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I am speaking about high school mathematics . Students have attended a mathematics course . By the end thereof , students are supposed to be able to:

  1. find the limit of a real function f as “x” approaches to a given “a” and write the equations of vertical , horizontal and oblique asymptotes to the function’s curve (but not curved asymptotes)
  2. use the squeeze theorem to find some limits
  3. find the derivative of a real function and deduce whether the function is monotonic or not (depending on the sign of the derivative)
  4. write the equation of the tangent straight line to a curve at a given point
  5. determine the number of the solutions to an equation of the form f(x) = m (m is a real number) using the mean value theorem
  6. determine whether a real function is continuous or not at a given point
  7. draw the curve of a given function in an orthogonal system
  8. decide whether a sequence is monotonic or not
  9. decide whether a sequence is an arithmetic one , a geometric one , or neither
  10. calculate the limit of a recursive sequence (of the form un+1 = f(un) )
  11. decide whether two sequences are adjacent or not
  • Students are familiar with the following functions: the square root function , the absolute value function , exponential and logarithmic functions , polynomial functions , and trigonometric functions

  1. decide whether three vectors are coplanar or not

  2. calculate the dot product of two vectors

  3. write a cartesian equation of a plane , sphere , cone and cylinder

  4. write a parametric equation of a straight line , a half line , a line segment

  5. find the barycenter of n weighted points

  6. solve a system of three linear equations

  7. write the algebraic, rectangular and exponential forms of a complex number

  8. solve a second degree equation with complex variables and/or coefficients

  9. describe each of these geometrical transformations (rotation , homothety , and translation ) using complex numbers

  • Those are NOT all the skills which the students must have developed throughout the course.

After the course have ended and the students have mastered the skills above, they are supposed to take a 3 hours test which is “the final exam for the entire course” .

The questions is : How the test is supposed to be designed?

The problems from which the test is composed , should they be routine, typical ones which mimic the ones in the students’ textbooks? Or new ones which need a lot of thinking and imagination, yet require the same knowledge provided by the students’ textbook?

Some students who are accurate and do not make “arithmetic error” would find no difficulty solving any of the routine problems they are used to such as

  1. Given a function f defined on a set I : the students would find the limits of the function , the derivative thereof, determine the number of solutions to the equation f(x)=0, draw the graph of the function in an orthogonal system
  2. Given two complex numbers , the students would write both numbers in the exponential form , found both the exponential and algebraic form of their product , deduce the trigonometric ratios of an angle (most probably , the argument of the product ) Etc.

However, what benefit did such a test give? Did the test reflect enough the mathematical thinking of the students who took it?

Would it be reasonable if ,for instance, out of 100 students , 10 students took a perfect score ? It is possible that there exists 10 math “geniuses” within a group of 100?

I hope the question became clearer after this edit.If still not clear, please notify in the comments.

Please note that the test is NOT a multiple choice test.

Aloz371
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    I disagree with your idea that only one student should achieve a perfect score. The goal of a test is not to trip up students it is to test how well they have understood and can apply the material learned. A good teacher should have been able to clarify the material in such a way that multiple students have achieved such clarity as to achieve a perfect or near perfect score. – Burt Sep 01 '20 at 16:19
  • @Burt Acheiving a high score is a mutual achievement (between the teacher who “clarified” the material and the student ). However , I see that acheiving a perfect score must be the student’s achievement and only his – Aloz371 Sep 01 '20 at 17:03
  • The questions of the test must asses the students’ own skills and abilities . The teacher’s job is to “clarify the material” . However , it is the student’s job to improve his /her mathematical tools and critical thinking . – Aloz371 Sep 01 '20 at 17:05
  • The results must reflect these three factors :

    1- How successful the teacher was in delivering the material

    2- How well the students studied

    3- How much have the students developed their own mathematical abilities

     – Aloz371 Sep 01 '20 at 17:11
  • @Burt It is all about “how well the can apply the material learned” I think that even if all students understand the material , there still exist students who are better at applying the material than their peers. The test must also discover these students . What I meant is that one student at most most take the perfect score (or three in the worst cases), the others can take 95/100, but not 100/100 – Aloz371 Sep 01 '20 at 17:17
  • I agree that not all students should be achieving a perfect score, and that would be a sign of a test that was too easy and not displaying any critical thinking. I simply feel that the number of students achieving a perfect or near perfect score should be more than one. – Burt Sep 01 '20 at 17:23
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    FYI, these scores look like scores on a gateway test to me. As for your several questions, they seem highly dependent on your specific local circumstances. If I were requested to provide advice (say, as an external math consultant), then I would need to have a fairly complete idea of texts used, syllabi followed, samples of in-class short quizzes and hour tests from several previous teachers of the course, the purpose of the course $(%$ of students is it a prerequisite for further study, $%$ of students is it a terminal course), etc. – Dave L Renfro Sep 01 '20 at 17:25
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    This really isn't a new topic, debate. Huge literature on testing design and objectives. – guest Sep 01 '20 at 17:39
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    It's important to differentiate performance measurement versus differentiation within the cohort. Too many people think the overall objective is/should be ranking. This means a poorly taught class, for example, still gets curved to pass people. However, more sophisticated design, actually has a specific standard and then verifies if people meet it. Think about carrier landings for instance. They don't curve those. – guest Sep 01 '20 at 17:42
  • In education, you have to do larger studies or year to year comparisons and the like. But you can still certainly come up with a performance standard. It won't be like a Euclid proof perfect, but will be way better than just curving. Like a war plan or an NPV. The effort to make it teaches you thinks about your topic. – guest Sep 01 '20 at 17:45
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    "One student at most would be considered “outstanding”" — grading on a curve sucks, agreed with guest above. – Rusty Core Sep 01 '20 at 20:11
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    The idea that only one student should achieve perfect score seems rather outlandish to me. Especially at the high school level, where many of the problems taught are routine exercises, there should be (in my personal experience, though of course this could vary for different people) many students who would be perfectly capable of handling the material well enough to achieve perfect score. In fact, for some classes most students should be able to achieve perfect score without much trouble, and the most common reason some of them do not is some kind of easy, careless computational error. – YiFan Sep 01 '20 at 22:14
  • @YiFan You have pointed at the issue I raise “routine exercises”. I think that the problems in tests shouldn’t be that routine . – Aloz371 Sep 02 '20 at 11:30
  • The issue is that the test problems are “routine exercises” . Thus, any student who has tried to solve a few of these typical problems and doesn’t make “careless computational error” would take a perfect ,or near perfect , score . – Aloz371 Sep 02 '20 at 11:34
  • Let me clarify , not all people comprehend mathematics with the same “deepness” . Some students tend to have a deeper comprehension of the mathematical concepts than their peers , others think nothing of mathematics but finding the limit at a , and how to “get rid “ of an indeterminate form while finding the limit. – Aloz371 Sep 02 '20 at 11:42
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    I also see that this applies to teachers. Not all teachers have the same “depth” in understanding math. Not all teachers show the same proffesionality while teaching math or writing a solution to a problem. Another example , some students/ teachers would insert the implication symbol at every moment without thinking if this symbol makes any sense within the context within which it was inserted. Such “routine” tests show nothing about the depth of students’ understanding . – Aloz371 Sep 02 '20 at 11:48
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    Even for routine exercises, it is unusual for large cohorts to get a several question test, perfect. Remember that they are "routine" in the sense of non-tricky or complicated, but are still a new topic for the trainee. – guest Sep 02 '20 at 13:39
  • I think not. Let’s consider the example I used in the question’s post . If students were taught throught a long period of time ( a year for instance) , how to calculate limits and how to find derivatives , and have caluclated dozens of limits ,throughout that period of time,have faced dozens of indeterminate forms, and have found the derivatives of dozens of functions after memorizing by hard the deriviatives of the basic functions and the rules of derivation (of composite functions for example). – Aloz371 Sep 02 '20 at 15:54
  • @guest The students were then asked ,in the test, to find the deriviative of a function f defined on I , to calculate the limits therof at (...) , to prove there exists n solutions and only n to the equation f(x) = 0 (a cliché which they memorize by hard) , and to draw the curve of the function in an orthogonal system. What kind of mental effort the students are supposed to exert while solving to the previous problem ? The students already know what steps they must follow in order to solve the problem , they only need to be careful not to make “careless computational error”. – Aloz371 Sep 02 '20 at 16:26
  • Maybe I was mistaken when I said “one student at most must take a perfect score” , “one student at most would be considered outstanding”. It is still “somehow” reasonable if three or four students scored perfect and were considered very good , but not more. – Aloz371 Sep 02 '20 at 16:28
  • I am relatively new to SE. Please tell me if I have violated the rules of the community. Is my question clear enough ? Does it need more clarification? It is considered “opinion-based” ? I ask this as I haven’t received any answers until the moment. – Aloz371 Sep 02 '20 at 16:32
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    I am thinking about why you haven't gotten answers. I would like to suggest that you put forth a premise (that the distribution of scores indicate that there is something wrong with the test) and then asked how to fix the test. The problem seems to be that people don't agree with your premise. Perhaps you should first ask about your premise in a question. – Amy B Sep 02 '20 at 16:38
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    I haven’t received any answers --- For me the problem in trying to address this is that I know nothing about the context of the "test". For example, you said "3 hours". The last 3 hour tests I have experience with were final exams for an entire course (not just a "test"), and this was back in the 1970s. Throughout the 1980s, 1990s, 2000s all the high-schools/colleges/universities I've taught at (10, unless I've miscounted) had final exams that were a maximum of 2 hours. I suspect you're not in the U.S., and furthermore, what you mean by "test" is not what many here have experience with. – Dave L Renfro Sep 02 '20 at 16:46
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    The reason I did not post an answer is that I have no idea what the question is. If the question is, as you titled it, "How should mathematics tests be designed," then it is too broad for an SE site. If it is not that, then spending effort on removing unnecessary paragraphs, asides, and ideas from your question would improve it significantly. Identifying which premises you are asking about, and which ones you are not willing to discuss, and distinguishing those clearly, would improve the question as well. – Chris Cunningham Sep 02 '20 at 19:05
  • I think the question is too narrow. Suggest broadening it to be "how to teach mathematics". ;-) – guest Sep 02 '20 at 22:43
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  • Even routine questions are somewhat new to the people. I mean, they didn't know them the year before! 2. Arithmetic errors or algebra errors are not trivial. Are an important factor in doing good work, processing mathematics. 3. In addition, remember the students are under pressure. 4. Students are often not well drilled on "routine" problems.
    • – guest Sep 02 '20 at 22:46
  • I'm sure you could construct some artificial example to pass your wickets. But I doubt that you would get 20% perfect scores with most trials of your case. For a tactical example, consider a final exam of medium level students (say level one state uni, already a level over what most here teach at) in ODEs, using the text by Speigel. It has HW in 3 grades. A (routine), C (hard) and B (in between, usually more multi-step). I don't think you'd get 20% perfect scores for a 15 question test (5 per hour) drawn from the A/B bank. Not even if all from the A bank. I will make it Bayesian and bet. – guest Sep 02 '20 at 22:54
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    Please clarify what you mean by "test." Different tests have different objectives. See this Wikipedia entry. Is the test you are thinking of a placement test, a formative test, a summative test, or a diagnostic test? – JRN Sep 03 '20 at 01:00
  • Here's the thing, there already is a test like this for Math majors. We gave it to our exiting majors to check on their progress. Sometimes we got 5 with a perfect score. It was useful data, because, often we got just one or none with the perfect score. Yes it just tests mostly the mindless routine stuff, but that stuff matters. That said, I do think we'd do better to have tests with a lot more uncertainty. Some outlier questions would help to discern whether or not some math departments are teaching more than just the drill part of math that some of us love so much to champion... – James S. Cook Sep 03 '20 at 07:58
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    You just edited your post to remove three words. Why didn't you edit it to clarify your post? What kind of test are you interested in? I'm voting to close your post as unclear. – JRN Sep 03 '20 at 10:04
  • @JoelReyesNoche The test can be considered a summative one. – Aloz371 Sep 03 '20 at 11:56
  • (I've retracted my close vote.) – JRN Sep 03 '20 at 13:16
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    There are a lot of comments and I may have missed the answer to this - if I did I apologize. Your question has the label "gifted students". I am wondering if this is because all the students in the class are gifted or are you wondering how to identify the gifted students using your test. – Amy B Sep 04 '20 at 13:36
  • @AmyB Frankly speaking , I have put this label for two reasons : a) I felt that two tags/labels are not enough , I thus had to add a third one. However , I failed to find another suitable tag , so I chose to use the “gifted students” tag, as it is the “most” suitable one. This is the main reason. b) Because I see that among the multiple purposes of the tests (the final exam for the entire course) is to identify the gifted students. – Aloz371 Sep 05 '20 at 07:54