Last semester, I formally assessed my honors precalculus students via summative assessments fourteen times: ten quizzes, three tests, and one final exam. Informal formative assessments occurred frequently and varied over time. Formal formative assessments occurred once to twice weekly via an online, Google form. The latter provided insight into student understanding in advance of any class, as I employ a hybrid flipped classroom model with these students.
Each quiz primarily covered a section in our textbook: Precalculus: Graphical, Numerical, Algebraic: Demana, Waits, Foley, and Kennedy. 7th Edition. 2007. Quizzes consisted of 10-15 free response questions taken from a test generator that accompanied the textbook; at the outset of the semester, the first three or so quizzes were multiple choice / multiple guess – however, given the latter name, students showed minimal work, which led me to question whether they knew how to work the exercise properly or simply knew how to work backwards eliminating possible answer choices. Questions in the test generator closely followed the exercises in the textbook, which were assigned as homework and discussed thoroughly in class, often with students working in groups and presenting their work to the entire class from the multitude of white boards within my classroom. Additionally, preceding each quiz, I provided a practice version with representative questions mirroring the upcoming quiz; often, the real quiz included questions from the practice quiz. Furthermore, one to two days after releasing the practice quiz, I posted completely worked solutions for the practice quiz with the intent to allay any student concern about how to work any question as well as to demonstrate how to show work following mathematical conventions. In summary, quizzes provided students the opportunity to display procedural fluency with familiar questions and to develop their proficiency writing mathematics in a clear, concise fashion.
On the other hand, I designed my tests to challenge students’ conceptual understanding, adaptive reasoning, strategic competence, and their productive disposition, four of the National Research Council’s (NRC) five strands of mathematical proficiency; the fifth strand consists of procedural fluency, which the quizzes cover extensively. Likewise, in the vernacular of the National Council of Teachers of Mathematics’ (NCTM) process standards for mathematics, my tests assessed students’ abilities to solve problems, to reason and to make proofs, to interpret and to express mathematics in a variety of representations, to communicate about the mathematics, and to make connections between concepts covered in the course, and beforehand; the Common Core subsumes these into their eight mathematical practices. Lastly, not to shirk my Common Core responsibilities to develop literacy within my subject area, my tests offered students opportunities for close reading of technical text. In many ways, all of these dimensions, whether espoused by the NRC, NCTM, and/or the various entities promoting the Common Core, rarely present themselves to secondary students of mathematics. Frankly, I believe deeply in them as necessary elements of a world class education. Each test presents students with ten questions steeped in these dimensions.
The comprehensive final exam, out of necessity, solely consisted of multiple choice (aka multiple guess) questions, 125 of them to be exact. One could call the final an über quiz, as questions on the final exam matched those on my quizzes in form and function; imagine all ten quizzes at one sitting. Additionally, I provided students with a review guide for the final comprised of 150 free response questions in a near identical sequence as those on the final exam to help them prepare for the two hour exam. Lastly, students did not need to show their work to receive credit on the final. I even encouraged guessing, as it cost no points to do so.
Example Honors Precalculus Test Question
Now, on to the topic of this post: an example assessment question from one of my tests. Creating my tests takes a fair amount of time, as I faithfully attempt to assess my students’ mathematical abilities as I describe above. At the same time, the tests do not benefit from a peer review process. Hence, this post.
Example #1: The Law of Cosines
The following background preceded the actual tasks required of a student. It provides peripherally interesting commentary as well as facts, terms, and definitions required to understand the various parts of the question.
Right below the background information, the following figure illustrates various parameters of importance in the question. Simply creating this figure in Microsoft Powerpoint consumed a fair amount of time.
Lastly, the specific questions followed, as shown below.
Soliciting Your Comments
Given the intent of my assessments I detailed earlier in this post, what are your thoughts on this question? Does it follow the spirit of my intent? Could your honors precalculus students handle this question without difficulty? Any other thoughts, comments, suggestions?
I purposely did not include my worked solutions to this question. I am open to doing so if readers request.