by David W. Henderson^{1}
Cornell University, Ithaca, NY 148537901, USA
# of students 
# who showed me new mathematics 
% who showed me new mathematics 
178  56  31% 
In this paper I give several examples of new mathematics (theorems and proofs) shown to me by my students. This will be followed by some reflections on the notion of proof which have helped me to make sense of why it is that I learn from my students. This will lead to more data from my students and connect the discussion to issues of multiculturalism in mathematics and to our descriptions of what is mathematics.
Mathematics Students Showed Me
There follows a few examples of mathematics I have learned from my students. I picked these examples because they were particularly powerful and clear proofs. It is possible that some of these new pieces of mathematics are known by others, but at the time I had not seen them before. As far as I know, none of these proofs have previously appeared in print. For conciseness and clarity, these examples are written in my words and not in the students' words:
Theorem 1: The sum of the angles of a (geodesic) triangle on the sphere is more than a straight angle.
Here is a proof that was discovered and shown to me by Mariah Magargee, a women first year student who was taking a course for "students who did not yet feel comfortable with mathematics" which is taught in the same style and using some of the same problems as the geometry course. In class I stressed that the students should remember that latitudes circles (except for the equator) are not geodesics and I urged them not try to apply the notions of parallel to latitude circles. Mariah ignored my urgings and came up with the following delightful proof:
First note that:
Two latitude circles which are symmetric about the equator have the property that every (great circle) transversal has opposite interior angles congruent.
This follows because the two latitudes have halfturn symmetry about any point on the equator.
Now we can mimic the usual planar proof:
We see that the sum of the angles of the "triangle" in the figure sum to a straight angle. This is not a true spherical triangle because the base is a segment of a latitude circle instead of a (geodesic) great circle. If we replace this latitude segment by a great circle segment then the base angles will increase. Clearly then the angles of the resulting spherical triangle sum to more than an straight angle.
You can check that any small spherical triangle can be derived in this manner.
Theorem 2: Stereographic projection from a sphere to the plane is conformal (i.e. preserves the size of angles). [If the sphere is tangent to a plane P at its South Pole S then stereographic projection is the projection of the sphere from the North Pole N onto the plane P.]
The following proof was shown to me by Lucy Dladla, a Mosotho woman from South Africa. Instead of focusing on the plane P she focused on the plane Q tangent to the North Pole:
The angle between two great circles on the sphere is the same as the angle between the tangent vectors to these circles at the point of their intersection x. Let the tangents to these great circles at the point x intersect the plane Q in points L and K (if a tangent vector does not intersect the plane then use the opposite pointing vector). Let S(x) be the projection of x on the plane P. Now connect the points K and L with the point N. Kx@KN because they are two tangents to the sphere drawn from the same point; and Lx@LN due to the same reason. DKLx @ DKLN by SSS, therefore ÐKxL @ ÐKNL.
Now the circles are projected onto plane P as two curves emerging from the point S(x), the angle between these curves being equal to that between their tangent vectors. These tangents S(x)K´ and S(x)L´ are projections of the tangents xK and xL and are therefore, intersections of the planes NKx and NLx with the projection plane P. But the planes NKx and NLx intersect the plane Q parallel to the plane P, along the straight lines NK and NL, therefore the straight lines S(x)K´ and S(x)L´ are parallel, respectively to the lines NK and NL. Thus ÐK´S(x)L´ @ ÐKNL @ ÐKxL and the size of the angle is preserved.
Theorem 3: Every isometry of the plane is the composition of one, two or three reflections and is either the identity, a reflection, a rotation, a translation, or a glide reflection.
The standard proof that I knew proceeded by using the Lemma (Every isometry is determined by its effect on any (nondegenerate) triangle) and then showing that the effect on a triangle can be accomplished by zero, one, two, or three reflections and proving that each of the these cases corresponds to the identity, a reflection, a rotation, a translation, or a glide reflection. This proof is experienced by many students as being indirect and convoluted. In particular, Jun Kawashima, an AsianAmerican man, looked for a proof that would show that each isometry was the identity, a reflection, a rotation, a translation, or a glide reflection more directly.
The following is an outline of Jun's proof which also uses the Lemma:
So the question emerges: Why is it that I am learning mathematics from my students? Clearly, a necessary condition is that I listen well to my students — if I did not listen there would be no chance for me to learn from them. But that does not answer why the students continue to come up with mathematics that is new to me. How can I make sense of this? I find that I can make some sense of this situation by reflecting on the notion of proof which is at the core of understanding, thinking, and listening effectively about mathematics.
Proof as a Convincing Communication That Answers — Why?
So, in this context, what is proof? A proof must do more than merely show that something is true. Here are some conclusions about proof that I have been led to by my experiences as a mathematician and teacher. Much of these understandings have come from listening to my students.
Why is 3 x 2 = 2 x 3 ? To say "It follows from the Commutative Law" or "It can be proved from Peano's Axioms" does not answer the whyquestion. But most people will be convinced by, "I can count three 2's and then two 3's and see that they are both equal to the same six". OK, now why is 2,657,873 x 92,564 = 92,564 x 2,657,873 ? We can not count this — it is too large. But is there a way to see 3 x 2 = 2 x 3 without counting? Yes:
Most people will not have trouble extending this proof to include 2,657,873 x 92,564 = 92,564 x 2,657,873 or the more general n x m = m x n. Note that for the above to make sense I must have a meaning for n x m and a meaning for m x n and these meanings must be different. So naturally I have the question: "Why (or in what sense) are these meanings related?" A proof should help me experience relationships between the meanings — it is not just an argument to show THAT something is true. In my experience, to perform the formal mathematical induction proof starting from Peano's Axioms does not answer anyone's whyquestions unless it is such a question as: "Why does the commutative Law follow from Peano's Axioms?" Most people (other than logicians) seem to have little interest in that question.
Conclusion 1: In order for me to be satisfied by a proof, the proof must answer my whyquestion and relate my meanings of the concepts involved.
As further evidence toward this conclusion, you have probably had the experience of reading a proof and following each step logically but still not being satisfied because the proof did not lead you to experience the answer to your whyquestion. In fact most proofs in the literature are not written out in such a way that it is possible to follow each step in a logical formal way. Even if they were so written, most proofs would be too long and complicated for a person to check each step. Furthermore, even among mathematics researchers, a formal logical proof that they can follow stepbystep is not always satisfying. For example, my shortest research paper [Henderson 1973] has a very concise simple proof that anyone who understands the terms involved can easily follow logically stepbystep. But, I have received more questions from other mathematicians about that paper than about any of my other research papers and most of the questions were of the sort: "Why is it true?" "Where did it come from?" "How did you see it?" They accepted the proof logically, they were convinced that it was true, but were not satisfied. To a large extent I was not able to answer these questions and I find the same phenomenon in my students — whyquestions are hard to find and hard to articulate.
Let us look at another example  the Vertical Angle Theorem: If l and l´ are straight lines then the angle a is congruent to the angle b.
The proof in most textbooks goes something like this:
If m(a) denotes the measure of angle a, then m(a) + m(g) = 180 degrees = m(g) + m(b), subtracting m(g) from both sides we conclude that m(a) = m(b) and thus that a is congruent to b.
This proof is most satisfying to persons for whom the meaning of congruence of angles is in terms of measure and who see that the usual operations of arithmetic can be applied (with some care) to measures. I used to be satisfied. But several years ago a student in my geometry course objected to this proof because to her an angle is a geometric object that is congruent to another angle if it is possible to rigidly move one angle until it coincides with the other. She offered the following proof:
Let h be a halfturn rotation about the point of intersection p. Since the straight lines have halfturn symmetry about p, h(a) = b. Thus a is congruent to b.
My first reaction was that her argument could not possibly be a proof — it was too simple and seemed to leave out important parts of the standard proof. But she persisted patiently for several days and my understandings deepened. Now her proof is much more convincing to me than the standard proof. You have may have had similar experiences with mathematics.
Conclusion 2: A proof that satisfies someone else may not satisfy me because their meanings and whyquestions are different from mine.
You may ask: "But, at least in plane geometry, isn't an angle an angle? Don't we all agree on what an angle is?" Well, yes and no. Consider this acute angle:
The angle is somehow at the corner. It is difficult to express this formally. As evidence, I looked in most of the plane geometry books in the university library and found their definitions for "angle". I found nine different definitions! Each expressed a different meaning or aspect of "angle" and thus, potentially each would lead to a different proof of the Vertical Angle Theorem. For example, if you see an angle as a pencil of rays, then reflection through the point p will directly take angle a to angel b. What would be the proof if you viewed an angle as a rotation (as many "modern" books do)?
Sometimes we have legitimate whyquestions even with respect to statements traditionally accepted as axioms. The Commutative Law is one possible example. Another one is SideAngleSide (or SAS): If two triangles have two sides and the included angle of one congruent to two sides and the included angle of the other, then the triangles are congruent. You can find SAS listed in some geometry textbooks as an axiom to be assumed, in others it is listed as a theorem to be proved and in still others as a definition of two triangles being congruent. But clearly, in any of these cases one can ask, "Why is SAS true in the plane?" This is especially true since SAS is false for (geodesic) triangles on a sphere. So one can naturally ask, "Why is SAS true on the plane but not on the sphere?" (Two sides and the included angle determine the endpoints of the third side but on the sphere there are, in general, two geodesic segments joining any two points.)
Learning From Students Who Differ From Me
The above four examples of new mathematics are all from students who are women or persons of color (or both). Is this only a coincidence? I think not. As I think back over the mathematics that I have learned from students, the most memorable mathematics is from students who differ from me (a White man) in gender or race. The data also supports this in another way:

# of students 
# who showed me new mathematics 
% who showed me new mathematics 
all students 
178  56  31% 
White men 
85  25  29% 
White women 
58  21  36% 
women of color^{1} 
22  5  23% 
men of color^{*} 
13  5  38% 




all Blacks 
10  4  40% 

Note that White women, men of color, and Blacks all have higher percentages who showed me new mathematics than the percentage of White men. This is significant because each of these groups is underrepresented in mathematics in the USA^{2}. I separated out the data for Blacks because it stands out, particularly given that there are so few Blacks participating in mathematics in the USA.
People Understanding and Communicating Mathematics
Multiculturalism and Our Views of Mathematics
But listening effectively is not automatic. When I started using this teaching method twenty years ago I felt threatened when I could not understand a student's questions or explanations — after all I was the expert in geometry. Gradually, after much persistence from the students, I began to realize that my old ways of understanding had blinded me from hearing alternatives. Among the old ways that interfered with my effective listening were the common views of mathematics that are embedded in most of our textbooks. These views emphasize precise, consistent assumptions and definitions from which theorems are proved by applying certain logical rules to the assumptions and definitions. According to these views, the end results are mathematics that is certain. (See, for example, [Henley 1991] and [Jaffe/Quinn 1993] for arguments by mathematicians who support of these types of views.) I see no way within these views of mathematics to even start to account for the data given in this paper. I agree with Thurston [Thurston 1994] that we must label these views as being inadequate descriptions of mathematics, especially mathematical experience and progress. We must proceed in directions that include social and individual aspects of human mathematical experiences into our descriptions of mathematics.^{3} Unless we proceed in these directions, our very descriptions of mathematics will continue to be obstacles to progress and the full participation of all peoples in mathematics.
[Henley 1991] J.M. Henley, A Happy Formalist, Mathematical Intelligencer, 13, 1218.
[MAA 1992] Heeding The Call For Change: Suggestions for Curricular Action (Lynn Steen, editor), MAA.
[MSEB 1990] Reshaping School Mathematics: A Philosophy and Framework for Curriculum. MSEB Report.
[MSEB 1994] Making Mathematics Work for Minorities, MSEB Report.
[NRC 1977] Women & Minority Ph.D.'s in the 1970's: A Data Book, National Research Council.
[NSF 1992] America's Academic Future, NSF report.
[Taylor 1972] Taylor, Creative Teaching: Heritage of R. L. Moore, University of Houston,
^{2} It is well known that Asians as a whole are not considered to be underrepresented in the United States, but it is not well known that AsianAmericans (nativeborn) are vastly underrepresented at least at the Ph.D. level. (See [NRC 1977], Table GWF19.) In 1975, of the 15,569 persons in the USA with a Doctorate in mathematics 14,222 were White, 121 Black, 22 Native American, 72 Hispanic, 351 foreignborn Asian, and only 28 nativeborn Asian. Since 1975 the Census Bureau has not distinguished between foreign born and native born.
^{3} In addition to the discussions above see, for example, [Thurston 1994], [MSEB 1990, 1994], [Fellows et al 1994], and the newsletter Humanistic Mathematics and in the mathematics education research literature, for example, [Confrey 1991].