300-Level Math: Introduction to Euclidean Geometry (Writing-Intensive)

WRITING GEOMETRY

Professor Joel Weiner finds it natural to ask his UH Mānoa geometry students to do a lot of writing. "Writing good mathematics is like writing good prose," Weiner believes. "Both require multiple drafts, guidance and revisions." Many upper-level mathematical students already possess advanced skills in manipulating symbols but Weiner aims to take them beyond symbol manipulation to a broader comprehension of what symbols and their manipulation mean.

Weiner’s course focuses on teaching students to read, speak, listen and write mathematics using everyday English in combination with mathematical symbolism. Weiner’s teaching strategies include:

  • Leading students through a sequence of mathematical problems that require an increasing level of skill
  • Focusing class discussions on techniques for speaking the language of mathematics
  • Guiding student practice in reading mathematics aloud
  • Frequent and repeated responding to student drafts

Each of these aspects of this course is described more fully below.

Sequence of Mathematical Problems

Weiner guides students through a sequence of assignments requiring increasingly complex composition tasks.

  1. A first set of assignments emphasizes increasing student awareness and comprehension of mathematical concepts. Students are required to rephrase definitions and mathematical symbols in terms that are more understandable to them. Students are also asked to take an idea that has been presented using English prose and restate it using as much mathematical symbolism as possible.
  2. A second series of assignments requires students to translate theorems and proofs into everyday language. The resulting documents must explain each step separately, showing an understanding both of its common sense meaning and of how it fits into the sequence of the theorem or proof.
  3. A third and final set of assignments asks students to write their own proofs, once again using everyday language that relies upon a combination of English and formal mathematical symbolism.

Earlier assignments provide practice in the skills students need in completing later assignments. Most of the problems have a computational component but all require English language answers rather than extensive computational or symbolic manipulation.

Weiner tries to present problems that are just beyond the boundaries of what students already know, thus encouraging students to make the jump to understanding the next level on their own.

The ten major problems Weiner assigned in his Fall 1996 class are available for you to read.

Speaking and Reading Mathematics

Like many other instructors (see, for example, Professor James Tiles), Weiner believes effective writing instruction requires simultaneous instruction in reading. During class, Weiner offers students practice both in speaking mathematics and in reading mathematics aloud.

Weiner models for students by translating into ordinary English simple and then progressively complex mathematical concepts, theorems and proofs. Once students have become familiar with his example, they too are asked to practice reading aloud in class, transforming mathematics into everyday speech.

As the semester progresses, Weiner increasingly asks students to converse with him and each other using the spoken language of mathematics they are learning.

Students at first often find the proof difficult to read and understand. They must be taught to identify separate parts of mathematical statements and helped to understand why these parts are sequenced in specified orders. Frequent reading aloud to the class and conversations with peers and the instructor help train students to recognize these features so that in the final weeks of the course they are able to speak and write proofs they compose on their own.

Responding to Student Drafts

Weiner communicates his expectations for student writing through frequent class discussions, private responses to drafts, and the distribution of a handout explaining his objectives for the class. This handout includes:

It is up to you to make yourself understood. Thus it is very important that your written work be done using sentences exclusively, combined into paragraphs as appropriate. These sentences should be carefully written so that they are grammatically and logically correct. I do not stand by ready to fill in missing explanations or incomplete arguments or to forgive improper use of terminology or symbolism. I have the same expectations for oral work

In earlier versions of this class, Weiner allowed students to rewrite their assignments as often as they wished. Now he still expects revisions but allows a maximum of three, all of which must be done within two weeks of receiving the assignment.

Weiner asks students to imagine that their audience is an intelligent computer that requires a specified syntax and understands only concepts that have been explicitly explained. Statements that do not meet these requirements are not understood. Weiner tries to read as if he were this computer, returning student drafts with responses that indicate where they failed to successfully use mathematical notation, use logic or write in unambiguous everyday language.

Weiner responds to all drafts, often with questions, guiding students to revise until their text present a solution to the problem in an effective combination of ordinary English prose and mathematical symbolism. Some sample student drafts with Weiner’s comments are included below.

Students are encouraged to conference with Weiner to discuss their revisions. These conferences continue the dialog in mathematics that takes place in class. Because his general goal is to teach students to communicate more effectively in the language of mathematics, Weiner occasionally accepts an oral revision in his office in lieu of a further rewrite.

Sample Student Drafts and Instructor Responses

Student draft:

f(A) is equal to a set of values in which are evaluated by the function f is evaluated as the set A.
Instructor response:
You’ve got the right idea but this statement is a little too long and definitely awkward. Make it sound better.

Student draft:

l & l’ are disjoint, therefore parallel.
Instructor response:
This means they have no points incommon but you just showed ONE point on each is not on the other.

Student draft:

Since none of the implications above satisfy the true value criteria of the negated implication in Table 1, then it is not possible.
Instructor response:
You assumed that the negation of an implication is an implication. Why should that be so?

Student draft:

The geometric description of the transformation TA ° ¡ q is "a rotation of q with center at the origin translated by A and translation of A."
Instructor response:
Are you saying TA ° ¡ q istwo different things? Or did you mean to say one thing follows theother; that is, describe it as a two-step process. But since this can be described as a 1-step process that is what I want to see. You will have to look at some specific examples to see what is going on. Choose the A and g to get these examples that make computation easy. Look for fixed points.