Use of Conceptual vs. Procedural Knowledge by Engineering Students
Studying Entry Level Mechanics
Edward E. Anderson
Department of Mechanical Engineering
Texas Tech University
Roman Taraban
Department of Psychological Sciences
Texas Tech University
Abstract
Engineering students need to use conceptual knowledge and procedural knowledge in the process
of successful problem solving. All too often, students apply the correct procedure to the wrong
concept or the wrong procedure to the correct concept. It is then important for engineering
educators to understand the role of these two forms of knowledge, how they are related and their
impact upon student performance. This paper examines these question based upon the results of
pencil-paper student productions and analysis of video recordings taken while students solved
typical Introduction to Mechanics homework problems. These were analyzed for frequency of
conceptual/procedural occurrences and the relationship of these occurrences upon student
performance. The results of these analyses are reported in the paper.
Introduction
Conceptual knowledge is that understanding of a knowledge domain needed to formulate and
ultimately solve problems. It differs from procedural knowledge which is used to solve a
specific problem through application of specific algorithms or processes. Conceptual knowledge
transcends a knowledge domain and can be applied to all problems in that domain. Procedural
knowledge applies to a specific problem. Both conceptual and procedural knowledge in the
domain of engineering statics problems involves forces, moments, couples, dimensions, and
angles. The successful problem solver needs to be proficient with conceptual and procedural
domain knowledge and processes.
Steif [1] divides engineering statics conceptual knowledge into four categories:
1. Forces act on bodies
2. Combinations and/or distributions of forces acting on a body are statically equivalent to a
force and couple
3. Conditions of contact between bodies or types of bodies imply simplification of forces
4. Equilibrium conditions are imposed on a body
The first of these is the concept of a free-body which is isolated for the purpose of problem
solving. The second one allows one to reduce the action caused by several forces to the same
action caused by a single force and couple. Although the second concept is important in other
knowledge domains such as dynamics, it is little used in engineering statics. The third concept
underlies the addition of contact forces and moments to the free-body to replace the interaction at
the contact points or surfaces between two bodies to an equivalent reaction. Concepts 1 and 3
are then the basis of the free-body diagram that successful problem solvers use to formulate and
solve a problem. The last concept is simply the fact that the vector sum of all forces and
moments acting on a free-body must be zero as required by Newton’s law of motion. These
concepts are the basis and an integral part of all recognized methods [2, 3, 4, 5, 6, 7] for solving
engineering statics problems.
Steif [1] recognized that these general concepts result in conceptual errors or “conceptual
lapses”. He identified those listed below as common errors or lapses:
1. Failure to be clear as to which body is being considered for equilibrium
2. Failure to take advantage of the options of treating a collection of parts as a single body,
dismembering a system into individual parts, or dividing a part into two:
3. Leaving a force off the free body diagram (FBD)
4. Drawing a force as acting on the body of the FBD even though that force is exerted by a
part which is included in the body of the FBD
5. Drawing a force as acting on the body of the FBD even though that force does not act
directly on the body
6. Failing to account for the mutual (equal and opposite) nature of forces between connected
bodies that are separated for analysis.
7. Ignoring a couple that could act between two bodies or falsely presuming its presence.
8. Not allowing for the full range of possible forces, or not sufficiently restricting the
possible forces:
9. Presuming a friction force is at the slipping limit even though equilibrium is maintained
with a friction force of lesser magnitude:
10. Failure to impose balance of forces in all directions and moments about all axes:
11. Having a couple contribute to a force summation or not having a moment summation
include a couple:
This list, which is the result of the analysis of many student problem solutions, is not exhaustive.
Additional misconceptions and conceptual errors will be introduced in this paper.
Higley, et al. [8] investigated the role of conceptual understanding in problem solving and
reported a self-explanation intervention which was demonstrated to improve student success with
solving statics problems. Steif [9, 10, 11] has used the idea of conceptual knowledge to develop
a standardized statics concepts inventory instrument which is used to assess student conceptual
knowledge as well as for research purposes.
For the purposes of this investigation, a concept is defined as any construct used to reduce a
problem to a procedure that can be followed to solve the problem. Selecting a free-body,
addition of forces and couples to replace actions at free-body-surrounding interfaces as well as
applied forces and couples, and knowing that some or all equilibrium equations need to be
written are examples of concepts according to this definition. Procedures are then those steps
necessary to reduce the concepts as applied to a given problem to a form suitable for problem
solution. The addition of dimensions and angles to a free-body diagram, reduction of forces and
moments to component form, correct assignment of signs to forces and moments, and algebraic
solution of the equilibrium equations are examples of procedures in this paper.
Experiment Description
The subjects for this experiment were 49 second semester engineering students selected from
two universities. Both student pools were enrolled in and had completed about three-fourths of
an engineering statics course as required in their respective curricula. They had completed the
topic of rigid-body equilibrium.
All subjects worked two typical engineering statics problems with pencil and paper while also
being video recorded. During the recording, subjects were asked to verbally state what they
were thinking. The proctor during the video recording only interfered if the subject had not
spoken for some time. Then the proctor would ask a question like “What are you thinking now?’
to stimulate subjects to verbalize their thoughts.
Both problems, taken from Beer and Johnson [12], were typical simple engineering statics
problems. The first problem asked the question “Neglecting friction, determine the tension in
cable ABD and the reaction at support C?” with respect to the graphic illustrated in Figure 1.
The FBD for the first experimental problem is also included in Figure 1. Although this is not the
only valid FBD for this problem, it is the one an expert would use and the one drawn by most of
the subjects. Practically all the subjects did not show the given information, F, l1, and l2, in
symbol form on the FBD as illustrated in Figure 1. Rather they choose to show numerical values
rather than symbolic variable names. Inclusion of the numerical values is viewed as a procedural
step, not a conceptual step.
Fig. 1: Problem statement graphic and FBD for experimental problem 1
The second experimental problem was also selected from Beer and Johnson [12] and read as
“The 10-m beam AB rests upon, but is not attached to, supports C and D. Neglecting the weight
of the beam, determine the range of values of P for which the beam will remain in equilibrium.”
and the graphic for this problem along with the problem FBD are shown in Figure 2. The shown
FBD with geometric dimensions was that used by all subjects as well as an expert who used
symbolic variables rather than numerical values
Fig. 2: Problem statement graphic and FBD for experimental problem 2. .
Inspection of the two problems reveals that they are not overly complicated, but are dependent
upon key concepts. The key concept for Problem 1 is the fact that the tension in the cable
attached at D equals the tension in the cable attached at A since the pulley is frictionless. The
key concept for the second problem is the fact that equilibrium requires that reaction C is zero or
positive and reaction D is also zero or positive when the beam is in contact with support C and
D. The limiting values of P are when reaction C becomes zero and the beam loses contact with
support C and when reaction D becomes zero and the beam loses contact with support D.
Without these key concepts, neither problem can be solved.
Subjects first completed their paper and pencil product while being recorded. The researchers
than compared a subject’s solution to the video recording to determine the order in which the
various steps were accomplished. The paper was then graded by one of the researchers with a
rubric used for grade assessment during a regular course offering. Subsequently, one of the
researchers inspected the pencil and paper product for the occurrence of conceptual and
procedural elements. These elements for problem 1 were:
1. (C) Free-body selection
2. (C) Add applied force F
3. (P) Assign value to F
4. (C) Add cable reaction T at D
5. (C) Add cable reaction T at A
6. (C) Recognition that T at A equals T at D
7. (C) Add horizontal reaction C
8. (C) Add vertical reaction C
9. (P) Add dimensions to FBD
10. (C) Write first valid equilibrium equation
11. (P) Solve first equilibrium equation
12. (C) Write second valid equilibrium equation
13. (P) Solve second equilibrium equation
14. (C) Write third valid equilibrium equation
15. (P) Solve third equilibrium equation
where C and P denoted conceptual and procedural elements respectively.
Since there are three unknowns, T, Cx, and Cy, to be found in this problem, three independent,
valid equilibrium equations are required. Although there are several valid choices for these
equations. The ones most commonly used by experts as well the student subjects were xdirection equilibrium, y-direction equilibrium, and moment about one axis equilibrium, not
necessarily in that order. Experts and high-performing students would use the pin at C for their
moment equation and it would be the first equation written and solved.
The conceptual and procedural elements for the second problem were:
1. (C) Free body selection
2. (C) Add applied force P
3. (C) Add applied Force F1
4. (P) Assign value to F1
5. (C) Add Applied force F2
6. (P) Assign value to F2
7. (C) Add vertical reaction C
8. (C) Add vertical reaction D
9. (P) Add dimensions to FBD
10. (C) Reaction C = 0 for one force range limit
11. (C) Reaction D = 0 for one force range limit
12. (C) Write first valid equilibrium equation
13. (P) Solve first equilibrium equation
14. (C) Write second valid equilibrium equation
15. (P) Solve second equilibrium equation
16. (C) Write third valid equilibrium equation
17. (P) Solve third equilibrium equation
This problem only involves 2 unknown values, Pmax and Pmin, and hence only two valid
equilibrium equations were required. As will be shown in the results section, most students
wrote three or more equilibrium equations. Items 16 and 17 are included in the above list for this
reason. Experts and high-performing students recognize that the problem is most easily solved
by setting reaction C to zero and then summing moments about point D to determine one of the
limits on force P. The other limit is most quickly found by setting reaction D to zero and then
summing moments about point C. Note that it is possible and efficient to solve the second
problem without force equilibrium.
Student subjects were divided into two groups for the purpose of correlation: a high-performing
group and a low-performing group. The high-performing group was defined as those students
who would be given a grade of A or B based on grader scores. This translated into a grade of
about 80% or more. The remaining subjects were classified as low-performing students although
many would have received a grade of C. There were of course some extremely low performers
who had scores of 30% and less as well as two students who could not do the problems.
Results
Problem 1 frequency count results for the two groups are shown in Table 1 on the next page.
Inspection of Table 1 reveals that high-preforming students completed practically all the
conceptual and procedural elements of Problem 1. Items 3 and 9 of Table 1 suggest that both
high- and low-performers performed rote procedural steps while setting up their problems. Items
14 and 15 of Table 1 suggest that the low-performers perhaps misread, did not completely read
or forgot the problem statement and thereby failed to finish the problem.
High-Performers (25 students) Low-Performers (24 students)
1. (C) Free-body selection
25
19
2. (C) Applied force F
25
19
3. (P) Assign value to F
25
19
4. (C) Cable reaction T at D
25
16
5. (C) Cable reaction T at A
25
17
6. (C) Recognition that T at A equals T at D
25
16
7. (C) Horizontal reaction C
22
5
8. (C) Vertical reaction C
25
14
9. (P) Add dimensions to FBD
14
12
10. (C) Write first valid equilibrium equation
25
20
11. (P) Solve first equilibrium equation
25
12
12. (C) Write second valid equilibrium equation
24
17
13. (P) Solve second equilibrium equation
24
15
14. (C) Write third valid equilibrium equation
22
7
15. (P) Solve third equilibrium equation
22
6
Table 1: Conceptual and Procedural Element Frequency Counts for Problem 1
Problem 1 low-performers had more difficulty with selecting a free-body than the high
performers as indicated by item 1 of Table 1. The most common mistake made by the lowperformers was to use a global free-body which included the pulley. They then had four
unknowns and only three equations. They failed to realize their FBD was under-specified and
were therefore unable to solve the problem. Item 2, 3, and 9 indicate that low-performers were
able to perform the low-order procedural steps of copying the problem graphic. It is interesting
to observe that 16 of the 19 low-performing students who selected a valid free-body also
understood the key concept (item 6) that the cable tension does not change magnitude when
passing over a frictionless pulley. Although some of the high-performers failed to include the xreaction force component at C (3/25), this conceptual mistake was more common amongst the
low-performers (14/19). It is not clear why this occurred but, it may be due to the problem
statement which asked for the reaction at C not the reaction components. If this was the case, the
low-performers have a deep misconception about vectors and how vectors are represented in
two-dimensional space by two items: magnitude and direction or orthogonal components.
Finally, low-performers who could write equilibrium equations had more difficulty with solving
the equations as shown by items 10-15 of Table 1.
The order in which the problem 1 equilibrium equations were written and solved is shown in
Table 2 on the next page.
The results of Table 2 indicate that high-performers tended to write the equilibrium equations in
the conventional Cx, Cy, and Mc order. Only 28% of the high-performers wrote their equations in
the same order as an expert would follow by starting with the Mc equation. This indicates that
they weren’t thinking strategically and conceptually, but rather were following a rote procedure.
But, although they wrote the Mc equation last, they tended to solve it first. Thus, as they were
engaged in the procedure of solving the equilibrium equations, they invoked the concept of the
order in which the equations should be solved. High-performers then engaged in conceptual
thinking as well as procedural thinking and 28% employed conceptual thinking before
procedural thinking. All of the high-performers knew that they needed to write equilibrium
equations and one of them choose to write the moment equilibrium equation about some point
other than point C.
High-Performers (25 students)
Low-Performers (24 students)
First Equation Written:
Mc – 7
Mc – 10
Cx – 12
Cx – 5
Cy – 6
Cy - 4
First Equation Solved:
Mc – 22
Mc – 14
Cx - 1
Cx – 0
Cy - 0
Cy - 0
Second Equation Written: Mc – 2
Mc – 3
Cx – 9
Cx – 1
Cy – 13
Cy - 10
Second Equation Solved:
Mc – 0
Mc – 0
Cx - 16
Cx – 2
Cy - 9
Cy - 10
Third Equation Written: Mc – 15
Mc – 3
Cx – 2
Cx –0
Cy – 5
Cy - 0
Third Equation Solved:
Mc – 1
Mc – 0
Cx - 6
Cx – 1
Cy - 16
Cy – 1
Table 2: Frequency of Problem 1 Equations Written and Solved
A higher percentage of the low-performers wrote the Mc equation first, but fewer lowerperformers (19/23) knew they needed to start by writing equilibrium equations. Only 3 of the 23
low-performers wrote 3 equilibrium equations and only 2 solved 3 equilibrium equations. Those
that attempted to solve their equilibrium equations started with the Mc equation followed by the
Cy equation. This a consequence of the fact that most low-performers failed to include the Cx
reaction force on their FBD. They missed a very important conceptual step by failing to include
this force component. Low-performers were also more likely to write and attempt to solve
moment equilibrium equations written about some point other than C.
The problem 2 comparison of low- and high-performers is presented in Table 3. Problem 2 was
more difficult than Problem 1. As a result, there were fewer Problem 2 high-performers then
Problem 1 high-performers.
High- and low-performers were able to set up the FBD correctly and followed the procedural
steps of assigning values to problem statement variables. High-performers recognized that one
limiting value of the applied force occurred when the reaction at C became zero. But, over onethird of them were unable to recognize that the second limiting applied force occurred when the
reaction at D became zero. Only 7 of the 33 low-performers had any concept of tipping and
what becomes of the supporting reactions when tipping impends. Some of the high-performers
(6/15) wrote 3 equilibrium equations not recognizing that only two equilibrium equations were
necessary while only 1 of the 6 solved the third equilibrium equation. A third, unnecessary
equilibrium equation was invoked by practically of all the low-performers. The results of Table
3 items 10, 11, 16, and 17 indicate that low-performers were following procedures without
understanding the underlying concepts or engaging in strategic thinking or planning. This was
also evidenced in about one-third of the high-performing students.
High-Performers (15 students) Low-Performers (34 students)
1. (C) Free body selection
15
33
2. (C) Applied force P
15
33
3. (C) Applied Force F1
15
33
4. (P) Assign value to F1
15
33
5. (C) Applied force F2
15
33
6. (P) Assign value to F2
15
33
7. (C) Vertical reaction C
15
31
8. (C) Vertical reaction D
15
31
9. (P) Add dimensions to FBD
15
30
10. (C) Reaction C = 0 for
one force range limit
14
5
11. (C) Reaction D = 0 for
one force range limit
9
2
12. (C) First valid equilibrium equation
14
27
13. (P) Solve first equilibrium equation
14
27
14. (C) Second valid equilibrium equation 12
23
15. (P) Solve second equilibrium equation 12
23
16. (C) Third valid equilibrium equation
6
30
17. (P) Solve third equilibrium equation
1
25
Table 3: Conceptual and Procedural Element Frequency Counts for Problem 2
Both low- and high-performers were capable of applying the conceptual and procedural steps
required to set up the problem FBD. But the low-performers did not understand the concept of
tipping. Rather they followed the standard procedure of writing 3 equilibrium equations and
then became confused as they attempted to solve their equilibrium equations. Although it is not
demonstrated in Table3, low-performers tended to write either meaningless equilibrium
equations or equations that were difficult to solve (e.g., moments about A and/or B). Since they
wrote and attempted to solve three equilibrium equations, they over specified the problem and
consequently were unable to solve the problem. This demonstrates another conceptual
misconception common among low-performers. This misconception is the understanding that
there are only a limited number of independent equilibrium equations that can be used to solve a
problem and that the inclusion of additional equations will not produce a problem solution and
will confuse the solution.
The order in which the problem 2 equilibrium equations were written and solved is shown in
Table 4. Item M/F in this table indicates either a moment or a force equation. For most of the
subjects, this was a vertical force equation, but several subjects wrote another moment equation
about either A or B.
High-Performers (15 students)
Low-Performers (34 students)
First Equation Written:
Md – 8
Md – 17
Mc – 1
Mc – 0
M/F – 5
M/F – 10
First Equation Solved:
Md – 3
Md – 3
Mc - 9
Mc - 7
M/F - 6
M/F - 21
Second Equation Written: Md – 1
Mc – 0
Mc – 9
Mc – 21
M/F – 3
M/F – 2
Second Equation Solved:
Md – 9
Md – 17
Mc - 1
Mc - 12
M/F - 0
M/F – 2
Third Equation Written: Md – 0
Md – 0
Mc – 0
Mc – 0
M/F – 11
M/F – 1
Third Equation Solved:
Md – 0
Md – 7
Mc - 0
Mc - 4
M/F - 6
M/F – 1
Table 4: Frequency of Problem 2 Equations Written and Solved
A significant number of the high-performers wrote a force equilibrium equation first, followed
by moment equations. The most common sequence in which high-performing students wrote the
equilibrium equations was Md, Mc, and Fy. But, most high-performance subjects did not use
(solve) the third force equilibrium equation. Just as in Problem 1, they tended to follow a
standard procedure of writing three equilibrium equations and then discovered the concept that
they did not need all three equations to solve the problem.
High- and low-performers wrote two moment equilibrium equations first indicating that they
understood the concept of moments and strategy for addressing this problem. But, highperformers followed that strategy by solving the moment equations first while the lowperformers tended to attempt to solve the force equilibrium equation first. Low-performers also
did not grasp the concept of tipping and were thereby unable to solve the force equation without
making some incorrect assumption about one or more of the forces. The low-performers were
quite capable of using concepts and procedures required to set up their FBD and did not
understand the concept of tipping sufficiently well to be able to solve this problem.
Conclusion
The results of this study indicate that high- and most low-performing students understood the
concepts and procedures for setting up a problem FBD. The most common misconception about
FBDs was the setting up of the forces at a pin-connection by omitting some required forces.
Many of the low-performers did not demonstrate knowledge of the concept that vectors in two-
dimensions require two variables to describe them. Low-performers as well as a small number
of high-performers failed to recognize key problem concepts that were required to solve the two
problems. This was particularly true for the second more conceptually challenging problem
where even the high-performers had difficulty. The greatest difficulty for practically all the lowperformers and a small number of the high performers was application of concepts and
procedures to write and solve their equilibrium equations. This could be expected since this step
is dependent upon the FBD and key problem concepts. Both high- and low-performers tended to
begin the problem solution by following a procedure of writing three equilibrium equations even
when a smaller number was required. A small number (about 14% of all subjects) of the subjects
demonstrated an expert’s solution to the problems.
High-performing students in this study had a good grasp on the conceptual elements required to
solve the problems. But, they were poor strategy planners and often discovered the solution as
they wrote and attempted to solve their equilibrium equations rather than reflecting on the
problems before attempting to solve them. Statics instructors should be able to improve the
performance of the high-performers by including problem solution strategies as part of their
instruction.
Both low- and high-performing subjects demonstrated a number of “conceptual lapses” that
significantly impacted their performance as indicated in their grades. It is the experience of one
of the researchers that concepts are typically not assessed in Statics courses. But the results of
this study indicate that it is a major factor limiting student performance. The majority of the
statics textbooks do not emphasize concepts and tend to focus upon problem solutions rather than
conceptual issues. Instructors can than help both low- and high-performing students with their
performance by including conceptual questions in their assessments and emphasizing them
during their instruction.
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