Thursday, October 27, 2011

Students are talking about assessment are we listening?

I recently went to a grade 12 math class where multiple choice, numerical and written response exams were abolished and replaced with open ended projects.  These projects were handed in and given back with written feedback only. Then students were allowed to make changes and re-hand in the project.  This could occur as many times as the student needs.  The teacher noticed an increase in both student achievement and, more importantly, student learning.  I am a full believer that students should have a voice in how they will be assessed.  Consequently, I gave these students a quick survey on their assessment in the class.  Below are the questions I asked, accompanied with the students’ answers. 

First off, a very scary statistic…I asked:
In your past schooling experience, how is your knowledge demonstrated in your classes?

Every single student in the class reported: “Multiple Choice, Numerical Response, Written Response Exams”
I am saddened by the truth that after 12 years of schooling, students are shown that the ONLY way to demonstrate knowledge is on a MC, NR and WR exam!
Now, if you believe that testing is a choice that STUDENTS would prefer, I strongly ask you to read the answers below. 

Comparing between open ended projects and traditional exams (multiple choice, and numerical response), which do you feel as you LEARN more as you complete the assessment?
88% Answered Projects, while 12.5% Answered Exams.

Explain why you don't feel as you learn as much from projects/exams (depending on your answer from the previous question). Start your sentence with: I don't learn as much from _____ because...
I don't learn as much from traditional exams because they don't allow you, as projects do, to use your creativity (which in my opinion should be much more important in school than it is now yet always takes a backseat to logic) or to be able to apply the information you've learned in a flexible way as opposed to the cookie-cutter approach of tests. Also, I feel that test marks are based partially on luck, as some tests might just happen to have easier questions than others, or people can make lucky guesses. Projects, on the other hand, offer a more general way to demonstrate knowledge of a subject.

I dont learn as much from exams because i cram the day before to learn everything just for the test and I get anxiety because of it anyways which then leads me to stress and worry about my mark way too much. Also, if i fail the exam because i simply dont care then i accept it and just move on. These assignments dont cause this because it gives me time to actually work on it and if i get something wrong it doesnt just end there. I can fix my mistakes and actually learn from them. Also if a exam is on a specific day and i am very busy and cannot study for it then that leaves me as being screwwed where as these assignments give you time to get them done and they also work around your life.
I don't learn as much from Exams because you write them and you finish regardless of how much you actually know. With the projects you finish them like an exam but instead of just getting the mark you would have gotten on an exam the project gives you questions of things that you need to know and gives you time to learnt them and complete them. If you didn't get 100 the first time around you get extra chances to learn the concepts until you are comfortable with the course.

I don't learn as much from traditional exams because I am less inclined to look at the questions I got wrong since it doesn't benefit my mark.
I dont learn as much from traditional exams because they assess what you remember... the projects allow me to look back at my notes are relearn things that I may have forgotten and having XXXXX checking it over and giving it back allows me a second chance to fix and understand what I did wrong

i feel that i dont learn as much from exams because i only look at what i do wrong and sometimes i just leave it because i cant get that mark back. compared to open ended projects, i can redo it and earn marks.
I don't learn as much from exams because if I don't understand something I can't ask. When I do open ended projects I can ask right then and learn right then and apply my knowledge right then.

I don't learn as much from exams because I don't get a chance to askk questions or work through any questions I don't understand. I get stressed out and hate having to prepare for them.
i dont learn as much from exams because i dont take the time to go over them. i just get upset over the mark and discouraged in my ability to understand the material. with XXXX projects, he gives us in the moment feedback to iron out any kinks and make sure we fully grasp the concepts before moving on. students that fail tests normally dont go back and look over them, so they never end up understanding the concepts.

I don't learn as much from exams because it is just memorization and when doing the projects I have to actually know how to do things and the steps involved.
I dont learn as much from multiple choice tests because they dont allow us to apply our knowledge and all it basically does is make us memorize how to answer questions instead of understanding how it works.

I don't learn as much from exams because I cant ask much questions and with the projects feel like i know the content and can be more confident more in my answers. With exama I find that they are good to an extent because we are thrown random questions that we need to know the basis of the question to asnwer. I find that both of the assessments, projects and tests are a way to assess our knowledge however I find that we need a balance of tests and projects..
i don't think i learn as much from exams because it gets marked and we don't really go over it and learn from our mistake because the teacher only goes through the question that most people did wrong

exams test my ability to regurgitate information and projects force me to get creative
I dont learn as much from exams cause they test you not teach you. But they do require you put more effort to prepare for them.

i dont learn as much the standard test because i usually dont usually get the chance to go over it and fix what i know
This is just because i get stressed out for tests and study for them so i get more practice but i still prefer the projects because they relieve the stress that tests cause and it still evaluates the extent of knowledge well.

I don't learn as much from the projects because I feel like on an exam I am being tested in a variety of ways on a certain concept instead of one or two ways in a project.
I dont learn as much fromExams because i do it then i dont really get to see what mistakes and how to fix it

I don't get to see a variety of different types of questions.
Exams are just answers and questions. If I get it wrong, I sometimes don't know time and the class moves on so its hard to ask.

I don't learn as much from exams because once I get a question wrong and I get the test back, I look over the answers but dont really care about why I got it wrong. With the projects, I am "forced" to learn the content, and I get to keep trying until I get it right. I love it.
exams because when i make a mistake on a exam i just get it back and then we move onto a different concept without me understand completely the previous one

tests, I do not get the corrections I need

17 comments:

  1. I love this! The student's thoughts, opinions, and feelings about assessment are very insightful. What a great experience for everyone - asking the question is a very powerful step to improve our teaching. I really think we need to give students more say in their learning - the process and the assessment. They know so much more about themselves as learners than we ever will - they just need to be shown that they do, in fact, have this knowledge. So much of how we teach and assess stifles the learning process and demonstration of learning.

    Thank you for asking the question. And, thank you for starting my day with this great post!

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  2. Great blog post. Now I need to work on convincing parents that exams are NOT necessary. Any ideas on how to do this?

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  3. Dave Martin suggested, on the AP Calculus Discussion List, that these student comments may be "...an indictment of how traditional exams are assessed...".

    I see other indictments, as well.

    Some of the indictments have little to do with the exams, but with failures to address the difficulties the students had with the exams. Sometimes that responsibility lies with the instructors, and sometimes it lies with the students, and we don't know, here, where it may lie. But it doesn't lie with the exams themselves.

    Many of the students indict themselves repeatedly for their own failures to prepare properly for exams, but they blame the exams instead of accepting responsibility for their own shortcomings.

    Many of the students indict themselves repeatedly for their own failures to use their exams as learning experiences, and they again blame the exams instead of taking any responsibility themselves.

    They indict exams for failing to ask for creativity. But, in my experience, they level much stronger indictments against exams that require creativity.  But, of course, the question that elicited the responses given in the blog was loaded against such exams---asking only about multiple choice or numeric response exams as it did.

    I have, myself, indictments to bring against multiple choice and numerical response exams. But they're very different from the indictments these students have brought---mostly against themselves.

    --Lou Talman
      Department of Mathematical & Computer Sciences
      Metropolitan State College of Denver

     

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  4. Poignant comments from those students. Reading it in their own words is very powerful. Thanks for sharing Dave!

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  5. Hey Lou Talman, I am one of Mr. Martin's current Math 31 students and I wrote the first posted comment to the last question. I do agree with you that students' learning isn't only affected by the type of exam they take, and that the teachers and the students themselves have a role to play. However, the difference in Mr. Martin's teaching style isn't only the fact that he gives us open-ended projects as much as he can as opposed to traditional exams. A teacher could easily assign such a project, mark it and return it to the students and that could be the end of the story. However, like other students have pointed out, Mr. Martin won't let us rest until there is no mistake remaining on the test and he makes sure we understand where we went wrong. If he didn't, I am sure that many of us wouldn't care too much about the things we got wrong and would move on with an incomplete knowledge of the subject.
    I find it strange that you consider the question about which tests we prefer 'loaded,' as the vast majority of assessments I have ever taken in math were multiple choice, numeric and written response. Is there another kind of math test that I've forgotten about?
    As for the creativity aspect, the reason that many students have strong indictments against it, in my opinion, is that allowing us to be more creative forces us to have a deeper understanding of the subject compared to answering standard questions. It takes more thought and effort on both the teacher's and the student's part, and students without much interest in the subject prefer to simply get it over with quickly without thinking as much. Making us use more freedom can actually make us more interested in the subject, and you may notice that the students who don't like the projects said it was because of the lack of variety (which I disagree with as our projects have always encompassed all the important things we've learned) as opposed to the creativity aspect. Those of us, like me, who want to become scientists or mathematicians have to be able to apply their knowledge and use it to find new things, not answer questions that have all the required information wrapped up in a neat little package and requiring a mere memorization of concepts. Even the great physicist Albert Einstein said "Imagination is more important than knowledge." In addition, Mr. Martin's final assessment is a project in which we have to explain what we've learned in the class to some judges, and a huge amount of people would agree with me that teaching something is one of the best ways of learning it.
    In essence, teaching using traditional exams makes us learn how to pass classes, while teaching using open-ended projects makes us learn how to apply our knowledge. The point of school, as I sometimes have to remind myself, is to learn, not to get certain numbers. I can punch buttons for that.

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  6. Dante wrote...

    "As for the creativity aspect, the reason that many students have strong indictments against it, in my opinion, is that allowing us to be more creative forces us to have a deeper understanding of the subject compared to answering standard questions."

    If this were true then the exams would be a piece of cake as they were for me and my peers that tackled both the technical and creative requirements of this subject and put it all to good use in our careers. This would be a non-discussion, since we have already been there, done that and are here right now.

    "The point of school, as I sometimes have to remind myself, is to learn, not to get certain numbers. I can punch buttons for that."

    The point of school, at least one major point, is to prepare you for work and when that time comes you will have to be able to apply what you have learned and you will find yourself competing, and losing, against those that hold both the technical and creative components in high regard. It is NOT one or the other. If you are aceing these exams and scoring 4's and 5's on the AP exam (not a hard feat) then adding creativity to the mix is icing on the cake.

    BUT IF, as the tone of Dave's blog repeatedly suggests, you are trying to replace technicality with creativity, then I suggest you go out in the real world and take a good look at those things your cherish for their innovation and creativity, even the bold and new things, and you look close enough you will find that when they are successful, they are not just bold and new, they are bold and new AND done very well. And that "done very well" part requires the technicality you incorrectly characterize as "pushing buttons".

    Innovation and creativity does not come instead of a solid fundamental technical understanding, it comes AFTER.

    Bob Hansen

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  7. I don't think I have explained myself properly then, I apologize for that. What I meant by my last paragraph is that school is most certainly about learning about things that will be useful in the life we choose, and that the grade we receive is a consequence necessary in our society for universities and employers to choose the best person they can get in their field; not the primary goal of us as students. I wasn't referring to technicality when I mentioned pushing buttons, I was referring to the mark we get for the course. I have come to believe that true objectivity doesn't exist, so a good teacher could assess a student based on their application of their knowledge with the same accuracy as with a standardized test.

    We are definitely still assessed on the technicality aspect and it most certainly isn't "replaced" with creativity. Though we are encouraged to be creative in any way we want, the mistakes that are pointed out to us are technical mistakes, not a lack of creativity. Our projects are not accepted unless they are, as Bob Hansen said, "done very well."

    Mr. Hansen wrote that "Innovation and creativity does not come instead of a solid fundamental technical understanding, it comes AFTER." I completely agree with this statement. That is why we learn and do "normal" questions out of our textbook as well as quizzes before we do the projects. However, though I may have misinterpreted this part, his comment seems to suggest that innovation and creativity shouldn't be used at all until we enter the workforce. Shouldn't they both be practised in the same way that fundamental understanding is practised? Artists and inventors have had to properly learn how to use their imagination and creativity and connect it to their knowledge, and this is exactly what we do in every project.

    I don't believe our opinions are actually that far off from each other, it's just that I may have accidentally implied that we shouldn't demonstrate our basic understanding of the subject. What I meant was that making a connection between our knowledge and our creativity, essentially between our left and right brains, greatly solidifies our knowledge.

    Dante Bencivenga

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  8. Robert, Dante summed it up well. I do not REPLACE anything with creativity except the scantron. I am a firm believer that a multiple choice test tells me very little about the knowledge of the student. Also with written response, I don't believe the mark gives the necessary feedback to the student. As Dante pointed out, students are given written feedback only on their technicality of their work and sometimes on the creativity aspect, if the math doesn't make sense in the scenario. Keep in mind the MATH is what comes first but creativity is a very close second.

    I guess if this way is so horrible, why do 88% of students prefer it? How do you assess your students Bob? Also, do they have input on the type of assessment?


    Remember, in my class the project is a choice and there is always a test which could be chosen as well.

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  9. Dante wrote...

    "As for the creativity aspect, the reason that many students have strong indictments against it, in my opinion, is that allowing us to be more creative forces us to have a deeper understanding of the subject compared to answering standard questions."

    First, let's start with an axiom. Students like exams they do well on and dislike exams they do poorly on.

    It isn't that students dislike "creative" exams, it is that they dislike "creative" exams that they do poorly on. This would include truly (mathematically) creative exams that require proofs or solutions to problems very unlike the textbook, such as the problems you find in competitive exams. Problems that make a student think mathematically to solve. Problems that many students do poorly on.

    But the example Dave posted a couple days ago (the Prezi derivative presentation) is not mathematically creative, it is what we call "busy work". It doesn't make you think math more, it replaces that time with non mathematical forms of creativity, the busy work. Imagine if you were watching a piano class, would you call it creative if they spent their time creating collages of famous pianists? That might be creative from a graphical arts perspective, but not in the context of playing the piano. Likewise, if you want to put the "creative" moniker on this stuff at least stick to the context at hand, mathematics.

    Also, mathematics is implicitly creative all by itself, just like music. If you get into the math, then you get that. Busy work is not a substitute for mathematical creativity. Some teachers that employ busy work actually agree with that statement but counter that busy work can draw a student into mathematics. I haven't seen that to be the case. If that were the case you would see a year start with busy work and then progress to actual mathematics, but instead, when I see teachers employ busy work I see it all year long.

    In any event, if you are going to say "creative" in mathematics then it should not mean "busy work".

    Bob Hansen

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  10. Dave asked...

    "I guess if this way is so horrible, why do 88% of students prefer it? How do you assess your students Bob?"

    First off, most kids would prefer not to go to school at all if they had the choice. That is why you are the teacher and they the student.

    I can tell you how I assess my son, with blunt honesty. If he is doing well in math then an exam is not going to trip him up. And I don't expect the exam to tell me why. My astuteness tells me why and to be honest, I know why before the exam even happens.

    Dave, if you are adding to really good exam results then maybe that should be your message. I have no problem with you saying there should be more than just exams. I certainly go far beyond exams with my son. But your message keeps coming across as "instead of exams" and to be precise "instead of failed exams". You have to be good in something to be creative in it. That goes for music, math, writing, you name it. That is my message. Being put to the test is part of that, an important part of that, but not all of that. I also pointed out in a previous post that exams help the student judge how well they are doing comparatively with others. Since this is the way the world operates what gain can there be to deprive them of that?

    The majority of your case seems to be a case against exams. But I need more than that. Are you saying that your students are doing well on the exams but there should be more? Or are you saying that your students are doing poorly on exams and you think exams are stupid? I am in the former category. Exams serve a very necessary and reasonably reliable service, and knowing the ins and outs of math as I do, I don't see how you could reliably provide the necessary coverage required in the field without them, but there should certainly be more from school than just that (necessary) component.

    Bob Hansen

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  11. This is a really exciting exchange! I love it, so I will put in my $0.02 worth.
    Just to clarify my perspective on this, I am Dante's dad, but more to the point, I have been teaching calculus, linear algebra and statistics at RDC for 23 years. So, probably I have a little experience with the issue at hand.

    I find that assessment is one of the most debated and least understood aspects of the teaching/learning process, especially in how it relates to so many different facets of a student's life, both academic and not.
    Mr Hansen stated an axiom he believes in, and I wish that axioms were that easy to identify. In fact, they are difficult to identify in math foundations and even more in the study of the psychological processes that are behind learning. So, I will limit myself to integrate my personal experience in this discussion, differentiating it from any objective truth that so far eludes all of us (the references to calculus were put there just for fun).

    In my experience, the traditional assessment process is vary disengaged from the learning process. Learning should be about growth, curiosity, excitement and passion. None of these are served well by traditional testing, which provides just a way to attach a label to a student at the end of a course, a label somewhat related to what and how much they learned.
    Unfortunately the system within which we have to work leaves little room for creativity WITH RESPECT TO TESTING METHODS. I too used to include a project in my assessment. My goal was the project, the marks being there just to motivate students to do it. But at some point I was told not to do them any more, since our transferring institutions did not do them and the administration (mind you NOT the students) saw them as an additional burden that may have led students to go elsewhere. This despite the fact that students found a lot of satisfaction in them. To reprise the musical analogy, one student planning to major in music, used calculus to analyze a Beethoven Sonata for piano!

    Now I use non-conventional testing methods, which also include options for corrections. The result? Good students take advantage of them and grow tremendously, poor students avoid them and do not gain as much from them, but it's their choice.

    And here is another issue to consider (and then I will close). Students come in all shapes and variety (intellectually as well as physically) and we cannot expect one method to fit all of them, to motivate all of them and to work for all of them. This applies to traditional and innovative methods. That is why it is so important to listen to the students and to offer them alternatives. Reasonable alternatives, of course, but that is why we call ourselves professional: because, among other things, we should be able to identify what works and what doesn't, both generally and specifically, and to be able to guide our students in the way that best fits them.

    Roberto

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  12. And BTW, Mr Hansen, are you aware of the amount of synthesis that is needed to create an accurate and effective Prezi presentation? How often are students challenged to reach the synthesis stage of their learning through traditional testing methods? Make work? I don't think so, although of course it can be used that way, but the outcome will make that clear.

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  13. Roberto wrote...

    "And here is another issue to consider (and then I will close). Students come in all shapes and variety (intellectually as well as physically) and we cannot expect one method to fit all of them, to motivate all of them and to work for all of them. This applies to traditional and innovative methods. That is why it is so important to listen to the students and to offer them alternatives."

    In the end however, success in this subject, or any subject looks the same. Even the little successes, the epiphanies along the way look the same. The same journey has been travelled by countless individuals. I have no issue with trying various methods to make this journey, until the methods make the result fuzzy and unrecognizable to those that know better. When I see that I see someone trying to change what we call success, not how we achieve it.

    Regarding assessments, I see nothing anywhere in this discussion why it must be either/or. I mean, if a project adds to the learning of mathematics as you suggest (and it very well might) then I see no reason here that you cannot have projects and exams rather than instead of. Exams are what they are, comparative assessments of how technically strong you are with the subject. No different than the exams given in music or art. We know what mathematics looks like and we know what it looks like when students are successful with it. We know what music looks like and we know what it looks like when students are successful with it. We know what writing looks like and we know what it looks like when students are with it. My point is simple. Since we are not fuzzy on these things and since this discussion is fuzzy I suspect that this discussion is not about these things nor is it about successful outcomes. It is instead about alternative outcomes. And that is where you get into strange waters. Why would there be an alternative successful outcome in a subject like mathematics or any of the other subjects I mentioned. People find out what they are good at and what they are not good at and that is a very important part of school as well. It is only recently that people started dictating (hopelessly) that every student be good at every thing and due to Newton's law of reaction, only recently that these odd theories of alternative outcomes emerged. Besides a whole lot of other uglies like schools and teachers cheating with student exams.

    There seems to be another motive here and it isn't pro-learning, it is anti-exam, it is anti-grade, it is anti-success and I have yet to get a reasonable explanation of why the same exams that we and our peers thrived on and became accomplished with are now some sort of problem. To a reasonable person, that would sound fishy, no?

    "And BTW, Mr Hansen, are you aware of the amount of synthesis that is needed to create an accurate and effective Prezi presentation?"

    Yes I do. Now explain to me why we need two creative writing classes rather than one creative writing class and one math class.

    Bob Hansen

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  14. Mr Hansen,
    it seems obvious that we view the world of math education from two different axiomatic systems, as I totally disagree with most of your statements. I will just give you a couple of examples and an anecdote that (coincidence? miracle?) I just received from a a friend of mine.

    You say: "In the end however, success in this subject, or any subject looks the same." My experience is TOTALLY different. Success comes in many different ways, from the student who manages to memorize enough steps to answer test questions without understanding much about them, to the student who understands the concepts, but has troubles keeping enough focus to do the algebraic steps correctly, to the student who does both, to the student who simply overcomes his/her fears and realizes that learning math is possible. In fact, I have no idea of what your common success would look like: passing tests?

    You state "We know what mathematics looks like and we know what it looks like when students are successful with it. We know what music looks like and we know what it looks like when students are successful with it." Do we? Really? Was Jimi Hendrix successful in playing guitar? Surely his method was not traditional. Was Mozart successful with it? He never became rich in his lifetime and was accused of writing "too many notes". And was Einstein successful when he failed a test for refusing to work out several questions all needing the same method? Once again, I don't know what this "success" you refer to is.

    You state "Yes I do. Now explain to me why we need two creative writing classes rather than one creative writing class and one math class." Because this gives me a better hope that future math teachers and mathematicians will be less boring and more effective than they are now. When I attend presentations at major conferences and see people still using whiteboards and illegible and uninspiring handwriting I want to cry. Creative writing classes in math and physics and chemistry? I pray they will become the norm soon.

    I could object to more, but I have a video to prepare for my calculus students. So here is the anecdote I promised. Remember this was given to me, not my original.

    During a dinner, a CEO, commenting on education, argued, "What's a kid going to learn from someone who decided his best option in life was to become a teacher?"
    To stress his point he said to another guest; "You're a teacher, Bonnie. Be honest. What do you make?"
    Bonnie, who had a reputation for honesty and frankness replied, "You want to know what I make?" (She paused for a second, then began...)
    "Well, I make kids work harder than they ever thought they could. I make them sit through 40 minutes of class time when their parents can't make them sit for 5 without an I Pod, Game Cube or movie rental.
    I make kids wonder. I make them question. I make them have respect and take responsibility for their actions. […] I make them show all their work in math. I make my classroom a place where all my students feel safe. Finally, I make them understand that if they use the gifts they were given, work hard, and follow their hearts, they can succeed in life. […] I MAKE A DIFFERENCE. What do you make Mr. CEO?"

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  15. Dante wrote...

    "I don't believe our opinions are actually that far off from each other, it's just that I may have accidentally implied that we shouldn't demonstrate our basic understanding of the subject. What I meant was that making a connection between our knowledge and our creativity, essentially between our left and right brains, greatly solidifies our knowledge."

    Dante, first I would like to say that you are very well spoken, more well spoken than I at your age. You are on a great journey and please don't take my posts too seriously, this is all part of the critical thinking process. And as you move forward in life you should seek criticism as much as agreement, but obviously you must make your own choices.

    Regarding creativity, I exhibit a lot of creativity in my work but it is technical and analytical creativity. I think math is an art all by itself. I do not think it needs PowerPoint or Prezi to make it an art. And that isn't that extraordinary, I also do not think music needs PowerPoint or Prezi to make it an art. So when I see a Prezi or PowerPoint math presentation, I will will be looking for the math, not the use of color or selection of typefaces. But don't get me wrong, I do many presentations and creativity there is useful as well, but it isn't math. It is left side, like you said, or which ever side it is supposed to be, I forget.

    And I agree, I don't think our opinions are that far apart. I am not against creativity, mathematical creativity in math I mean. And I am not against bringing other creative elements into a presentation, though I wouldn't assign a math grade based on those elements, not in a math class. But my stance here is not anti-creativity, it is actually pro-creativity. My contention is that the technical skills which exams assess and the creative skills that creative problems assess are both elements of the same product, like they are with any other art. You need them both and the technical skills take a lot of work. Thus, I disagree with Mr. Martin's contention that exams are bad and that removing them enhances creativity. Of course, I see math as an art, not as a subject of a PowerPoint presentation.

    Bob Hansen

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  16. Roberto said...

    "You say: "In the end however, success in this subject, or any subject looks the same." My experience is TOTALLY different. Success comes in many different ways, from the student who manages to memorize enough steps to answer test questions without understanding much about them, to the student who understands the concepts, but has troubles keeping enough focus to do the algebraic steps correctly, to the student who does both, to the student who simply overcomes his/her fears and realizes that learning math is possible. In fact, I have no idea of what your common success would look like: passing tests?"

    No, I don't mean tests, I mean actual realized success, like if you were to have 5 supposed musicians play music you would know in very short order which ones can play and which ones can't. It isn't fuzzy at all. Likewise, we know when people are good at math and when they are not. Hell, the people even know themselves. And yeah, a teacher's job is to encourage, but in the end you still have to be honest. And whether you like it or not, exams correlate with all this, music exams, writing exams and math exams. With absolute certainty, no, there will be outliers no matter what system, but more than reliably enough. And your examples above, I guess except the one that overcame, are not successful, so I think we are on the same page.

    I should also think that after my clarification you realize that I would classify Hendrix, Mozart and Einstein as very successful and technically competent in their domains. I am not playing any tricks here, I am calling a duck a duck. It isn't about money or fame, just a level of mastery of the subject.

    Regarding the rest of your post, I don't recall ever saying that life is math. I just think that math is math. As far as life goes, I think you find what you're good at and that will most certainly be also what you enjoy the most.

    And back to the subject of exams, I posed this question a few posts back, are we talking about students that do well on the exams and taking them further, or are we talking about students that do poorly on the exams and doing something else? I think it is a fair question and appropriate to this critical discussion.

    Bob Hansen

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  17. This is really great. Keep it up the good going. Really very great blog this has given me all the information that i needed, good for visiting daily it will increase our knowledge. Best luck for future.

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