Conceptual and procedural knowledge in undergraduate mathematics

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Conceptual and procedural knowledge in
undergraduate mathematics

• Johann Engelbrecht
• University of Pretoria

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General concerns and opinions

• Our students ask for food we give them
  medicine (G-C Rota)
• high school teaching of mathematics tends to be
  fairly procedural.
• Although most students arriving at university
  have well developed manipulation skills in
  mathematics and can do procedural chemistry, few
  of these students have really been exposed to
  deeper conceptual thinking.
• Training of engineering students?

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Working definitions
Procedural (mechanical) knowledge in
mathematics the ability to physically solve a
problem through the manipulation of mathematical
skills, with the tools used in mathematics such
as pencil and paper, calculator, computer, etc.
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Working definitions
Conceptual knowledge in mathematics the
ability to show understanding of mathematical
concepts by being able to interpret and apply
them correctly to a variety of situations as well
as the ability to translate these concepts
between verbal statements and their equivalent
mathematical expressions a connected network in
which linking relationships is as prominent as
the separate bits of information.
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The experiment

• First year students in life sciences
• 93 students
• 10 item instrument 5 procedural, 5
  conceptual

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Panel of colleagues Nature of item
Procedural kp Conceptual kc ( 1 kc)
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Procedural knowledge

• Example 1
• Solve for x
• A. x 1.85
• x 2.24
• x 0.45
• x 3.12
• None of the above

Panel kp 0.91, kc 0.09
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Procedural knowledge

• Example 2
• For the function f(x) x³, what is the equation
  of the
• tangent line at x 1?
•     
• A. y x 2
• B. y 3x 2
• C. y x
• y 3x 2
• None of the above

Panel kp 0.89, kc 0.11
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Conceptual knowledge
Example 3 The slope of the line in the sketch
is approximately 3000. The function P f(t)
gives the population after t years. Choose
from statements A D which of the statements
is NOT true or choose E.

• the average rate of change of population over the
  
• interval 4 lt t lt 12 is 3000 people per year.
• B. the rate of change of the population at t 4
  is more than 3000 people per year.
• C. the rate of change of the population at t 10
  is more than 3000 people per year.
• D. the function is increasing.
• E. they are all true.

Panel kp 0.18, kc 0.82
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Conceptual knowledge
Example 4 Which of the following graphs f(x)
satisfies both the given conditions (i) f '(x)
gt 0 on (- 8, 0) and f '(x) lt 0 elsewhere
(ii) f ''(x) gt 0 on (- 8, -1) and f ''(x) lt
0 elsewhere


Panel kp 0.18, kc 0.82
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Conceptual / Procedural?

• Example 5
• For a function f(x), the graph of the function
  g(x) f(x-1) - 1 will be
• higher and to the right of the graph of f(x)
• lower and to the right of the graph of f(x)
• higher and to the left of the graph of f(x)
• lower and to the left of the graph of f(x)
• none of these possibilities.

Panel kp 0. 75, kc 0.25
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Conceptual / Procedural?

• Example 6
• Money is deposited into an account paying
  interest compounded continuously. What is the
  annual interest rate if the amount doubles every
  5 years?
• 20.0
• 13.86
• 12.34
• 14.68
• None of the above

Panel kp 0. 58, kc 0.42
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Conceptual / Procedural?

• Example 7
• If for some function f(x) it is given that
  f(20) 12 and f '(20) - 2, then the best
  approximation for f(19.5) is
• 10
• 14
• 11
• 13
• None of the above.

Panel kp 0. 60, kc 0.40
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Confidence of Response
Degree of certainty the student has in his/her
own ability to select and utilise
well-established knowledge, concepts and laws to
arrive at an answer.

• On completion of each of the items
• completely certain
• fairly certain
• not certain
• total guess

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Confidence of Response
Hasan et al
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Conceptual (CPI)
Procedural (PPI)



with ? 0 or 1
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Procedural (PCI)
Conceptual (CCI)
 



CCI
with ? 1 or 2 or 3
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Comparisons

• conceptual vs procedural performance
• conceptual vs procedural confidence
• conceptual confidence vs performance
• procedural confidence vs performance

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Analysis of results
Conceptual
Procedural
 
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Results
Comparison 1 Performance Conceptual vs
procedural
93
0.34
36
18
15
31
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Results
Comparison 2 Confidence Conceptual vs procedural
93
0.70
38
10
9
43
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Results
Comparison 3 Procedural confidence vs
performance
93
. 42
32
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students that have high confidence, low
performance
14
35
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Results
Comparison 4 Conceptual confidence vs
performance
93
.34
33
17
16
34
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Conclusions

• Students in this course do not perform
  better in procedural
• problems than in conceptual problems
• Students are not more confident about their
  ability to answer
• procedural problems than conceptual
  problems
• Students do not have more misconceptions
  about conceptual
• issues than about procedural mathematics

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So?