2. 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 Mathematics2

3. Without Gestalt Individual Parts Only Two Gestalts for Mathematics 3

4. With Gestalt More than the Sum of the Parts Two Gestalts for Mathematics 4

5. 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. Two Gestalts for Mathematics 5

6. 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. Two Gestalts for Mathematics6

7. Methodology The development of measuring instruments for our two gestalts involved 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. Two Gestalts for Mathematics7

8. 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 Math books, respectively. Two Gestalts for Mathematics8

9. 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. Two Gestalts for Mathematics9

10. Word/GroupBooks Avg StdFREQ Weight theorem/lemma/proposition25439.519.12 let25416.217.81 proof/prove25341.213.58 function/map25315.112.11 set25281.310.21 hence/thus/therefore25235.97.65 definition/define25212.46.33 show24194.55.32 follow24174.24.18 since24165.53.69 TOTAL100.00 Two Gestalts for Mathematics10 Logical Math Gestalt

11. 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). Two Gestalts for Mathematics11

12. Word/GroupWeightStdFREQ LMATH Scale theorem/lemma/proposition19.12355.267.9 let17.81*557.199.2 proof/prove13.58*476.164.7 function/map12.11358.843.5 set10.21*532.154.3 hence/thus/therefore7.65231.917.7 definition/define6.33298.818.9 show5.32219.611.7 follow4.18254.710.6 since3.69153.75.7 TOTAL 394.2 Two Gestalts for Mathematics12 LMATH Scale Values Bloch -- Proofs and Fundamentals

13. 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. Two Gestalts for Mathematics13

14. Word/GroupBooks Avg StdFREQ Weight problem25389.119.28 method/algorithm24346.016.40 solution/solve25314.314.29 value/variable24267.111.14 equation20265.211.02 function/map24263.410.90 model20223.58.24 system23167.24.48 condition/constraint25163.84.25 TOTAL100.00 Two Gestalts for Mathematics14 Computational Math Gestalt

15. 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). Two Gestalts for Mathematics15

16. Word/GroupWeightStdFREQ CMATH Scale problem19.28*631.0121.7 method/algorithm16.40*633.3103.9 solution/solve14.29*475.467.9 value/variable11.14221.032.4 equation11.0267.47.4 function/map10.90291.024.1 model8.24200.216.5 system4.48151.86.8 condition/constraint4.25218.59.3 TOTAL 390.0 Two Gestalts for Mathematics16 CMATH Scale Values Pardalos -- Handbook of Applied Optimization

17. Computational Math Gestalt 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? Two Gestalts for Mathematics17

18. Word/GroupWeightStdFREQ CMATH Scale problem19.28*1005.3193.8 method/algorithm16.40-- solution/solve14.29*402.057.4 value/variable11.14-- equation11.0257.96.4 function/map10.90-- model8.24-- system4.48-- condition/constraint4.25189.88.1 TOTAL 265.7 Two Gestalts for Mathematics18 CMATH Scale Values Polya -- How to Solve It

19. 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. Two Gestalts for Mathematics19

20. LMATH vs. CMATH Scales Two Gestalts for Mathematics20

21. Math Areas and Gestalts We used the LMATH and CMATH scales to measure the preferred gestalt 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 averages because these areas contain both theoretical and applied books. Two Gestalts for Mathematics21

22. Two Gestalts for Mathematics22 Mathematical Area Avg LMATH Avg CMATH Books Mathematical Logic *323.652.06 Topology *311.151.04 Analysis *302.681.58 Number Theory *279.789.06 Probability *264.691.68 Calculus 227.4123.64 Algebra 204.190.47 Geometry 146.451.22 Traditional Math Areas LMATH and CMATH Scores

23. Two Gestalts for Mathematics23 Mathematical Area Avg LMATH Avg CMATH Books Operations Research 38.8*292.95 Optimization 152.3*274.84 Numerical Analysis 91.7*256.33 Applied Math 115.7227.79 Differential Equations 169.9227.26 Math Modeling 47.3226.26 Computational Math 114.6212.311 Simulation 44.9146.16 Statistics 102.6120.79 Graphics 48.2117.72 Applied Math Areas LMATH and CMATH Scores

24. Which Math Gestalt for IS? The Realistic Math Education program in Holland, based on the work of 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 develops abstraction skills—by constructing models to solve problems and by manipulating symbolic objects in proofs. Ultimately, it is the instructor who must choose an appropriate blend of problem-solving gestalt and theorem-proving gestalt in IS courses. Two Gestalts for Mathematics24

25. Two Gestalts for Mathematics25 Real World Problems Requirements Data Math World Models Algorithms Computer World Solutions Software Databases Computational Math Gestalt Viewed Horizontally in Three Worlds

26. Two Gestalts for Mathematics26 Real World Irrelevant Math World Definitions Theorems Proofs Computer World Unnecessary Logical Math Gestalt Viewed Vertically in One World

27. 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. Two Gestalts for Mathematics27