The Final, Final Exam?

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Mark Coté, Issaquah, WA,

As the end of the first term of the 2020-21 school year rapidly approaches, many teachers are faced with several challenging decisions regarding summative assessments. Do I give a final exam? And if I do, what does that look like? How will this impact student learning? How high should the stakes be? What if my students fail? Will they cheat? If I have the freedom not to give a traditional final, what are the alternatives?

Compounding the stress and uncertainty of schooling during a pandemic, many of America’s secondary mathematics students will face an additional hurdle – a one shot, high-stakes, comprehensive final exam resulting in a single, irrevocable score that will have a significant impact on overall course grades and future mathematics learning pathways. Indeed, about half the states require high stakes end-of-course
(EOC) tests, which by default become the final exams of the year in many classrooms. A few states actually require that the EOC test be factored into a student’s final course grade, no matter what the consequences. Is such an exam an effective summative assessment of a student’s learning? How will the outcome impact the educational future of each student?

A brief literature survey coupled with some informal polling of CPM teachers revealed several common reasons for the practice of employing final exams as a standard summative assessment instrument. 1) The exam serves as a vehicle for course closure, tasking students to review and pull together all that has been learned during the term. 2) As a form of pedagogical temperature taking, exam results provide data for evaluating student proficiency and overall program effectiveness. And, 3) many teachers are required to give a final, or feel pressure to do so, as a result of the testing herd mentality that exists at all levels of the US educational system. When summative assessments are collaboratively developed by teachers and coherently aligned with learning goals, meaningful information can be gathered to inform students, teachers, and administrators (NCSM 2008). So, for many, a traditional final exam crafted with best practices in mind is a productive activity. But how high should the stakes be?

Attitudes and beliefs about high-stakes testing have a significant impact on teacher decision-making regarding instructional practices. In an educational policy study, Amrein and Berliner (2002) listed a summary of traditional arguments used to promote high-stakes tests: students and teachers need high-stakes tests to know what is important to learn and to teach; students work harder and learn more when they have to take high-stakes tests; and scoring well on the test will lead to feelings of success, while doing poorly on such tests will lead to increased effort to learn. Do these sound familiar? I recall hearing such justifications during heated conversations with colleagues as we made decisions about the weight of final exams. Should one test determine who passes the course, who gets tracked into remedial classes, or, in extreme cases, who graduates?

Unfortunately, these beliefs do not hold up under scrutiny. Amrein and Berliner found evidence from a study of 18 states that student learning remained the same or actually went down when high-stakes testing policies were instituted. They also cited numerous reports of unintended consequences associated with high-stakes testing policies, including increased drop-out rates, teachers’ and schools’ cheating on exams, and teachers’ defection from the profession. Additional pitfalls of high-stakes tests include a narrowing of the curriculum in an effort to “teach to the test,” coupled with a loss of instructional time as some teachers amp up disconnected skills practice. Note that “…research indicates that standardized test scores are lower in schools where teachers spend large amounts of time engaged in ‘test prep’ activities.” (NCTM 2014) Finally, for many struggling students who are forced to stare down the high-stakes gun barrel, the end of the term is a fortnight of dread that produces anxiety and bad feelings about learning mathematics precisely at a time when students should be able to close out the course with a sense of accomplishment and positive aspirations for future learning.

One way to balance the impact of a final exam is to distribute its weight with other summative assessments employed during the term. For this approach, a final exam would have the same structure and roughly the same impact on the overall grade as a chapter test. CPM’s Assessment Handbook offers helpful recommendations for employing balanced summative assessments, including: 1) being involved in developing the exam, 2) reading and working through all items carefully before giving them to students, 3) including only content with which your students have been meaningfully engaged, and with which they have had ample time to make sense of, 4) keeping the grading flexible enough to allow for variations when students master the mathematics, such as an exam with the possibility of descriptive, effective feedback and revisions, and 5) offering cumulative testing with some emphasis on what has recently been learned and more emphasis on what has previously been learned.

The suggested structural balance of approximately 50%current and 50% review material (or 40% current and 60% review) allows you to assess basic understanding of new but not yet mastered, topics and intermediate or advanced understanding of review topics after students have had weeks, not days, to work with them. This structure also allows for the use of problems that require multiple ideas in their solutions, emphasizing the connectedness of mathematics. Lastly, it is neither necessary nor desirable to assess every concept the students have seen on every test during the final exam.

For those deciding to employ a traditional final exam this academic year, remote testing will certainly provide a number of challenges. Topping the list of concerns are academic integrity and the aftermath of failure. On the issue of cheating, the current director of teaching and learning at Singapore American School and the consultant for ASCD, Andrew Miller, offers some fresh advice, “Instead of a deficit-based approach to assessment—expecting that students will cheat—we need to have an asset-based approach where we trust them to do the right thing and engage them in teachable moments around academic honesty. Teacher expectations matter.” Andrew advocates that there is no time like the present to incorporate such lessons into the curriculum. He offers additional helpful strategies in his article Summative Assessment in Distance Learning (Edutopia, April 28, 2020).

And what if they fail the exam? Jill Barshay, staff writer and editor of the Hechinger Report, recently journaled on an innovation introduced at the college level that may cut the typical exam failure rate significantly. In the article, “Proof Points: Improving college exams during remote learning” (The Hechinger Report, September 28, 2020), Jill documents the use of two-stage exams by a team of professors at Harvard University and the University of California, Merced. The CPM Nation will recognize this strategy as similar to our Team Tests, but with a couple of modifications that fit well with the conditions of remote teaching. In a nutshell, “Students, working from their homes, took the individual test directly on the computer on a specified date and time — synchronously — for an hour and a half. Then, in groups of three to four students, they had 24 hours to retake the test over a video app like Zoom, FaceTime, or SnapChat. Some even completed the group exam during an old-fashioned telephone conference call. Students scheduled the collaborative stage at their own convenience — in effect, retaking the test asynchronously.”

Following this remote-testing experiment, students reported a reduction in stress, enjoyed the freedom to schedule as much time as needed (within a 24 hour window) for the group work, and noted the importance of getting such prompt feedback about what they had just done individually. The professors have noted the powerful learning associated with this form of student engagement in the assessment process. Students gain valuable insights from their own mistakes, enjoy the process of arriving at consensus about a solution, and seem deterred from cheating since peers are potential witnesses.

If you have made the decision to step away from a traditional final exam, you may wonder if there are other ways to gather information about what your students have learned. An unproductive belief that persists in our profession is that, “Only multiple-choice and other ‘objective’ paper-and- pencil tests can measure mathematical knowledge reliably and accurately.” (NCTM 2014) As CPM Founding Director Judy Kysh points out, “Numerous summative assessment activities have value and provide students with an opportunity to put all the learning together. Such activities help students see the whole mathematical picture and actively participate in a summary that can be carried forward to support what a student needs for the following year.” With thousands of hours of teacher observation during her career, Judy has witnessed the powerful learning possible when a summative assignment is a collaborative effort between the student and teacher. By choosing mathematical work that interconnects learning accomplished over time, students have a say in how they will be assessed.

CPM Teacher Researcher Laura Bell ( offers her students just such an opportunity. In an effort to differentiate content according to interest and need at the end of the term, her students make a choice about learning targets that have not yet been mastered. Each student selects needed targets and provides evidence of mastery to demonstrate their understanding along with completing a CPM closure project. She has found that this approach, coupled with ongoing formative assessment, reduces the need for a comprehensive final. “I didn’t get to be a part of the discussion regarding a 20% multiple choice final exam in my previous school, so I decided to make some changes. This approach provides choice and does a good job of preparing them for high school,” Laura summarized.

As another alternative, many teachers follow CPM’s portfolio suggestions. Portfolio entries could showcase students’ understanding of key ideas, concept progressions, reflections on learning from mistakes, mastery of a mathematical idea or group of ideas, and/or other struggles they have overcome. Activities from each Chapter Closure section make natural portfolio entries. Also for consideration are self-evaluation and reflective assignments, well-written solutions to rich problems, results from eBook and Desmos lessons, and selections from daily student work.

Tatiyana Webb (, another CPM Teacher Researcher, has implemented a culminating project as the final summative assessment. She moved beyond a traditional final exam because the results “…didn’t tell me anything that I didn’t already know about my kids.” Since state testing is held in early April, Tatiyana has her students complete the year with a unit on financial literacy entitled Real World – Can You Handle It? She tries to replicate real life using a student-driven economy that involves jobs, currency, vendors and consumption. During this simulation, students respond to a variety of realistic scenarios by applying the math they have learned during the year to survive economically. To date, she has not had a student request a traditional final instead of the project.

If the primary purpose of assessment is to gather data that support the teaching and learning of mathematics, then it makes sense to choose meaningful summative activities that have value for both the student and the teacher. When students have the opportunity to gather evidence of their learning and see the course content “big picture,” they internalize a summary experience that can be carried forward to support future learning. By monitoring progress, offering descriptive, effective feedback, and responding to student revisions in real time, teachers send a strong message about the importance and usefulness of the end-of-term assessment task. (NCTM 2014)

So how will you conduct final exams this year? Hopefully, this article has offered some food for thought and alternatives for discussion. If you do decide to go traditional, heed the advice of Dr. Jay Heubert, professor of education and law at Columbia University and author of the book High Stakes: Testing for Tracking, Promotion, and Graduation. He said, ‘‘One test does not improve learning any more than a thermometer cures a fever.’’


Amrein, A.L. & Berliner, D.C. (2002, March 28). High-stakes testing, uncertainty, and student learning. Education Policy
Analysis Archives, 10(18). Retrieved on 10.10.20 from

Barshay, J. (2020, September 28). PROOF POINTS:
Improving college exams during remote learning. Hechinger Report.

Miller A. (2020, April 28). Summative Assessment in Distance Learning. Edutopia.

Nichols, S. L., & Brewington, S. (2020). Preservice teachers’ beliefs about high-stakes testing and their working environments. Education Policy Analysis Archives, 28(30). Retrieved on 10.10.20 from

NCSM. (2008). The Prime Leadership Framework. Solution Tree.

NCTM. (2014). Principles to Action. Reston, VA: The Council.

NCTM. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: The Council.

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Algebra Tiles Session

  • Used throughout CPM middle and high school courses
  • Concrete, geometric representation of algebraic concepts.
  • Two-hour virtual session,
  •  Learn how students build their conceptual understanding of simplifying algebraic expressions
  • Solving equations using these tools.  
  • Determining perimeter,
  • Combining like terms,
  • Comparing expressions,
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  • Use an area model to multiply polynomials,
  • Factor quadratics and other polynomials, and
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  • Support the transition from a concrete (manipulative) representation to an abstract model of mathematics..

Foundations for Implementation

This professional learning is designed for teachers as they begin their implementation of CPM. This series contains multiple components and is grounded in multiple active experiences delivered over the first year. This learning experience will encourage teachers to adjust their instructional practices, expand their content knowledge, and challenge their beliefs about teaching and learning. Teachers and leaders will gain first-hand experience with CPM with emphasis on what they will be teaching. Throughout this series educators will experience the mathematics, consider instructional practices, and learn about the classroom environment necessary for a successful implementation of CPM curriculum resources.

Page 2 of the Professional Learning Progression (PDF) describes all of the components of this learning event and the additional support available. Teachers new to a course, but have previously attended Foundations for Implementation, can choose to engage in the course Content Modules in the Professional Learning Portal rather than attending the entire series of learning events again.

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Building on Instructional Practice Series

The Building on Instructional Practice Series consists of three different events – Building on Discourse, Building on Assessment, Building on Equity – that are designed for teachers with a minimum of one year of experience teaching with CPM instructional materials and who have completed the Foundations for Implementation Series.

Building on Equity

In Building on Equity, participants will learn how to include equitable practices in their classroom and support traditionally underserved students in becoming leaders of their own learning. Essential questions include: How do I shift dependent learners into independent learners? How does my own math identity and cultural background impact my classroom? The focus of day one is equitable classroom culture. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for Chapter 1 that creates an equitable classroom culture. The focus of day two and three is implementing equitable tasks. Participants will develop their use of the 5 Practices for Orchestrating Meaningful Mathematical Discussions and curate strategies for supporting all students in becoming leaders of their own learning. Participants will use an equity lens to reflect on and revise their Chapter 1 lesson plans.

Building on Assessment

In Building on Assessment, participants will apply assessment research and develop methods to provide feedback to students and inform equitable assessment decisions. On day one, participants will align assessment practices with learning progressions and the principle of mastery over time as well as write assessment items. During day two, participants will develop rubrics, explore alternate types of assessment, and plan for implementation that supports student ownership. On the third day, participants will develop strategies to monitor progress and provide evidence of proficiency with identified mathematics content and practices. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

Building on Discourse

In Building on Discourse, participants will improve their ability to facilitate meaningful mathematical discourse. This learning experience will encourage participants to adjust their instructional practices in the areas of sharing math authority, developing independent learners, and the creation of equitable classroom environments. Participants will plan for student learning by using teaching practices such as posing purposeful questioning, supporting productive struggle, and facilitating meaningful mathematical discourse. In doing so, participants learn to support students collaboratively engaged with rich tasks with all elements of the Effective Mathematics Teaching Practices incorporated through intentional and reflective planning.