Homework Assistance and Conceptual Understanding in Pre-Calculus

Samuel B. Simpson, Mathematics Teacher,  All City High School.   Samuel.Simpson@RCSDK12.org


I am a math teacher at All City High School in the Rochester City School District in Rochester, New York.   I am an electrical engineer who, after 15 years in the telecommunications industry changed his career. I have taught math for the past 10 years ranging from 7th grade math to AP Calculus.  I started my teaching career as a middle school math teacher for 7th grade. When the district decided to close Frederick Douglass Middle School, I came to John Marshall High School (Marshall). I chose Marshall because my teaching mentor and the teacher I observed with (I did not student teach) where located at Marshall. I was selected to teach a Math 7th grade Honors class and later selected to teach Math 8th Honors as well.  With the success my students from those classes received, the administrative team saw fit to make me the middle school Math Coach and later adding high school responsibilities as a part of my role, while I continued to teach students. I began the CCTE Fellows Program as a member of the teaching staff at Marshall. It is with many of these students, students I had taught, that I sought to work with while in the CCTE Fellows Program.

This year, the year of my action research, Marshall closed. All City High opened on Marshall’s campus, and is comprised of students from five failing high schools that were closed. I chose to become a staff member at All City High because many of my former students elected to remain at the Marshall campus and I wanted to do my action research with them.

I am a member of the teaching staff at All City High (ACH). ACH was established as an alternative high school within the Rochester City School District. The primary focus of ACH is not only to meet the challenges of 21st Century learning, but also to facilitate a transition of five distinctive cultures into a single family. ACH students speak 27 different languages at home. ACH has the most Advanced Placement courses offered of any high school in RCSD.  ACH has a non-traditional, schedule, which includes 12 hour days, Saturday school, academic and social supports systems, rich extra-curricular activities, community partnerships and all-day food service providing three meals per day.

As a middle school and high school math teacher many of my students have achieved success. Most have past their Regent exams and or have taken higher level math courses. The results my students achieved easily surpass their peers at Marshall and in the Rochester City School District. While students have achieved high grades there appears to be a disconnect between their grades and their understanding of complex concepts. In their most challenging math courses, Algebra 2 and Trigonometry, many of these students barely pass the Regents exam. I believe if students could solve more complex problems correctly, their understanding of the subject matter would in increase. As a result, my action research is based on students completing more complex homework.  Previously, my students completed 75% to 95% of their homework. The focus of my action research was to answer the question, “If students had assistance completing their homework, would their homework accuracy and their understanding of complex concepts increase”?

Inquiry Design

I taught three classes the first semester, an Intro to Integrated Algebra, a Back on Track Intro to Integrated Algebra and a Pre-Calculus class. The first two classes were comprised of struggling, over-aged, under-credited students with attendance issues and a wide variety of social issues that would make action research precarious. I chose to conduct my action research with the third class, Pre-Calculus. Forty percent of the Pre-Calculus class was comprised of students I had taught in middle school at Marshall. The remaining 60% of the class were students from three other schools. While many did not have the appropriate pre-requisites, this class consisted of students with a wide range of capabilities. I believe that if students solved complex homework problems correctly their understanding of complex mathematical concepts would increase. I realized that many of my students would not attempt the more complex problems (in correcting Regents and state exams that fact bore out). I analyzed how students had performed on the pre-requisite class regent exam, Algebra II and Trigonometry.

The class had the following profile:  low score =36, Q1=48, mean=57, median=62, Q3=66, high score=84. With only three students passing the exam by more the 3 points, the range of scores indicated to me that this would be an excellent class to test my inquiry question.

Inquiry Question and Action Plan

If students had assistance completing their homework would their accuracy and their understanding of complex concepts increase?

To get students to solve complex homework, my action research would have two focuses.

1) I would employ collaborative learning, (in addition to my teaching they would need to teach each other) and

2) I would provide them with homework assistance.

I elected to utilize collaborative learning to capitalize on students’ knowledge, ideas, resources and skills. I wanted each student to depend on and be accountable to each other.

To determine the academic topics I would focus my action research on I gave the class an assessment based on topics covered in both Algebra 2 and Trigonometry and Pre-Calculus. I used the results of the assessment as a benchmark. After a thorough analysis I chose Logarithmic Functions and Trigonometric Functions as the topics I would focus my action research on. These two subjects had the lowest scoring averages of all the topics. After teaching these topics, I again assessed the class.  (Click image to enlarge).

Simpson-initial survey-2

As I began teaching it became apparent to me that the classroom did not have the kind of interactions and rapport conducive to everyone achieving success.  I spent the first marking period (10 weeks) establishing rapport with the students and having them establish it with each other; the class had to be a family. I conducted a number of surveys to determine how best to pursue my action research. I conducted an initial survey to see how prepared they were for the course, and later a survey to see how they felt they would perform in this class. Based on the two surveys and how they performed on the pre-requisite Regent exam, I place them in collaborative groups. Each group member would receive the same grade and I would have the opportunity to request any member of the group to articulate or defend an answer or position.  (Click image to enlarge).

Simpson-how will you-3


To support collaborative groups, seating arrangements and the classroom environment was modified.  In their groups students would perform classwork, quizzes and an exam. They spent class time as well as non-class time working together.

With students working together and rapport established, I sought out the best way for students to access me for feedback on homework.  Initially I thought I would utilize Facebook, however because of district policy this was ruled out. I considered Pencast, an application that captures my audio with my text however, because of district software constraints this also was challenging and disregarded. After yet another survey, I provided students with my school and personal email address, and my personal cell phone number and I encouraged them to contact me.  I wanted students to be able to contact me in person, live over phone, text, or via the internet.  (Click image to enlarge).

Simpson-what is the best way-4

Because of the feedback from the survey I started after school tutoring on Tuesdays and Thursdays and some Saturdays. I encouraged students to participate and I provided snacks. In conjunction with the tutoring I launched a website. The website included a pacing guide, my contact information, daily class agendas, daily homework problems, detailed homework solutions, homework hints and helpful website links.

I was encouraged with the student feedback and interactions I received.

  • I averaged about 1 ½ phones calls a day and approximately 1 email every other day.
  • I averaged 5 students per each after school tutoring session and 4 on the weekend sessions.
  • The website averaged 10 targeted visits per day and over 400 site visits in 30 days.

My school day typically extended through the evening. After school I would tutor students for an hour and a half. I would follow that up with a couple of hours of planning for the next day. Prior to leaving the school I would update the website, creating/modifying and uploading documents. I would then answer any emails or phone calls I had received. My day was still not done, that evening I would again check emails and return phones typically shutting it down around 9:30.

What I learned

If students have assistance completing their homework their accuracy and their understanding of complex concepts will increase.  The key to understanding complex concepts is predicated on a student’s willingness to solve challenging problems. Motivation is critical, students must have the desire to work hard, either they are self-motivated or are challenged by their teacher, family or friends.  My role is to create an environment where hard work is the norm and is expected.

While the amount of homework remained relatively consistent around 85%, more complex problems were attempted and solved correctly compared with past classes.

The impact of collaborative learning on my classroom vastly exceeded my expectations. Many of the lessons and assignments in the Logarithmic Functions unit were based on collaborative learning.  The test for that unit had the highest class average of any unit test I have given in Pre-Calculus. The class average went from 56% percentage on the benchmark assessment to 84%, a 50% increase. No student received a grade below a “B”!

The most challenging topic that I chose, Trigonometric Functions, had only a 33% passing average on the benchmark assessment. After teaching the subject and providing review materials to the students, the class average increased to 62%, an 88% increase. While the increase is significant, I need to drive the pass rate higher.

The impact of my involvement in this program precipitated that I modify my classroom teaching. The results positively impacted student confidence and their mathematics problem-solving abilities. Toward the end of my research I conducted a second survey of “How will you perform in this class?” The results of that survey demonstrated that students showed an increase of confidence in their abilities. Students thinking they would receive a D or F declined from 8 to 1 and those students thinking they would receive an A or B increased from 5 to 9.  The results pleased me.

Reflections and Future Practices

The CCTE Fellows program has impacted my teaching practice. The use of technology, 21st century technology learning techniques and collaborative learning has made me a more effective teacher. I foresee utilizing these techniques in all of my classes going forward.

My classroom philosophy is that “Every student will develop a conceptual understanding of mathematics and be able to solve rigorous problems quickly and effectively.” As an educator my goal is to stimulate and encourage all students focusing on reaching down and pulling up struggling students.

Although students have increased their understanding of complex mathematical concepts, still questions arise.

  • What other attributes can foster increased understanding of complex mathematical concepts besides understanding complex homework?
  • What other teaching tools would be effective for me and my students?

I will seek to deploy collaborative learning in all my future classes, yet, I need to determine the negative aspects of collaborative learning and how to compensate for them. The impact of having a website was significant.

  • How can I quantify the impact it made with the students that use it?
  • How can I improve upon the website and drive usage?

One aspect of the website I would seek to incorporate in the future is a homework blog.

The power I received from surveying my students and listening to them allowed me to tailor my support services to their needs.

  • In what other ways might surveying my students impact my instruction?

I realize it is difficult to quantify student understanding, yet, students, educators, and schools are measured on standardized test. Of the twelve students that retook the Algebra 2 and Trigonometry exam at the end of the course, 10 increased their scores. In conclusion, I would like to share the results of two– one student’s score went from 66 to 86(from barely passing to mastery), and the other student’s score went from 47 to 70, (from failing badly to competent understanding). For many of the students who actively participated, their understanding of complex concepts increased.


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Gokhale, A. (1995). Collaborative Learning Enhances Critical Thinking. Journal of Technology Education. Vol. 7, 1:1045-1064.

Usher, A. & Kober, N. (2013). Student Motivation – An Overlooked Piece of School Reform. The Education Digest, (January): 9-16.

Wiggins, G. (2011). Giving Students a Voice: The Power of Feedback to Improve Teaching. Educational Horizons, Vol. 89, 23-26.

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