Intro to Mathematical Reasoning (Math 300)
Information for instructors


Last revised: January 20, 2005.

Note: This page is intended for instructors of math 300, not for students. While there is no great harm in students seeing this page, it is better not to share this URL with students.

Contents of this page

  • Course purpose
  • The students
  • Course coordination
  • Exams
  • Grading standards
  • Syllabus (latex) (pdf)
    (Note: this syllabus is written for instructors and not students; instructors can prepare their own modified version to distributed to students.)
  • Homework
  • Warnings about the book
  • Resources

    Course purpose

    The purpose of Math 300 is to develop the thinking and writing skills of students (primarily mathematics majors) in order to prepare them for upper level mathematics courses, especially 311, 350 and 351. Most importantly this means that students be able to read and understand proofs and to write correct and readable proofs.

    The mathematical content (basic set theory, relations, functions, and the axioms of the real numbers) is very elementary but essential for future courses. Return to table of contents

    The students

    This course is intended for math majors, but, for some reason, it is a surprisingly popular elective with students from a number of departments taking it. The abilities of the students is also highly variable (and not particularly correlated with whether the student is a math major or not).

    During the academic year 2003-2004, of the 270 students initially registered for the course in all sections, 48 withdrew and 39 failed, which means that only 2/3 of the registered students succeeded in even the most generous sense. (On the positive side, there were also 39 A's). It is important to impress upon students early that the course is difficult and may require extensive effort on their part. Return to table of contents

    Course coordination

    It is unusual to have a 300 level course with a course coordinator. During the past few semesters I have served as a ``loose'' coordinator for the course.

    My role as coordinator is to ensure a ``base'' amount of uniformity between sections, to encourage interaction between instructors, and to be a resource to instructors concerning the teaching of the course. The uniformity expected of all instructors is:

  • A minimal common set of material to be covered (see the syllabus ).
  • A roughly uniform set of expectations for students and standards for grading.

    I will provide some supplementary handouts , that fill some gaps in the text. I think it's worth spending class time on them, but you may choose to just let students read them on their own.

    I hope that instructors in the course will talk to me and each other about the many pedagogical issues that arise in teaching this course. Return to table of contents

    Exams

    Standard practice in the course is to give two midterm exams and a final exam. You are not compelled to follow this, but please inform me if you don't.

    We will not have common exams. A file of exams from previous semesters will be made available. Also, you are asked to add your own exams to this file. Information about the location of this file will be sent by email.

    Grading standards

    It is important that all instructors in this course have roughly the same standards in evaluating students. The guiding principle in assigning grades is that grades should reflect the instructors expectation of the students potential for success in later math courses that require proofs, e.g., 311,350 and 351. Thus a grade of:

    A indicates a student who is clearly well prepared for future courses

    B indicates a student who is prepared for future courses,

    C indicates a student who is marginally prepared for future courses.

    Prior to 2003-2004, the overall grade average in 300 was about 2.6, but there were huge differences among the average grades in different sections. During AY 2003-2004, the instructors all agreed to the above guiding principle, and the overall average was somewhere around 2.3, with the average grade per section in a fairly tight range.

    At the extremes, out of 134 students in the 6 sections in spring 2004, there were 18 A's, 16 F's and 26 W's.

    Your grading standards should be such that given an ``average'' class you would expect an average grade somewhere in the 2.1 to 2.5 range. Of course, you might be blessed (cursed) with more good (bad) students than an ``average'' class so this should not be taken as a firm requirement to have this average. Also, a section where several students withdrew might have a higher average than a section where few withdrew. Return to table of contents

    Homework

    Homework is an essential part of this course. Ideally, students should be doing homework every week. Good feedback on the homework is crucial especially early in the course, because students make serious mistakes that are often pretty subtle. I devoted substantial class time to discussion of errors made in the homework.

    When I have taught the course, I gave pretty heavy assignments, but this meant that the grading was quite arduous. In order to keep grading chores within reason, I suggest that the students be assigned about 4-6 problems to be handed in, and given other supplementary problems and exercises for which they are responsible, but which won't be graded. Such problems and exercises can be the basis for class discussion. The book contains some exercises that are suitable for this purpose, but not so suitable for graded homework.

    Especially early in the course, I think that the choice of problems assigned is important; problems should be chosen to advance the main course goals and anticipate things that come later.

    I recommend that you provide the students with fairly specific instructions about the presentation of homework. Here is a copy of my requirements in pdf and latex . Here is Charlie Sims' version .

    During AY 2003-2004, as coordinator of this course, I prepared suggested homework assignments. I did this because many sections of the previous book were lacking good problems. I will not do this this year. However, you might find some of the problems from these assignments useful. So here they are:
    Homework 1 pdf latex 1/14/04 version Notes
    Homework 2 pdf latex 1/19/04 version Notes
    Homework 3 pdf latex 1/28/04 version Notes
    Homework 4 pdf latex 2/5/04 version
    Homework 5 pdf latex 2/16/04 version
    Homework 6 pdf latex Version 2/22/04
    Homework 7 pdf latex Version 2/29/04
    Homework 8 pdf latex version 3/9/04
    Homework 9 pdf latex version 3/29/04
    Return to table of contents

    Warnings about the book

    This is the first time we are using this book for this course. The book was selected after an arduous and frustrating search. There are many books that aim to serve the purpose of this course, but all of them have problems. The chosen text is no exception.

    Here I will maintain a list of comments/warnings about the text. I will alert people whenever something is added to the list. Feel free to send your own comments to be added.

  • (Saks, 8/24/04) One of the hardest things for both student and instructor in the course is understanding what known results can be assumed and what needs to be proved. This is especially true because we often ask students to prove things that are ``obvious'' such as ``Prove that every integer number is odd or even'', students are often confused about this. Of course we know that what you assume depends on the context and the audience, but this is a subtle thing for the students. One weakness in the text, particularly in the first chapter, is that the level of the proofs jumps around. In section 1.5, they do an example ``Prove that the product of two odd integers is odd'' (which is quite appropriate for our students at this point in the course), but also they do an example ``Use the law of cosines to prove that a triangle with side lengths $a,b,c$ is a right triangle if and only if $a^2+b^2=c^2$.'' I suggest that you explicitly discuss the issue of what can be assumed in proofs with the students. (See section 5 of Supplement 3 for a discussion of this.
  • (Saks, 8/24/04) Continuing the previous point The example at the bottom of page 45 is inappropriate for our students at this stage of the course. One of the big difficulties students have is knowing what known facts they are allowed to assume in doing their proofs. Here the text uses, as an early example of a proof, a proof that relies on the fundamental theorem of algebra and the complex root theorem. This is far above most students.
  • (Saks, 8/24/04) Problem 6, section 1.1: This is an example of a problem that is both too trivial and too difficult. Students are asked to prove some trivialities without having any idea what constitutes a valid proof.
  • Exercise 1.3, part m. You are asked to translate a sentence into a symbolic sentence with quantifiers, using a given universe. This part says ``For every nonzero complex number, there is a unique complex number such that their product is pi. The universe to be used is the real numbers, not the complex numbers.'' This is tricky, and the answer book gets it wrong. What you'd like to do is to quantify over R^2 rather than R: ``For all (a,b) in R^2, if not a=0 or not b=0, there exists unique (x,y) in R^2 such that ax-by=pi and ay+bx=0.'' But the problem asks for quantification over R. The first quantifier can be split: ``For all a in R, for all b in R, if not a=0 or not b=0 there exists unique (x,y) in R^2 such that ax-by=pi and ay+bx=0.'' But you can't do this with the second quanitifier. Thus, the sentence: ``For all a in R, For all b in R, if not a=0 or not b=0 there exists unique x in R such that there exists a unique y in R, such that ax-by=pi and ay+bx=0'' is not logically equivalent to the others. (This is essentially the answer given in the answer book, and it's wrong.)

    To see the fallacy, consider the following simpler example: The following is clearly false: ``There exists a unique (x,y) in R^2 such that y^2=x.'' Using the incorrect transformation above, we get:``There exists a unique real x such that there exists a unique y such that y squared equals x'', which is true!

    The best way I see to correctly answer the given question is the following horrible sentence: ``For all a in R, for all b in R, if not a=0 or not b=0 there exists x in R, there exists y in R, such that ax-by=pi and ay+bx=0, and for any x in R,y in R, u in R, v in R, if ax-by=pi and ay+bx=0 and if au-bv=pi and av+bu=0 then x=u and y=v.''
  • (Sussmann, 1/20/05): 1.7: prob 9. The statement to be proved is false since $1$ is not prime.
  • (Saks, 3/20/05):3.2, 11. The definition of T_R, while correct uses a shorthand notation that students are bound to have trouble with. A clearer (though still hard) definition T_R={(x,y) in A x A: there is a natural number n and a list a_0,a_1,...,a_n of elements of A satisfying x=a_0, y=a_n, and a_0 R a_1, a_1 R a_2,....,a_{n-1} R a_n.
  • (Saks, 3/20/05): 3.4:prob 8. The definition of lexicographic order is scrambled. It should be x_1x_2<=y_1y_2 iff (i) x_1 < y_1 or (ii) x_1=y_1 and x_2 <= y_2. Return to table of contents

    Resources on the Web

    The URL for the public 300 web page for Fall 2004 is:

    http://www.math.rutgers.edu/courses/300/300-f04/.

    You will notice that each section is listed. Some of you may wish to have your own course web page accessible from this page. You can place files in the course directory ~www/courses/300/300-f04/, but please use file names that identify the files as yours. (Be sure to set the permissions to the file using ``chmod 755 [filename]'' so that students can read it.)

    You may find it helpful to see what other instructors have posted for their section.

    Hector Sussmann has taught Math 300 many times and has a web page with extensive supplementary material that you might find useful.

    Here is a set of supplemental notes I prepared for the course.
    Return to table of contents