I am speaking about high school mathematics . Students have attended a mathematics course . By the end thereof , students are supposed to be able to:
- 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)
- use the squeeze theorem to find some limits
- find the derivative of a real function and deduce whether the function is monotonic or not (depending on the sign of the derivative)
- write the equation of the tangent straight line to a curve at a given point
- 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
- determine whether a real function is continuous or not at a given point
- draw the curve of a given function in an orthogonal system
- decide whether a sequence is monotonic or not
- decide whether a sequence is an arithmetic one , a geometric one , or neither
- calculate the limit of a recursive sequence (of the form un+1 = f(un) )
- 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
decide whether three vectors are coplanar or not
calculate the dot product of two vectors
write a cartesian equation of a plane , sphere , cone and cylinder
write a parametric equation of a straight line , a half line , a line segment
find the barycenter of n weighted points
solve a system of three linear equations
write the algebraic, rectangular and exponential forms of a complex number
solve a second degree equation with complex variables and/or coefficients
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
- 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
- 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.