Two Frameworks for Discrete Mathematics Kirby McMaster Brian Rague Steven Hadfield.

Two Frameworksfor Discrete Mathematics

Kirby McMaster

Brian Rague

Steven Hadfield

Two Frameworks for Discrete Mathematics 2

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.

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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 over emphasis 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.

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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.

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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.

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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.

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Word/Group Books AvgStdFreq

Weight

theorem/lemma/corollary 25 428.2 18.34

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

TOTAL 100.00

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LMATH Scale


Word/Group Weight StdFreqLMATHScale

theorem/lemma/corollary 18.34 214.8 39.4

let 17.78 *413.1 73.4

set/subset 13.03 *962.3 125.4

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

TOTAL 377.6

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LMATH CalculationsRoyden – Real Analysis


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.

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Word/Group Books AvgStdFreq

Weight

problem 25 394.3 18.56

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

TOTAL 100.00

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CMATH Scale


Word/Group Weight StdFreqCMATHScale

problem 18.56 *483.8 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

TOTAL 303.7

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CMATH CalculationsHillier & Lieberman – Operations Research


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?

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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.

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Word/Group Books Avg

StdFreqset/subset 25 427.0

proof/prove 22 217.0

definition/define 17 210.6

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

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LMATH Word/Groups25 Discrete Math Books


Word/Group Books Avg

StdFreqfunction/map 22 195.0value/variable 22 155.1

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

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CMATH Word/Groups25 Discrete Math Books


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.

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LMATH and CMATH Distributions25 Discrete Math Books

17

0

2

4

6

8

10

12

14

0-49 50-99 100-

149

150-

199

200-

249

250-

299

300-

349

350+

Gestalt Score

Bo

oks

LMATH CMATH


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.

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Word/Group Weight StdFREQLMATHScale

theorem/lemma/corollary 18.34 *689.8 126.5

let 17.78 -- --

set/subset 13.03 *586.9 76.5

proof/prove 12.85 *536.1 68.9

function/map 11.23 146.8 16.5


definition/define 6.78 *353.2 23.9

show/shown 5.10 96.4 4.9


since 3.40 -- --

TOTAL 348.9

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LMATH CalculationsNievergelt – Foundations of Logic and Mathematics


Word/Group Weight StdFREQLMATHScale

theorem/lemma/corollary 18.34 *422.8 77.5

let 17.78 103.0 18.3

set/subset 13.03 *411.0 53.6

proof/prove 12.85 *716.3 92.0

function/map 11.23 192.9 21.7


definition/define 6.78 *342.7 23.2

show/shown 5.10 101.1 5.2


since 3.40 108.6 3.7

TOTAL 316.5

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LMATH CalculationsGries – A Logical Approach to Discrete Math


Word/Group Weight StdFREQCMATHScale

problem 18.56 206.5 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 -- --

TOTAL 175.2

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CMATH CalculationsRosen – Handbook of Discrete and Combinatorial Math


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.

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LMATH vs. CMATH Scatter Plot25 Discrete Math Books

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50

100

150

200

250

0 50 100 150 200 250 300 350

LMATH Score

CM

AT

H S

core


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.

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Two Frameworks for Discrete Mathematics Kirby McMaster Brian Rague Steven Hadfield.

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