Why Should that Convince Me?: Teaching Toulmin Analysis Across the Curriculum

Experienced provers employ a host of skills when assessing the validity of a justification, often without names for those skills. This paper offers an introduction to a lens called Toulmin analysis that can help make sense of this process. Then this paper describes both an in-class module to help students learn to apply Toulmin analysis and multiple ways this module can be integrated into courses for mathematics majors and non-majors. Toulmin analysis has helped us bring human dimensions of proving to the surface in our classrooms, and it helps us build stronger and more inclusive entryways for students into mathematical communities. Learning about Toulmin analysis helps our students connect mathematical reasoning to other thinking, supporting transfer in both directions, and it has changed our thinking about the teaching and learning of mathematics even in contexts where we do not devote class time to this model overtly.


INTRODUCTION
The public discourse around mathematics implicitly, and sometimes explicitly, asserts that our discipline is concerned with truth and correctness, and many assessment and pedagogical practices bake this assumption into students' experiences. As a result, many of our students are surprised to learn that when professional mathematicians evaluate a justification, we are looking for more than a list of true and correct statements; we are looking for the ways the ideas connect and build on one another. The essential question for the work of mathematical justification is not "Is this claim true?", but rather "Is this justification valid?".
Much of the work that experienced provers do to read and write mathematical justifications both is outside of many of our students' previous experiences with math and is internal, unnamed, and invisible to an external observer (e.g., [1,4]). Moreover, experienced provers tend to do much of this work on autopilot, which makes it even harder for instructors to effectively communicate what is happening to a student. Failing to scaffold student learning of these critical skills is a mechanism of exclusion from mathematics communities. We have seen instructors and curricula work to make some of these invisible habits visible by facilitating detailed discussions of what constitutes a mathematical claim, of patterns in proof structures, and of general principles of logical deduction. In this paper we propose introducing Toulmin's model of argumentation, which extends and complements these kinds of efforts, to support students in learning the internal and often invisible steps we take to analyze the validity of a mathematical justification.
Toulmin analysis is a model of argumentation developed originally in the context of public rhetoric for analyzing the validity of an argument [9] that has been applied to mathematical justifications (e.g., [5]). This paper is a response to a call by Weber and Alcock [11] for mathematics instructors to teach validation explicitly in their courses. Toulmin models validation as a line-by-line analysis in which the reader connects claims with both the general principles used to justify them and the established specific information required to apply those principles.
The work here is split into two different parts: Section 2 describes Toulmin's model for argumentation and how students can benefit from using this model to critically analyze the mathematical justifications they write and encounter. Section 3 describes how we have implemented activities designed to get students started working with Toulmin's model and both integrated and leveraged this learning in a variety of mathematics courses for majors and non-majors. Section 2 of this paper is written to students but is meant for both students and instructors. We open in Section 2.1 with motivation for a focus on validity, as opposed to truth or correctness, in mathematical justification. Our goal is to make explicit, for students and instructors, the shift in thinking required for students to take up these habits of analysis, which in turn creates a need for a model of argumentation like Toulmin's for both audiences. We follow in Section 2.2 with a description of the process of Toulmin analysis along with an example -also written to a student, but with instructors in mind as well. Finally, in Section 2.3 one author (Vanessa) describes her experiences learning to use Toulmin analysis as a student and how internalizing the model and its inherent messiness impacted her thinking about mathematical justification and proof. Taken together, Section 2 is intended to be an accessible introduction to Toulmin analysis. Section 3 is written to instructors and begins with an overview of how we have implemented classroom activities designed to support students in learning to use Toulmin analysis. Section 3.1 details our implementation of a 1.5-3 day module for teaching Toulmin analysis (which includes asking students to read Section 2 of this paper) adaptable for a wide variety of mathematics courses. We follow in Section 3.2 with a discussion of the collective wisdom we have accrued teaching this subtle and complex topic -in particular in response to the inherent messiness of proof validation. And because this kind of learning is most impactful when it is motivated, leveraged, and situated in the larger course, we finish Section 3 with detailed descriptions in Section 3.3 and 3.4 of how and why we have integrated the work of Toulmin analysis into a Math for Liberal Arts (MLA) course and several proof-based courses. Sections 3.3 and 3.4 can be read independently.

TO STUDENTS
Section 2 describes Toulmin's model for argumentation and how we can use this model to critically analyze mathematical justifications we write and encounter.

What's Truth Got to Do with It?
It turns out, mathematics is not just about truth, if that word even means anything. In particular, when your teachers are looking at your reasoning and justifications, they are rarely interested in whether the conclusions are "true" -they are looking for something else. We hope that this paper will help you better understand what else beyond "truth" someone might look for in another's reasoning, why they might be doing it, and how you might do it yourself.
First, let's consider two different examples of justifications to try to convince you that truth is not the goal. These examples should be mildly unsettling.

Example 1:
Consider two non-zero numbers a and b and suppose that a = b. Then a 2 = ab, by multiplying both sides of the previous equation by the same value. So a 2 − b 2 = ab − b 2 , by subtracting the same value from both sides of the previous equation. Which can be restated as (a − b) * (a + b) = (a − b) * b using factoring. Therefore a + b = b, because we simply divided both sides by the same value. However, using the fact that a = b, we get b + b = b, which is equivalent to 2b = b. Because b is non-zero, this means that 2 = 1.
Claim 2a: 7 is prime. Claim 2b: If 7 is prime, then 107 is prime. Claim 2c: Therefore 107 is prime. Example 1 appears to be a justification that 2 = 1, which is likely something that you believe is false. What's worse: every single phrase of it is "true" in a reasonable sense, so how can that be? If you're not familiar with this trick, there's a problem in the connection between the fourth and fifth lines (which is hinted at in the last line): we are only able to divide by non-zero numbers, but (a − b) = 0 because a = b. In the context of algebra, we make these kinds of moves all the time without checking that they are valid. In fact, dividing both sides of an equation does preserve the equality, but we were not able to do that division in this case. The point is that each individual phrase we actually wrote (or said out loud) seems true in isolation but produced something that seems false together. So, when we evaluate a justification, we must look at the statements in context, and we must attend to the connections between the statements.
Perhaps the issue with zero in Example 1 feels like a trick or a pedantic technicality, but it turns out there is a lot going on here. For example, consider two activities, one that takes 3 hours and one that takes 9 hours, and imagine doing each one 4 times in a row. Just looking at the starting and ending times on a clock, both of these scenarios would start and end at the same clock position. So there is a sense in which 4 * 3 = 4 * 9; you cannot "cancel" the 4 on both sides of this equation either, and in this situation there are no zeroes! [If you study Number Theory or Abstract Algebra, you'll think more about this.] We hope this example helps you notice a shift away from the question "Is this claim true?" and toward "Is this justification valid?." In particular, reading Example 1 only for truth ends with a vague sense that something must have gone wrong because the last line is false, so we need a different way to read justifications like this example.
In contrast, Example 2 is likely to feel like it contains only true claims that are applied appropriately but that it does not actually justify the conclusion that 107 is prime. In our experience, people take issue with the second line. Before we delve into that reaction, let's consider a similar justification.

Example 2 :
Claim 2 a: Socrates is a human. Claim 2 b: Humans are mortal. Claim 2 c: Therefore Socrates is mortal.
If we represent "Socrates is a human" with P, represent "Socrates is mortal" with Q, and narrow the scope of the middle claim, we get an abstracted version.

Example 2 :
Claim 2 a: P. Claim 2 b: P implies Q [Or: If P, then Q.] Claim 2 c: Q. Example 2 is sometimes called a syllogism or modus ponens. If you accept an If-Then statement "If P, then Q." and you accept that P is true, then you must accept that Q is true. In the case of Example 2, we do accept that 7 is prime, but people take issue with Claim 2b: If 7 is prime, then 107 is prime. People usually see a reason to accept Claim 2 b based on the belief that humanity causes mortality but see no such causality for Claim 2b.
So, what does it mean for a claim of the form "If P, then Q" (or equivalently "P implies Q") to be true? You are probably familiar with theorems from your previous math work. For example: "If a quadrilateral has all four congruent sides, then it is a parallelogram." We generally say that a theorem (If P, then Q) is true when we can deduce the conclusion (Q) using only hypotheses (P), previously accepted truths, and logic. Stepping back a little further, there are four possible situations for "If P, then Q" depending on whether P and Q are each true or false. Let's use a promise as an analogy to analyze If-Then statements. Consider this promise: "If we have class on Saturday, I'll bring cookies." I brought cookies to class.
I did not bring cookies to class. We had class on Saturday.
My promise is fulfilled. My promise is broken. We did not have class on Saturday.
My promise is fulfilled (and more). My promise is fulfilled because the context never came up.
There are lots of things to notice about this table, but three pop out. First, if I bring cookies to every class, my promise will definitely be fulfilled. Returning to the abstracted forms: If Q is true, then "If P, then Q" is true without having to think about P at all! It's perfectly fine to feel that the promise was a little silly if I was planning to bring cookies every day, but it would definitely preserve the honor of my promise. Second, if we never have class on Saturday, my promise is also fulfilled. Returning to the abstracted forms: If P is false, then "If P, then Q" is true without having to think about Q at all! The only way for me to break my promise is to have class on Saturday and fail to bring cookies. Third, this interpretation of If-Then statements is a little different from how they are often interpreted in casual conversations. For example "If you pick up your toys, you can play video games." is often informally taken to mean both (i) picking up the toys guarantees access to the video games and (ii) failing to pick up the toys will deny access to the video games. In other words, in casual language "If P, then Q" is sometimes used to mean "P if and only if Q," but in more formal mathematical contexts, we mean only (i). Yes, one of us has a 6-year-old child.
Summarizing, here's how we interpret the truth of the statement "If P, then Q." in formal mathematical contexts: Q is true.
Q is false P is true.
"If P, then Q" is true. "If P, then Q" is false. P is false. "If P, then Q" is true.
"If P, then Q" is true.
Returning to Example 2, according to this analysis, Claim 2b is true because its conclusion ("107 is prime") is true. So Example 2 has the structure of a syllogism/modus ponens: P is true. If P is true, then Q is true. Therefore Q is true. The point here is that just looking at "truth" does not explain why almost everyone finds this justification unacceptable.
Weber and Alcock [11] gave (essentially) Example 2 to several mathematics faculty, who also rejected it based on validity. In particular, they looked at Claim 2b and asked themselves if there was a reason to believe it, considering statements such as "If p is prime, then p + 100 is prime." This potential statement is not true because 5 is prime but 105 (= 5 * 21) is composite, so they rejected the whole justification from Example 2. In other words, three true statements that are logically connected in a syllogism to conclude the final statement was rejected as not a proof. So the truth of these statements cannot be what the mathematicians were assessing as they read! Amusingly, the evidence for this claim about the mathematicians' thinking is actually stronger than the previous paragraph indicates. Weber and Alcock [11] actually gave the mathematicians a version of Example 2 with the numbers 7 and 1007 . . . but 1007 (= 19 * 53) is composite, not prime. So the mathematicians (and the researchers) all wrongly believed that all three statements were true, but the mathematicians rejected the justification based on the gap of a missing reason to believe Claim 2b instead. They could have rejected this justification based on truth/falsity, but instead they rejected it based on the absence of a reason to believe the claims that they accepted as true without checking.
These interview responses do not mean that the mathematicians do not care at all about truth. In fact, it's reasonable to view the results of a valid justification as true, so validity supports truth. However these interviews indicate that the professional mathematicians focused on validity before truth. We believe that these kinds of validation behaviors/skills are used by experienced provers to make sense of both formal and informal justifications, that students are (often implicitly) expected to learn these skills that can be invisible to external observers, and that developing habits related to these skills will support students in future success. Therefore, we believe that instructors should intentionally design learning experiences for students around these skills that make the skills explicit and visible, that help students learn to employ them consciously and critically, and that support students in turning these skills into productive habits and ways of knowing.
Taken together, these examples indicate that experienced provers evaluate justifications based on whether each claim is supported by an accepted, general principle or prior knowledge, not whether it is made of true statements. In the next section, we'll talk about what this means in practice.

An Example of Toulmin Analysis
We think that the examples above will resonate with the thinking of many readers, but in our experience many professional mathematicians still do not have words to describe what they are doing when they evaluate justifications. Thankfully, a scholar named Stephen Toulmin developed a model of argumentation [9] that offers good vocabulary in the field of rhetoric, a model that was adapted by Götz Krummheuer [5] for thinking about mathematical justifications. Weber and Alcock [11] summarize it as follows. [Note: We have chosen to use the word "claim" below and in our discussion of Toulmin analysis to separate it from the "conclusion" of an If-Then statement and because "claim" suggests that the statement is still under review by the community.] According to Toulmin [9], an argument consists of at least three essential parts called the core of the argument: the data, the warrant, and the claim. When one presents an argument, one is trying to convince an audience of a specific assertion. In Toulmin's framework (see Figure 1), this assertion is called the claim. To support the claim, the presenter typically puts forth evidence or data. The presenter's explanation for why the data necessitate the claim is referred to as the warrant. At this stage, the audience can accept the data but reject the explanation that the data establishes the claim -in other words, the authority of the warrant can be challenged. If this occurs, the presenter is required to present additional backing to justify why the warrant, and therefore the core of the argument, is valid. (p. 35, edited) This is the big idea of the previous section: evaluating mathematical justifications focuses on validating by checking whether or not the claims are warranted, not on determining whether the claims are true. Weber and Alcock [11] go further to describe the process of evaluating a mathematical justification: We conjecture one's line-by-line verification of a proof might proceed like this. Each line is interpreted as an argumentation whose claim is the statement being asserted. The reader of the proof identifies the data and the warrant used in this proof, inferring them if necessary. If the warrant for the argumentation is socially agreed upon by the mathematical community, this line is accepted as valid. If the warrant is false, the line and the entire proof are declared to be invalid. If the warrant of an argumentation is plausible, but not socially agreed upon by the mathematical community, backing for the warrant is needed and the proof is said to have a "gap" in it. (p. 37, edited) Let's analyze Example 1 to practice applying Toulmin's model (i.e., "doing a Toulmin analysis").

Example 1:
Consider two non-zero numbers a and b and suppose that a = b. Then a 2 = ab, by multiplying both sides of the previous equation by the same value. So a 2 − b 2 = ab − b 2 , by subtracting the same value from both sides of the previous equation. Which can be restated as (a − b) * (a + b) = (a − b) * b using factoring. Therefore a + b = b, because we simply divided both sides by the same value. However, using the fact that a = b, we get b + b = b, which is equivalent to 2b = b. Because b is non-zero, this means that 2 = 1.

Claim
Data Warrant C1: "Then a 2 = ab" D1.1: "a = b," D1.2: a and b are numbers (both assumed in Line 1) W1: "by multiplying both sides of the previous equation by the same value" Performing the same arithmetic/algebraic operation to two (potentially different) representations of the same number produces two (potentially different) representations of a (potentially) new number, in this case multiplication by a.
D2.1: "a 2 = ab" (C1), D2.2: b is a number (assumed in Line 1) W2: "by subtracting the same value from both sides of the previous equation" Performing the same arithmetic/algebraic operation to two (potentially different) representations of the same number produces two (potentially different) representations of a (potentially) new number, in this case subtraction of b 2 . If we wanted to be VERY careful, we might warrant the fact that b 2 is a number too.
C3: "Which can be restated D3.2: all of the terms in this expression are numbers (based on the assumption in Line 1) W3: "using factoring" Factoring, which is the distributive property used backwards, is an assumed property of arithmetic/algebra.
is a number (based on the assumption in Line 1) W4: "because we simply divided both sides by the same value" Performing the same arithmetic/algebraic operation to two (potentially different) representations of the same number produces two (potentially) different representations of a (potentially) new number, in this case division by (a − b). However, (ab) is actually the number 0, and division by 0 is not an allowable arithmetic/algebraic operation, so this is a serious flaw in the justification. As a result, this justification is deemed invalid.
D5.1: "a + b = b" (C4), D5.2: "using the fact that a = b" (assumed in Line 1) W5: We see the validity of substitution as an implicit warrant here. Substituting a (potentially different) representation for a number into an expression produces a (potentially different) representation for that same expression because arithmetic is well-defined.
C6: "which is equivalent to 2b = b" D6.1: D6.2: "b is a number" (assumed in Line 1) W6: We see factoring and potentially other basic arithmetic/algebraic properties as an implicit warrant here. The claim of "equivalence" is warranted by the fact that factoring and distributing are reversible processes.
C7: "this means that 2 = 1" D7.1: "2b = b" (C6), D7.2: "Because b is non-zero" (assumed in Line 1), D7.3: "b is a number" (assumed in Line 1) W7: Implicitly, performing the same arithmetic/algebraic operation to two (potentially different) representations of the same number produces two (potentially) different representations of a (potentially) new number, in this case division by b. Because b is non-zero, this is a valid operation.
This three-column structure might remind you of two-column proofs from a Geometry course because both are organizational schemes intended to distinguish between various elements of a mathematical justification. However, many students experience two-column geometry proofs as an algorithm for proof-writing that obscures the connections and has little to no meaning for learners. Our goal with Toulmin analysis is to share a model to clarify and support thinking that illuminates the critical connections in a mathematical justification.
We hope this example gives you a sense of the analysis process, and we can also see its power here. The flaw in this justification becomes clear: Warrant 4 requires as data that (a − b) not be zero, which is false in context. So we do not accept Claim 4, either (i) because Data 4 is missing the (false) information about (a − b) not being zero or (ii) because Warrant 4 is false as a general principle if 0 is not excluded. Both interpretations fit the writing in Example 1, so if a particular person presented this example as a justification, we would need more information from them to determine which aligns better with their thinking. Returning to the language in Example 1, Toulmin analysis helped us see that between the fourth and fifth lines, we "canceled" (a − b) from both sides of the equation, but because a = b we do not have the data we need to use cancelation as a warrant. Note that in this example, a key piece of data for each claim came from the previous claim; in more complex justifications you should expect to have to look further back for your data.
The analysis in this subsection is a particularly clean example where the warrants are familiar, general principles and theorems from algebra -moreover we leverage deep understanding of those theorems to clearly and precisely describe each warrant. In other contexts, the process can get messier. For example, in online supplemental Appendix A we include a proof of the Pythagorean Theorem for Isosceles Triangles along with a Toulmin analysis of the proof written for students in a general education mathematics course. The warrants are described more loosely because they reference ideas from geometry that did not come from a shared geometry course experience and because the course is more concerned with the process of validation than the geometry context. For instance, the whole proof relies on rearranging four identical triangles to find two equivalent formulas for their area, so an essential warrant is identified in the analysis: rearranging triangles will not change their area. We see the utility of Toulmin analysis in this second, messier example as well: it brings this key idea in the proof to the surface even though this warrant is not codified as a theorem or formula. Most Toulmin analysis work is messy, but the ability to help us make meaning out of messier justifications is exactly why Toulmin analysis is so powerful.

Perspective from a Fellow Student, Vanessa
As learners in a proof-based course, reading and understanding proofs can be challenging -much less producing your own proof. It's normal to be confused about proofs. This is indicative of your desire to understand the many working pieces that build towards a conclusion. As a past student, I share these experiences and recognize the struggles that we all inevitably go through. I will be telling you about my past experiences with proofs, discussing why certain struggles may occur, reflecting on how the Toulmin model has developed my interactions with proofs, and sharing some of the inherent challenges in this learning. I hope these experiences will connect to your own.

Prior to Learning About Toulmin Analysis
Looking back at my experiences, before learning about the Toulmin model, I found proof-writing to be confusing because I did not have the skills that proofs heavily rely on, such as conceptual understanding and proof structure knowledge. As a result, I proceeded through my proof-based courses with the lingering thought that there were inconsistencies in my understanding of argument construction. By the time I got to Real Analysis (an upper-division course in my math major), these inconsistencies had caught up to me as I realized that some of my arguments were not thoroughly communicated as a proof. In addition, I did not have the language to even begin to pinpoint exactly what was causing me to be confused when interacting with proofs. These struggles almost resulted in me not wanting to pursue math anymore because I felt so lost. In hindsight, I believe that if I had learned the Toulmin model earlier, I would have been more prepared to interact with the challenging concepts and more complex proof structures in Real Analysis.
I used to think of proofs as a list of true statements that led to the reader accepting the final claim; however, this interpretation undermines what proofs really are. After having internalized Toulmin analysis, I realized that there often exist subtle components in proofs that I had not accounted for. What has been happening in the background is that those true statements in that list are actually interacting and building upon one another in order to propel the proof forward. I believe that many students have learned the direction and structure of arguments, but in my experience, connections between claims seem to be missing.
Admittedly so, as a student, I have read through proofs and accepted them simply because it "seemed true," with only a surface-level analysis of how the claims actually tied in together. There are many problems with this, one of them being that it promotes memorization instead of cognizance. In addition, I would also argue that this surface-level understanding encourages a lack of appreciation for proof. We fail to recognize the beauty behind developing a proof when we reduce it to a procedure.
Why do we suppose this type of reduction happens? One reason is that perhaps some of our past experiences have encouraged us to think about arguments at this superficial level. As a result, these experiences will impact how we interact with proofs. I have noticed this within my non-proof math courses, where an obsession with getting a "right answer" hindered my thought processes on the reasons behind why such an answer becomes accepted as "right." Proof-writing is different because, although we still want our claims to be true, we also need our justifications to convince us to wholly believe the proof. According to Weber and Alcock [11], "If an individual cannot reliably determine if an argument proves a theorem, some other statement, or nothing at all, that argument cannot legitimately convince the individual that the theorem is true." This signals the importance for students of interacting with proofs in a more nuanced and intentional manner, such that the truth of the theorem is not the entity of most value but dually your comprehension of the entire argument.

Advantages of Toulmin Analysis
One particular advantage of using Toulmin analysis is that we develop the rationale behind arguments when analyzing and composing proofs. For example, within proofs, one must know where they want to go and be cognizant of what must be shown to prove a particular claim. This means that, with purpose, you take the results you currently have access to (data) and ask yourself what exactly (warrant) allows you to reach your destination (claim). Toulmin analysis gives us this language to describe the many pieces of proof and the structure that we use to assemble them. Before I learned about this tool, I searched for "data" and "warrants" within proofs, though I did not refer to them as such. By using this vocabulary, I tend to purposefully identify these pieces as I read through proofs to convince myself that the proof is valid. In addition, as I am writing a proof, I also practice this piecing together to avoid "holes," which are a likely common experience for students.
In relation, communication, which is an essential aspect of proof-writing, tends to develop as well with the Toulmin model. In particular, I notice that I place myself in the readers' perspective to check the validity of each statement along the way. This practice is similar to how, if we can explain a concept to a peer, then we, in turn, deepen our very own understanding of that concept. I also noticed that I gained reassurance that each of my sub-claims connected to the subsequent one.
Practicing Toulmin analysis also allows for the subconscious thinking that is essential for proof insight to transition into our consciousness. This type of thinking metamorphosizes through metacognitive practices, habits of thinking about my own thinking, which serve as somewhat of a thought dissecting tool. First, we begin to train ourselves to ask questions, such as: Do we actually know this? Can I do this? Is this clear to the audience? How can we possibly get to where I want to go? And, do these statements fit together as a data-warrant-claim? Eventually, asking yourself questions while reading or writing proofs becomes a self-assessing practice. Asking questions, in turn, helps you become more independent in your interactions with proofs. I noticed that this boost in metacognition also happened because I now had a strategy to analyze proofs in a more intentional manner. Rather than checking off if I believed each statement was true, I began to instead look at the connections between the statements. I also felt more in control of my learning because I could monitor it with the Toulmin language.

Challenges and Messiness That Were Worth the Struggle
The Toulmin model presented a few challenges for me, the main one being that it can become quite messy. One source of messiness occurs when we analyze a statement and then realize that it serves more than one function at different times, such as when a previously supported claim will be used as data in a subsequent claim. For example, say we had the appropriate data and warrant to support a claim that two triangles are congruent. After this, then the congruence of these triangles will no longer just be a claim. Now, it can be used as data, along with some principle of congruence as a warrant, to support a new claim that these two triangles have corresponding congruent angles and congruent sides. Though it takes some getting used to, this will occur often, and after some practice, you will identify these instances of messiness and use them to recognize the connections that exist from one statement to the next. Hopefully, this adds some appreciation for both the entirety and the minuscule pieces of the proof.
A second source of messiness happens when one is producing a proof and has to decide how far down to go in their claim/warrant/data search. This usually has to do with wondering how much detail is necessary and when it is enough to stop. It turns out that this largely depends on the community we are in and what is accepted by everyone at that moment. The group as a whole also determines the level of detail needed since one may want more sub-arguments with further backing to accept the proof. A few questions we can ask to guide us are: Would my peers accept this? Does this statement clearly connect to the next one? Are there any pieces of missing information? Neither the Toulmin model nor interacting with proofs in general are by any means a clear-cut path that one can easily follow. As a result, sometimes there is no right or wrong answer. Disagreements and convincing others is another type of messiness you will probably experience. However, this is not to say that these are impossible tasks for us, but rather that there is room for growth in our reasoning, communication, and argumentative skills, which is very exciting! In addition, this reveals the counter-productive belief that math and proofs are absolute. Often, amongst your peers there will be opportunities for discussions on the inner workings of a proof, where disagreements may arise. Just because there are disagreements does not mean that it is not useful nor that it is going to be a terrible experience. Instead, it is an opportunity to practice (1) listening and understanding your peers' perspectives and (2) justifying your rationale on why you agree or disagree, which allows us to improve our argumentative and proof skills. The Toulmin model is a lens that we can use to maneuver through these discussions. One of my favorite things that happens when there is a disagreement is that we now have an opportunity to address the confusion of our peers and, as a result, uplift the classroom as a whole. So, do not be afraid to ask for clarification, as it will probably be a learning moment for everyone involved.
For any prospective teachers, these moments of messiness will prepare you for when you have your own classroom and group of students. Though your students may not be writing proofs in particular, reasoning and justifications are essential in just about any course. Since learning is a process, there will be many moments where students will seek additional clarification about their thinking. You might begin by asking questions to assess their understanding at that moment. After practicing the Toulmin model, you are now equipped to analyze your students' arguments. Perhaps they are confused about what a claim is asking, or they may not be incorporating a suitable justification, or they may not have sufficient data that support a claim. After identifying the leaps in your students' arguments, by emphasizing connections from the Toulmin model you can help students construct stronger arguments. Without even mentioning Toulmin, you will engage students in more meaningfully practicing their reasoning and communication skills.

After Learning About Toulmin Analysis
The way that I think about proof has changed after learning the Toulmin model. I once thought the only outcome that mattered was the final conclusion. As a result, I placed a higher value on the final claim and, at times, neglected the arguments leading up to this claim. After using the Toulmin model, I think of a final proof as similar to the process that goes into constructing a physical structure. To begin this construction, we first need blueprints. Though these will not be a physical part of the final structure, they serve as a guide for us to follow. Think of this as our scratch work or discussions with peers that occur prior to the actual proofwriting. Next, we begin the actual construction, so we will need strong building materials (data), which we will continually use during the project's construction. We must also remember to have the correct permits and adhere to building codes (warrants) to ensure the safety of the structure. There is, of course, the reasoning behind why certain permits and building codes are to be used for this particular structure (backing), perhaps determined by the city and construction committees. A finished layer can be thought of as a valid claim. If we are lacking in building materials or not adhering to permits and building codes, we risk the next layers falling apart. Our structure may require multiple resilient layers in order for it to be completed. Since the structure is built upon itself, each new layer is actually working with the previous one (claims become data) as we get closer to our objective. We can even be decorative and add our own esthetic with a style of architecture, such as Gothic or Romantic, or we can keep it straightforward, like Modern architecture. We each have a different style of writing that comes out in our proofs. When building a structure it is necessary to have knowledge of construction and of all of the aforementioned pieces, which we can think of as our experience with proof and our knowledge of proof construction, which will grow over time.
To a passerby, this completed structure may seem like it suddenly appeared without much effort. However, the workers are well aware that this was not the case and that much planning, significant knowledge, and multiple intentional moves were required to create this structure. Similarly, other construction workers will appreciate the structure, as much as the process of constructing, because they are aware of the production that goes into such structures. Professional mathematicians and proof-writers have a great appreciation for a well-constructed proof, partly based on our experiences with creating our own proofs, but also because we can see the construction of each argument as well as the connections made between arguments, which are all working together so we can reach the final result. Keeping all of this in mind, our goal is not to create a picture of a building (a string of true statements), but rather to construct a building that is connected and holds together (warranted).

TO INSTRUCTORS
Collectively, we have taught students to use Toulmin analysis in courses for nonmajors, introduction-to-proofs courses, proof-based courses (including Abstract Algebra, Real Analysis, and Modern Geometry), capstone courses for future secondary teachers, and even high school Calculus courses. In Section 3.1 we describe a course module that is similar across all of these contexts, then share in Section 3.2 some of the collective wisdom and observations we have accrued about the messier elements of teaching Toulmin analysis. Then we detail the divergent ways that we have integrated this module into different course contexts. In Section 3.3 Elizabeth shares considerations for integrating Toulmin analysis into a MLA course as a support for strengthening and transferring reasoning skills. And in Section 3.4, BK contrasts the integration of the Toulmin module in proof-based courses, depending on whether the primary goal is to teach proving skills, support other challenging learning, or reflect on the nature of proof. Sections 3.3 and 3.4 can be read independently.

Suggested Implementation in Class: A Module for Learning Toulmin Analysis
There are at least four challenges in helping students learn to use Toulmin analysis. First, there are some big ideas that require both quiet processing time and supported practice and discussion. As part of this, we have, in the past, asked students to read the Weber and Alcock [11] paper, but this research paper is quite challenging for many readers; one purpose of this piece is to replace that reading with the more accessible sections of this paper! Second, starting to make sense of Toulmin analysis requires that the learner take the stance that mathematical knowledge is constructed through communication, a stance that is itself founded on subtle philosophical ideas that are not often discussed overtly in mathematics classrooms, at least as reported by our former students. Third, applying Toulmin analysis is like listening to a different register or subtext of a conversation, and just like with other subtext layers of communication, it can be hard for a person to identify them consistently as they are just starting to listen for them. And fourth, it can be quite challenging for a person to apply Toulmin analysis to a justification that relies on many concepts with which they are struggling to build a stable understanding. Conversely, we have found that students also struggle to apply Toulmin analysis to justifications of ideas that they find to be obviously true without evidence. These two forces create a need to balance learners' familiarity with the concepts in the justifications they use while learning to apply Toulmin analysis.
Here is the structure of our module that addresses these challenges. We suggest that this module should take 1.5-3 class periods, some of which could also make progress toward other course learning objectives.
Summary of the module: • Day 1: Prime students to see some of the challenges of mathematical justification and communication. • Day 2: Work together on applying Toulmin analysis as a response to those challenges. • Day 3: Use Toulmin analysis as a lens to organize ongoing and future work with other course objectives.
Goal for Day 1: Position learners so that they notice the difference between true and warranted and can see something like Toulmin analysis as a productive tool for addressing a real challenge.

Task 1a:
Ask students about their initial reactions to Example 2 above (the "cursed proof" that 107 is prime). Ask students about their reactions to the cookie promise above ("Promise: If we have class on Saturday, I will bring cookies."), suitably adapted to a day when you do not have class. Also, ask students whether the statement "If unicorns exist, then today is Tuesday." is true, with the day adapted to both match and not match the current class session. You will likely need to include more example statements in response to students' ideas in order to complicate their efforts to make unwanted generalizations about the truth of If-Then statements.
Task 1a brings students' ideas about truth to the surface, especially the truth of If-Then statements. This will be a messy conversation, but it does not need a clean resolution for the rest of this module to progress; in fact, the goal is for students to be more concerned than at the start of the module about what truth means to them. The prompts leverage the lived and linguistic expertise of the students rather than trying to forestall the ways of thinking about truth that do not match the ways professional mathematicians often work. The "promise" structure is useful for helping students accept a statement as true when its hypotheses are not satisfied. Temporality is useful because it allows students to consider the same statement as having different truth values in different systems. Conditions involving measurement (such as claims about the lowest elevation point on a moon) may bring up ideas of both the perspective required to make a measurement (What units will we use for elevation?) and the distinction between considering a measurement and implementing that measurement (Do we need to find all of these lowest points to make logical claims about them?), and measurements of potentially-changing quantities may cue temporality. Discussion of unicorns will likely bifurcate, with some students saying that this is false and some saying that it's probably false but unknown or unknowable. Students may make distinctions between conditions that feel like external descriptors and those that feel like identities (e.g., sporty vs. an athlete) because of how we can know these conditions, and these can contrast with terms that are part of a formal or abstract structure (e.g., captain of the team).
In our experience, at some point during this discussion, students will suggest that they consider all possible "cases" for statements. Representing the ideas that follow will produce a 2 × 2 table like the analysis of the cookie promise in Section 2.1. It is important in our work that this be a reaction to and representation of student thinking rather than the framing that corrals student thinking in advance. As a result, all of Task 1a should be done live in order to get at students' reactions.
Task 1b: Ask students to read one or both of the block quotes from Weber and Alcock [11] in Section 2.2 of this paper. From that reading, ask students to (1) explain the terms claim, warrant, data, and (possibly) backing in their own words and (2) give an example claim-warrant-data triple from their previous work in the course.
Task 1b helps students get oriented in this discussion. In our experience, some students will generate examples that align with our interpretation of Toulmin analysis and may notice that their warrants are mostly definitions, theorems, axioms, or other implicitly-held general principles. Our other students have generated reasonable examples that differ from our interpretation in subtle and idiosyncratic but important ways; the biggest struggle has been when students' examples are not specific enough to interpret. For example, a student might say that their claim is anti-derivatives, their data are derivatives, and their warrant is the Fundamental Theorem of Calculus. This captures the idea of Toulmin analysis at a metaphorical level but does not focus on claims and their justifications. Both of these types of student examples, those that align and do not align with expert conceptions, are worth discussing with the full class, likely after small group discussion. We suggest that you sequence the first example shared to be a clean and compelling one that aligns with Toulmin analysis, if students generate one, and then open the discussion for students to share other possible examples focused either on how they are similar to those that came before or on ways in which they are unconvinced by their own example.
The paragraph-reading and example generation can happen in preparation for class. In our experience, this allows students a more appropriate amount of time to think about the new terms and results in better student-generated examples that are more aligned with previous course justifications. However, if out-of-class work time is too scarce or if your students are likely to get too stuck and frustrated with this preparation, this whole task can be done in class.
Tasks 1a and 1b can comfortably fill a 50-75-minute class session (or longer), but if Task 1b is started in preparation for class, this discussion can be compressed to about 35 minutes if needed. There is no particular reason to do Tasks 1a and 1b on the same day, but we have found that classes pick up some momentum for discussing these kinds of philosophical issues, making this discussion more efficient when paired. It is also reasonable to reverse the order of these two tasks.
Goal for Day 2: Engineer some successful experiences for students analyzing mathematical justifications, and start discussing the more subtle issues with and implications of Toulmin's model.

Task 2a:
In preparation for this class day, ask students to read about Toulmin analysis and then attempt their own Toulmin analysis on a mathematical justification.
Historically, we have asked students to read an annotated version of Weber and Alcock's [11] paper, but this paper is quite challenging. We have written Section 2 above with the intention of instructors being able to ask students to read this instead. One particular challenge of the Weber and Alcock paper is that the example analyses are done in the context of Real Analysis (bounded monotonic sequences and function continuity), which is a stretch for most course populations. We have developed reading guides that help students read this paper, but we have often instead replaced sections of that paper with examples of Toulmin analysis that we wrote for the class. We have provided several of these as appendices to this paper (Appendices A-D).
Years ago, the students in one of BK's courses made a video entitled "Dear Math Majors: A letter from the upper-class students" [6] about analyzing mathematical justifications (https://youtu.be/_8q9OX7toq0?t=694, Toulmin analysis starting at 11:34), and we sometimes also ask students to watch this video along with the reading, as long as the modular arithmetic context will not be a problem for them. The video does not perfectly align with the intended perspective, but it is very powerful for students to see the Toulmin analysis done live and by other students. The video contains discussions of two other lenses for analyzing proofs that will be mentioned briefly below in Section 3.4 as well as some testimonials from students encouraging younger learners to keep going with this difficult work.
Task 2b: Ask the students to discuss and share Toulmin analyses in class.
At times, we have used the first attempts that students write outside of class for Task 2a for the discussion in Task 2b, but in our experience students need to try to do an analysis and then start fresh on a new one with peer support in groups to be ready to share. Either way, we recommend that all students work on analyzing the same justification for Task 2b. This makes it much easier for them to focus during the discussion on the analysis rather than understanding the justification, and it makes their analyses comparable. Similarly, we recommend that this common justification be something from earlier in the course that the class has generally agreed is justified but that still feels challenging. We have returned to justifications of the isosceles Pythagorean theorem, the claim that rational numbers have finite or repeating decimal representations, the infinitude of primes, the claim that generated subsets are subgroups, the well-definedness of modular multiplication, previous epsilon-N proofs, and many others. It is important that students analyze a fixed written artifact; if they are analyzing something more ephemeral, they are likely to produce an analysis that does not attend to the justification, possibly by changing the pieces of the justification as they try to think about analyzing it.
Many interesting subtleties can arise in this discussion or from reading about Toulmin analysis. For example, Weber and Alcock [11] point out that there is no well-defined algorithm for determining if a claim is warranted and therefore that this piece of the process may be inherently subjective, that the transition words in proofs pop out as useful indicators in the Toulmin analysis, and how hard it can be to convince a student that a string of true statements can fail to be a proof. We discuss some of the subtleties that might arise in this discussion in Section 3.2.
Task 2c: Ask students to apply their learning to produce a thorough Toulmin analysis of a polished proof (revised if necessary as part of this assignment).
In many of our classes, students do some PostWork after class or weekly Reflections. Either format would be appropriate for this follow-up to the second-class session about Toulmin analysis. If you have the time, you can also ask students to reflect on their learning and new thinking as part of this task.
In support of the task of applying their learning, you might ask students to look at another Toulmin analysis, perhaps one of the appendices to this paper. Similarly, (re)watching the "Dear Math Major" video [6] or reading the upcoming Section 3.2 offers them a second pass with ideas that might have felt like they slipped past too quickly the first time in reading or class discussion.
Whatever choice you make for Task 2c, it is important that you help students connect the learning about Toulmin analysis to the rest of the course work. We want students to see Toulmin analysis as a useful tool for understanding the work they have already done and a key component in the rest of the work they will do in the course. In other words, this module should not end suddenly, or they will see Task 2c as a waste of time rather than a springboard for new learning. Sections 3.3 and 3.4 below will discuss some ways that we connect this module to the rest of our course designs.
Goal for Day 3+: Leverage Toulmin analysis to make progress toward the other course learning goals.
There is no specific Task 3 because it will depend on your particular context. As an example, in a recent capstone course for future secondary teachers, BK's students had built a definition of sequence convergence and then constructed proofs focused on how the quantifiers in the definition structure the proof. After learning about Toulmin analysis, BK's students re-analyzed these proofs through the new lens, often finding them more compelling. In a different context, Elizabeth's students analyzed public science writing about the COVID-19 pandemic using Toulmin analysis as a lens to see how these quantitative justifications were structured. We will discuss the ways that learning this module is integrated into our courses in Sections 3.3 and 3.4.

Anticipating Messier Elements of Teaching Toulmin Analysis
Once you start getting your hands dirty with Toulmin analysis, a host of subtle issues will arise. Each subsequent paragraph in this section discusses one of these issues.
We hope this will help instructors anticipate some of the messier elements of the discussion in Task 2b of the module above. In particular, we encourage instructors to listen for these ideas in the discussion and use facilitation strategies to make these into explicit topics of conversation for the group so that all students can grapple with these subtleties directly. Students may also benefit from reading this section after that discussion to keep grappling with these ideas and to make time for any ideas that slipped past too quickly in the messy discussion. It has also been our experience that students enjoy reading about their anticipated learning because it normalizes the struggles and helps them feel connected to other learners.
Scale: Toulmin analysis involves picking a scale at which to analyze a justification. If we were analyzing the proof of a theorem, we could call the hypotheses the data, the conclusion the claim, and the whole proof the warrant. This is broadly an appropriate interpretation of Toulmin analysis, but it will not help us see how the smaller claims build up to a larger justification. Instead, we should follow the process outlined by Weber and Alcock [11] above of reading line-by-line. At this scale, most of our warrants will be definitions, axioms or assumptions, previous theorems, or general principles. When a previous theorem is used as a warrant, the backing for that warrant is the whole previous proof! Similarly, in some sense, the last claim at the end of a proof is that the whole proof warrants the theorem. So this coarse scale is always nested in the background of the more fine-grained work.

Claims First/Conjunctions:
We identify the claims first. If you read Weber and Alcock's [11] summary, you see that the data and warrant are used to support a claim, so it's not possible to call a piece of the text "data" or "warrant" without a particular claim in mind. However, English is flexible with the order of phrases, so claims do not always appear before their data or warrants. As Weber and Alcock point out, keywords like "so," "therefore," and "by" indicate different roles in the analysis. Our students have often reported that they were baffled by why their instructors choose different connecting words in their justifications, but after learning to use Toulmin analysis they can see these choices as purposeful and start to make them for themselves.
Statements/Identifying Claims: One hard part of doing a Toulmin analysis, for most of our students, is identifying claims. In the analysis of Example 1 in Section 2.2, each equal sign corresponded to a claim, but in many justifications the claims will not be symbols, and many symbols will not be claims. When we assert that something exists, that we can do something, that two things are the same/different, or that an object has a special property, we are making a claim (unless we are making an assumption). But more generally, claims are stand-alone statements, sentences, that could be true or false. You may already have strategies or course materials for helping students learn to distinguish statements from other kinds of phrases, but here is another way that makes use of Example 1. The phrase "red" is not a statement and therefore cannot be a claim. Similarly, the phrase "is red" is not a statement. However, the phrase "This apple is red." is a statement that could be true or false. Translating this example back into the analysis above, "b" is not a statement, and "= b" is not a statement, but "b + b = b" is a statement that we claim is true in context. Also, the phrase "b is a number" is a statement in the analysis above, but it was part of the assumptions, so it does not function as a claim that needs a warrant in this analysis.

Data vs. Warrant:
Another similar challenge for students is distinguishing between data and warrants. Part of the confusion here comes from the fact that both data and warrants support the claim, but the distinction between concrete properties and general properties can be subtle (e.g., "this area is (a + b) 2 " vs. "area is length * width") and we often do not make a distinction between them or do not make general principles explicit in informal contexts. In the Toulmin analysis of Example 1 above, the warrant boxes contain both the quotation from the text indicating the warrant and a more thorough statement of the general principle in order to support students with this struggle. This confusion may also come from experiences proving using generic examples (e.g., any geometry proof that relies on one diagram) because the specific properties of our example are imagined to be general, mixing the specific and general. Broadly, warrants are statements that transcend the particular justification while data are specific to objects in the justification. Competing conceptions of variable may also compound the situation: the use of variable is in itself a small-scale generalization, even when it is done to pin down a quantity in a specific way [10], and students often struggle to see formulae as relationships between quantities. Making the distinction between a general principle and specific information is a subtle skill that professional mathematicians use all the time, but it is not overtly taught in many curricula.
Strings of (In)Equalities: One common structure you will see in proofs is something like this: We interpret this as three claims. First "A = B," which is warranted by X; second "B = C," which is warranted by Y; and third, "C = D," which is warranted by Z. This format makes the warrants very clear, but students sometimes struggle to identify the claims cleanly (as seen in the video, [6]). Interpreted this way, each of A/B/C/D is an expression, and the claims are that each is equal to the next, represented by the three equality signs. If warrant X were, for example, distribution, then the data would be that the symbols in the expression A are elements of an algebraic structure that has distribution, like the integers. Sometimes a justification structured like this will be followed by the observation that "A = B = C = D" implies that "A = D" by transitivity, which might otherwise be an implicit warrant. Students will also often instead interpret this as the string of claims "A = B," "A = C," and "A = D"; while this is reasonable, the warrants usually do not align very well because they would require repeated use of transitivity at each step.
Multipurpose/Unused phrases: Another common challenge for our students has been the belief that each phrase has exactly one role in the analysis, when in practice most phrases have multiple roles or get used repeatedly while some do not contribute to the logic of the justification. Looking at the example analysis in Section 2.2, each claim gets used as data once subsequently, but that's not always the case. Looking at the other data entries, we see that the first line gets used repeatedly. In general, when considering a claim, any previously warranted claims or assumptions in the justification can be used as data. Most of the sentences contained a single claim. However, the first sentence does not contain a claim (it was only used as data), and the sixth line contained two separate claims. Moreover, sometimes lines or sentences will not fit into your Toulmin analysis of a justification. In our experience, the most common content that does not fit into the analysis is there to foreshadow the structure of the justification without putting forth any new claims.

Formatting:
We are aware of four methods of formatting a Toulmin analysis. (i) Weber and Alcock [11] do their Toulmin analysis by numbering the lines of the justification and discussing the roles of each. This can work well, but individual lines can be made up of multiple pieces with different roles, so when using this format you may want to break the lines into smaller pieces before numbering or use a system that allows you to sub-number pieces of a line. For a second example, see online supplemental Appendix C. (ii) When students are working with a copy of the justification on which they can draw or highlight, they often color or underline the pieces of the analysis. This is powerful, but when a phrase gets used for a new purpose, when data are far from their claims, or when a warrant is implicit, this can get confusing. For an example, see online supplemental Appendix B. (iii) Some researchers use nested and complex versions of Figure 1 to record their Toulmin analyses; this approach is powerful for research but very hard to build during a first pass analyzing a justification. For more examples, see [7] and [2]. (iv) Our threecolumn analysis of Example 1 in Section 2.2 breaks away from the written form of the justification to identify the elements of the analysis more explicitly, and tables are easy to manage in digital learning environments. We recommend this format for analyzing justifications with more than 2-3 claims; the main challenge is that it obscures the flow of the proof and tempts us to insert more of our own ideas rather than sticking to what was actually written in the justification. For more examples, see online supplemental Appendices A and D.
Subjectivity: As Weber and Alcock [11] point out, "[d]etermining whether an implication is warranted is not such a well-defined task. . . . In fact, determining whether a warrant would be acceptable by the mathematical community may inherently involve a degree of subjectivity." Perhaps there is an algorithm that would check, in some context, whether each claim in a proof is true, but it is not possible to check if warrants are socially acceptable with an algorithm. This vision of justification helps humanize mathematics. And while this analysis is easier to perform carefully on written justifications, we assert that it is the same process that happens during live mathematical discussions and oral justifications.

Integration into Elizabeth's Mathematics for Liberal Arts Course
In this section, I (Elizabeth) share ways that Toulmin analysis can directly address some of the particular challenges of justification for students in a MLA course, how this work fits into my course, and how my students have leveraged this lens to analyze argumentative essays on a wide range of topics, from magazine articles on gender inequity in automotive safety standards to blog posts on the high price of french bulldogs.
A primary goal of my MLA course is to create opportunities for students to engage in authentic mathematics, and justification is central to that work. However, students in this course often have little, if any, experience with constructing mathematical justifications that are meaningful to them, and, given their prior experiences with procedure-focused mathematics, often do not see how this kind of work is related to mathematics! In an MLA course, Toulmin analysis has great potential for making the often-invisible work of mathematical justification more transparent. Operating in the new register that Toulmin analysis necessitates makes it possible for many of my students who have been marginalized by mathematics to begin to see and make sense of what mathematical justification can entail.
Part of the challenge of emphasizing justification in an MLA course is that we are not in a position to build an axiomatic system from the ground up for several reasons. The mathematical ideas are often motivated by investigation and intuition, rather than by formal definitions and axioms. The students have not been enculturated to communicate in the idiosyncratic ways that professional mathematicians communicate (e.g., see Data vs. Warrant in Section 3.2 above). The course touches on a wide range of topics and contexts, from counting problems to geometry to probability. And all of these topics are to some degree familiar to the students, but they do not have a shared prior experience with these ideas or common ways of thinking about them. We would lose too much momentum trying to fully formalize all of these disparate ideas into definitions and axioms. Moreover, quite reasonably, many of these students do not see an intellectual need for formalization.
Working without a clearly established set of axioms also greatly exacerbates a challenge that Vanessa describes in Section 2.3: it is unclear how far down the justifications need to go. Do we call back to the distributive property or just call it "algebra"? If working through an example touches on all of the ideas necessary for the general case, do we need to follow up with the general justification? Or is it enough to write the warrants generally in the analysis? In my experience, this muddiness has always been there when MLA students start to justify their reasoning, and Toulmin analysis brings it to the surface so we can address it explicitly. My response to questions of how far down to justify always depends on the students.
If pushing for more clarification has the potential to be fruitful, I will. If a student has already done significant mathematics and asking for more will undermine their confidence or risk them losing interest, I let it go. Similarly, the justifications vary in style and approach and we have to work together to unify them. Warrants end up being messy (see online supplemental Appendix A), and I find myself figuring out how to break down the subclaims and tease out specific data from general warrants right alongside my students.
Context can go a long way towards addressing some of the inherent fuzziness of justification in MLA courses. Counting and then probability provide a good context flow for early Toulmin analyses in MLA because they build on one another in concrete ways. The questions can be framed naturally, and there is a clear foundational warrant that we can build from the multiplication principle. I have found that because we keep circling back to this same warrant, it helps students tease out the difference between warrant and data. Once counting formulas are established, they can be used as warrants for solving other more complex counting problems. In contrast, graph theory was challenging for MLA students because the warrants vary so much and, without the careful definitions that ground graph theory, it was often not clear what constituted a warrant and what kind of data were necessary to support each claim.
In my MLA course, students make and justify an assertion related to the day's investigation after every class meeting, and this work serves as draft material to be revised for a Midterm Portfolio and again for a Final Portfolio. We do a version of the Toulmin module described in Section 3.1 a week or two before students submit their Midterm Portfolios. This first pass at Toulmin analysis helps create the intellectual need for such a lens, but many of the more subtle ideas are not developed deeply yet. It has also been productive to pair this introduction to Toulmin analysis with a careful discussion of the different ways people interpret If-Then statements in casual language compared to the specific ways we read them in formal mathematics. Each Portfolio is a revision of a previous justification that includes a Toulmin analysis of that revised justification. The goal is for students to use the process of analysis to improve their previous justifications. This first independent run is usually bumpy and primarily serves to draw attention to the parts of analysis and justification that are causing students to struggle. In the week after submitting their Midterm Portfolios, students work in teams on a project that also requires a justification with Toulmin analysis. I have found that my MLA students really begin to get a handle on the expectations and value of Toulmin analysis during this second run at the work with the support of a team. Students return to this kind of work again at the end of the semester to write another Portfolio and work on another Team Project.
The function of Toulmin analysis in my MLA course is to provide students with a framework for analyzing justifications, and I see it as a tool for improving their reasoning. Building facility with Toulmin analysis itself was never the goal, so I do not worry too much about them getting it "right" (not that there is one correct analysis of a justification). I do not spend much time assessing their analysis beyond checking that they've got the idea and they're doing something that will help them think through their justifications. Because it is a support for their reasoning, I assess it as one might assess a rough draft in a writing course, by giving feedback and guidance for revisions or future justifications. I want my students to come away from my MLA course with skills and dispositions that will help them analyze justification in all kinds of contexts. Mathematics is a particularly good domain for learning about justification for several reasons. The structure is often easier to follow because the ideas in mathematical justifications tend to build linearly. Mathematical justifications often make assumptions and warrants more explicit than in other domains. And it is often cheap and fast to consider hypotheticals and ask questions whose answers we genuinely do not know. Toulmin analysis makes the work of justification explicit, and I use this explicit discussion to support the transfer of this learning to new domains. This past spring I had students practice using Toulmin's framework to analyze an article on COVID-19 vaccines, and then they had the opportunity to analyze an article of their choosing for their Final Team Projects. Because these students each selected a domain that's meaningful and important to them, their background knowledge supported the reasoning and this, in turn, supported the transfer of learning.
One thing I learned from working with my students on these projects is that using Toulmin's model to analyze public, persuasive essays and news articles is much messier than the analysis we had done of their mathematical justifications in the course (despite the fact that Toulmin's model was developed for these public contexts). It was often hard to tell if a new claim was building linearly on or running parallel to the prior claim towards a larger assertion of the essay or article. And the warrants were much more likely to be implicit, and they often drew upon much broader assumptions about the world. For example, many of the justifications my students analyzed were made on the implicit basis that one option being more expensive than another makes it less likely to be chosen. This warrant certainly holds true for many scenarios but not all, and as a result using it as a warrant forces us to engage with a specific worldview when we are reading and when we are analyzing.
I am still building strategies for supporting students in this learning. In my experience, the challenge of analysis varies greatly with scope and style of the article or essay, so there is work in just choosing the piece to analyze. I also think it would be helpful for instructors to talk with students about their own struggles analyzing these kinds of justifications, focusing on the nonlinear nature and hidden warrants of an example article. Despite all of these challenges, students were enthusiastic about the work, and it was really cool to see these skills transfer to domains that were important to my students.

Integration into BK's Proof-Based Courses
I (BK) discuss Toulmin analysis in almost every course I teach that focuses overtly on proofs using the module in Section 3.1, but it fits into these courses differently based on the students' prior experience with proving and serves a different function.

Building Toward Toulmin Analysis to Teach Proving Skills
In a course whose primary goals are teaching students to construct proofs and to think about proof productively, I build up to Toulmin analysis for two reasons. First, Toulmin analysis requires some skill with using general warrants in mathematical contexts. In mathematics, we treat our definitions as objects rather than as descriptions of objects and we operationalize the quantifiers in these definitions for both using and proving; similarly, we apply general theorems in particular contexts by building a coherent correspondence between the specific and general to check the hypotheses and then use that same correspondence in reverse to interpret the conclusions in context. Second, Toulmin analysis is applied to whole proofs, even if it focuses on line-by-line reading. In order to manage the cognitive load of this work, students need to have developed some large-scale frameworks for thinking about proofs based on experiences writing proofs. And in my experience, students are most likely to see value in Toulmin analysis if they are using it to think about proofs that they helped to construct but about which they do not feel completely settled.
My Introduction-to-Proofs courses usually have three small units. The first unit focuses on having students participate in defining and conjecturing so that they start to see justification as a response to the ambiguity of the human work of mathematics. I find graph theory to be productive for this learning because it is easy to draw new examples whose properties are obscure. As part of this unit, students learn that the statements we are trying to prove tell us a lot about how we should structure the beginnings and endings of our justifications [8]. The second unit focuses on parsing mathematical statements by attending to the quantifiers "for all" and "there exists." I find number theory to be a useful domain for this learning, and Elizabeth has done similar work using set theory. At the end of this second unit, the students have enough skill with general warrants, they have ways of thinking about the structures of whole proofs, and they have a large collection of proofs that they built and that they find mostly convincing.
The Toulmin analysis module is set between the second and third units in the course. This placement allows us to look back on proofs from both previous units. Applying Toulmin analysis to proofs that mostly leverage operationalized definitions (like those in "elementary" number theory) is often very compelling for the students because these proofs feel quite "clean" from their perspective, and, as a result, the Toulmin analysis is not very controversial and has a lot of intuitive validity for the learners. Returning to some of the messier proofs (such as those from graph theory), students often feel more powerful: having names for the elements in Toulmin's model focuses their attention, and this can help them feel like they have greater control over justifications that were at the edge of what they could handle initially.
Toulmin analysis also provides a vocabulary in the subsequent course activities, helping students focus their discussions of justifications on high level analysis of the use of warrants. The content of this third unit has been more flexible for me, often touching on sets and functions through combinatorics. In the case of combinatorics, for example, the language of Toulmin analysis can help convert implicit ideas about additive/multiplicative counting principles into explicitly-stated and collectivelyaccepted axioms that can serve as warrants. Similarly, in a unit about cardinality, the lens of Toulmin analysis can help us break out of the spiral of using unstated, implicit notions of size.

Leveraging Toulmin Analysis When Working with a Challenging Concept
In other proof-based courses like Abstract Algebra and Real Analysis, I engage similar ideas about proving, but my students bring experiences with quantifiers and proof frameworks to these courses that can be reengaged quickly. This allows me to use the Toulmin analysis module earlier in the course. As a result, Toulmin analysis becomes a key tool for focusing our justifications when the learning gets challenging, especially when learning to reason about challenging definitions.
For example, here is a summary of a Real Analysis unit I recently taught; because there is less lead-up, I can give more details of the integration into this course. On Day 1, the students discussed potential definitions for "sandwich." This activity encourages students to view definitions as human choices that separate a set of potential sandwiches into categories differently, depending on those choices. The students also start practicing finding (non)examples and boundary cases of definitions as well as examples that separate two potential definitions (by satisfying one but not the other). On Day 2, the students did a guided reinvention of a definition of approaching (i.e., sequence convergence) that is closely related to [3]. On Day 3, the students discussed quantifiers and then used them to build and analyze a justification that claimed to show sequence convergence. On Day 4, the students discussed relationships between statements and the structure of their proofs [8] and used this lens to construct new justifications and re-analyze the previous justifications from a second perspective. On Day 5, the students considered other potential definitions of sequence convergence (often with the variables and quantifiers changed slightly) and then did the first day of the Toulmin analysis module, looking at the terms claim, data, warrant, and backing. On Day 6, the students engaged in Toulmin analysis of some more challenging proofs about sequence convergence, such as a proof that the sum of two convergent sequences is convergent; these proofs use the definition as both a known warrant and the goal, so their Toulmin analyses support deep engagement with this challenging concept. In this particular example, the remainder of the Real Analysis unit connected to teaching high school mathematics, but I have done something similar for the beginning of a semester-length Real Analysis course.
And while it's not a proof-based college course, I did something similar when I taught a proof-based unit in high school a few years ago in which my students engaged the epsilon-delta definition of function limits. The ideas about quantifiers and proof frameworks were not an explicit part of the discussion in this course, but these lenses did inform my strategies for supporting students' justifications. The students read about Toulmin analysis and used it to understand our definition of function limits and a handful of proofs about it. Though it was challenging for them, this group of high school seniors did read a scaffolded version of Weber and Alcock [11].

Using Toulmin Analysis to Reflect on Proof and Proving
Finally, the Toulmin analysis module can be used as a capstone or reflective activity. For many years, I have taught a Modern Geometry course that is often the last proof-based course for the students and hence a capstone for future teachers. For the students who are still struggling to construct proofs, this module helps them solidify their skills, but this module can also be used as a jumping-off point for discussions about proof and the nature of mathematical knowledge.
As a first example in this context, I have leveraged Toulmin analysis to ask students to look back at their learning about mathematical justifications (in previous courses or this course) to identify and process ideas that initially did not make sense but now do. Many of my students have reported finally understanding why their previous professors cared about subtle details that seemed mystifying to them a few years ago, such as their choices of conjunctions or the difference between "similarly" and "without loss of generality." In particular, this module has supported productive discussions about the teaching and learning of proof.
As a second example, learning Toulmin analysis can open doors for discussions about the nature of proof. What is a proof? Is it the written artifact or something else? What are its purposes and how does it achieve those purposes? Toulmin analysis makes it clear that warranting claims is a fundamentally human, social, and subjective endeavor. This realization often disrupts more absolute and universal views of mathematics, and many of my past students have appreciated the opportunity and time to think about and discuss with others the implications of this disruption for their identities, views on mathematics, and future actions as mathematicians and educators.

CONCLUSION
Mathematics is a human endeavor. Toulmin analysis has helped us bring that humanity to the surface in our classrooms, and it helps us build stronger and more inclusive entryways for students into mathematical communities. Learning about Toulmin analysis helps our students connect mathematical reasoning to other thinking, supporting transfer in both directions, and it has changed our thinking about the teaching and learning of mathematics even in contexts where we do not devote class time to this model overtly. We hope you consider adapting the module from this paper to meet the needs of the students you support.

DISCLOSURE STATEMENT
No potential conflict of interest was reported by the author(s).

FUNDING
Work on this project was partially funded by the Richard D. Green Endowment at California State University, Long Beach.