Math 221 C
Discrete Mathematics for Computer Science
Fall 1999

Information on Tests and Exams

General Guidelines and Information

Format

The format of most questions on exams will be open-ended and free-form. That is, you should not expect to find lots of multiple choice, true/false or the like. Rather, you should expect to see questions that require a written response, often requiring some sort of support for your claims.

Short answers. Many types of questions fall into this category, and the questions you have seen on reading assignments and quizzes are good examples. You may be asked to make a list, to give a definition, to provide examples, to compare or contrast two concepts, readings or positions, to explain, etc. The most important thing to remember on short answer questions (and other questions too, but especially on short answer questions) is that I am not looking merely for "correct" answers, I am looking for quality answers. Your answers will therefore be graded on such criteria as truth, accuracy, importance, conciseness, coherence, and supporting evidence. In particular more is not always better. Don't tell me everything you know about the topic in question; show me that you can distill the most important issues or most essential features into a precise, coherent response. If asked to give an example of something, try to give the best possible example; one very good example is usually better than several poorer examples.

Problems and Exercises. You will be required to do some mathematics on each test. Hopefully, some of the tasks on each exam will be exercises for you and some will be problems, but exactly which are which may vary somewhat from student to student. In particular, you should be prepared to do tasks that are similar to any exercises or problems asigned, worked on in groups, presented in readings or discussed in class. If you have understood these tasks from class, this type of task should be an exercise for you on the exam, even if it was a problem when first encountered in class.

It is also possible that you may be given a task that is a genuine problem for you. Of course, if you have forgotten how to do something, a task of the type described above may also be a problem for you. But I may also design some tasks with the intent that they be problems rather than exercises. These will usually require only background knowledge and problem solving skills that have been used in other tasks in the course, but may require you to do some experimenting and use problem solving heuristics before you will be able to come to a solution.

Solutions to either type of mathematical task should be presented clearly using words as well as symbols, explaining your reasoning, etc. This is especially important if your solution is "incomplete". Significant progress, good ideas, reasonable attempts, realization of errors, etc. as well as quality of presentation will all be considered when evaluating your work.

Essays. More rarely I may give you an essay question. Essays are just longer version of short answers. Typically the difference will be in the scope of the question. An essay may cover more or the issues may be more involved or just more difficult to present, so that a longer answer is required. Nevertheless, the same criteria as for short answers apply, with an additional criterion: organization. Essays should be written using good writing skills so that the organization and flow of your answer are readily apparent.

Content: Test 1

Readings: Important Topics -- Big Ideas. Obviously a one-hour exam cannot cover everything we have read in this class so far. In fact, it cannot even cover every important topic. Thus the exam will include questions covering a sample of (some of) the most important topics. Part of your job in preparing for the exam is to identify the major themes and learn the big ideas. One good way to this is to go through the questions on the readings and mark the questions from each chapter that you think are most important. For each chapter or reading assigned, ask yourself questions like: What are the big ideas here? What are the key examples or supporting evidence? Do I understand the key terms used to describe the big ideas? (Could I give a concise definition or a good example for each?) Class discussions can give some indication as to what kinds of topics I think are important, but do not neglect a topic simply because it was not discussed at length in class.

In particular, for the first exam, most of the material you have read falls into the category of foundational work for the rest of the semester. Chapters 3 and 4 of Van de Walle are especially important in this regard, and you should be sure you understand the ideas presented there (and our discussions in class pertaining to those ideas) particularly well.

Mathematics Content. Remember, one of the goals for this course is that you have the mathematical competance to teach K-8 mathematics. Thus part of the exam will be devoted to making sure that you all have the necessary knowledge, skills and background for this. Note that this includes both procedural knowledge and conceptual knowledge and the ability to relate the two. For each exam you should make a list of the content areas covered. (For example, for the first test, the specific content areas from the K-8 curriculum that we have covered include attributes, patterns, sets and set operations. The list will be larger for future exams.) This may be tested using exercises, problems or short answer questions.

Activities. You should be familiar enough with the activities we have done in class (like the A-block activity) that a very brief description recalls the activity to your mind. You may be asked to do portions of an activity, to answer questions about the methods or goals of a particular activity, or to use activities as examples in answers to other types of qeustoins.

Teaching Methods. This will become more important later on, as we have more examples from elementary school curriculum, but you should prepared to answer questions about how to teach mathematics, advantages and disadvantages of various methods, typical errors students make and how to help them avoid or unlearn them, etc.

Synthesis. Avoid the temptation to overly compartmentalize your learning. We can only discuss so much at a time in class, and we must read things in some order. But once we have covered a number of things, don't forget to watch for connections. For example, how does what Van de Walle says about problem solving fit in with the three articles (R1, R2, and R3) that you read, with your experience solving problems in class, with the video Double-column addition, with the text book you looked at for your curriculum project?

Later in the semester we will repeatedly connect the material from this early part of the course with specific content areas throughout the curriculum.


Content: Test 2

All tests in this course are cummulative, but they will generally emphasize the most recent material. The general guidelines and some of the general comments made under the the heading "Content: Test 1" still apply. But here is an outline of important topics that are new since the first test. Use them to help you prepare for the types of questions mentioned above.

Counting:
components of (phases in learning)
Number Concepts:
4 key relationships (and why important)
Number Systems:
types (tally, additive, multiplicative, place value), Roman numbers and modifications
Place Value:
types of models, different bases, place value language
Meaning of Addition/Subtraction:
categories of problems, use of models, translations among representations, number sentences (computational form), important properties
Meaning of Multiplication/Division:
categories of problems, models, number sentences (computational form), interpretations of multiplication (repeated addition, equal groups), interpretations of division (missing factor, repeated subtraction, fair-share), remainders, important properties
Basic Facts:
identification, importance, VdW's fact mastery program, thinking strategies (you will not be required to name them, but you should be able to describe the thought process you are using to figure out a basic fact)


Content: Test 3

All tests in this course are cummulative, but they will generally emphasize the most recent material. The general guidelines and some of the general comments made under the the heading "Content: Test 1" still apply. But here is an outline of important topics that are new since the second test. Use them to help you prepare for the types of questions mentioned above.

Algorithms:
characteristics (how to compare algorithms), prerequisites, developmental teaching, models, other bases, be able to explain how and why algorithms work
you should be familiar with
Mental Computation and Estimation:
Be able to use and explain any of the strategies we have seen so far. Also be aware of issues related to selection of mental math, paper-and-pencil, calculators, etc.
Role of Calculators in the Classroom
You should be able to give arguments in favor of and against the use of calculators in schools. You could also be asked to give your own opinion.

Content: Test 4

All tests in this course are cummulative, but they will generally emphasize the most recent material. The general guidelines and some of the general comments made under the the heading "Content: Test 1" still apply. But here is an outline of important topics that are new since the third test. Use them to help you prepare for the types of questions mentioned above.

Factors and Multiples:
definitions of factor, divisor, multiple, prime, composite, prime factorization
GCD (greatest common divisor), LCM (least common multiple)
divisibility tests and why they work
how to determine if a number is prime
how to get prime factorization and use it to determine greatest common divisor, least common multiple, number of factors
realationship between GCD and LCM of two numbers
locker problem and solution
Fractions
meaning of fractions
models for fractions (3 types with examples of each)
conceptually based fraction comparison
Fraction operations
meaning/models of operations with fractions
algorithms for addition, subtraction, multiplication, division and why they work
common denominators
Mental Computation and Estimation:
Be able to use and explain any of the strategies we have seen so far. Also be aware of issues related to selection of mental math, paper-and-pencil, calculators, etc.
Role of Calculators in the Classroom
You should be able to give arguments in favor of and against the use of calculators in schools. You could also be asked to give your own opinion. (from VdW and Dialogues handout)

Content: Test 5

All tests in this course are cummulative, but they will generally emphasize the most recent material. The general guidelines and some of the general comments made under the the heading "Content: Test 1" still apply. But here is an outline of important topics that are new since the fourth test. Use them to help you prepare for the types of questions mentioned above.

Fractions
Most of the introductory material on fractions was covered on the last test, but since it is intimately conected to decimals, percents and ratios, and since the division algorithm was not covered last time, you should be sure to review fractions for this test as well.
Conceptual teaching of new material
the big triangle (models, symbols, words); big star (models, symbols, words, real world, pictures)
connections (to other mathematics topics, to "real world", to other disciplines, etc)
attention to procedural/conceptual knowledge involved
Decimals and percents
meaning, connections between fractions, decimals and percents
models and place value
algorithms for basic operations (how and why)
word problems, especially percentage change problems
converting between fractions, decimals and percents, including repeating decimals
Ratio and Proportion
difference between ratio and fraction
word problems
three solution strategies (unit rate, scaling, cross-multiply) and why they work
Integers
models, situations
operations

Final Exam

The final exam will cover material since the beginning of the course. To prepare, look over your old tests and quizzes and the review materials for each test. As you do so, in addition to making sure you can do all the problems listed there, ask yourself questions like:

Note: There has also been some new material since fifth test:

Number Systems: Counting Numbers, Whole Numbers, Integers, Rationals, Reals
closure properties
density
how to represent rationals (as "fractions", certain kinds of decimals)
how to represent reals
examples of reals that are not rationals (and why they are not rational)
Pythagorean Theorem (careful statement, uses)
representing irrationals on a geoboard
Graphs
situations -> graphs, graphs -> situations
role of axes (independent and dependent variables)
continuous vs. discrete
Algebra
Using variables and equations to represent number relationships
Solving algebraic equations for unknown quantities (2 rules)
Models for algebra

Last Modified: Thursday, 11-Jan-2001 16:03:25 EST
Course Home Page: http://www.calvin.edu/~rpruim/courses/m221/F99/
Maintained by: Randall Pruim