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.

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The singular and important role of the classroom teacher in getting students to learn is well established in the research literature (see here and here). I have no quarrel…]]>

The essence of my teaching methods depends upon me successfully cultivating student attitudes and habits that help them take responsibility for their learning…just as Dr. Cuban states in the last sentence of his post.

The challenge is being successful at said cultivation as most of the students I have encountered in my six years of teaching secondary mathematics have mostly been acculturated to being passive recipients of knowledge rather than active seekers of knowledge…This is the crux of the dilemma facing our nation’s secondary schools in preparing students for post-secondary success…the world is not fill in the blank or a series of highly scaffolded worksheets…

Larry Cuban on School Reform and Classroom Practice

The singular and important role of the classroom teacher in getting students to learn is well established in the research literature (see here and here). I have no quarrel with that frequent finding (whatever the metrics) to confirm that teachers are instrumental to student learning. What is far less clear is what part do five to 18 year-old students play in the chemistry of learning.

It is a question that I have puzzled over in my many years teaching high school and graduate courses. And I have no certainty in answering it.

For some teachers, as one told me after I observed his mediocre lesson, “I was selling but the students weren’t buying,” students bear the lion’s share of the responsibility. They are expected to come to class, obey the rules, do the homework, participate in discussions, and do well on tests. Those are students’ responsibilities. Other teachers (and…

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Anyone with children in public schools in the US should read the letter from the US DOE at this blog. It is accountability run amok, and borderline fascism. Bureaucrats just love…]]>

Reflections of a Second-career Math Teacher

Anyone with children in public schools in the US should read the letter from the US DOE at this blog.

It is accountability run amok, and borderline fascism. Bureaucrats just love to make threats from their bully pulpits, whether at local, state, or federal levels.

While monitoring some relevant measure(s) over time in return for receipt of federal funds earmarked for specific use is prudent, coercively mandating near census level approaches constitutes federal overreach, tramples on the rights of parents, and ignores less onerous, more effective statistical methods.

Is this what we want from our federal government?

It is accountability run amok, and borderline fascism. Bureaucrats just love to make threats from their bully pulpits, whether at local, state, or federal levels.

While monitoring some relevant measure(s) over time in return for receipt of federal funds earmarked for specific use is prudent, coercively mandating near census level approaches constitutes federal overreach, tramples on the rights of parents, and ignores less onerous, more effective statistical methods.

Is this what we want from our federal government?

It appears the DNC nominee heir apparent has little idea about public education in spite of the NEA and AFT endorsements…what does that say about those organizations?

“There are few things more dear to corporate reformers than the idea that schools with low test scores should be closed. There are several reasons for this. On the ideological front, the possibility of closure (and the threat of displacement and unemployment for the teachers and administrators that work there) is seen as an essential element of accountability. One of the indictments “accountability hawks” level at public schools is that while failing businesses must go bankrupt and close their doors, “failing schools” can continue to function unless the rules force them to close. According to this way of thinking, people are not properly motivated without the threat of unemployment hanging over their heads.”

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Scores improved significantly on the final for this cohort. I believe there are a variety of factors contributing to the improvement:

- I taught 80% of these students last year in honors precalculus,
- I created practice quizzes (with fully worked solutions) and tests to aid them in preparing for all assessments to include the final, and
- I lessened use of the “flipped classroom” providing more direct instruction before students engaged with their homework.

Net-net, I am pleased with the results for this cohort.

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Updated post with following data addressing a reader’s comment re: relative distribution of former v. new students in 2015 results.

I suspect that if I ran a t-test on the former v. new data it would yield a statistically significant improvement in many of these descriptive statistics.

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The professor’s plight is worth reading as he appears to be an excellent mathematics lecturer who cares tremendously for his students at Berkeley. Like me, he rejects calls for doing things as they’ve always been done if how they’ve always been done is not working. Also, like me, he has suffered emotionally from the stress of staying true to his calling and beliefs while being castigated for not teaching like everyone else.

I hope the UC Berkeley administration steps in to support this outstanding lecturer.

It will be a travesty if they do not.

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Hi Mr. Math Teacher, hope you are having a great school year!

Just wanted to let you know that I am loving my Calculus class this semester (it’s my favorite class!) and to thank you for providing me with such a strong foundation. My professor reminds me of you (to some extent) **because he is so so passionate about our learning and consistently reminds us that learning is more important than grade performance (which I remember you often talking about too)**.

My professor was recently fired because of department issues and his unusual, eccentric teaching style and abnormally high student ratings, and has written a response that you may find interesting:

http://alexandercoward.com/BlowingTheWhistleOnUCBerkeleyMathematics.html

He also wrote another article regarding strikes and education in 2013 (which you may have seen before – I think it went viral):

I guess I am writing this and sending you these articles because he really does remind me of you. **The two of you are the best teachers I’ve had thus far in school, and are why I love math and especially why I value my education so much.** Thank you for everything!

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]]>Dear Mrs. Laurene Powell Jobs: I commend you for initiating a national challenge to transform the comprehensive high school into a Super School and putting $50 million on the…]]>

Larry Cuban on School Reform and Classroom Practice

Dear Mrs. Laurene Powell Jobs:

I commend you for initiating a national challenge to transform the comprehensive high school into a Super School and putting $50 million on the stump for experts, parents, practitioners, and academics to compete for in creating better high schools than exist now. Reinventing the high school should generate an enormous range of suggestions for your expert panel to consider after the national round of open meetings end in November. What you are launching is worthwhile especially if it were to spark a national conversation about the goals of tax-supported public schools in a democracy where the economy has shifted from industrial-based to an information-driven one. Whether that conversation (and debate, I hope) will occur depends greatly, I believe, on you and your associates knowing about how high schools have, indeed, changed over the past century and, of equal importance, the checkered history of efforts to…

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Posting my comments to Larry Cuban in response to his post titled “Evidence: The Case of the Common Core Standards.” Larry shines a spotlight on the lack of research behind the…]]>

Republishing given the recent revelations regarding the questionable validity of psychology research, and the similar challenges facing its distant cousin, education research, which this prior post addresses.

Reflections of a Second-career Math Teacher

Posting my comments to Larry Cuban in response to his post titled “Evidence: The Case of the Common Core Standards.” Larry shines a spotlight on the lack of research behind the widespread adoption of Common Core. I agree with his assessment, and offer additional perspective, in my comments below.

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As someone now aboard the Common Core train, I believe the CCSS effort embodies the “bandwagon effect” at a national scale. In many ways, it parallels NCLB as a cause that at the sound bite level sounds right and just, but at the classroom, school, and district level results in distortions of staggering proportion. Sadly, such fallacies hypnotize education policymakers into institutionalizing illogic, which takes decades to reverse.

For what its worth, I find the use of “research-based” in education as more of a code word intended to signal the efficacy of a new policy, process, program, and etcetera…

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There are three ways to reduce the kind of burnout that so many K-12 teachers, particularly in low-income minority schools such as Spanish teacher Alli Baugher at Ballou High…]]>

“Teaching is energizing but also exhausting work. Each day teachers spend the rich intellectual, physical, and emotional capital that they have accumulated over the years on their students. Because of that loss in capital, teachers need to re-invest in themselves by doing what expert gardeners do with favorite potted plants.”

So true, sadly. I suspect the nature of the teaching profession will continue to demand more from many teachers than they are capable of supplying in a sustainable manner. At the outset of this, my fifth, year, I relinquished my desire to affect change on a grander scale to return to my classroom with renewed vigor. Here’s hoping the re-potting takes.

Larry Cuban on School Reform and Classroom Practice

There are three ways to reduce the kind of burnout that so many K-12 teachers, particularly in low-income minority schools such as Spanish teacher Alli Baugher at Ballou High School in Washington, D.C. experienced. Change the work conditions or change yourself (or both).

**Change working conditions.** The age-graded school was a mid-19^{th} century innovation imported from Prussia and planted in the U.S. Within a half-century, the innovation slowly and irrevocably replaced the one-room schoolhouse throughout the nation. Erecting a “grammar school” housing eight grades with separate classrooms where teachers teach six year-olds in one room and ten year-olds in another reorganized the very nature of schooling in the U.S. The principal and teacher would determine whether each student had learned that portion of the curriculum allotted to that grade in one year’s time most often through tests. If the student passed the various tests he or she advanced to…

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