Bold, New Higher Education Agenda

Much of what I read in various popular, or special interest, education-centric publications, whether research-based or not, appears to offer palliative measures at best, frequently approaches pablum, often is misinterpreted or misapplied, and at worst is harmful in one manner or the other. Also, as someone educated in the natural sciences, and steeped in the scientific method, I find myself struggling to see the broader applicability for much of the educational research that I’ve read since entering the education field. Yet, as a result of my experiences as a high school teacher, I know that much must change in how we educate students today, especially in mathematics.

Furthermore, given the inherent dependencies in mathematics, a strong foundation in numeracy must be established early in life, in some ways prior to a child entering any formal schooling environment, such as preschool. Similarly, developing literacy requires immersing children in all aspects of one’s native, or parent-chosen, language starting at birth. To do otherwise on either front could severely limit a child’s ability to maximize their potential in life, as most professional or vocational careers reward individuals who possess advanced abilities in both areas: numeracy and literacy. Any delays doing so complicate the efforts of our educational system to prepare the most capable, learned graduates ready to contribute to the furtherance of society.

Hence, it was with great excitement that I read the commentary: California’s next governor has opportunity to set a bold new higher education agenda by William G. Tierney, university professor and Wilbur-Kieffer Professor of Higher Education in the USC Rossier School of Education and the co-director of the Pullias Center for Higher Education.

He strikes the proverbial nail on the head in the following excerpt from his commentary.

“In addition, too many of our students are not educationally ready when they arrive to college, and they are not prepared for the workplace after college. They take too long to graduate, and take on too much debt. Too few who attend community college transfer to a four-year institution.

All of these problems are well-known, but past measures have been modest.

To truly make California a prosperous state, the new governor must set transformative goals. He must spearhead a plan that ensures college hopefuls are college-ready upon graduation from high school. He must allow community college students to transfer seamlessly to four-year institutions. And he must enable bachelor’s degree candidates to graduate within six years, ready for the job market and unburdened by crippling debt.”

As someone re-energized to improve outcomes in our nation’s educational system, but from the outside this time, I hope to connect with Mr. Tierney to collaborate on how best to contribute to moving his ideas forward so that we start to graduate students prepared for college and/or career. Today’s system, common core infused or not, falls far short of the mark.

* Success in differential equations requires success in calculus which requires success in algebra and trigonometry which requires success in arithmetic and geometry, and not as linearly as described. Success in any and all of these areas requires much more than a superficial treatment, which is what most students of mathematics receive today in most classrooms across America. The reasons why vary. Mathematics curriculum and pedagogy must change if our nation wishes to remain a leader in a highly competitive, global economy.

Posted in Pedagogy, Political | Tagged , , , | Leave a comment

Back in the Saddle, Again…

After a lengthy hiatus from posting to this blog, I’ve decided to break out my pencil sharpener and eraser to pen some prose, if that’s even possible using an iPencil…

It’s been an amazingly challenging few years since I posted regularly. However, due to the good fortune of recent events, I have time now to describe many of my experiences, up and down, from that time period primarily through the lens of a second-career, math teacher.

The essence of most of my posts, past and future, is captured nicely in the title of the article: Why are Americans so bad at math? from which the following quote is taken.

“Without much of a surprise, it turned out that the memorizers were the least likely to achieve high scores and understanding. The United States ranked in the top three for this learning method. A more in-depth look showed that memorizers were about a half year behind students who used either relational or self-monitoring strategies.”

Be sure to read the short article for sufficient context to understand the quote more fully.

While I had planned to include a link to Aerosmith’s version of Back in the Saddle, its lyrics, which I must admit I never knew*, are highly inappropriate for any educational setting. So, here’s Gene Autry from the movie Back in the Saddle.

* …along with most broadcast or recorded songs of which I’ve listened to and enjoyed in my life, for that matter…hence, why you should never listen to me while I’m singing along with any song!

Posted in Uncategorized | 1 Comment

Showing Work in High School Mathematics

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.


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.

  1. Unsupported answers will not receive any credit. Even the simplest of questions require some form of justification unless otherwise stated.
  2. Never copy work from another student on any assessment. Doing so has dire consequences as detailed on my green sheet.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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.)
  14. 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.
  15. 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))
  16. Include proper units, if provided in the question. There is no need to state “units” otherwise.
  17. 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.
  18. 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.
  19. 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.”
  20. 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.
  21. As appropriate, consider including transition words such as “hence,” “thus,” or “therefore” to help the reader follow your work.
  22. Simplify your final answer
  23. Time permitting, check you answer and show it as such.
  24. 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.
  25. 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.




Posted in Mathematical Proficiency, Pedagogy, Uncategorized | Tagged , , | 25 Comments