Two Frameworks
for Discrete Mathematics
Kirby McMaster
Brian Rague
Steven Hadfield
Frameworks and Gestalts
We use the term gestalt to refer to a mental framework that allows
students to organize course topics into a unified whole (greater than
the sum of the parts).
 Davis and Hersh: People vary dramatically in what might be called their cognitive
style, that is, their primary mode of thinking.
 Bain: Students bring paradigms to the class that shape how they construct
meaning.... Even if they know nothing about our subjects, they still use an existing
mental model of something to build their knowledge of what we tell them.
Two Frameworks for Discrete Mathematics
2
Gestalts for Mathematics
Which gestalts are emphasized in Mathematics?
 Davis and Hersh: The definition-theorem-proof approach to mathematics has
become almost the sole paradigm of mathematical exposition and advanced
instruction.
 Courant and Robbins: There seems to be a great danger in the prevailing overemphasis on the deductive-postulational character of mathematics. The element of
constructive invention, of directing and motivating intuition, remains the core of any
mathematical achievement.
 Polya: Mathematics has two faces. Mathematics presented in the Euclidean way
appears as a systematic, deductive science; but mathematics in the making appears
as an experimental, inductive science.
Two Frameworks for Discrete Mathematics
3
Measuring Mathematical Gestalt
In previous research, we were able to characterize and measure
two frameworks for mathematics:
 Logical Math gestalt – based on proving theorems.
 Computational Math gestalt – based on solving problems.
Our methodology assumed that:
 The words people use are suggestive of their mental state.
 In particular, the words used frequently in a book indicate the gestalt of
the author.
Two Frameworks for Discrete Mathematics
4
Purpose of this Study
In this study, we apply our Logical Math (LMATH) and Computational
Math (CMATH) scales to a sample of 25 Discrete Math books.
 Our primary purpose is to assess the level of emphasis these books give to
each gestalt.
 Our findings have relevance in the development of approaches for teaching
mathematical topics in computing courses, especially Discrete Math courses.
Two Frameworks for Discrete Mathematics
5
Logical Math Gestalt
The LMATH scale consists of 10 word/groups and weights.
 As expected, theorem, proof, and definition are on the scale.
 Scale words used to convey logical order in proofs include let, since,
follow, hence, and therefore.
 Two general math concepts on the scale are set and function.
The LMATH weights for each word/group can be used to calculate
an overall weighted-average score for a specific book.
Two Frameworks for Discrete Mathematics
6
LMATH Scale
theorem/lemma/corollary
25
Avg
StdFreq
428.2
let
25
418.1
17.78
set/subset
25
333.1
13.03
proof/prove
25
329.9
12.85
function/map
25
300.9
11.23
hence/thus/therefore
25
230.3
7.28
definition/define
25
221.4
6.78
show/shown
25
191.3
5.10
follow/following
24
175.4
4.21
since
24
160.8
3.40
Word/Group
Books
TOTAL
Weight
18.34
100.00
Two Frameworks for Discrete Mathematics
7
LMATH Calculations
Royden – Real Analysis
Weight
StdFreq
theorem/lemma/corollary
18.34
214.8
LMATH
Scale
39.4
let
set/subset
17.78
13.03
*413.1
*962.3
73.4
125.4
proof/prove
12.85
188.7
24.2
function/map
11.23
*527.0
59.2
hence/thus/therefore
7.28
226.8
16.5
definition/define
6.78
230.0
15.6
show/shown
5.10
237.4
12.1
follow/following
4.21
127.7
5.4
since
3.40
187.3
6.4
Word/Group
TOTAL
377.6
Two Frameworks for Discrete Mathematics
8
Computational Math Gestalt
The CMATH scale consists of 10 word/groups and weights.
 As expected, problem and solution are on the scale.
 Scale words model and algorithm describe how problems are to be solved.
 The words variable, equation, and constraint represent components of
models.
 The word function is on both scales.
The CMATH weights for each word/group can be used to calculate
an overall weighted-average score for a specific book.
Two Frameworks for Discrete Mathematics
9
CMATH Scale
problem
25
Avg
StdFreq
394.3
method/algorithm
23
343.1
15.33
solution/solve
25
322.4
14.03
value/variable
24
271.7
10.83
equation/inequality
21
265.9
10.46
function/map
24
257.0
9.90
model/modeling
19
222.3
7.71
point/line
24
175.8
4.78
system/subsystem
23
168.7
4.33
condition/constraint
25
164.3
4.06
Word/Group
Books
TOTAL
Weight
18.56
100.00
Two Frameworks for Discrete Mathematics
10
CMATH Calculations
Hillier & Lieberman – Operations Research
Weight
StdFreq
problem
18.56
*483.8
CMATH
Scale
89.8
method/algorithm
15.33
262.7
40.3
solution/solve
14.03
*496.9
69.7
value/variable
10.83
*474.4
51.4
equation/inequality
10.46
69.3
7.2
function/map
9.90
114.4
11.3
model/modeling
7.71
268.3
20.7
point/line
4.78
--
--
system/subsystem
4.33
137.4
5.9
condition/constraint
4.06
180.7
7.3
Word/Group
TOTAL
303.7
Two Frameworks for Discrete Mathematics
11
Gestalts in Discrete Math Books
Our analysis of mathematical gestalt data from 25 Discrete Math
books seeks to answer the following questions:
1. How often do LMATH and CMATH word/groups appear in the sample
books?
2. What is the distribution of LMATH and CMATH scores?
3. What is the contribution of individual word/groups to the overall LMATH
and CMATH scores for selected high-scoring books?
4. What is the relationship between LMATH and CMATH scores?
Two Frameworks for Discrete Mathematics
12
LMATH and CMATH Word/Groups
The following tables summarize how often LMATH and CMATH
word/groups appear in the sample Discrete Math books.
 Nine of the 10 LMATH word/groups appear in the concordance of more
than 20 books.
 Two of the 10 CMATH word/groups appear in the concordance of more
than 20 books (including function/map, which is on both scales).
 Only one Discrete Math book listed model as a frequent word.
 Discrete Math books contain substantially more Logical Math words than
Computational Math words.
Two Frameworks for Discrete Mathematics
13
LMATH Word/Groups
25 Discrete Math Books
Word/Group
Books
set/subset
25
Avg
StdFreq
427.0
proof/prove
definition/define
22
17
217.0
210.6
hence/thus/therefore
21
199.2
function/map
22
195.0
follow/following
24
194.9
let
22
191.1
theorem/lemma/corollary
23
180.3
since
22
144.6
show/shown
24
132.4
Two Frameworks for Discrete Mathematics
14
CMATH Word/Groups
25 Discrete Math Books
function/map
value/variable
22
22
Avg
StdFreq
195.0
155.1
point/line
14
145.4
method/algorithm
17
145.3
solution/solve
14
139.2
problem
18
127.4
system/subsystem
10
87.2
equation/inequality
9
80.2
condition/constraint
3
60.5
model/modeling
1
52.8
Word/Group
Books
Two Frameworks for Discrete Mathematics
15
LMATH and CMATH Distributions
LMATH and CMATH scores vary widely across the 25 Discrete
Math books in the sample.
 LMATH scores range from 94.3 to 348.9.
The mean is 200.3.
 CMATH scores range from 36.8 to 175.2.
The mean is 83.8.
A graph of the LMATH and CMATH distributions is shown on the
following slide.
Two Frameworks for Discrete Mathematics
16
LMATH and CMATH Distributions
25 Discrete Math Books
14
12
Books
10
8
6
4
2
0
0-49
50-99
100-
150-
200-
250-
300-
149
199
249
299
349
350+
Gestalt Score
LMATH
CMATH
Two Frameworks for Discrete Mathematics
17
LMATH and CMATH Calculations
Eleven of the Discrete Math books have LMATH scores above 200.
The two books with highest LMATH scores are:
 Nievergelt (2002), Foundations of Logic and Mathematics (LMATH=348.9).
 Gries (1993), A Logical Approach to Discrete Math (LMATH=316.5).
No Discrete Math book has a CMATH score above 200. The book
with highest CMATH score is:
 Rosen (1999), Handbook of Discrete and Combinatorial Math
(CMATH=175.2).
The LMATH calculations for Nievergelt and Gries, along with the
CMATH calculations for Rosen, are shown in the following tables.
Two Frameworks for Discrete Mathematics
18
LMATH Calculations
Nievergelt – Foundations of Logic and Mathematics
Weight
StdFREQ
theorem/lemma/corollary
18.34
*689.8
LMATH
Scale
126.5
let
set/subset
17.78
13.03
-*586.9
-76.5
proof/prove
12.85
*536.1
68.9
function/map
11.23
146.8
16.5
hence/thus/therefore
7.28
273.7
19.9
definition/define
6.78
*353.2
23.9
show/shown
5.10
96.4
4.9
follow/following
4.21
279.3
11.8
since
3.40
--
--
Word/Group
TOTAL
348.9
Two Frameworks for Discrete Mathematics
19
LMATH Calculations
Gries – A Logical Approach to Discrete Math
Weight
StdFREQ
theorem/lemma/corollary
18.34
*422.8
LMATH
Scale
77.5
let
set/subset
17.78
13.03
103.0
*411.0
18.3
53.6
proof/prove
12.85
*716.3
92.0
function/map
11.23
192.9
21.7
hence/thus/therefore
7.28
164.8
12.0
definition/define
6.78
*342.7
23.2
show/shown
5.10
101.1
5.2
follow/following
4.21
220.0
9.3
since
3.40
108.6
3.7
Word/Group
TOTAL
316.5
Two Frameworks for Discrete Mathematics
20
CMATH Calculations
Rosen – Handbook of Discrete and Combinatorial Math
Weight
StdFREQ
problem
18.56
206.5
CMATH
Scale
38.3
method/algorithm
15.33
*341.3
52.3
solution/solve
14.03
105.4
14.8
value/variable
10.83
182.1
19.7
equation/inequality
10.46
55.3
5.8
function/map
9.90
241.3
23.9
model/modeling
7.71
52.8
4.1
point/line
4.78
249.6
11.9
system/subsystem
4.33
101.3
4.4
condition/constraint
4.06
--
--
Word/Group
TOTAL
175.2
Two Frameworks for Discrete Mathematics
21
LMATH vs. CMATH Relationship
The relationship between LMATH and CMATH scores for the
sample books is displayed on the next slide as a scatter plot.
 The relationship between the two gestalt scales is slightly negative
(correlation coefficient = -0.343).
 Books with high LMATH scores have relatively low CMATH scores.
(The converse is not true.)
 For 24 of the 25 Discrete Math books, the LMATH score is higher than
the CMATH score.
Two Frameworks for Discrete Mathematics
22
LMATH vs. CMATH Scatter Plot
25 Discrete Math Books
CMATH Score
250
200
150
100
50
0
0
50 100 150 200 250 300 350
LMATH Score
Two Frameworks for Discrete Mathematics
23
Which Framework for Discrete Math?
Our intent in this study was not to determine which framework is
"best“ for Discrete Math, since each gestalt provides value for the course.
 Both Logical Math and Computational Math encourage abstraction:
 By manipulating symbolic objects in proofs, and
 Through constructing models to solve problems.
 The Discrete Math instructor should provide an appropriate blend of
problem-solving and theorem-proving in the course.
 The difficult task will be finding a textbook that supports both gestalts,
since most Discrete Math books favor Logical Math.
 Programming assignments are one way to support Computational Math in
a Discrete Math course.
Two Frameworks for Discrete Mathematics
24