What is Gestalt? Wiki: Gestalt refers to a structure, configuration, or pattern of physical, biological, or psychological phenomena so integrated as to

What is Gestalt?

Wiki: Gestalt refers to a structure, configuration, or pattern of physical, biological, or psychological phenomena so integrated as to constitute a functional unit with properties not derivable by summation of its parts.

Gestalt Psychology describes how people tend to organize visual elements into groups or unified wholes.

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 Gestalts for Mathematics 2

Without GestaltIndividual Parts Only


With GestaltMore than the Sum of the Parts


Which Gestalt for Mathematics?

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.

Henrici: Dialectic mathematics is a rigorously logical science, where statements are true or false…. Algorithmic mathematics is a tool for solving problems.

Body and Soul (Sweden): a reform program that combines the constructive /computational (body) aspects with the symbolic (soul) aspects of mathematics.


Purpose of this Study

The primary purpose of this study is to develop measuring scales for two mathematical gestalts: Logical Math gestalt – based on proving theorems. Computational Math gestalt – based on solving problems.

Our methodology assumes that the words people use are suggestive of their mental state. In particular, the words used frequently in a textbook indicate the gestalt of the author.

The gestalt scales can be used to evaluate textbooks and provide frameworks for teaching math topics in IS courses.


Methodology

The development of measuring instruments for our two gestaltsinvolved the following steps:

1.Sampling: Select a diverse sample of Traditional Math and Applied Math books having an Amazon concordance.

2.Measurement: For each concordance word, record the book code, word, and frequency (FREQ).

3.Conversion: Change nouns, verbs, adjectives, and adverbs to a consistent form (e.g. singular nouns, present tense verbs).

4.Transformation: Rescale word frequencies (StdFREQ) within a book so that the average concordance word receives a score of 100.

5.Grouping: Combine relevant synonyms into word groups, summing the StdFREQ scores for each group.


Methodology

6.Scale Construction: Build the Logical Math (LMATH) and Computational Math (CMATH) scales using an iterative process:

Look for words that are used frequently within each book and consistently across similar books.

Build a tentative scale, and calculate scores for every book. Remove books with low scale scores, and repeat the process. Stop when the scales and list of remaining books stabilize.

Starting with 56 Traditional Math and 56 Applied Math books,we obtained (after 3 iterations) LMATH and CMATH scales constructed from 25 Traditional Math and 25 Applied Mathbooks, respectively.


Logical Math Gestalt

The LMATH scale consists of 10 words/groups and weights. As expected, theorem, proof, and definition are on the

scale. Scale words used to convey logical order in proofs include

let, hence/thus/therefore, follow, and since. Two general math concepts on the scale are set and

function.

Each scale word appears in at least 24 of the 25 Traditional Math

books used to construct the LMATH scale.


Word/Group BooksAvg

StdFREQWeight

theorem/lemma/proposition 25 439.5 19.12

let 25 416.2 17.81

proof/prove 25 341.2 13.58

function/map 25 315.1 12.11

set 25 281.3 10.21

hence/thus/therefore 25 235.9 7.65

definition/define 25 212.4 6.33

show 24 194.5 5.32

follow 24 174.2 4.18

since 24 165.5 3.69

TOTAL 100.00




When calculated for a specific book, the LMATH scale provides weights and frequencies for each scale word, in addition to an overall weighted-average score for the book.

Block’s Proofs and Fundamentals: A First Course in Abstract Mathematics received the highest LMATH score (394.2).

The most frequent words in this book are: let (StdFreq = 557.1) set (StdFREQ = 532.1) proof/prove (StdFREQ = 476.1).


Word/Group Weight StdFREQLMATHScale

theorem/lemma/proposition 19.12 355.2 67.9

let 17.81 *557.1 99.2

proof/prove 13.58 *476.1 64.7

function/map 12.11 358.8 43.5

set 10.21 *532.1 54.3

hence/thus/therefore 7.65 231.9 17.7

definition/define 6.33 298.8 18.9

show 5.32 219.6 11.7

follow 4.18 254.7 10.6

since 3.69 153.7 5.7

TOTAL 394.2


LMATH Scale ValuesBloch -- Proofs and Fundamentals

Computational Math Gestalt

The CMATH scale consists of 9 words/groups and weights. As expected, problem and solution are on the scale. Scale words model and method/algorithm describe how

problems are to be solved. The words variable, equation, and condition/constraint

represent components of models. Note that function is on both scales.

Each scale word appears in at least 20 of the 25 Applied Math

books used to construct the CMATH scale.


Word/Group BooksAvg

StdFREQWeight

problem 25 389.1 19.28

method/algorithm 24 346.0 16.40

solution/solve 25 314.3 14.29

value/variable 24 267.1 11.14

equation 20 265.2 11.02

function/map 24 263.4 10.90

model 20 223.5 8.24

system 23 167.2 4.48

condition/constraint 25 163.8 4.25

TOTAL 100.00




Pardalos’ Handbook of Applied Optimization received the highest CMATH score (390.0).

The most frequent words in this book are: method/algorithm (StdFreq = 633.3) problem (StdFREQ = 631.0) solution/solve (StdFREQ = 475.4).


Word/Group Weight StdFREQCMATHScale

problem 19.28 *631.0 121.7

method/algorithm 16.40 *633.3 103.9

solution/solve 14.29 *475.4 67.9

value/variable 11.14 221.0 32.4

equation 11.02 67.4 7.4

function/map 10.90 291.0 24.1

model 8.24 200.2 16.5

system 4.48 151.8 6.8

condition/constraint 4.25 218.5 9.3

TOTAL 390.0


CMATH Scale ValuesPardalos -- Handbook of Applied Optimization


In comparison, Polya’s classic How to Solve It has a CMATH score of 265.7.

The most frequent words in this book are: problem (StdFREQ = 1005.3) solution/solve (StdFREQ = 402.0).

However, the concordance for this book includes neither model nor method/algorithm. Why?


Word/Group Weight StdFREQCMATHScale

problem 19.28 *1005.3 193.8

method/algorithm 16.40 -- --

solution/solve 14.29 *402.0 57.4

value/variable 11.14 -- --

equation 11.02 57.9 6.4

function/map 10.90 -- --

model 8.24 -- --

system 4.48 -- --

condition/constraint 4.25 189.8 8.1

TOTAL 265.7


CMATH Scale ValuesPolya -- How to Solve It

LMATH vs. CMATH Scales

LMATH and CMATH scores varied widely both between and within the Traditional Math and Applied Math book groups. Traditional Math LMATH scores ranged from 95 to 394. Applied Math CMATH scores ranged from 59 to 390. No book scored higher than 200 on both scales.

The relationship between LMATH and CMATH scores for all books in our sample is displayed on the next screen as a scatter diagram.

The relationship between the two gestalt scales is negative. Books with high LMATH scores have low CMATH scores, and vice versa.

In most of our sample books, one type of Math gestalt predominates.


LMATH vs. CMATH Scales


Math Areas and Gestalts

We used the LMATH and CMATH scales to measure the preferredgestalt in different areas of mathematics. We classified each of our sample books into a Traditional Math

area (e.g. Algebra) or an Applied Math area (e.g. Numerical Analysis)

For each Math area, we calculated the average LMATH score and average CMATH score.

Math areas with an average LMATH score > 250 are Logic, Topology, Analysis, Number Theory, and Probability.

Math areas with an average CMATH score > 250 are Operations Research, Optimization, and Numerical Analysis.

Algebra and Differential Equations have relatively low averagesbecause these areas contain both theoretical and applied books.



Mathematical AreaAvg

LMATHAvg

CMATHBooks

Mathematical Logic *323.6 52.0 6

Topology *311.1 51.0 4

Analysis *302.6 81.5 8

Number Theory *279.7 89.0 6

Probability *264.6 91.6 8

Calculus 227.4 123.6 4

Algebra 204.1 90.4 7

Geometry 146.4 51.2 2

Traditional Math Areas LMATH and CMATH Scores


Mathematical AreaAvgLMATH

AvgCMATH

Books

Operations Research 38.8 *292.9 5

Optimization 152.3 *274.8 4

Numerical Analysis 91.7 *256.3 3

Applied Math 115.7 227.7 9

Differential Equations 169.9 227.2 6

Math Modeling 47.3 226.2 6

Computational Math 114.6 212.3 11

Simulation 44.9 146.1 6

Statistics 102.6 120.7 9

Graphics 48.2 117.7 2

Applied Math Areas LMATH and CMATH Scores

Which Math Gestalt for IS?

The Realistic Math Education program in Holland, based on the workof Treffers, encourages two types of "mathematization": Horizontal mathematization, where students solve a problem

located in a real-life situation [Computational Math]. Vertical mathematization, where students reorganize concepts

within the mathematical system itself [Logical Math].

Both forms of mathematical activity are of value in IS. Each developsabstraction skills—by constructing models to solve problems and bymanipulating symbolic objects in proofs.

Ultimately, it is the instructor who must choose an appropriateblend of problem-solving gestalt and theorem-proving gestalt inIS courses.



Real World

Problems

Requirements

Data

Math World

Models

Algorithms

Computer World

Solutions

Software

Databases

Computational Math GestaltViewed Horizontally in Three Worlds


Real World

Irrelevant

Math World

Definitions

Theorems

Proofs

Computer World

Unnecessary

Logical Math GestaltViewed Vertically in One World

Current and Future Research

1. We have applied the LMATH and CMATH scales to a sample of Discrete Math textbooks to determine which type of gestalt is emphasized in these books.

2. We are currently defining scales for measuring gestalts in Software Development. Three scales are being developed—Programming, Database, and Software Engineering.

3. We plan to relate software development gestalts to the mathematical gestalts described in this paper. Our goal is to find ways to combine mathematics with software development in computing courses.


Documents

What is Gestalt? Wiki: Gestalt refers to a structure, configuration, or pattern of physical, biological, or psychological phenomena so integrated as to