How should mathematics tests be designed? [closed]

Question

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

12. decide whether three vectors are coplanar or not

13. calculate the dot product of two vectors

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

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

16. find the barycenter of n weighted points

17. solve a system of three linear equations

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

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

20. 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.

Answer 1

"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 text?"

I recommend to go with the "routine" questions. (Scare quotes intentional!) (1) Don't underestimate the difficulty of "routine" to many students. (1.5) ESPECIALLY under test conditions. (2.) You're covering a lot of material and some of it will be several months old. (3.) There is a value to the routine questions. (4.) Separating the stars (small minority) from the above average (larger amount, but still a minority) should not replace the objective of measuring basic competency for the majority of the kids. (4.5) And you'll get it anyway, since in a 3 hour test, there's room for stumbles. It won't be so basic that the top 3 of 30 can't be differentiated from the next 6. They'll separate. Look at the literature on psychrometrics. Look at "easy" tests like SAT-M.

In fact, there's a real danger of running acrux if your questions are too tricky and non routine. You're basically testing problem solving savvy more than mastery of the curriculum and even the sharpies can just get unlucky about figuring out some Euler trick or the like under exam conditions. I think a routine 20 question test (one per objective) will be fine.

Oh...and a lot of those objectives are pretty conceptual. Not sure I agree with so much stress on that. It is important to have kids that can do manipulations to support what they will need in physics and engineering homeworks. It's not awful and there are some tactical objectives. And maybe it's just written that way to look fancy and the questions are more recognizable. Not a strong objection, just a little hair on my neck. ;-)

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It's interesting to look at your MSE question, where you ask about making silly mistakes.

https://math.stackexchange.com/questions/3796357/how-can-i-prevent-silly-frustrating-mistakes-while-solving-math-problems

It's one thing if you just want hard questions. But hard questions (concept emphasis) when you don't have mastery of the basics? Sorry, that is a big problem. You NEED to have automaticity of the basics if you are going to do advanced math or physics or engineering (even decent undergrad level homework, especially junior year on, let alone research).

Look at Olympic level gymnasts (I worked out with some back in the day). They spent their time working on very hard D-level and above skills (full twisting double backs and such). Remember asking one how to progress and he told me...you need to be solid on your basics...you don't have a 30 second handstand, you don't have good circles on horse, and your backhandspring has no distance.

The movie The Karate Kid is an exaggeration of this concept, but it really is true in spirt in martial arts. Well, at least boxing. You need to have basic stance and protection and the like down solid. And jab, jab, jab. Yeah, you'll want those fancy combinations. But learn to jab, baby. It will keep you safe.

Here is what Richard Feynman advised CALTECH students regarding mathematics (section 1-3 of Feynman Lectures Physics):

So, this guy comes into my office and asks me to try to make everything straight that I taught him, and this is the best I can do. The problem is to try to explain the stuff that was being taught. So I start, now, with the review. I would tell this guy, “The first thing you must learn is the mathematics. And that involves, first, calculus. And in calculus, differentiation.”

Now, mathematics is a beautiful subject, and has its ins and outs, too, but we’re trying to figure out what the minimum amount we have to learn for physics purposes are. So the attitude that’s taken here is a “disrespectful” one towards the mathematics, for sheer efficiency only; I’m not trying to undo mathematics.

What we have to do is to learn to differentiate like we know how much is 3 and 5, or how much is 5 times 7, because that kind of work is involved so often that it’s good not to be confounded by it. When you write something down, you should be able to immediately differentiate it without even thinking about it, and without making any mistakes. You’ll find you need to do this operation all the time—not only in physics, but in all the sciences. Therefore differentiation is like the arithmetic you had to learn before you could learn algebra.

Incidentally, the same goes for algebra: there’s a lot of algebra. We are assuming that you can do algebra in your sleep, upside down, without making a mistake. We know it isn’t true, so you should also practice algebra: write yourself a lot of expressions, practice them, and don’t make any errors. Errors in algebra, differentiation, and integration are only nonsense; they’re things that just annoy the physics, and annoy your mind while you’re trying to analyze something. You should be able to do calculations as quickly as possible, and with a minimum of errors. That requires nothing but rote practice—that’s the only way to do it. It’s like making yourself a multiplication table, like you did in elementary school: they’d put a bunch of numbers on the board, and you’d go: “This times that, this times that,” and so on—Bing! Bing! Bing!

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I would even say that you can use some of the advanced problems as an OPPORTUNITY to hone your rusty "routine" skills. For example if you do a maximization problem (not the most conceptually hard, but still a little non routine in that it is multistep and may be a "word problem") and you have the method down, but mess up a calculation. Now, when you check your answer and recognize the mistake, DON'T just say no biggie and move on. Instead, put the solution (or answer) away and do the problem again from scratch as if it were new. This will quickly drill you to start NOT making so many errors and it will give you more practice in both the method and the manipulations. And it will tend to improve your test speed.