Thursday, January 19, 2012

Learning can occur on a final exam

Historically, I would say that the least amount of learning occurs during exam week.  This year, however, I have tried to make this statement false through giving my students a new final exam.  When the first group presented, not only did I see, hear and was taught about unique examples of how to use calculus, I can actually say the group still LEARNED through the assessment. 
Below is a video of their PowerPoint presentation, which they used as well as supplemented with dialogue. 

After the presentation they opened the floor to me by giving me 10 minutes of questioning.  Most of the questions they answered swift and correctly, until I asked “Do all functions, on a closed interval, have an absolute maximum?” . Their answer was “Yes”.  Of course this is incorrect.
Now here is how the learning occurred… First I will address the traditional way of assessment:
If this was a traditional final exam, I would have marked this question wrong and moved on to the next question and continued marking.  These students would never have received any feedback, as in the past, I have yet to see many students come back to see WHAT they did incorrectly on a final exam.  These students would have then gone on to university/college with this false knowledge.
How it has changed this year…
I didn’t let this false information continue.  I then asked, “What if I told you, I could draw a function on a closed interval without an absolute maximum?”  The girls looked at each other with confused eyes, and pondered the idea.  After some passing minutes, one replied accusing me of a liar.  I wanted to ensure they didn't continue on with this false information, so I then asked, “Is there any kind of function that continues upward on forever?”.  One quickly answered, “WAIT! A function could have a vertical asymptote and therefore have no absolute maximum.  I guess you weren’t lying”, the other girl smiled and agreed. 
After this, I wanted to ensure they had a true understanding and therefore I asked, “Can a function, without any asymptotes, on a closed interval, not have an absolute maximum?”  The enlightenment has occurred!  Both girls whispered quietly, and then turned and replied “If the curve has an open point at the highest point, then it would not have an absolute maximum”.
Learning had occurred, and yet it was a final exam.
I continued with my questioning, which they answered correctly, and I am happy to say that this experience has been a success with the first group!


  1. Dave, this is what a normal (good) day in class looks like. Socratic dialog and such. I don't see that you changed the final exam, you simply didn't do one. The message I am getting (and have gotten) is that you do not believe in testing, even if for just for one day out of a year, with all of the rest of the days devoted to teaching.

    Your problem is trying to convince a competitive world not to test. To not compare students' performance against their peers. You might be able to compare your students to each other but how are we supposed to compare your students to another teacher's students? And maybe you think we should not be comparing them at all. But that goes back to the very competitive nature of the world. If ten of us apply for two teaching positions, only two of us will be teaching. The other eight will be doing something else.

  2. btw dave, you might be interested in this...

    iBooks has been updated to support interactive textbooks and there is a new (free) app for the Mac to create them.

  3. Robert, I love the second comment. I have already been following this and am super excited about the capabilities of the new iBooks 2.

    To your first comment, I know you support competition in schools but you must remember that competition is only for the strong while public education is for everyone. This is a point you and I have argued greatly, and I love the debate but I must still disagree which the notion of ranking students in a class.

    I refuse, and will always refuse, to compare my students against each other.

    Sure, outside of school there is competition, but I feel we should be teaching this idea in schools without punishing students (with lower grades) who are not motivated by competition.

    I am aware that you disagree with me here, but I do appreciate you engaging in the conversation with me!

  4. For my Geometry class I had something similar set up: i.e. a presentation instead of a final. The idea was, my students were NOT ready to show me they understood the material, but together they could help each other succeed in understanding most of the math that we had done through the year. The set-up took about a full week where - instead of assigning review problems - I had the students write up (in pairs) three different problems that covered the different chapters.

    During the two hour block designated for the final, I used the problems they had created and had them come up one at a time and "teach"/"present" the answer. The grade breakdown came into the following categories:
    - The questions they created (creativity, solutions accurate, drawings included, demonstrates an understanding of the subtleties of the topic)
    - Their participation during the presentation (polite, did they offer help, were they correct in the answers they provided, number of times they raised their hands)
    - Their presentation (could they recognize errors from the other students, did they engage the students, could they organize the solution so it was clear afterwards?)

    Not only was this a success for the midterm, it allowed me to spot those students who still didn't have the clearest understanding of the material and yet do so in a way that gave them the freedom to make mistakes without it affecting their grade through a bad Final Grade.

    The best part is that, after the midterm, when students came back and we returned to the lesson at hand, they carried with them the confidence of a job well done, a better understanding of the math they knew and a willingness to take more risks during new lessons. I have been able to call them to the board on a regular basis and "test" them with the homework questions in ways similar to the "final" that we did.

    The GOAL of this, though, was to boost their knowledge and ability so that they could find better ways to study and connect the material for future exams. I am with Robert in that my goal is to present these students with the opportunity to demonstrate what THEY know personally (I don't believe in ranking students, a C to a student who normally scores lower means something completely different than the C to a student who normally scores much higher; grades are not accurate ranking tools). I believe school is there to prepare students for the world, which means getting students ready for doing the best THEY can do on their own AND in teams. That means both regular testing (as has been discussed in previous posts) and projects.

  5. I agree 100% with Robert. You didn't give a final exam. The students just gave a review project (full of erroneous/incomplete ideas) and then had a normal class discussion about it. Your "final exam" is what most good teachers do every day. (Although the physical placement of the teacher and students are reversed.)

    And as to: "These students would never have received any feedback, as in the past, I have yet to see many students come back to see WHAT they did incorrectly on a final exam." In my class, we go over nearly every problem of every test/quiz/final exam the day after they have taken it. They get feedback on everything.

    This is a performance based subject. I want them to perform for me. I want to see what mathematics they understand on their own. Given a problem, can the student solve it?
    Why can't you accomplish what you felt you've accomplished here during normal class time? And then, after you've asked great questions about functions on closed intervals etc, have them prove that they understand that concept a week later while sitting for a final exam?

  6. @Anonymous, You go over the final exam with the students? Is this optional? If so, and the students show up then I believe you are living an anomaly. Our final exams are given during exam break when classes are over, is this the same for you?

    When you go over the other exams, which I assumed you have marked and reported the mark on them, I guarantee that the learning is not as great if you did not mark it and instead gave written/verbal comments. (This is backed by research)

    You mentioned, “prove it on an exam”, what does this mean… Does it mean I ask them a question and if they answer correctly then they have proven it? Why does this have to occur on a written part? I would argue, by them answering my questions verbally it is equivalent as circling in a bubble on a scantron.

    1. The first 20-40 minutes of the next day's class period are spent going over the test/final. So it's not optional for the students. For regular tests, the "next day" is the next day. For final exams, the "next day" might be 5 days later.
      I do grade the exams with some minor written comments. The verbal comments come the next day. I can find the common mistakes that the class made and make my comment to all of them all at once.
      I want them to prove to me (and themselves) that they undertand Calculus. Often times, they say, "I think I've got this stuff." Then I give a problem for them to solve on the test, and I/they learn that they don't understand the material/concept as deeply as they should. Only about 20-25% of my tests are multiple choice. Most of their problems require them to justify themselves with words/mathematics. Could I have them prove their understanding with a conversation? Sure. But I want to see if they can solve 10 problems. Each problem requires about 5-10 minutes to solve. It is a much more efficient use of my time to have them all take a test than sit individually with all 30 students and verbally ask them questions about these problems.
      I don't see how that Power Point presentation + 10 minutes of question time (no matter how good your questions are) could possibly give me a good sense as to how well those girls know that material. The power point shows a very, very basic understanding of Calculus at best. (And that is something that shouldn't be overlooked. That particular Power Point is not a good product at all. Maybe it looks nice: who cares? It shows little understanding. They copied their review notes. What does that show/prove?) I don't see how 10 minutes of probing questions could tell me more than what 2 pages of my final exam would.
      And I'd like to reiterate: These verbal Q/A's are happening daily in my classroom. I just don't try to pass them off as a final exam.

  7. Dave... I absolutely love this. Some thoughts and questions for the discussion:

    What is a final exam?
    Who gets to define this?
    What is the purpose of the final exam?

    The last question is where we need to start... do we actually think that a final exam is for learning? Do we think that we can get a good assessment of how a student did through the entire year? (I am not saying that we need to completely end final assessments but we need to rethink the way we do it).

    IS the purpose of ranking and sorting? Do we really think that an exam can actually be completely standardized so we get an accurate assessment of how our students are doing relative to each other (there are way more variables than the student so this is never a fair test).

    A few things that we are seeing that will help move us away from the traditional final exam to more of a final assessment:
    - Performance standards. Yes, these are standardized but it is based on skills and learning rather than tests and bubble sheets. For SOME subjects, there are standards/criteria in which students are aware of and teachers can help guide.
    - Focus on AFL - assessment is becoming less of an event or something in which we catch students to see what they have learned but more about ongoing dialogue about learning and improvement.
    - the movement away from Provincial Exams. In BC, we have moved away from grade 12 provincial exams - made the optional and then nobody took them. Universities are now looking at thing beyond exam marks.

    Also, to comment on the 2/10 teachers will get a job... part of my job is to hire teachers. I do not look at marks. I check out their portfolio, talk to references and do interviews - sounds a little like what you have done. I even provide feedback in the interviews... and these are a conversation, not a pop-quiz style interview. I also can see the comment coming: but teachers needed good marks to get into universities and they were compared to others. My response to this is that we all know that good exam marks have very little to do with good teaching and some of the best teachers (including my previous principal who went back to her passion - the classroom) struggled in school.

    When we stick our necks out there and create change when few people are doing it, those who are comfortable with the current system will be critical. I am not saying that your way is the BEST way (as I am still learning new ideas too) but I will say that the method of assessment you have described is focused on learning conversations based curricular outcomes. This is a direction we need to be looking.

    As this is far different than what we are used to, we need to bring people along by asking powerful questions. Robert brigs up some great questions that are very common in our current system. Without having these discussion, we do not know the mindsets that may be opposed to where we want to go.

    Lastly... I will say - why WOULDN'T a teacher try this? If a teacher is given the autonomy to use their professionalism to assess a child based on the prescribed curriculum... why not? If it does not work, then we are no further behind because we know that the traditional final exam is not effective. I have a feeling your final exam will be effective and I cannot see you going back.... keep pushing us with your ideas around assessment!

  8. By the end of your exam, your students understood the content and were able to meet the learning outcomes. Brilliant! It doesn't matter if they came to the exam with this knowledge or not. They now understand the concepts you were assessing and were able to make connections between previous knowledge and the new information you provided. This demonstrates higher level thinking and indicates that they came in with a strong understanding which increased as you conversed.

    We really need to move away from the notion that a "final exam" must take place in a silent room with a set of scantron sheets and pencils. An interactive exam decreases anxiety for the students and allows them to demonstrate their learning in a variety of ways. I have students who can explain their understanding of course concepts well, but have told me that they freeze up on tests in other classes because they feel that a test is an "all or nothing" situation. By allowing students to learn from this exam, then demonstrate their knowledge again, you allow them to continue learning.

    The job of a teacher is to help students to meet learning outcomes in their own time, not to force them to compete with one another on a specific date. It sounds like you're doing a great job in your class!

  9. It is obvious that there is an irreconcilable difference of approach between those who, Like Dave, see what happens in school as learning and those who, like Robert, see it as preparation for a competitive world. It is both and the current emphasis is on the competitive nature, while real learning gets pushed away. But nobody will change their minds, if they are set. Hopefully those who visit here with an open mind will develop better ways.

    Two points on separate issues.

    1) The presentation skills displayed by these students do not seem very high. But then again, have they ever been taught how to present mathematics? I have seen my fair (actually unfair, as in "excessive") share of university lecturers who could not explain something clearly or in an engaging way if their life depended on it, yet had lots of experience. Why should we expect more from students who have always been asked to just answer and shut up? Are we really preparing students for a competitive world when they do not learn how to make presentations? Methink not, and this is a far worse failure than their not getting some fine points of calculus.

    2) I wonder if you (Dave) had a chance to discuss with students the fact that their understanding function is very close to what Newton and Leibniz had in mind, with their implicit assumption that all functions be continuous. Such a discussion may have opened their mind to the fact that some calculus issues are intrinsically tricky and counter-intuitive and, because of that, intriguing. We keep forgetting that these students are learning the very basic notions and that those of us who learned more benefited also from the work of the geniuses who followed Newton and Leibniz. The extreme value theorem may be obvious and clear to us, but is it such to anyone at first glance? And again we get to the same point: are we stimulating their interest and further learning or are we just making a selection of the "successful" students by using math?

  10. It's a little more than that Roberto. If we were to democratize music in the way Dave is trying to democratize math, music wouldn't be worth pursuing. I think students should be challenged in order to reach their true potentials. If you do not challenge students they will not know what they are capable of. Competition is inevitable. If you kill that then all you are doing is killing potential. I do not find Dave's curriculums challenging at all. And don't confuse my critical opinion with being closed minded. I have seen classes that don't have grades and in fact no one (including myself) even considering asking why. The classes were very challenging. Just going up to the board and doing your mathematical recital was enough. It is easy to do to. Your students just have to be somewhat matched.

  11. Another problem with this argument is that I am coming from the point of view that these subjects have no real purpose other than to be good at and for that reason I think they must be challenging to have a purpose. You can't find what you're really good at if the classes aren't taught that way. But the reality that Dave must operate under is that these subjects are mandated and students will be herded through like sheep regardless. Given those conditions it is Dave's class and his decision as to how to teach his class. And a couple of his assignments were challenging, but I find most of them not.

  12. Robert I will respond but swamped right now. But for now can you define challenging? And if I was to use an exam should ALL the questions be under this definition?

  13. In a final exam (the exam at the end of the year) all of the problems should be full problems that require the student to pull together a solution from all that they learned. They should represent a healthy level of acumen in the subject. They should be thoughtful but not necessarily ingenious, like AMC or Math Olympiad problems, though the student should be exposed to those as well. I don't see any mathematical acumen in most of the problems Dave presents. Having discussions is nice but there is much more to the experience of mathematics than that. If I were to apply what I have seen here so far to the teaching of piano, the first thing I would have to do is to make sure that there is no piano in the class. Which is perfectly fine I suppose if the students aren't actually piano players.

  14. Very Good Information about Learning can occur on a final exam.

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