ROBIN THOMAS INTRODUCTION TO SHORT WRITING ASSIGNMENT
The short assignment is for Psychology 373 Introduction to Mathematical Psychology, a course that I am reorganizing with the hope of re-titling it ‘Computational and Modeling Tools for the Psychological Sciences.’ Its focus, as I have developed it, is to allow students to acquire the analytical and computer skills to successfully apply mathematical models to psychological phenomena. Even in the case in which students have had significant mathematics experience, they have rarely seen these ideas altogether or applied to the discipline of psychology. One of the early topics in the course is how the psychological construct of similarity is formally modeled. Students have an intuitive idea of what the word ‘similarity’ means when it is used in everyday language but are often surprised to discover the hidden complexity associated with the concept when we discuss formal models of it. This topic is very useful to get them thinking about how mathematical entities can be used to represent a psychological concept. From the writing workshop sponsored by CWE, I obtained the idea of using short assignments to set up a problem or to get students to ponder a problem prior to discussing it in class. The purpose of this short assignment is to lay the groundwork for developing the connections between model properties and observable phenomena. In the past, I have simply lectured to the students what the consequences of assuming a distance-based representation of similarity are for observed data. In contrast, this assignment, which is primarily a brainstorming exercise, requires them to generate insights, correct or not, so that they have some context with which to receive a more formal presentation of the deeper constructs later. Robin D Thomas ROBIN THOMAS SHORT WRITING ASSIGNMENT FOR STUDENTS Psy 373 Think about Similarity Brief assignment (10 pts) Relation to Course Objectives: 1) to understand how mathematical entities (e.g., formulas, symbols, operations) can be used to represent psychological constructs 2) to connect formal properties of a mathematical model to their empirical implication; specifically, what does this property of the model mean for observable behavior? 3) to appreciate the (often hidden) complexity and assumptions underlying psychological constructs Specific Learning Objective of Assignment This is a “brainstorming” assignment to get you to begin thinking about how one works from specific mathematical statement to their meaning as a model for similarity between objects Process In class, we introduced the geometric model of similarity and defined a familiar distance function, the Euclidean metric. I mentioned that other functions can also serve as legitimate distance functions. One such is the city-block metric. For simplicity, assume the objects are two dimensional (x,y) with object p = (x1 , y1) and object q = (x2, y2). Their city-block distance is d(p,q) = | x1 - x2 | + | y1 - y2 | where | ? | denotes the absolute value. 1. What does this equation say in words? Draw a picture illustrating the computation of this distance to help you answer this question. What is a more intuitive description of this distance? (hint: another name for it is “taxi-cab” metric, why do you think is it called city-block or taxi-cab?) (3pts) 2. This brings us to consider what we mean by a “distance function.” Without consulting any formal definitions in your readings, write down all of the properties that you believe are specific to a distance function; that is, what properties should a mathematical function between two points have in order to be called a distance function? (As a hint, in one dimension, p = x1 and q = x2 , the function d(p,q) = | x1 - x2 | is a legitimate distance function but sin( x1 - x2 ) is not. In what ways are these two functions different from each other? Try graphing these in Excel or Matlab and let p be fixed and q vary). (4pts) 3. For the properties you think of as being part of what constitutes a distance function, what are the implications for similarity ratings? Specifically, given a particular property of distance, what should we observe in a person’s evaluation of the similarity between objects? (3pts) For the above, I expect roughly 2-3 pages of text. This is due ________________ Evaluation The primary purpose is to get you engaged in thinking about mathematical constructs so I will be looking more toward effort rather than absolute correctness (we will focus on that later). Hence, the following criteria will be used to evaluate your work: 1) comprehensiveness of interpretation of symbols and operations - do you address all of the elements in the equation? How many properties did you identify in 2? 2) clarity of writing - even if your answer is not technically correct, are you clear in your attempt? Is the writing good in terms of grammar, syntax, spelling and connectedness? |