My thoughts on Final Exams in Math

We (meaning math department) have been questioned lately on the necessity of final exams.  

"Are they necessary?"
"If a student demonstrates mastery earlier why do they need to write them?"
"Can you find an authentic task instead?"

These are just a few questions we've been asked the past couple of years.  Math is likely one of the last subjects where a comprehensive final is still given.  Now, having said that, we have changed our procedures and approaches to writing the final exam.

Instead of calling it a final exam, it can be looked upon as a summative assessment.  I believe that all of the assessments, whether termed formative or summative, that are done during the course of the semester are really formative in nature as the students are able to and encouraged to improve their level of understanding on any outcome at any time until the "summative assessment" (final exam) is given.  Even at that point, if a student is still not meeting the basic level of understanding (level 2), they are given opportunities to improve.  We have also moved our summative assessment to about three weeks prior to the end of the semester.  There will be one or two outcomes that are not on the summative assessment, as that is what will be done in the last three weeks.  We also spread the summative assessment out over two - three days and the students are told which outcomes will be on each chunk.  Our summative assessment consists of questions at each level of understanding (2, 3, and 4) for every outcome.  By moving the assessment up we have accomplished a number of things.  1)  They are not stressed about all their other final "projects" that need to be done in a short period of time.  They can focus on math.  2)  We are still testing understanding over time and at the completion of the semester (as stated in our curriculum!)  3)  We have time following this assessment for those who still have not met the basic level of understanding on an outcome to continue learning it  4)  it gives teachers time to evaluate and enter levels of understanding for all outcomes

Some will question why a student who got a level 4 two months into the semester should have to write that outcome on the final.  When I think about the 10 000 hours needed before you master a skill (Malcolm Gladwell, The Outliers) I shake my head at labeling a student "master" of a skill they have practiced for maybe 5 hours!  Most outcomes are less than 10 hours of in class time!  And yet we expect a student who has "mastered" the outcome at the time will actually remember this in a few months after new learning has been done!  Maybe our level 4 questions/activities need to be stepped up a notch, but at this point I am not comfortable that being a level 4 two months in will mean you are a level 4 at the end of the semester.  I only had a few students who demonstrated a level 4 more than once on an outcome and of those, most, but not all, were able to achieve a level 4 at the end.  Until my data shows that they will ALL remain at a level 4 I feel they still need to complete the summative assessment.

I try to teach the more complex outcomes or ones that are so critical for the next class, early in the semester so that we can spiral back through them over the course of the semester.   We do this through entrance slip questions.  I believe this really helps students improve their understanding of the outcome.  In fact, many students improve their level of understanding on the summative assessment at the end of the year!  This year I had 50% of my math 9 class score level 4 on outcomes on the final exam!  The majority of students improved their current levels of understanding so learning was achieved over the course of the year.  I was super impressed!

We do not average the scores of all of their assessments, we use professional judgement and look at most recent and most consistent levels.  The students know that if they improve on their summative assessment then that will be their current level.  I am always impressed at how well the students do at writing these assessments, when they know it can impact their grade in a positive way.  In the past, when a final was worth only 25% of their mark, the students would know what they needed to pass the class and thus I marked many crappy final exams!  Or they would know that a great score on the final would only boost their mark a few percent so it wasn't worth their time to prepare for it.  Now the students know they must demonstrate a basic level of understanding (a little bit of pressure, but this is truly a BASIC level, if they don't know these questions they aren't ready for the next level), so they actually try and don't leave entire pages blank!  They also know that they can improve their score and overwrite previous "mistakes" so there is motivation to prepare.

Math is built upon prior knowledge and I think it is really important that a student demonstrates at least a basic understanding of the outcomes at the end of the semester.  I want to know that they are ready for the next class.  If they have a  basic understanding it means that I might only have to do a quick refresher of the skill instead of having the student look at me as if I'm speaking Greek and they've never seen what I'm doing before!   This occurs in students that have not even met the basic level of understanding.  I want these students to do additional learning before I see them at the next level.

So my question is "Is math a unique subject where some sort of a summative assessment needs to take place towards the end of the semester?"   Does this have to be a written assessment?   Does anyone have a project/task/assessment that they do instead of a sit down written assessment that will test each outcome?    Should all students have to complete all outcomes or should we allow "recommends" or "opting out" if previous levels of understanding were 4's?  I would love to hear thoughts and suggestions from other math teachers!  I'm always willing to learn and share ideas!