The enigma of "Exemplary"

How an attempt to recognize excellent work can create more trouble than it’s worth

Photo by Joanna Kosinska on Unsplash

In a recent post, I wrote about the four-level rubric that I use as a starting point in most of my classes: Exemplary, Successful, Revisable, and New attempt needed or Not yet.

Today I want to examine the top end of this rubric – Exemplary – and look at some ways where a desire to recognize excellent work can lead to unintended consequences.

First, a bit more about the rubric. Everything is organized around Successful, which I usually define as:

Correct, complete, and demonstrates a thorough understanding of the relevant topics with at most minor errors that are not central to the topics.

Individual objectives or specifications add concrete explanation about what Successful means in that context.

Most recently, I’ve defined Exemplary to build on Successful:

All of the qualities of Successful, but with no noticeable errors. In addition, your solution is particularly notable in some way, such as: conciseness, clarity, simplicity, an especially well-chosen approach, etc.

Let’s pull this idea of Exemplary apart and see what’s hiding inside.

The mechanics of Exemplary

The reason I have a mark called Exemplary is pretty straightforward: I want to recognize work that is not just satisfactory, but indeed exemplary – literally, work that could be used as an example in a textbook or for students to review.¹ This lets me recognize work that goes above and beyond, shows exceptional understanding, or otherwise does something noteworthy in terms of learning. These differences are often clear to me as I read student work, and this mark gives me a way to call that out.

When I use Exemplary, it comes into my final grade requirements only at the highest grade level. In one recent class, Exemplary only appears in the criteria for an A and doesn’t appear at any other grade level. To earn an A in that class, I required (among other things) “All homework problems at Successful or higher, with at least two at Exemplary.”

Note that I said “when I use Exemplary”, because I don’t always include it as an option. I say more about this below.

Exemplary can be motivational

Students definitely understand my purpose in using Exemplary: It’s concrete recognition for excellence. Many students appreciate and are motivated by the chance to earn Exemplary.

Exemplary is especially useful in classes where I’m interested not just in content knowledge, but in communication quality. For me, this usually means upper-level elective classes where students are refining and polishing their understanding of mathematical communication.

In this context, Exemplary lets me distinguish between levels of correctness. Some work is logically correct yet convoluted, long-winded, notation-heavy, inelegant, or otherwise difficult to follow. This work earns Successful, which is the basis for earning passing grades. But work that is both correct and demonstrates skill in clarity, elegance, conciseness, and so forth, can be acknowledged with Exemplary.

This mark also allows me to give personalized challenges to students. Based on a student’s past work, I can often identify a particular aspect of communication that they need to work on, and so I can challenge them to practice with that: “To earn Exemplary, you need to find a simpler approach so you can avoid the need for 5 separate cases” or “make this more concise, while keeping all necessary logic”, or so forth… whatever that student particularly needs to work on. Exemplary can be a carrot that encourages students to practice with difficult aspects of communication.

Exemplary can be distracting

In some classes – especially introductory ones – students are just beginning to work on clear communication. In these cases, I want students to focus on showing a fundamental understanding of the class topics in their work, while following very basic rules for communication. These communication rules are usually so essential (“state each question”, “include at least one complete sentence of explanation”) that if a solution doesn’t follow them it falls into Revisable or even Not Yet territory.

In this case, Exemplary can distract students from the real goals. Much like “extra credit” in traditional grading, Exemplary can be a shiny distraction. It can grab students’ attention and cause them to spend their time thinking about punctuation and spelling, when that time would be better spent focusing on other topics they don’t yet understand. A distraction like that actually hurts student learning.

Here’s a concrete example from my world as a math teacher. I often teach an introduction-to-proofs class, which – as its name suggests – is a class where students first begin to learn about written mathematical communication. I never use Exemplary in this class. That’s because, by design, this is a class where students are just starting to understand what professional mathematical communication looks like, and I want them to work to build a broad but basic understanding of it. A student who earns an “A” in an introduction-to-proofs class shouldn’t be writing beautiful proofs, rather they should be consistently writing correct proofs that follow a basic style book. That’s the goal of the class, and Exemplary would distract from it.²

Compare this to the motivation that students can find in upper-level electives, like I described above. Here students have advanced beyond their initial lessons on mathematical writing. Indeed, it’s appropriate for students to hone high-level mathematical writing skills in such a class, and so it makes sense to distinguish between Successful and Exemplary in such a context.

But even in high-level classes, I’ve noticed that students often want to revise Successful work up to Exemplary rather than revising work that has more fundamental problems. There’s something shiny about Exemplary that can distract students from the most important ideas. I’ve tried to limit this by requiring students to revise any Revisable or Not Yet work first, which more or less works – but I need to watch carefully and enforce this rule.

Exemplary can be confusing

Above, I’ve argued that the philosophy behind Exemplary makes more sense in some contexts than others. But there are also more practical issues: Exemplary marks can actively confuse or mislead students.

Occasionally, I’ve tried to avoid some of the distraction provided by Exemplary by eliminating it from grade requirements. In this case, Successful and Exemplary count exactly the same way in final grades; Exemplary is just a form of feedback that lets me say “job well done”. There’s no need to revise or improve Successful work up to Exemplary because they count in exactly the same way.

But as you can easily imagine, this distinction-without-a-difference can be very confusing for students. Students end up thinking that Exemplary is somehow “worth more”. I mean, it’s a different mark, it clearly means something better – can you blame them? This ends up costing me a lot of time explaining that earning Exemplary is exactly the same as Successful in grade terms. Some students still want to revise Successful work up to Exemplary, which can be OK as far as it goes – as long as it’s not distracting from more critical issues in other work – but I suspect students doing this don’t fully understand that. Some students inevitably maintain that misconception and end the semester expecting that Exemplary will somehow help their grade.

All of this has convinced me that it’s not worth the effort to have a separate Exemplary mark if I’m not going to give it concrete grade value. Instead, if I want to acknowledge excellence without a grade consequence, I give students personalized comments where I call out the specific reasons that their work is excellent. I sometimes even do this in person, one-on-one before or after class, or in an office hour.

Finally, whether or not Exemplary counts in final grade requirements, it makes those grade requirements harder to communicate. For example, “All homework at Successful” has to become “All homework at Successful and/or Exemplary” or “All homework at Successful or better” or something similar. Clarity is essential in an alternative grading system, and no matter how hard I’ve tried to make my wording clear, students do seem to get hung up on this.

Exemplary can be norm-referenced

A central feature of alternative grading is that student work is assessed against clearly defined standards, not against other students’ work. Assessing against a clearly defined standard is called “criterion-referenced assessment”. Setting clear and consistent standards is critical because it ties grades directly to learning rather than the relative level of student work compared to others.

When a student’s grade depends on what others do, that’s called “norm-referenced assessment”. “Curving” is a classic example of this, as is any system in which we categorize student work relative to other student work, to the class average, to historical averages, and so on.

Now go back and take a look at my description of Exemplary – it’s kind of tenuous, isn’t it? They key part – “especially notable in some way” – just about begs me to compare each student’s work to others. Indeed, while I’m grading it is quite easy for me to think “oh, this is the clearest proof that I’ve seen so far” or “wow, nobody else thought of this clever approach!”, at which point I’m tempted to assign them (and only them) Exemplary.

That’s contrary to the spirit of clearly defined standards and marks that indicate progress towards those standards. One major negative consequence of norm-referenced assessment is that it introduces competition into assessment, in which students compete against each other for a limited number of high marks. I don’t want that – I would love it if every student earned Exemplary – but without a clearer definition it’s hard to avoid comparisons.

To try to address this, I have tried creating more concrete and limited definitions of Exemplary. For example, I’ve tried emphasizing the need for excellent notation, spelling, grammar, and punctuation. This generally leads to very nice revisions that certainly meet my criteria, but these aren’t “excellent” in the sense that I really want. You could reach this kind of excellence in a purely mechanical way. This might be a fundamental flaw that I can’t avoid.

I’ve also found that the potentially norm-referenced aspect of Exemplary means that it’s hard to give useful feedback on how to revise Successful work to become Exemplary. I described above how I sometimes give feedback that includes personalized challenges. What if I can’t think of a useful challenge? Sometimes I’m stuck trying to figure out why I don’t think a solution is Exemplary. This risks becoming an “I know it when I see it” situation, which is not a standard at all.

Summary

I have ambivalent feelings about Exemplary marks. I go back and forth about using them. Grades create incentives, and the incentives that I want aren’t always the ones that Exemplary creates.

My advice to new alternative graders is to avoid Exemplary entirely. It is much more valuable to focus on what a student can do if their work isn’t yet Successful, such as by distinguishing between work that can be fixed up (Revisable) and work that needs to be reattempted from scratch (New attempt needed).

It’s also worth remembering Linda Nilson’s advice in Specifications Grading: The bar for Successful should be quite high, something like 85% or a high B in a traditional grading scale. This doesn’t require perfection, but it does mean that Successful work should show more than a minimum level of competency. Successful work on its own should be something to be proud of. Feedback loops and reassessments can help raise student work up to this high standard.

If you have ideas about how to use Exemplary more effectively, I’d love to hear them in the comments.

1. Students also seem to like and understand the idea of “An example that could be used in a textbook”.

2. Not everyone may agree with me about this goal for an intro-to-proofs class, but it’s something I believe strongly. Learning to write good mathematical proofs is a lifelong endeavor, certainly one that extends throughout the undergraduate curriculum for a math major. Expecting students to emerge from their first proof-writing class as fully formed expert writers is impractical at best.