Department of Mathematics & Statistics

Swarthmore College

Swarthmore PA 19081--1397

smaurer1@swarthmore.edu

I contend that these two meanings really are different. The second sort of
reasoning (henceforth called **general reasoning**) is much broader than
the first (henceforth called **mathematical reasoning**). I further
contend that we have something to say about teaching both, and ought to
teach both, because both are relevant to the education of our future
majors and the future citizenry who come to our courses for part of their
liberal education.

One more definition. ** Deductive scientists** are those people to whom
logical reasoning means mathematical reasoning. Most mathematicians and
computer scientists are deductive scientists, as are some engineers,
natural scientists, economists and people in operations research. There is
one other discipline in which many people are deductive scientists:
philosophy. In fact, is not most direct teaching of logic at the college
level (i.e., courses titled Logic) given by Philosophy departments? I have
consulted with a friend in Philosophy at Swarthmore who teaches logic, and
he assures me that there is much discussion among philosophers about what
to teach in logic.

Henceforth "we" means deductive scientists, and more particularly mathematicians, since that's the group I really know about.

How are we teaching reasoning? Are we doing it well? My impression is that we do a lot of teaching of mathematical reasoning, usually indirectly. In fact, I applaud an indirect approach (so long as it is consciously indirect), but I do have some suggestions about how to make the indirection more interesting. As for teaching general reasoning, I don't feel we do much of that. I will suggest how we can do more.

The rest of this paper proceeds as follows. The next section, my longest, explains what I mean by general reasoning. Then I overview how we currently teach reasoning, what the weaknesses are, and my own projects that relate to these issues. I finish up with brief sections on indirection and conclusions.

It would be useful if someone did a comprehensive study of this belief, scouring the literature in all fields to see just what each discipline actually does. I made only a very brief study. I looked for approaches to logical reasoning in a writing guide written by two English professors [1], in a study guide for the Medical College Admissions Tests (MCAT) [3], and in the official information booklet for the Law School Admission Test (LSAT) [4].

The LSAT is very helpful for contemplating the scope of general reasoning. It has three types of questions: reading comprehension, "analytical reasoning", and "logical reasoning". Analytical reasoning questions are in fact combinatorial puzzles; e.g., restrictions are given on the order in which 7 musicians will play and you must deduce the actual order. The logical reasoning section is best explained by an example (p. 51).

During the construction of the Quebec Bridge in 1907, the bridge's designer, Theodore Cooper, received word that the suspended span being built out from the Bridge's cantilever was deflecting downward by a fraction of an inch. Before he could telegraph to freeze the project, the whole cantilever arm broke off and plunged, along with seven dozen workers, into the St. Lawrence River. It was the worst bridge construction disaster in history. As a direct result of the inquiry that followed, the engineering "rules of thumb" by which thousands of bridges had been built around the world went down with the Quebec Bridge. Twentieth-century bridge engineers would thereafter depend on far more rigorous applications of mathematical analysis.What we are really being asked is: which conclusion is most reasonable based on general knowledge; which one does not assume too much or interpret the passage too broadly. We are not being asked to find the sole correct conclusion in terms of logical deduction, for there are no if-then statements in the passage. We have to supply one or more missing implications, but in fact, each of A-E follows from the passage by some if-then statement. The question is: which if-then connector assumes the least.Which of the following statements can be properly inferred from the passage?

- A.
- Bridges built before about 1907 were built without thorough mathematical analysis and, therefore, were unsafe for the public to use.
- B.
- Cooper's absence from the Quebec Bridge construction site resulted in the breaking off of the cantilever.
- C.
- Nineteenth-century bridge engineers relied on their rules of thumb because analytical methods were inadequate to solve their design problems.
- D.
- Only a more rigorous application of mathematical analysis to the design of the Quebec Bridge could have prevented its collapse.
- E.
- Prior to 1907 the mathematical analysis incorporated in engineering rules of thumb was insufficient to completely assure the safety of bridges under construction.

Another sample passage claims that the Surgeon General's warning on cigarette packages has had no effect because millions of smokers are still unaware of the dangers of smoking, and readers are asked to select the assertion that, if true, would refute this argument.

A third passage says that electrons orbit nuclei the same way planets orbit suns, so we should expect that gravity is the explanation for atomic orbital structure. The question is to identify the type of reasoning just used: specialization, generalization, analogy, etc. (Note that none of these types of reasoning, except specialization, is deductive.)

The MCAT was less interesting for our purposes. It has no section called logical reasoning per se, though it does ask participants to draw conclusions from reading passages. However, to draw conclusions, the MCAT asks you to use scientific knowledge from pre-med courses, as well as the information in the passages. Such problems would not make useful examples here.

The writing guide by the English professors has a chapter on "Persuasion", where much of the discussion is about "argument". There are brief discussions of induction, deduction, and syllogism. There is lengthier discussion of faulty arguments, such as false authority, generalization from insufficient evidence, circular reasoning, and ad hominem arguments. Finally there is a section on organizing your arguments: should you give arguments in order of "increasing strength" or vice versa? (The whole idea of giving parallel arguments in hopes that each reader will find one of them convincing is foreign to deductive science.) The chapter concludes by printing several example essays (e.g., Woody Allen arguing against colorizing old movies) followed by discussion questions.

Am I belittling what these other fields view as reasoned discourse? Not at all. We all do this sort of reasoning outside of our professional work. Although I am bothered by my lack of formal understanding of general reasoning, it is a form of reasoning. I believe that training in mathematical reasoning helps one to do a better job at general reasoning. In our minds we try to recast general arguments closer to if-then form, and thus we identify what is really being assumed and and can better judge how plausible it is.

Moreover there is plenty of need for general reasoning within deductive science as well. We must reason when we motivate, when we pick our notation, when we set the order of your material, when we organize the presentation of a lengthy argument, when we work out examples. We must provide various levels of confirmation for different purposes; we don't just prove theorems, but verify, show, and illustrate items. Finally, we use general reasoning when we argue the significance of some results, or an approach, or some subfield.

Another approach is to take a course called transition to higher mathematics. The usual subject matter for this course is either set theory, (including functions and relations done set-theoretically), or foundations of the number system.

So usually math students learn proving my osmosis in the process of learning some factual subject matter However, there are a few books devoted specifically to teaching how to prove. I recommend Solow's How to Read and Do Proofs[6] and Velleman's How to Prove It [8]. These books show how to take a mathematical claim and break it down, working backwards and forwards to create subgoals, until the task of proving it is divided into manageable pieces. They also take you through proving each piece (including what to write down) and then show you the final proof with the pieces put together. They give lots of small theorems for you to prove that slowly build up your proof muscles.

About indirection I have no objection; I'll say a bit more in a later section.

About pedagogy, I tend to agree with a speech by Wilf [9], where he says you've got to attack argumentation interactively, seminar style, with students writing and presenting proofs, students and teachers critiquing them, and students rewriting them. I strongly agree with Wilf's emphasis on learning through writing. I disagree with him that books will make almost no difference, and that a week or two of this regimen help enough.

Wilf also said that a computer won't solve the problem, nor a CD nor a video. But I do think software can help, just like a good book can.

As for subject matter, the problem with using real analysis and abstract algebra is that the theorems are hard and long to prove. The problem with transition courses is that the usual subject matter is too dull for most students. What one needs is an interesting subject matter with many proofs in easy gradations.

As for books on how to do proofs, they are quite good at getting students
to the point where they are competent at short proofs. But what if a proof
is longer? And how do you organize a whole paper? In what order should you
present things? How do you make transitions? How do you keep your reader
informed where you are at all times? What goes in each paragraph? How much
computation should you show, and how do you explain it? Should you use a
figure; if so how? Should some parts of your results be separated out as
preliminary results? I have not seen such issues addressed much in books
for undergraduates, and they are very much proof issues in the general
sense. They are also writing issues; talking about writing in deductive
sciences *is* talking about general reasoning.

The first project is a transition book with a more attractive topic (I believe): a linear space approach to convex sets and elementary topological notions, primarily in 2 dimensions. The theorems and approach are different from what appear in most monographs on convex sets. Though of little importance in itself, the material has a visual aspect that grabs students and sometimes surprises them: Is the sum of two convex sets convex? Is the sum of two closed sets closed? Is the convex hull of a closed set closed? (The sum of $S$ and $T$ is $\{s+t\,|\, s\in S, t\in T\}$.)

I run the course as a seminar and students devote much time to devising and critiquing proofs. I have given this course several times, to advanced high school students through college juniors. Currently I give it to first-year students at Swarthmore who have placed out of 3/4 year of calculus. They begin with 1/2 semester to finish calculus and then, if they wish, take this course for 1/2 semester.

At present my manuscript consists of many pages of exercises. My intention is to add a few examples of proofs and some brief commentary. I intend it to remain a problem book around which a seminar may be based.

I will bring the manuscript to the conference. The issues (in terms of
teaching proving) are such things as: Are the exercises graded, starting
with one and two liners, and do they have sufficient topic variety, so the
instructor can adjust the thrust depending on student interest? Is there
a lot of emphasis on translating between words and "analytic statements"

%

($S$ is convex $\Longleftrightarrow$ if $\lambda \in [0,1]$ and
$p,q\in S$, then $\lambda p + (1{-}\lambda)q \in S$)?

%

Do students work enough with key concepts like the generic particular (if
you want to show that $S+T$ is convex, you have to show that for
*arbitrary* $p,q\in S{+}T$, something happens)? I think my material
does well on these scores,
but so could a lot of other things. For instance, the discrete math book
by Epp [2] has nice material on learning proofs using
number theory.

As for the second issue, general reasoning, I have suggested that this is properly viewed as a writing issue. Writing in mathematics courses has already increased, but much of it is informal, even emotive, "process" writing. What should help most with reasoning is structured writing, fairly close to what mathematicians do professionally. For 10 years, Swarthmore College has required that, in some courses in each discipline, students write "in the style of the discipline". Based on this, I am currently preparing a guide to writing mathematics for undergraduates [5]. Several excellent mathematics writing manuals have been published in recent years, but they are all aimed at professionals or graduate students.

Two sections of my Guide deal with reasoning directly, one section on global issues, one on local. I try to complement books like Solow and Velleman, dealing with the issues I said earlier they leave out. Other sections deal with reasoning as well, e.g., a section on theorems and proofs. (Again, I emphasize that such sections do not deal directly with the nuts and bolts of mathematical reasoning, but rather with the clear presentation of material.) I will bring appropriate sections to the conference.

Of course, reading a book is too passive. My suggestion is that materials like this be used in various courses in the deductive sciences -- courses in other fields have used ancillary texts for years. I further suggest that deductive science students at various levels be asked to write in the style of the discipline, that their work be critiqued, and that the critiques be consciously stated in terms of improving general reasoning.

I took both Latin and geometry, and I do believe they helped me to be logical. The help was rather indirect, though. And we all know that many people have trouble making transfers --even between two weeks of the same math course!

So, is an indirect approach to teaching logic enough, or should one study
syllogisms and symbolic logic as well? I can't say from personal
experience, since I did both. I found the language of symbolic logic
extremely helpful in expressing (and hence solidifying) my knowledge of
various mathematical ideas. For instance, the commutative law $x+y=y+x$
is really $(\forall x)(\forall y) [x+y=y+x]$; and, in induction, the
reason the inductive step is not itself enough is because the principal of
induction is of the form

$[P(1) \wedge (\forall n)[P(n{-}1)\Rightarrow P(n)]] \Rightarrow
(\forall n) P(n)$.

However, I suspect a student could learn to do mathematical proofs very successfully without study deductions within logical systems. Learning some of the logical symbolism may be enough. Indeed, for many students, a formal approach will be too abstract, and no transfer will be made. Moreover, for the broader issues of general reasoning, no symbolic system I know of is really relevant. What is relevant is writing, and lots of discussion based around that writing.

One possibility for teaching logic directly is to do only a small part of it, perhaps well before college. I once taught elementary logic to 7th graders (from Suppes-Hill [7]). It has a good combination of elementary symbolic deductions and translations in and out of words. Also, finite math books and discrete math books always contain some logic.

- Sylvan Barnet & Marcia Stubbs, Practical Guide to Writing with Additional Readings, 6th ed., Scott Foresman, Glenview IL, 1990.
- Susanna S. Epp, Discrete mathematics with
applications, 2nd ed.,
PWS Pub. Co., Boston, 1995.
- James L. Flowers, M.D.. \& Theodore Silver, M.D.,
*Flowers & Silver MCAT 1996 Edition*, The Princeton Review Series, Random House, New York, 1995 - LSAT/LSDAS on and Information Book, 1995-96,
Law School Admission Council, Newtown PA.
- Stephen B. Maurer,
*A Short Guide to Writing Mathematics*, The Short Guide Series, Addison-Wesley-Longman (formerly HarperCollins), in preparation. - Daniel Solow, How to Read and Do
Proofs, 2nd ed., Wiley, New York,
1990.
- Patrick Suppes \&Shirley Hill,
First Course in Mathematical Logic, 2nd ed.,
Ginn and Co., Boston, 1965.
- Daniel J. Velleman, How to Prove
It, Cambridge Univ. Press, New York,
1994.
- Herbert S. Wilf, Epsilon
Sandwiches
(speech given in Orlando, at the Awards Ceremony of the AMS-MAA
Joint Meetings, January, 1996),

http://www.cis.upenn.edu/ wilf/MAASpeech,

reprinted in*FOCUS*, April, 1996, pp. 24-25, MAA, Washington DC.