At the beginning of the course, for practice students complete a couple of stand-alone explications of documents provided by me. Then, once they begin work on their papers, similar but different explicated documents that they discover become the core of those papers. In concentrating on explication, I am tapping into our mathematics students’ aptitude and interest in precise, technical work, the nuts-and-bolts of mathematical tasks. As well, I am tapping into their problem-solving inclinations and imagination in the form of detective work to discover what the author wrote. I tell them they have both the advantage and disadvantage of hundreds and in some cases thousands of years of mathematical progress and training in their background beyond that of the author they are reading. The advantage is that they can often “see through” what the arguments are leading to, as they jot notes in the margins using modern symbolism and graphs, and so on. The disadvantage is that by engaging that modern perspective they entirely miss the cultural context of the time, and in fact fail to “see” what the writer has written. Their modern understanding eclipses their attention to the details of what is on the page. We work on stripping back their “overview” to discern what is said and what is not said, what is assumed in that time period and what is anachronistic and ahistorical.

Explications must be “just in time.” It is annoying and loses the reader’s attention to read a long passage which makes little sense to them, only to be followed by a long explication. Such an explication is too late, and often hard to precisely align with the points in the passage where it is meant to apply. It also tempts the reader to ignore the original passage and just read the explication, which denies them the true understanding of the context as well as the substance of the arguments. I will tell students “The author is the focus, not you. With the assistance of your gentle guidance, your light touch, we want to be able to read the author’s work without pause.” I share with my students what the *Oxford English Dictionary* writes:

Explicate, verb:

- a. To unfold, unroll; to smooth out (wrinkles); to open out (what is wrapped up): to expand ...

c. To spread out to view, display. - a. To disentangle, unravel.
- To develop, bring out what is implicitly contained in (a notion, principle, proposition).
- To unfold in words; to give a detailed account of … .
- To make clear the meaning of (anything); to remove difficulties or obscurities from; to clear up, explain.

To explicate is to cleverly, subtly answer questions before they are asked, so the reader, when he or she raises these questions, realizes the answer has already been handily provided. Thus, apprehension of the material and moments of clarity arrive more quickly because the reader’s intellectual struggle has been diminished by the explicator's apt remarks. An explication is not provided in order to train readers; it is instead to ease their path to understanding what the author wrote. Examples of helpful insertions include:

- A single word, or short phrase.
- An intermediary calculation.
- A contextual remark.
- A longer, separated discourse on its own line, where needed.
- Added diagrams or pictures, often in the form of a film strip of images leading to the desired configuration. This is helpful in geometry papers where only a single picture might be included in the original document that is better understood in fact as a sequence of pictures.
- Domains of variables.
- How extreme cases are handled, omitted, or overlooked.
- Adding enough links in the chain of reasoning that the mathematically knowledgeable reader is willing to trust any few pedestrian operations omitted.

In the beginning, students are not sensitive to where these additions may be needed. For instance, they often include long explanations of parts well-written by the author (as an unconscious act, I think, to prove to themselves that they understand), and neglect spots of confusion that they “saw through” but which desperately require explanation. My job is to generously red ink their early attempts and point out where there is real need for explanatory assistance. I remind them, “It is harder to be clear than you think!”

Here’s the sheet of advice I give my students for those short practice explications in advance of and separate from writing their papers, although of course the advice is universally useful. As I mentioned above, papers will then center on similar explications, but not consist solely of them. Note that their first task is to accurately type up the original source document, which seems easy, but often reveals a novice inattention to details like unusual or archaic spelling and capitalizations, grammatical oddities, and so on. This too turns out to be good practice for them.

**Historical “Proof” Explication**

- Read the given argument, proof, or theorem and proof combination. I have photocopied them from an original historical document, or faithful English translation. The assignment is designed to be more-or-less self-contained.
- Explicate this result, that is, write an expository version. Your version will usually therefore be longer than the original. Remember that a “proof” is a narrative, telling the story of (proving) why the theorem is true. Your job is to make that story transparent.
- Stay as close as possible to the style and form of the argument, preserving the historical flavor and ideas of the author. Do not substitute a faster, modern statement and proof.
- You will be graded on the clarity of your exposition.
- You will also be graded on how critically you have read the result, whether you found all the confusions, omitted or overlooked arguments, and so on, even if you were not able to settle all puzzles to your satisfaction.
- Your work may require any or all of the following:
- Clarify words, definitions, and statements. For instance, "line" may be used where "line segment" is meant, "equation" confused with "expression", or "equal" with "congruent" or "equivalent"; the same letters or words may be used for several different objects; out-of-date terminology and phrasing may need to be updated, or just made more precise.
- Is the result properly stated as a Theorem, Proposition, Lemma, Corollary, etc.? Is the Proof so named, and clearly delineated?
- Add as many pictures as you like to clarify the argument. These include "idea" pictures, as well as the usual graphs, diagrams, constructions, etc. A detailed “movie” or “film strip” of images is often needed.
- Include omitted arguments, or other details. Some arguments may be long enough to be stated (by you) separately as a Lemma. Do so, if you like. Other arguments may be knowledge assumed common by the author, but not clear to you or your modern readers. Tell us. This is vital to good exposition.
- Correct any mathematical errors or omissions you may find. For example, if a variable suddenly appears in a denominator, did the author consider the case when that variable might be zero? Are there other omissions of cases we would today include? Are there typographic errors? Are the calculations really correct? Take nothing for granted.
- Modernize the mathematical notation in your notes if needed, but again, stay close to the history.

**When you read my Explications (and when you review your own)**

- What changes did I make? Mark or circle them.
- Why do you think I made them?
- Was I reasonably faithful to the author's ideas?
- Are the pictures sufficient?
- Is the explication clearer than the original?
- Would you do anything differently or better? What exactly would you do?
- What did you learn about the author's strategy, ideas, methods, understanding?