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Leslie Aspinwall
Florida State Univesity

Kenneth L. Shaw
Florida State University
Panama City 

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Visualization in the Calculus

Research on Teaching and Learning of the Derivative

NCB Deposit  # 21

This section . . . 

presents research on
how students think
about the derivative
of a function.

basic calculus curves
Can you take theses 
derivatives?  See below. 

The notion of derivative is one of the most important ideas in calculus.  Any subject that uses calculus -- and most subjects do --  uses applications of the derivative.

In September, 2002 Aspinwall and Shaw published an article in "Mathematics Teacher" on the contrasting styles of two students learning calculus concepts.  One student's preference for analytic representations became evident while the other clearly preferred graphic representations.

This web page is an informal, but highly insightful, presentation of research on university students' learning of the derivative from the calculus. In the left column is the graph of a given function.   To its right are responses from students for the graph of the derivative.  Click on the idea image  for their feedback to the investigators.

Task One
Function # 1
parabola graph
linear segment


A Group of Students
linear graph


Task Two

Function # 2
sine graph

Skewed cosine graph

not a cosine graph
What is the graph of the
cosine function?

Why are Betty and Al
both wrong?

Task Three

Function # 3
Trig function

Derivative at endpoints?

Another step function excluding endpoints.

Al's explanation
Two well-known triangles for help with the derivative.

Task Four

Function # 4
Slopes: Negative, positive,neg but at y=0?

Slope at y = zero ??????

No slope of the derivative at y = zero.

Al's explanation
Slope at y = 0 is non existent.

Both Betty and Al could easily take the derivatives of the following three functions.

Thus, we conclude both had mastered the mechanical aspects of finding a derivative.

Can you calculate these derivatives?

(1.) ƒ(x) = x2 sin x
y ' = 2 x cos x + sin xx2
    = x ( 2 cos x + x • sin x )
y ' = 2 x sin x - cos xx2
    = x ( 2 sin x - x • cos x )
y ' = 2 x sin x + cos xx2
    = x ( 2 sin x + x • cos x )
(2.) ƒ(x) = ( x2 + 2 x ) 3
y ' = 3 ( x2 + 2 x )
    = 6 ( x2 + 2 x ) 2 ( x + 1 )
y ' = 3 ( x2 + 2 x ) 2
y ' = 3 ( 2 x + 2 ) 2
(3.) ƒ(x) = 3 x / (2 x + 1 )
y ' = 3 / ( 2 x + 1) 2
y ' = 3 / ( 2 x + 1)
y ' = - 3 / ( 2 x + 1) 2

For the Instructor:

Traditional instruction in Calculus concentrates primarily on manipulating symbolic or analytic representations.  Reform efforts over the past decade have promoted the understanding of both analytic and graphic representations of functions and derivatives. 

The research literature on teaching and learning clearly indicates that having multiple ways - for example,  graphic and analytic - to represent mathematical concepts is beneficial. 

Cognitive psychologists urge instructors to provide a variety of tasks that allow students to become proficient in several areas. 

Briefly, the three main types of mathematical processing by individuals are described as analytic, geometric, and harmonic

List of terms

  [Mathematics instructors often hear the complaint that students can do the mechanics of finding the derivative but have no idea of when and where it is applied.]


Printed References

Aspinwall, Leslie, Kenneth Shaw, and Norma Presmet.  "Uncontrollable Mental Imagery:  Graphical Connections between a Function and Its Derivative."  Educational Studies in Mathematics 33 (September 1997)" 301 - 17.

Walter Zimmermann and Steve Cunningham (eds.), Visualization in Teaching and Learning Mathematics, 9-24, MAA, Washington, DC, 1991. 

If you enjoy comparing a variety of student responses, be sure to take a look at the following publication: 

Friedberg, Solomon, et al., "Teaching Mathematics in Colleges and Universities:  Case Studies for Today's Classroom" Issues in Mathematics Education, Vol. 10, CBMS Conference Board of the Mathematical Sciences, AMS-MAA, 2001.

Rule of Three

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