statements, the status of assertions (is a claim a definition, an axiom, a theorem, a hypothesis, or a deduction from prior steps?), justification of individual steps, identification of proof structure and method, summarizing high level ideas, generating examples, and transferring the method.  The related paper “Comprehension tests in mathematics” by Conradie and Frith provided additional examples of questions to be used for assessment.  Authors noted that a disadvantage to questions of the form “State and prove theorem _____” can lead to students who “try to commit to memory the entire proof, word by word, with possibly no understanding at all” (p. 225), and moreover, are difficult to evaluate. 

Recognizing that many of my proof questions on assessments from past semesters focus primarily on proof construction, my main takeaway is the need to develop exercises that also require critical reading of proofs with more detailed prompts for use in-class, on homework, and in assessments.  In particular, I aim to:

• Develop explicit prompts to direct students’ comprehension of “local” statements (such as the meaning of terms and statements, the role of a statement as a definition, hypothesis, or justification of a particular claim) as well as “holistic” understanding (ability to summarize the key ideas, illustrate with examples/diagrams, or transfer methods to another context).

• Add proof comprehension questions to weekly homework assignments (along the lines of the “comprehension tests on ‘unseen’ proofs”, Conradie and Frith p. 233)

A huge thanks to Debbie Borkovitz for launching this group to explore mathematical education research on teaching proofs. Summer is a great time to reflect on what is working in courses and what needs to be tweaked or improved. As always, I am grateful to colleagues in the NE-Commit community for sharing their ideas, their challenges, and their experience.

Rethinking Proof Teaching by Maria Fung

 

This is the third in a series of five blog posts from members of NE-COMMIT’s Summer 2026 Proofs Literature Reading Group, written in response to the call: For those of us who were in this summer’s group reading proof literature, write a bit about what you got from the group, such as articles you found interesting or plans for the fall. Our audience is NE-COMMIT and anyone else curious about improving teaching mathematics.

After a spring semester in which her students’ gaps were hard to see, Maria is rebuilding her Real Analysis assessments around a local/holistic framework for proof comprehension, including a 60/40 split between analyzing and writing proofs and an oral final exam.

The series: The posts move from how the group started, to changes in proof courses, to ideas for non-proof courses, and finally to the theory behind it all.

1. Proofs Literature Reading Group, by Debbie Borkovitz
2. Teaching Proofs through Critical Reading, by Dana Rowland
3. Rethinking Proof Teaching: Reading Proofs as Much as Writing Them, by Maria Fung (this post)
4. Possible Uses of Comprehension Questions in Non-Proof Service Courses, by Michael Nathanson
5. What “I Don’t Understand This Proof” Means: A Cross-Walk of Four Theories, by Ileana Vasu

In the Spring 2026 semester I taught the transition to higher mathematics course at Worcester State University, Theory of Proof, for the first time in three years. And it felt different. The students seemed to be struggling with background knowledge that previous groups had been comfortable and familiar with. For example, a lot of them were confused about the closure properties of integers that appear in many of the elementary proofs (for example showing that the sum of two odd integers is even). It was very difficult for me to know where gaps in their knowledge would show up. This was most obvious when students were commenting on their peers’ work on the board. Often it would take serious prompting from me to point them to the sticky parts of their work. So I welcomed the Proofs Group that Debbie Borkovitz put together this summer as an opportunity to get ideas from the mathematics education research of how to improve the learning of proofs for my students.

The article that made the strongest impression on me was” An Assessment Model for Proof Comprehension” by  Mejia-Ramos et al. I loved the framework presented in this article. It has two distinct parts--local and holistic. The local part includes several subparts--identifying the meaning of terms and statements; overall structure of the proof (is it direct, by contradiction, by induction or a combination of these methods); and how each step follows from the previous and determines the next. This way of thinking and decomposing proofs provides a very useful map to student understanding. Thinking back to my experiences from the spring semester, I believe students had most trouble with the first part. As I go forward I intend to spend a lot more time emphasizing definition, axioms, assumptions, and properties that are often implicit in proof writing. 

The holistic part is divided into first being able to summarize a proof or some of its subparts; then being able to identify the distinct parts or modules that make up the proof; and finally transfer the main ideas of the proof to other contexts. This holistic part of the understanding requires a higher level of thinking and I suspect it would take time for students to cultivate the skills necessary to be able to do this type of analysis. Yet asking students to summarize a proof or provide an example of where it does and does not apply provides valuable insight into their thinking and potential misconceptions.

In my Real Analysis course this fall, in class, I plan to invite students to perform this type of analysis on proofs that are both generated by other students or perhaps even by the AIs. I anticipate that we will discuss all the elements of this analysis on multiple occasions, so that students get experience with this type of work.  I am hoping to get students to read mathematics in this fashion--pausing and thinking about each word, each step, each logical connection, and each big idea. 

This means that I am restructuring my assessments to include 60% of analysis of proof using the above framework, and 40% constructing proofs. I believe this shift will give me much more information about students’ thinking and comprehension of the proof process and allow me to decide which ideas require more time and discussion.

Moreover,I would also like to incorporate an oral portion of the final exam in the course, where I will invite students to choose a proof from the course from the big list of important results and perform the holistic part of the analysis on it.  I am really looking forward to incorporating these changes into my course and seeing in practice how this framework hopefully leads to deeper understanding and learning.

Possible Uses of Comprehension Questions in Non-Proof Service Courses by Michael Nathanson

This is the fourth in a series of five blog posts from members of NE-COMMIT’s Summer 2026 Proofs Literature Reading Group, written in response to the call: For those of us who were in this summer’s group reading proof literature, write a bit about what you got from the group, such as articles you found interesting or plans for the fall. Our audience is NE-COMMIT and anyone else curious about improving teaching mathematics.

Michael brings the comprehension-question frameworks out of the proof course and into business math. He shares a linear programming test question you can borrow, along with an honest note on how it went this fall.

The series: The posts move from how the group started, to changes in proof courses, to ideas for non-proof courses, and finally to the theory behind it all.

1. Proofs Literature Reading Group, by Debbie Borkovitz
2. Teaching Proofs through Critical Reading, by Dana Rowland
3. Rethinking Proof Teaching: Reading Proofs as Much as Writing Them, by Maria Fung
4. Possible Uses of Comprehension Questions in Non-Proof Service Courses, by Michael Nathanson (this post)
5. What “I Don’t Understand This Proof” Means: A Cross-Walk of Four Theories, by Ileana Vasu

I teach introductory math courses for first-year students, including introductory calculus and a business math course surveying an array of elementary topics. My students do not write any formal proofs, but they do work to make their thinking visible, and our online conversation about Conradie and Frith's “Comprehension Tests in Mathematics”  resonated with the desire to promote mathematical communication and understanding in all my courses. The authors’ specific focus on comprehension seems especially urgent at the moment. Geillan Aly and Carly Briggs led an excellent NE-COMMIT workshop last year on mathematical literacy, which in part focused on students’ declining abilities in close reading (which includes mathematics), and this feels like as big an impediment as any to students’ mathematical  success. 

When we ask students to produce a proof or calculation on an exam, we are simultaneously testing two abilities–students need to read and understand what they are being asked for before they can demonstrate their ability to produce the desired result. If a student is unsuccessful answering a question like this, it may not be immediately clear which step was the stumbling block. The authors of “Comprehension Tests in Mathematics” describe targeted efforts to assess understanding in such a way that student errors illuminate where the stumbling blocks are and allow for useful feedback. 

The first part of my business math class covers linear models followed by linear programming. The standard LP questions on my tests mostly come in two flavors: (a) Set up an LP problem given a written description of a situation (but don’t solve it); and (b) Given a partially-completed graphical solution to an LP problem, finish answering it. Separating the process into two questions is more manageable for students, and I get more meaningful results. But currently there is not much opportunity to consider a problem in its entirety.

In questions where students are asked to set up an LP problem, this comment about traditional proof-writing questions really resonated:

The weaker student may receive very little feedback from being tested in this way: if the answer provided is sufficiently garbled the marker will find it very difficult to indicate in any reasonable way those areas in which the student has shown lack of understanding. (Conradie and Frith, p. 226)

Inspired by the article, I am imagining possible questions that provide a solved LP problem and ask students to comment or interpret. What would set these apart from questions I already ask is that they are clearly asked from outside of the situation described–they are meta-questions about the process of solving problems. Or at least they aspire to be. I have included a draft version of such a question after the final paragraph. 

In “An Assessment Model for Proof Comprehension in Undergraduate Mathematics” by Mejia-Ramos, Fuller, Weber, Rhoads, and Samkoff (2012), the authors build on the framework of Conradie and Frith to classify different typers of comprehension questions, in particular distinguishing local from holistic questions. I have used their language in commenting on the sample problem. 

As pointed out by the authors (page 229), these questions will take time for students to complete–though why the authors seem surprised by this is unclear. Students need to read, understand, and before writing anything significant. Having a set of focused, short-answer questions will help with this, but only after they have taken time to familiarize themselves with the problem. Likewise, the authors’ admonishment to be precise is crucial–if we ask students to do things outside of their mathematical experience, we need to help them see what we are looking for. 

As is so often the case, the lessons from one specific context (proof-based courses) are more generally applicable than might be initially assumed, and the article and discussions about proofs reinforced my commitment to support student reading comprehension in general and suggested a path to bring this into tests and in-class activities. I will continue to think about ways to use the specificity of the authors’ thinking to sharpen my test questions. 

SAMPLE TEST QUESTION #1 (Linear Programming)

Read the following constrained optimization problem. Do not solve it! Instead, answer the comprehension questions that follow. 

Problem:

Fiona Apple and Chuck Berry have a business selling fruit salad. They have two different recipes: The Banana Boat and the Grape Train Robbery. The required ingredients are listed below, with all numbers given in pounds of fruit. They can sell a batch of Banana Boat salad for $15 and a batch of Grape Train Robbery for $12. Currently in their storerooms, they have 20 pounds of bananas, 16 pounds of strawberries, and 13 pounds of grapes. 

	Banana Boat	Grape Train Robbery
Bananas	2 lbs	1 lbs
Apples	1 lbs	0.5 lbs
Grapes	1 lbs	2 lbs

How much of each type of salad should they make if they want to maximize their revenue?  

Solution: 

1. The goal is to maximize P = 15x + 12y

Subject to the constraints:

2. 2x + y ≤ 20

3. x + 0.5y ≤ 16

4. x + 2y ≤ 13

5. x≥0 and y≥0

6. By plugging into an online solver, we find the solution that P = 159 at the point (9,2).

Questions:

1. This solution neglects to define its variables. What do the variables x and y represent here (including units)? How do you know which one is which? 

2. Translate the given solution (6) into an English sentence about fruit salad, using appropriate units. 

3. If Fiona and Chuck make the optimal amounts of fruit salad, how much of each type of fruit is left in their storeroom? 

4. Why is it necessary to include line (3)? What aspect of the problem is it reflecting? 

5. Why is it necessary to include line (5)? What aspect of the problem is it reflecting? 

6. If they raise the price of each serving of Banana Boat for $16 each, which line(s) in the solution need to be changed? Explain clearly. 

7. Is it feasible for them to make no Banana Boats and 10 servings of Grape Train Robbery? Answer clearly in a way that demonstrates that you know what feasible means. 

8. Suppose you did not have an online solver available. How could you find the optimal solution to this problem by hand, taking advantage of the fact that there are only two types of salad? List three most important steps you would take. 

Comments: It was harder to restrict myself to comprehension questions than I thought it would be. Questions 1 - 3 and 6 might be classified as holistic questions – they challenge students to consider aspects of the whole solution. #4 and 5 are local questions, focusing on interpreting a specific step. #7 and #8 are more typical types of questions that do not really respond to the provided solution and thus don’t fit into Conradie and Frith’s framework. But they resonate with the first six questions, I think. 

Follow-up comment from Fall 2026: In practice, I asked some very light versions of my aspirational questions. I did include the worked-out solution to the problem including a graph; but the questions I asked about it were pretty mundane. This is a work in progress.

What “I Don’t Understand This Proof” Means: A Cross-Walk of Four Theories, by Ileana Vasu

 This is the fifth in a series of five blog posts from members of NE-COMMIT’s Summer 2026 Proofs Literature Reading Group, written in response to the call: For those of us who were in this summer’s group reading proof literature, write a bit about what you got from the group, such as articles you found interesting or plans for the fall. Our audience is NE-COMMIT and anyone else curious about improving teaching mathematics.

Students can follow every line of a proof and still not be able to say what it proves. Ileana asks why, drawing on four learning theories to show that “I don’t understand this proof” can mean very different things, each calling for a different kind of help, with practical classroom moves for Real Analysis and beyond.

The series: The posts move from how the group started, to changes in proof courses, to ideas for non-proof courses, and finally to the theory behind it all.

1. Proofs Literature Reading Group, by Debbie Borkovitz
2. Teaching Proofs through Critical Reading, by Dana Rowland
3. Rethinking Proof Teaching: Reading Proofs as Much as Writing Them, by Maria Fung
4. Possible Uses of Comprehension Questions in Non-Proof Service Courses, by Michael Nathanson
5. What “I Don’t Understand This Proof” Means: A Cross-Walk of Four Theories, by Ileana Vasu (this post)

I keep coming back to a moment from Conradie and Frith's "Comprehension Tests in Mathematics" (2000). They gave students a full proof that a bounded, decreasing sequence is Cauchy, and asked questions about it. Most students could follow every line: why line 4 follows from line 3, why the epsilon works. But asked the simplest question of all, what does this actually prove, a surprising number got it wrong. Some named a stronger, more general claim than the one actually proved. Some just couldn't say.

This is a story about what "understanding a proof" is made of. And it turns out to have layers most of us don't usually pull apart. Our reading group spent time on three papers that each poke at those layers differently: Pirie and Kieren's (1994) "onion" model of how mathematical understanding grows, Conradie and Frith's (2000) comprehension-testing method, and Mejía-Ramos and colleagues' (2012) model of what it means to comprehend a written proof. I went further down the rabbit hole to other frameworks that belong in the same conversation: Dubinsky's APOS theory (Dubinsky & McDonald, 2001), Tall's Three Worlds (Gray & Tall, 1994), and empirical work by Selden & Selden (2003) and Weber & Alcock (2004) on how students validate and write proofs. None of these researchers were in conversation with each other when they built their frameworks, but lined up side by side, they're clearly circling the same thing.

Here's what I think is going on: "doesn't understand the proof" isn't one failure. It's a level, and these theories already give us names for the levels. A student who can't follow individual steps needs different help than one who follows every step but can't say what it proves, and both need different help again from a student who writes a flawless, correct proof with basically no idea what any of it means. Most theory courses have one move for all three: reread the proof, or reproduce it is like prescribing the same treatment for different problems just because they all present, at a glance, as falling short of some ideal.

While reading Pirie and Kieren we ran into Herscovics, who had a related but different idea: rather than naming a student's current level, he named the specific obstacle blocking the next one,  a concept he borrowed from the philosopher Bachelard. Level tells you where a student is; obstacle tells you what's actually stuck. Both turned out to be words I needed by the end.

The frameworks

Here's a quick summary of the frameworks I've crossed paths with, some from this reading group, some from my own frolic in math ed.

• Pirie & Kieren's (1994) "onion" model. Understanding grows in nested layers, from concrete (a physical action or mental image) to abstract (a formal system reasoned about on its own terms). The key claim isn't just that there are layers — a learner can always "fold back" to an earlier one, and can't skip straight to the outer layers without passing through the inner ones.

• Dubinsky's APOS theory (Dubinsky & McDonald, 2001). A more mechanical four-step version: Action (needs an external cue), Process (runs in your head now), Object (the whole process becomes one thing you can operate on), Schema (objects and processes organized into one structure). Popular in university math education because it's concrete enough to design a lesson around.

• Tall's Three Worlds (Gray & Tall, 1994). Less a ladder, more three coexisting modes of thinking: embodied (things you picture or act out), symbolic (things you manipulate with notation), and formal-axiomatic (things defined purely by properties and proof). A skilled mathematician moves between all three depending on the task rather than "graduating" out of the earlier ones — which Tall argues is exactly what separates novices from experts.

• Mejía-Ramos et al.'s (2012) proof comprehension model. The most hands-on of the four, built to answer "what does it mean to understand this proof I just handed you?" It splits comprehension into knowing what terms mean, justifying steps, understanding logical structure, summarizing the big idea, and applying the method elsewhere.

Looking at these side by side, a pattern jumps out: every one of them treats turning a process into a thing you can operate on as the hardest, most different-in-kind step in the whole progression. Sfard (1991) calls it reification, Dubinsky calls it encapsulation, Pirie and Kieren call it formalising, Tall calls it exploiting a procept.. Here's roughly how they line up, using the plain-language stage names I propose as the left column:

Stage	Pirie & Kieren	APOS	Tall	Mejía-Ramos et al.	Example
Example	Image making / having	Action	Embodied	Meaning of terms	Tanya covers fraction pieces
Procedure	Property noticing	Process	Embodied to Symbolic	Justifying one step	Follows the Cauchy proof, misses the whole
Object	Formalising	Object	Symbolic (procept)	Works for any case	Tanya: "works for them all"
Structure	Structuring	Schema	Formal-axiomatic	Summarizes the big idea	States what the Cauchy proof establishes

Footnote: One thing that's bugged me the whole way through: every time I want to talk about "row three," I borrow someone else's word: formalising, encapsulation, procept, and none of them belong to all four rows equally. I wanted one plain, framework-neutral name per row, the kind a mathematician would reach for without remembering whose theory it's from.

My attempt was to basically borrow from Dubinski  and come up with: 1. Example, 2. Procedure, 3. Object, 4. Structure.  This mirrors how mathematicians already talk about growing familiarity with something: a worked example, fluency with a procedure, a genuine object you can hand to someone else, and eventually a structure it sits inside. A leaner alternative: 1. Instance, 2. Recipe, 3.  Concept, 4.  Theory.  Neither has been tested on a reader who isn't me: take it as a proposal, useful mostly as shorthand for the rest of this post, not a claim that it improves on what anyone else already built.

It's one thing to lay out a table like that, and another to watch it happen in someone's actual answer. So let's walk up the ladder with real examples.

At the bottom, understanding is inseparable from the action that produced it. Tanya's early strategy for adding fractions: "if I can find a piece the addends fit on, I've found the sum"  only works because fractions are, for her, literally pieces of paper. A student reading a proof at this level can tell you what line 4 says because they can re-run it, not because they've pulled anything general out of it.

One level up, the student has noticed a pattern but it's still glued to the procedure that made it, where most of Conradie and Frith's "follows every step, misses the whole" students sit. They can say why line n follows from line n − 1, but haven't turned the sequence into one thing they could summarize or reuse.

Then comes the jump the Cauchy-proof story is really about: a student's methods start working "for all" cases, independent of whatever image got them there. 

Tanya can say this herself. Bart, the eight-year-old in Pirie and Kieren's (1994) fraction study, is a stranger and more interesting case. Working a missing-fraction problem, he freely adds and subtracts numbers, some not even from the class's kit, until he lands in range, and when asked to explain, all he offers is "I would just add things and subtract things until I got a little below three fourths." What makes the authors call it formalising is almost entirely negative: he never refers to a physical image, and nobody else in the class can follow what he's doing. Bart hasn't told anyone the big idea. Conradie and Frith's "what does this prove?" question is really aimed at Tanya's kind of formalizing, the articulable kind, which is why it's such a good diagnostic: it catches students at Bart's stage too, just for different reasons.

The table can still mislead us: work that slots neatly into the Object row doesn't reliably tell you where a student's thinking sits. Weber and Alcock (2004) watched students construct proofs and found two styles. In semantic production, the student pictures an actual instance of the objects while writing - a concrete group, checked at each step. In syntactic production, the student pushes symbols around by valid rules, never asking what they stand for. A syntactic proof can be completely correct, grammatically fluent, every line following logically, while the student who wrote it has no real picture of what any of it means. This is where Herscovics's (1996) obstacle framing earns its keep: the problem isn't an inability to manipulate symbols, it's a specific, nameable gap, never grounding the symbols in a concrete example that fluent symbol work can hide rather than fix. There's a real parallel to Bart here, though I shouldn't overstate it: the paper never tells us whether Bart had built grounded understanding before he stopped needing it, or jumped straight to formal manipulation. What's clear is that a syntactic prover and Bart can look identical from the outside, while whether there's real understanding underneath stays open in both, fluency alone doesn't answer it.

If that doesn't unsettle you, this might. Selden and Selden (2003) gave undergrads four "proofs" of the same theorem, only one valid, and asked them to sort out which. Students judged mostly by surface feel - does it look rigorous - rather than logical structure; some accepted a proof of the converse as if it proved the theorem itself. Stranger still: when Inglis and Alcock (2012) and Weber (2008) gave the same borderline proofs to actual mathematicians, they didn't fully agree either,  in one version, five of twelve called a flawed proof valid, and five of twelve called a valid proof flawed. Even the top of this ladder isn't the clean, universally-agreed finish line it looks like.

How the frameworks compare

It would be misleading to present the table as four groups who independently discovered the same theory. They're answering related but different questions, and it's worth being precise about how.

Pirie-Kieren and APOS are closest in spirit: both trace one learner's understanding of a single concept growing over time, and both build in regression. Pirie and Kieren call it "folding back"; in APOS, a student's Object-level fluency can break down under pressure, forcing them back to re-running the underlying Process. Where they differ is grain size: Pirie and Kieren name eight layers to Dubinsky's four, so APOS is really a coarser-grained version of the same climb. This useful precisely because it's easier to hold in your head while you're watching a student work.

Tall's Three Worlds is a different kind of theory altogether. It is not a ladder climbed once, but three modes that coexist and get invoked according to the task. That's a real difference from Pirie-Kieren and APOS, not just a stylistic one: the same student can look like they're at different rows depending on the topic, which the strictly nested models don't build in as cleanly. A mathematician can be embodied about geometry and formal about analysis in the same afternoon without that being a contradiction in Tall's framework, where it would be an odd case of "folding back" in Pirie and Kieren's.

Mejía-Ramos and colleagues are narrower than all three: not a theory of how understanding grows over months of teaching, but a model of what it means to comprehend one specific, already-written proof someone hands you. Its dimensions still map cleanly onto the other frameworks' rows -  that's this whole post's argument,  but it was built from a different starting point entirely: practical assessment design. That the two routes converge on the same rough structure is part of why I think the convergence is worth noticing rather than assuming.

So the table is a useful simplification I'm genuinely fond of, not a claim that these are secretly the same theory in different clothes. I also went back and forth on whether Sfard (1991) deserved her own column, since reification is arguably the earliest clean statement of the whole process-to-object idea. I left her out: the other four each name three to seven distinct steps that map onto four rows, while Sfard's theory is really about one transition, operational versus structural, process versus object. She's doing something narrower, and perhaps more foundational, than a rung on someone else's ladder.

What they say about circling back

None of these theories treat growth as a one-way climb, and it's worth being specific about how each one builds in reversal, since the mechanisms are genuinely different from each other, not just restatements of "sometimes people regress."

Pirie and Kieren are the most explicit about this, and they give it a name: folding back. When a learner hits a task that outstrips their current understanding, they don't just fail, they drop back to an earlier, more concrete layer, rework it, and climb out again, usually with a stronger grip than before. Folding back isn't a detour from the theory; it's built into the model as a normal, expected, even necessary part of growth. A student who suddenly needs to re-draw a picture to handle a proof they thought they'd formalised isn't losing ground. On Pirie and Kieren's account, that's what real growth looks like from the inside.

APOS has a quieter version of the same idea, usually called de-encapsulation: a student who has compressed a process into an Object can, when the task demands it, unpack that Object back into the underlying Process to work with it directly. A function a student normally treats as a single thing you compose or differentiate might need to be unpacked back into "the rule that takes an input and produces an output" to answer a question the Object-level view doesn't easily support. Dubinsky treats this as a sign of a robust Schema, not a fragile one,  the object and the process that built it stay connected and available in both directions.

Tall's coexisting worlds, discussed above, are really a stronger version of the same claim: not just that a person can fold back or de-encapsulate under pressure, but that fluent movement between embodied, symbolic, and formal thinking is the normal, ongoing state of expert practice, not a fallback reserved for when something goes wrong.

Herscovics's obstacles add one more piece, and it's a slightly unsettling one: an obstacle a student has cleared in one context can resurface in a new one. This is a direct import from Bachelard, who observed that epistemological obstacles in the history of science don't get permanently defeated so much as out-grown for a specific range of problems, only to reappear when the range expands. A student who has cleanly separated "definition" from "assumption" in the context of irrational numbers may make exactly the same conflation again the first time they meet a definition in group theory. 

Using this pragmatically in the classroom

The practical upshot is that circling back needs to be built into how a course runs, not treated as a symptom to eliminate.

• Say the quiet part out loud to students. Tell them explicitly, early in a proof-heavy course, that reaching for a picture, a small example, or a concrete case mid-proof isn't a sign they've failed to "get it", it's Pirie and Kieren's folding back, and it's what strong students do more, not less, when a formal argument gets away from them.

• Build de-encapsulation into practice, not just assessment. Occasionally ask students to unpack an object they're fluent with:  "walk me through what this theorem actually says step by step, as if you were about to derive it from scratch", even when you already trust they can use it. A student who can't do this on demand has an Object that isn't robustly connected to its Process, and that's worth catching before an exam does it for you.

• Teach the switch itself, not just each side of it. Following Sfard, give students the same problem twice: once asking for the step-by-step operational account, once asking them to reason about the object as a whole. The ability to move between the two on request is a skill in its own right, and most courses never name it, let alone practice it directly.

• Re-test old obstacles in new contexts. If a misconception showed up once (say, defining √20 as "m/n") and got corrected, don't assume it's gone. Herscovics's account predicts it can reappear the moment the surrounding content changes: a supremum, a group element, a limit in a new setting, so it's worth a quick, low-stakes check the first time a familiar-shaped obstacle has a new place to hide.

I've leaned on the Cauchy-sequence example this whole time, but real analysis might be the best subject for watching this ladder in action:  it's the course where students first have to treat things they used to just do (take a limit, check continuity) and transform them to objects with formal definitions they can reason about, prove things about, and doubt.

Take the limit itself. At the bottom, a student's sense of "the limit of a_n is L" is just "the numbers get closer and closer to L"  an image, not a formal object. That's Tall's embodied world, or any of the other three frameworks' equivalent.

The epsilon-N definition is the formalizing move: for any epsilon you hand me, I can find an N that works. Plenty of students can recite this and fill in a specific epsilon-N proof by pattern-matching to one they've seen, property noticing, still glued to the template, without "for any epsilon" yet feeling like a claim about all epsilons. You hear the difference: "I plug in epsilon and solve for N" (a procedure) versus "I need N to work no matter how small epsilon gets" (the object has clicked). The classic mistake here is quantifier order:  a student who picks N first and then tries to make it work for "an" epsilon, rather than starting from an arbitrary epsilon and building N from it, is running a memorized procedure rather than reasoning about what the claim says. It's an error that survives correct-looking algebra, because each step can still be valid even once the logical order that gives the whole thing meaning has quietly flipped.

If you're teaching a proof course 

• Don't only test local steps. It's the easiest comprehension question to write ("why does line 4 follow from line 3?") and the least diagnostic. Add a "what does this actually prove, in your own words" question: cheap to write, and it catches a failure mode step-by-step questions miss entirely.

• Occasionally ask students to validate, not just comprehend. Give them two or three "proofs" of the same result, only one correct, and ask which really proves it. Selden and Selden's (2003) finding, students grade by surface polish rather than logical structure, and this catches something a normal comprehension test never will.

• Watch for suspiciously clean written proofs. A student who reliably produces correct, well-formatted proofs isn't necessarily further along than one who's struggling. Ask them to explain a step out loud or apply their method elsewhere. If they can't, that's the syntactic-production trap (Weber & Alcock, 2004), more practice writing proofs won't fix it; they need the concrete picture they skipped.

• Tell students what you're testing for. Conradie and Frith (2000) found students overwhelmingly preferred comprehension-style testing once they understood its purpose, even though they found it harder. Explaining why probably goes a long way toward students taking it seriously instead of treating it as a trick question.

Where this is headed

I think there's a real paper in here - the Cauchy-proof puzzle that started this off deserves an actual answer, not just a table. Keep in touch, and if you're interested in collaborating, let me know: ivasu@smith.edu

References

Conradie, J., & Frith, J. (2000). Comprehension tests in mathematics. Educational Studies in Mathematics, 42(3), 225–235.

Dubinsky, E., & McDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton (Ed.), The Teaching and Learning of Mathematics at University Level: An ICMI Study (pp. 275–282). Kluwer.

Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141.

Herscovics, N. (1996). The construction of conceptual schemes in mathematics. In L. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of Mathematical Learning (pp. 351–379). Lawrence Erlbaum.

Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43(4), 358–390.

Mejía-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79(1), 3–18.

Pirie, S. E. B., & Kieren, T. E. (1994). Beyond metaphor: Formalising in mathematical understanding within constructivist environments. For the Learning of Mathematics, 14(1), 39–43.

Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4–36.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.

Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39(4), 431–459.

Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(2–3), 209–234.