A BALANCE BETWEEN CONCEPTUAL
UNDERSTANDING AND PROCEDURAL
SKILLS IN MATHEMATICS
- A PROPOSED MODEL
by
Paul Vaz
Following the presentation
Revisions to MAT 270
by
Jay Abramson and Pat Thompson,
School of Mathematical & Statistical Sciences,
ASU
Fabio Milnor, Director, Mathematics for STEM
Education, School of Mathematical & Statistical
Sciences, ASU, sent out an email asking anyone
interested, to send:
COMMENTS/OPINIONS ON THE FOLLOWING:
 A balance
between conceptual understanding and
procedural skills in the math classes;
 How
 How
to achieve this balance;
to assess the accomplishment of reaching
such a balance.
THE DISCUSSION IN THIS PRESENTATION
LEANS TOWARD CALCULUS
CURRENT STATUS OF CALCULUS IN
THE DEPARTMENT
Calculus classes offered in the department are
 Calculus with Analytical Geometry
(MAT 270, 271, 272)
 Calculus for Engineers
(MAT 265, 266, 267)
 Calculus for Business
(MAT 210, 211)
 Calculus for Life Sciences
(MAT 251)
More than 75% of our calculus students
are not mathematics majors.
DESCRIPTION IN THE ASU CATALOG
OF FIRST –SEMESTER CALCULUS
 MAT 210, MAT 251:
Differential and Integral Calculus of
Elementary functions and Applications
 MAT 265:
Limits and continuity, differential calculus of
functions of one variable, introduction to
integration.
 MAT 270: Real numbers, limits and
continuity, and differential and integral
calculus of functions of one variable.
CORE TOPICS: FIRST–SEMESTER CALCULUS
 Compute
limits
 Determine
whether
a
function
is
continuous at a given point.
 Determine the interval(s) of continuity of a
given function.
 Determine
the derivative and antiderivative of standard functions, like
polynomials, trigonometric, exponential,
logarithmic, hyperbolic, and the standard
inverse functions.
 Geometrical and Physical Applications of
the derivative and the integral.
NEEDS OF THE CLIENT
 Client
Departments stress more on the need for
functional understanding or procedural fluency
(how to do) than on conceptual understanding
(what to do).
 Client
Departments seem to prefer accelerated
programs and in some Universities, have designed
their own Mathematics Classes.
WHAT RESEARCH HAS TO
 Investigations
SAY
have shown that conceptual
knowledge has positive effects on student
learning.
 Students exposed to conceptual knowledge
had greater retention and were more likely
to use the ideas in new situations.
 Conceptual and procedural knowledge
developed inter-dependently
A REFERENCE ARTICLE
“The conflict between conceptual and
procedural knowledge: Should we need
to understand in order to be able to do,
or vice versa?”
By Lenni Haapasalo
University of Joensuu
The author utilizes a recent analysis of conceptual and
procedural knowledge, to present models for planning and
evaluating of learning processes based on a large empirical
project.
THE DISTINCTION BETWEEN
CONCEPTUAL UNDERSTANDING &
PROCEDURAL SKILLS
- AS STATED IN THE REFERENCE ARTICLE


Conceptual knowledge denotes knowledge of and a
skillful “drive” along particular networks, the elements of
which can be concepts, rules (algorithms, procedures,
etc.), and even problems (a solved problem may introduce
a new concept or rule) given in various representation
forms.
Procedural knowledge denotes dynamic and successful
utilization of particular rules, algorithms or procedures
within relevant representation forms.
CLOSING REMARKS IN THE ARTICLE
 An
appropriate assumption seems to be that
doing should be cognitively and
psychologically meaningful for the pupil
(team).
 A more important thing than to do is to
understand what, why and how you are
doing.
 This basically calls upon and considers the
following questions: do I know that, do I
know why, and do I know how I know
MAIN GOALS IN THIS PROPOSAL
 Foster
conceptual understanding without
sacrificing procedural fluency and viceversa.
 That the learner achieves an optimum
understanding of the concepts.
 That the learner can eventually apply the
concepts productively.
A MODEL FOR DELIVERY & ASSESSMENT
This model proposes the consideration of
the following elements in the plan of
teaching and assessment:
a.
b.
c.
d.
e.
f.
g.
Conceptual Understanding
Procedural skills
Collaborative learning
Multiple Representations
Interdisciplinary connections
Reading and Comprehension
Challenge
AN EXAMPLE OF A LESSON PLAN
TANGENT LINES
CONCEPTUAL UNDERSTANDING/PROCEDURAL
SKILLS
Definition of the tangent: Let y = f (x) be a
differentiable function.
A straight line is said to be a tangent to the
curve at the point (c, f (c)) on the curve if
 the line passes through the point (c, f (c)) on
the curve and
 The line has slope f '(c) where f ' is the
derivative of f.
NOTE
 You
require the knowledge of the derivative to
find the tangent line – this generates a large
number of procedural problems in this unit.
 Differentiability (smoothness) implies the
existence and uniqueness of the tangent line
 The tangent line as the limit of secant lines is
motivational in defining the derivative.
LESSON PLAN CONTINUED
CONCEPTUAL
UNDERSTANDING/PROCEDURAL SKILLS
CONTINUED
Finding the tangent line to a curve:
 Secants on a curve to find slopes
 Slope of a curve using the derivative
 Equation of a line – Review (PS & SI)
 Equation of a tangent at (c, f (c)):
y
f (c) f (c)( x c)
WHERE DOES THE METHOD FAIL?
The method assumes the existence of the
derivative and so fails where the derivative
does not exist:
 At points where the tangent is vertical
 At corner points
 At discontinuities
“PROCEDURAL”
IS NOT ENOUGH
by
B. Abramovitz, M. Berezina,
A. Berman, L. Shvartsman
4th MEDITERRANEAN CONFERENCE
ON MATHEMATICS EDUCATION

University of Palermo - Italy
28 – 30 January 2005
PROCEDURAL FORM
Let
g ( x) 3 sin 5 x and f (u ) e
(4 u 12) / 5
Find the tangent line of
h( x)
f ( g ( x)) at x 0
.
CONCEPTUAL FORM
2. Let
y 5x 3
be the tangent line of the function
Let
f ( g ( x)) e
4x
g ( x) at x 0.
for every real number
Find the tangent line of the function
x.
f ( x) at x 3.
LESSON PLAN CONTINUED
MULTIPLE REPRESENTATIONS (RULE OF 4)
 Estimating the slope of a curve using a graph paper
 Numerical calculation of the slope of a curve
 Definition of the slope of the curve – the derivative
 Connection between slope of the curve and the rate-ofchange of the function.
COLLABORATIVE LEARNING
 In-class activities: These activities reflect a mixture of
exercises to assess the conceptual understanding and the
procedural fluency of the student
LESSON PLAN CONTINUED
INTERDISCIPLINARY CONNECTION
 The connection between the slope of the curve and ‘rateof-change’ of the function opens the doors to a wide
range of applications in various fields
 Linear Approximations: The tangent line at a point on
the curve is the best straight-line approximation to the
curve at that point.
CHALLENGE
 “Differentiable implies continuous”
 “Discontinuous implies non-differentiable”
 Existence of the slope of a piecewise defined function at
a dividing point.
 Constructing a piecewise defined function that is
differentiable (smooth) at the dividing point
ASSESS WHETHER WE HAVE
ACCOMPLISHED THE GOAL OF REACHING
SUCH A BALANCE
By way of:
 In-class Group Activities
 Quizzes
 Online Homework system
 Midterms
 Final Exam
THINGS TO BE CONSIDERED
 Is
class time sufficient for learning that
fosters conceptual understanding and
procedural skills without sacrificing one for
the other?
 Some
learners end up with a concept
different from the intended one. Do we need
to consider time for the refinement or
correction of the learner's understanding of
the concepts