After four plus years teaching upper level high school mathematics, it is clear to me that students do not understand what it means to show one’s work in mathematics. In spite of creating close to two hundred completely worked examples clearly demonstrating how to show work for each of my courses (accelerated algebra 2, honors precalculus, and AP Calculus AB), students continue to write fragmented, disjointed, oftentimes illegible work accompanying their answer.

After stressing the necessity and the specifics of showing one’s mathematical work hundreds, if not thousands of times over the years, I decided to publish a short treatise on the matter, which I recently shared with my AP Calculus students.

I have yet to discuss the collection with them, however, I will do so soon.

Until then, and on an ongoing basis, I am interested in others’ perspectives on my requirements below. My communication with students mentioned that the list was not exhaustive. I thought of a few others shortly after I emailed it out. However, it captures many important aspects of effectively showing mathematical work.

Please share with me any that you believe I may have omitted, and there are many I am sure. Also, if you disagree with any, I would like to know that as well as why.

Here is what I sent my students.

**Showing Work – Best Practices**

In mathematics, showing work is how you make your mathematical thinking visible. Well-written mathematical work has a clear starting point, often restating given information along with what one is asked to find or to do frequently accompanied by a sketch in the form of a diagram or graph, followed by clearly delineated steps in a logical progression from the givens to the solution. Along the way, separate, side work may be required to determine information needed to complete the requested task, especially when that information was not directly stated in the question. This is often where prerequisite knowledge comes into play where one applies what they know to what they are given in order to proceed in determining what they are asked to find or to do.

Similar to writing an essay in a language arts course, where proper use of punctuation is required, as is adherence to proper rules of grammar, so too in mathematics one must follow established conventions. The following requirements detail many of the conventions typically required in higher-level courses of mathematics such as accelerated, honors, or AP courses. Many of these requirements should be familiar, as most students have encountered mathematics teachers who required students to follow these rules at one time or another. If they are not familiar to you, please see me for a detailed explanation and demonstration with examples.

__Requirement__

Students are required to adhere to the following requirements for showing their work on all quizzes and tests in this course unless otherwise indicated. Failure to do so will result in a deduction from the question’s point value (as an example, ½ point for a 2 point question). Repeated failure to follow these requirements may result in a 50% deduction or no credit. Accordingly, students should follow these requirements for their homework as a means to develop these as a natural habit.

- Unsupported answers will not receive any credit. Even the simplest of questions require some form of justification unless otherwise stated.
- Never copy work from another student on any assessment. Doing so has dire consequences as detailed on my green sheet.
- Follow the directions completely – Part I. Pay close attention to the text in the directions / instructions. Incompletely addressed questions will receive partial credit or no credit, depending upon the amount of tasks omitted.
- Follow the directions completely – Part II. If a particular method is specified, you must use that method in our work. You will not receive credit otherwise. You may use alternate methods as a means to check your work.
- Follow the directions completely – Part III. If multiple items need to be determined, clearly demonstrate how each is found and mark them clearly and separately.
- Write neatly. If your work cannot be read easily, you will likely not receive full credit and may not receive any credit depending upon the difficulty reading your work, or finding your answer.
- Start your work with an algebraic expression that represents what you were asked to do, to find, etc. If there is room following the provided, printed expression, you may use it as your first statement in your work followed by an appropriate relationship symbol, typically an equal sign.
- Do not mix work where algebraic expressions are written with variables with numeric expressions where specific values for those variables are used to evaluate the expression or equation. Clearly separate the algebraic statement from the evaluation of the algebraic statement.
- Include equal signs whenever you are stating equivalence, which is often the case when you are provided instructions such as solve, simplify, find dy/dx, etc. followed by an expression upon which to perform the required task.
- Write your work in a logical flow where it is clear to the reader why you performed a specific operation or applied a specific property in the work.
- If multiple intermediate segments of work are required to arrive to your final solution, group this work in logically separate areas and connect them as appropriate with some symbol to include ==> or arrows.
- It is highly recommended to write out any formula associated with the question as your initial work with the next step including the relevant values from the givens in the question.
- State answers as rational numbers unless given values were in decimal form. It is OK to leave the rational number as an improper fraction unless the question is dealing with units of measure (e.g. feet, seconds, meters per second, square feet, cubic feet, etc.)
- If a decimal form is appropriate, always provide the requested level of precision (e.g. three decimal places or thousandths) if specified. If not specified, use your best judgment.
- When asked to sketch or to graph, always do so neatly. Also, label the axes with the independent variable and the dependent variable. Also, provide a proper scale for both axes. When applicable, include a description of the parameters, with units, represented by each variable (e.g. the independent variable, t, could represent time in seconds and the dependent variable, h(t), could represent the height of a projectile in feet. If multiple functions are shown on the same graph, label each function appropriately (e.g.
*f*(*x*) and*g*(*x*)) - Include proper units, if provided in the question. There is no need to state “units” otherwise.
- If showing your work in vertical form, do not rewrite the left-hand side (LHS) expression, simply leave the LHS blank, however, you must include an equal sign for each line.
- If there is not enough space to continue showing your work in a vertical fashion, use an arrow from the last line of your work to the first line of a new column. Make it clear to the reader how to follow your work. Use these arrows sparingly. Work that includes arrows crossing all over the workspace is not presented clearly, and will not receive full credit.
- If you introduce new variables in your work, you must define them by stating “Let “new variable” = “appropriate algebraic expression, numeric value, written description, etc.”
- If asked to explain anything, always state your explanation as a complete sentence or set of sentences using the provided nomenclature. Use proper grammar, punctuation, etc.
- As appropriate, consider including transition words such as “hence,” “thus,” or “therefore” to help the reader follow your work.
- Simplify your final answer
- Time permitting, check you answer and show it as such.
- Only provide one answer with associated work. Multiple solutions will result in no credit for the question even if one of the answers provided is correct. Cross out any work you do not wish to be considered. At the same time, be careful not to cross out or erase work prematurely, as it may permit you to obtain partial credit; hence, carefully select which work you wish to keep.
- Box or double underline your answer(s). In other words, make the location of your answer clear. If it is not clear, you will not receive full credit.

I am really interested in why students fail to do this. Independent of how best to convince them, the fact that the resistance is so common seems really important.

Both my children are notorious for this. My younger used to argue ‘but I know the answer, why should I have to explain how I know it?’

To make matters worse, when he was younger, the younger one was able to do multi digit multiplication in his head but actually would fail when trying to write out the steps. In a sense, the goal of that process was arguably to get them to a point where they could do these things in their head. If they’ve already reached that point, it feels odd to put so much focus on the steps that are supposed to lead there.

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As with many things in life, there are nuances to the requirement to show one’s work in mathematics. The primary audience for my requirements is high school mathematics students, perhaps starting with those in an algebra 1 course, or even a pre-algebra course, where not only the computational elements are important but also the procedural structure in how the work is presented.

To facilitate greater skill in showing work in high school, I believe it would be beneficial if primary school students started to learn how to show their work under a less onerous set of requirements, yet, sufficient to help them develop the ability to communicate their mathematical thinking in a conventional manner. Would this apply to every single question asked at the primary school, or even middle school, level? Likely not. However, there need to be a set of questions asked of students where they not only complete the task at hand, whether in their head or not, but also show the significant step(s) in arriving at their solution.

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Thanks Dave. My comment was by no means an implication that your requirements should apply to primary students as well. I was only pointing out that the resistance is fundamental, starts early, and is almost universal. My anecdote only attempted to reinforce those points.

Personally, I think this points to a failure in the level of consistency with which we prioritize school tasks, but that’s just me. But if it’s true, it’s probably as or even more useful to figure that part out, and much earlier than high school. 🙂 thx.

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“…resistance is fundamental, starts early, and is almost universal.” I could not agree more.

At the primary school level, it has developed into an ingrained response that may never be completely overcome given the scaffolding of much of the tasks; the need for showing work is nuanced and requires deft application by practitioners skilled in the art, which is asking too much of primary school teachers, IMO. BTW, I took no offense at your comment. It was spot on. I just wanted to clarify that my post focused on secondary students. Cheers!

On Mon, Jan 25, 2016 at 10:42 AM, Reflections of a Second-career Math Teacher wrote:

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As I recall, you have an incredibly skewed grading system, in which a 40 counts as a passing grade. Doesn’t it seem as if you are knocking off points for things that you’re going to turn around and give back by a much more generous grade curve?

These are high school kids, which means almost none of them are going to read that long document. All you’re really doing is giving them the justification for how you will grade it. They’re going to remember the utterly meaningless things like box your answer (really? You’ll knock off points if they haven’t boxed it? That strikes me as insanely picayune.)

I’m amused if anyone ever thinks your list is too short. This list is 25 items too long.

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I no longer employ the system to which you refer. It is old-school now.

Please keep in mind, the document is intended for students of upper level mathematics, many of whom care deeply about their grades and whatever rules they must follow to maximize same. While few may read the document in detail, it now exists as a compendium of practices they have certainly heard from their math teachers over time and is easily referenced. The number of which and the extent to which they were enforced likely varied.

On the picayune, and utterly meaningless, front, if I or another student grading the work cannot locate their answer in a reasonable amount of time, I suspect the student would prefer a method by which they signal its location for easy identification rather than receive no credit.

If their answer is clearly visible, then I likely would not deduct points at the outset of enforcement period, yet I might over time as it is the required style in my course, like it or not.

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I agree I thought the document was a little long. I might show an example of a properly formatted answer with arrows pointing out the features described in your 25 rules.

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Yes, there is a fine line between too much and too little when it comes to communicating requirements. I have a separate document I am creating with one to two examples for each. It is a work in progress right now as there is an even greater challenge at balancing depth and breadth.

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That last sounded meaner than I meant it–I’m not trying to be disrespectful. I just am more, er, realistic about what it will accomplish. But I do think you should consider the list in light of the grading curve (unless you’ve changed that?). Because you’re taking more points away, but the impact is discounted.

Would you fail a kid who basically did the test correctly, got everything accurate, but didn’t box the answers, answered correctly to questions that really needed no support (and you were sure he wasn’t cheating), had messy work that didn’t follow your rules but clearly demonstrated understanding? If he didn’t use the right process, you wouldn’t just send him back telling him to read the instruction?

I definitely wouldn’t.

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ed: I would not fail anyone for not following my requirements with certain exceptions as I explain shortly. However, they likely will not receive full credit, especially if it is made clear that these are my requirements for showing work; of course, there are nuances and subtleties in all of them, which is open to discussion on a case by case basis.

A student might fail if they omitted work where no human could make the leap from question to answer or if they omitted a majority of a question with multiple parts.

My document is meant to serve as a style guide of sorts for my students somewhat akin to Strunk & White’s Elements of Style, but far, far from it in comprehensiveness, details, rationale, etc. But that is the difference between three pages of bullets versus 100 pages of text.

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That sounds reasonable. I like the idea of a style guide. I’d edit it down dramatically, though!

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one of your most relevant and succinct posts

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As I recall in my own math class, we were never instructed on how to format our answers, though it would have been welcome. Mostly, I ended up imitating the solutions my teacher would draw on the board. In particular, I learned to line up my equal signs. It’s always lovely and clear when those equal signs are all in a column. Also, to this day with my MathCounts team, when finding the are of a triangle, even though everyone knows the standard formula, I begin with A = 1/2* b* h. Always. And then I line up another equals sign and proceed with the solution. My own kids are homeschooled, and when they were in elementary if they got the problem right, they didn’t need to show their work. If it was wrong….well. Now their problems are too difficult to solve in their heads, lol.

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Good for you! Many of my students continue to forget to state what they are finding / solving / simplifying / etc followed by an equal sign as their first move even though I have shown them countless times…

FWIW, equation editor in MS Word confounds me no end when it does not automatically align the equal signs in vertically displayed work…I need to add spaces manually, which is beyond tedious when typing up mathematical solutions is difficult enough.

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Well, that list is one approach but I have a feeling it will have zero effect. Kids know they are supposed to show their work. Most just do not care enough to do so. Now you expect them to care enough to pay attention to a 25 point document about something they already know. I have been fighting this same battle for about 30 years. Telling them what they are supposed to do does not seem to work to well. Finding out why they do it is the true path to peace. That “why” can be quite revealing. It does not solve the problem but it does give some strategies for those kids that are willing to accept them. Remember most of these kids, even the advanced ones, are thinking (probably mistakenly) math is not going to play a huge part of their future so most are trying to do the minimum to survive their high school math courses. A few will be influenced by a change in the grading schema to stress the process and not the answer. Most will try and find the new minimum. I do not mean to sound pessimistic and I hope your strategy, both with the list and the grading schema change, works but 30 years has made me a realist. Keep us readers posted on the results. If it works I will borrow the list and give it a whirl. I am also an optimist.

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As I’m only in my fifth year of teaching, I’m much more hopeful. I believe most of my students truly want to show their work properly and appreciate having a set of guidelines as a check list of sorts. At least that is how I rationalize things for the moment. I will update folks on whether I see progress or not.

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I believe most of my students truly want to show their work properlyWell, you believe something that is not true.

Honestly, do you box all YOUR answers on the board when you do working? I box some of mine, when the answer is not clear from context, but not all. Sometimes the answer is so obvious boxing is totally unnecessary.

And I really, really doubt you check your answers all the time either. I bet you assume they are right, just like they do. Checking answers sounds good, but often actually isn’t because of time constraints. In exams they are better off answering more questions than checking ones that are likely correct. (I advise only my weaker students to check answers because they usually finish early and need to get all the ones they can do right.)

You’d be better off trying to persuade them, IMO, than force. Properly worked examples on the board help. Mocking them by doing it their way once or twice often has a salutory effect (especially doing differentiation without separating equations e.g. x^2 + 5x + 7 = 2x + 5).

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Your list is very thorough but my guess is that students will not bother to read a list of 25 small paragraphs that explain how to show their work. It might be good to figure out how to condense this to maybe a list of 10 key phrases. Maybe something like the 10 Commandments of showing work. Those are just my thoughts

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I agree a more concise version might be more easily digested. I am working on an expanded version with specific examples of each at the moment. I will reverse the process later in an attempt to condense requirements and examples into something akin to your suggestion. Thanks!

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Dave,

I concur with your thoughts about focusing the students on a standardized methodology for showing how they derived their answers. I agree with other posters that 25 points may be too long, and that the requirement be reduced to a few key common processes.

What the students may not grasp is that the work shown gives you a clue as to where they failed to grasp a key concept in class, or where they made a simple mathematical error. The double underline with ANS: clearly shows what the student is displaying as their final answer.

I have a good idea where these rules originally came from, but they are tried and tested.

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Hey Chris! It’s been a very long time, my friend. Now that you mention it, I should dig up my old USMA department of mathematics guidelines to see what I left out! 🙂

FWIW, I explain frequently the why’s of showing work to include a student troubleshooting their own work to enabling the diagnosis of misconceptions, etc.

Yet, they have been permitted not to show work for so long, it is a tall order to help them change their ways…

I agree a shorter, more condensed version could be more digestible.

Hope all is well.

Dave

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The document is too long. Also, never give a laundry list of 20+ items. Always group your ideas by topic into lists of no longer than 7 items (ideally 2-5) and order them within the groups with some rationale (importance, location, etc.) These are basic writing composition ideas for any document: work world, syllabus, etc.

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I’m sad to see that it’s been nearly a year since your last post. Do give us an update! Many of your fans are rooting for you.

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My apologies for the extreme delay in my response…I was consumed by life…but I’m back!

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