,From mixed angles to inﬁnitesimals, The
College Mathematics Journal l 39 (2008), no. 3,
,Analyseinﬁnitésimale- Lecalculus redécou-
vert, Editions Academia Bruylant Louvain-la-Neuve
(Belgium) 2008, 189pages. D/2008/4910/33.
 J. L.Bell,Continuityandinﬁnitesimals,Stanford
Encyclopediaof Philosophy, revised 20 July 2009.
 E.Bishop, Mathematics s as aNumerical Language,
1970 Intuitionism and Proof Theory (Proc. Conf.,
Buﬀalo, NY, 1968), pp. 53–71. North-Holland,
,Thecrisis incontemporarymathematics. Pro-
ceedings of the American Academy Workshop on
the Evolutionof ModernMathematics (Boston,Mass.,
1974), Historia Mathematica2(1975), no.4,507–517.
 E. Bishop,Review: H.Jerome Keisler,Elementary
Calculus, Bull. Amer. Math. Soc. 83(1977), 205–208.
In ErrettBishop: ReﬂectionsonHim and His Research
(San Diego, Calif., 1983),1–32, Contemp. Math., vol.
distributed in 1973].
 P. Błaszczyk, M. Katz, , and
their debunking, Foundations
of Science e 18
(2013), no. 1, 43–74. See
Ksiegi V–VI, Tłumaczenie i komentarz [Euclid, Ele-
ments, Books V–VI, Translation and commentary],
Copernicus Center Press, Kraków, 2013.
 A. Borovik and M. . Katz, , Who o gave e you
the Cauchy–Weierstrass tale? The dual his-
calculus, Foundations of
(2012), no. 3, 245–276. See
for History of ExactSciences 14 (1974), 1–90.
T. Hawkins, and K. Pedersen, From the Calculus
to Set Theory, 1630–1910. An Introductory History,
Co. Ltd., London, 1980.
Introduction,UniversitéHenri Poincaré, Nancy, 1947.
 K. Bråting, Anew w look at E.G.Björling andthe
Cauchy sum theorem.Arch.Hist.Exact Sci.61(2007),
no. 5, 519–535.
of Fermat, Archive for History of Exact Sciences46
(1994), no. 3, 193–219.
 A. L. . Cauchy, Cours s d’Analyse de L’École Roy-
Polytechnique. Première Partie. Analyse
algébrique. Paris: Imprimérie Royale, 1821. On-
line at t http://books.google.com/books?id=
,Théorie dela propagationdesondesà la sur-
face d’un ﬂuide pesant d’une profondeur indéﬁnie,
(published 1827, with additional Notes), Oeuvres,
Series 1, Vol. 1, 1815, pp. 4–318.
Italian school, Revue d’Histoiredes Mathématiques
13 (2007), no. 2, 259–299.
 G. Cifoletti, La méthode de Fermat: son statut
et sa diﬀusion, Algèbre et comparaison de ﬁgures
dans l’histoire de la méthode de Fermat, Cahiers
d’Histoire et de Philosophie des Sciences, Nouvelle
Série33, SociétéFrançaised’Histoire des Sciences et
des Techniques, Paris, 1990.
Press, Inc., San Diego, CA, 1994.
theory. In(Bos et al. 1980), pp. 181–219.
Memoirs of the National Academy of Sciences
82 (2003), 243-284. Available at the addresses
pdf and http://www.nap.edu/catalog/10683.
 A. De Morgan, On the early y history of
inﬁnitesimals in England, Philosophical Mag-
azine, Ser. 4
4 (1852), no. . 26, 321–330.
edition, Oxford, TheClarendon Press, 1958.
diagrams: Microscopes in non-standardand smooth
analysis, Studies in Computational Intelligence(SCI)
 P. Ehrlich, Therise of non-Archimedean mathe-
matics and the roots of a misconception. I. The
emergence of non-Archimedean systemsofmagni-
tudes, ArchiveforHistoryof ExactSciences60(2006),
no. 1, 1–121.
 R. Ely, , Loss of f dimension in n the history y of cal-
culus and in student reasoning, The Mathematics
Enthusiast 9 (2012), no. 3, 303–326.
 Euclid, Euclid’s s Elements of Geometry, , edited
and provided with a modern English transla-
tion by Richard Fitzpatrick, 2007. See e http://
primus, SPb and Lausana, 1748.
,Introductionto Analysisof theInﬁnite. Book I,
translated from theLatin and withan introduction
by JohnD. Blanton, Springer-Verlag, New York, 1988
[translation of (Euler 1748 )].
,InstitutionesCalculi Diﬀerentialis, SPb, 1755.
,Foundations of Diﬀerential Calculus,English
translation of Chapters 1–9 of (Euler 1755 ) by
D.Blanton, Springer, New York, 2000.
in the Eulerian foundations of thecalculus, Historia
Mathematica 31(2004), no. 1, 34–61.
nary of Scientiﬁc Biography, ed. by C. C. Gillispie,
vol. 3, CharlesScribner’sSons, New York, 1971, pp.
 C. I. Gerhardt (ed.), , Historia et t Origo calculi dif-
ferentialis a G. G. Leibnitio conscripta, Hannover,
(ed.), Leibnizens mathematische Schriften,
Berlinand Halle: Eidmann,1850–1863.
Notices of the AMS
Volume 60, Number 7
ennes, Revue de Métaphysique et de Morale 86e
Année, No. 1(Janvier-Mars 1981), 1–37.
 Hermann Grassmann, , Lehrbuch h der Arithmetik,
Enslin, Berlin, 1861.
Scientiﬁc Papers of Isaac Newton, Cambridge Uni-
versity Press, 1962, pp. 15–19, 31–37; quoted in I.
BernardCohen, RichardS. Westfall, Newton,Norton
CriticalEdition, 1995, pp. 377–386.
inAlterthum und Mittelalter, Teubner,Leipzig, 1876.
University Press, Cambridge,1897.
I, Trans. Amer.Math. Soc. 64 (1948),45–99.
 J. Heiberg (ed.), Archimedis s Opera a Omnia cum
CommentariisEutocii, Vol. I.Teubner, Leipzig, 1880.
 A. Heijting, Address s to o Professor r A. . Robin-
son, at the occasion of the Brouwer memorial
lecture given by Professor A. Robinson on
the 26th April 1973, Nieuw Arch. Wisk. (3)
(1973), 134–137. MathSciNet Review
zurEnthüllung desGauss-Weber Denkmals, Göttin-
gen, Leipzig, 1899.
,Les principes fondamentaux delagéométrie,
Annales scientiﬁques de l’E.N.S. 3
série 17 (1900),
 O. Hölder, Die e Axiome e der r Quantität t und d die
Lehre vom Mass, Berichte über die Verhandlungen
der Königlich Sächsischen Gesellschaft der Wis-
senschaften zu Leipzig, Mathematisch-Physische
Classe, 53, Leipzig, 1901, pp. 1–63.
guage, secondedition, Cambridge University Press,
mals: Strategies for ﬁndingtruth inﬁction, 27 pages.
In Leibniz on theInterrelations between Mathematics
leyand David Rabouin, Archimedes Series, Springer
the factorizationof thesinefunction into an inﬁnite
product, Russian Mathematical Surveys43 (1988),
jects,andchimeras:Connes ontheroleof hyperreals
in mathematics, Foundations of Science(onlineﬁrst).
9316-5 and http://arxiv.org/abs/1211.0244
Axiomatically, Springer Monographs in Mathematics,
Springer, Berlin, 2004.
model of the reals, Journal of Symbolic Logic c 69
(2004), no. 1, 159–164.
 K. Katz z and M. . Katz, Zooming in n on inﬁnitesi-
mal 1 1 :9:: in a post-triumvirate era, Educational
Studies in Mathematics s 74 (2010), no. 3, 259–273.
,When is:999::: less than 1? The Montana
Mathematics Enthusiast 7 (2010), no. 1, 3–30.
on Science 19 (2011), no. 4, 426–452. See
,Meaning in classical mathematics: Is it at
odds with intuitionism? Intellectica56(2011), no. 2,
, A Burgessian critique of nominalistic
tendencies in contemporary mathematics and
its historiography, Foundations of Science e 17
(2012), no. 1, 51–89. See http://dx.doi.org/
 M. Katz and E. Leichtnam, Commuting and
noncommuting inﬁnitesimals, American Mathemat-
ical Monthly y 120 (2013), no. 7, 631–641. See
andbeyond,PerspectivesonScience21 (2013), no. 3,
 M. Katz and D. Sherry, Leibniz’s inﬁnitesimals:
Their ﬁctionality, their modern implementations,
and theirfoes from Berkeley to Russell andbeyond,
Erkenntnis78 (2013), no. 3, 571–625; seehttp://
,Leibniz’s lawsofcontinuity and homogene-
ity, Notices of the American Mathematical Society
59 (2012), no. . 11, 1550–1558. See http://www.
ams.org/notices/201211/ and http://arxiv.
Foundations of Science 18 (2013), no. 1, 107–123.
9289-4 and http://arxiv.org/abs/1206.0119
ters acrossmathematics, Contemp. Math., vol. 530,
pp. 163–179, Amer. Math.Soc., Providence, RI, 2010.
Philosophers, CambridgeUniv. Press, 1983, §316.
Standpoint. Vol. I, Arithmetic, Algebra, Analysis.
Translation by E. R. Hedrick andC. A. Noble [Macmil-
lan, New York, 1932]from the thirdGerman edition
[Springer, Berlin, 1924]. Originally published as El-
ementarmathematik vom höheren Standpunkteaus
inﬁniteseries, AequationesMathematicae34 (1987),
,Deﬁnite valuesofinﬁnitesums: Aspects of
the foundations of inﬁnitesimal analysis around
1820, ArchiveforHistoryof ExactSciences39(1989),
no. 3, 195–245.
,Early deltafunctions andthe use of inﬁnites-
imals in research, Revue d’histoire des sciences s 45
(1992), no. 1, 115–128.
 G.Leibniz Cum Prodiisset…mss “Cumprodiisset
atque increbuisset Analysismea inﬁnitesimalis:::”
in Gerhardt [41,pp. 39–50].
,Letterto Varignon, 2 Feb 1702, in Gerhardt
[42, vol. IV, pp. 91–95].
,Symbolismus memorabilis calculi algebraici
et inﬁnitesimalis in comparatione potentiarum
Notices of the AMS
et diﬀerentiarum, et de lege homogeneorum
transcendentali, inGerhardt [42,vol.V, pp.377-382].
,Lanaissanceducalcul diﬀérentiel,26 articles
des Acta Eruditorum. Translated from theLatin and
with an introductionand notesby Marc Parmentier.
Witha preface by MichelSerres. Mathesis, Librairie
PhilosophiqueJ.Vrin, Paris, 1989.
,De quadratura arithmetica circuli ellipseos
et hyperbolae cujus corollarium est trigonome-
tria sine tabulis. Edited, annotated, and with a
foreword in German by Eberhard Knobloch, Ab-
handlungen der Akademie der Wissenschaften in
Göttingen, Mathematisch-Physikalische Klasse, Folge
3 [Papers of the Academy of Sciences in Göt-
tingen, Mathematical-Physical Class. Series 3], 43,
Vandenhoeck and Ruprecht, Göttingen, 1993.
sur les classes déﬁnissablesd’algèbres, inMathemat-
ical Interpretation of Formal Systems, pp. 98–113,
North-Holland Publishing Co.,Amsterdam, 1955.
persinthe foundationsof mathematics, American
Mathematical Monthly y 80(1973),no.6,partII,38–67.
and the inﬁnitesimal world: Unveiling and optical
diagrams in mathematics, Foundations of Science10
(2005), no. 1, 7–23.
ematical Practice in the Seventeenth Century, The
ClarendonPress, Oxford University Press, New York,
tory ofcomplex numbers, American Mathematical
Monthly 30(1923), no. 7, 369–374.
ler’s summation of the reciprocals of the squares,
 H. Meschkowski, Aus den Briefbuchern Georg Can-
tors, Archive for History of Exact Sciences2 (1965),
of neo-Kantian philosophy of science, HOPOS: The
Journal of the International Society for theHistory of
Philosophy of Science, to appear.
 D. Mumford, Intuition and rigor and Enriques’s
quest, Notices Amer. Math. Soc. . 58 (2011), no. 2,
sion, The Method of Fluxions and Inﬁnite Series
inﬁnitesimal,TheMonist 35 (1925), 633–666.
discussion between BennoKerry and Georg Cantor,
Historyand PhilosophyofLogic29(2008),no. 4,343–
 P. Reeder, Inﬁnitesimals s for r Metaphysics: : Conse-
quences for the Ontologies of Space andTime, Ph.D.
thesis,Ohio StateUniversity, 2012.
Wetensch. Proc. Ser. A64 =Indag. Math.23 (1961),
432–440 [reprinted inSelected Works;see(Robinson
1979[96, pp. 3–11]).
,Reviews: Foundations of ConstructiveAnaly-
sis,American Mathematical Monthly y 75 (1968),no.8,
,Selected Papersof AbrahamRobinson, Vol. II.
Nonstandard Analysis and Philosophy, edited and
with introductions by W. A. J. Luxemburg and S.
Körner, Yale UniversityPress, New Haven, CT, 1979.
 B. Russell, The e Principles s of Mathematics, Vol. I,
Cambridge Univ. Press,Cambridge, 1903.
 D.Sherry, Thewakeof Berkeley’s s analyst:Rigor
mathematicae?Stud. Hist. Philos.Sci.18(1987),no.4,
of Science 55 (1988), no. 1, 58–73.
ideals,andﬁctions, Studia Leibnitiana, toappear.See
Books, together with the Eleventh and Twelfth. The
Errors by which Theon and others have long ago
vitiated these Books are corrected and someofEu-
clid’sDemonstrations arerestored.Robert & Andrew
Foulis, Glasgow, 1762.
gen Charakterisierung der Zahlenreihemittels eines
endlichen Axiomensystems, Norsk Mat. Forenings
Skr., II.Ser. No. 1/12(1933), 73–82.
, Über die Nicht-charakterisierbarkeit der
Zahlenreihe mittels endlich oder abzählbar un-
endlichvieler Aussagen mit ausschliesslich Zahlen-
variablen, Fundamenta Mathematicae 23 (1934),
,Peano’s axioms and models of arithmetic,
in Mathematical Interpretation of Formal Systems,
pp. 1–14, North-Holland Publishing Co.,Amsterdam,
 R. Solovay, A model of f set-theory in which ev-
ery set of reals is Lebesgue measurable, Annals of
Mathematics (2) 92(1970), 1–56.
Teubner, Leipzig, 1885.
conceptions of function, continuity, limit, andinﬁni-
tesimal, withimplications for teaching thecalculus,
Educational Studiesin Mathematics, to appear.
 P. Vickers, , Understanding Inconsistent t Science,
OxfordUniversityPress, Oxford, 2013.
 J.Wallis,TheArithmetic ofInﬁnitesimals(1656).
Translated from the Latin and with an introduc-
tion by Jacqueline A. Stedall, Sources and Studies
in the History of Mathematics and Physical Sciences,
Springer-Verlag,New York, 2004.
ﬁnitesimals, Lecture Notes in Mathematics, 2067,
Springer, Heidelberg, 2013.
Notices of the AMS
Volume 60, Number 7
Advancing Mathematics Since 1888
Applications are invited for the position of Associate Executive
Director for Meetings and Professional Services.
The Associate Executive Director heads the division of Meetings
and Professional Services in the AMS, which includes
approximately 20 staff in the Providence headquarters.
Departments in the division work on meetings, surveys,
professional development, educational outreach, public awareness,
and membership development. The Associate Executive Director
has high visibility, interacting with every part of the Society, and
therefore has a profound effect on the way in which the AMS
serves the mathematical community. It is an exciting position with
much opportunity in the coming years.
Responsibilities of the Associate Executive Director include:
• Direction of all staff in the three departments comprising
• Development and implementation of long-range plans for all
parts of the division
• Budgetary planning and control for the division
• Leadership in creating, planning, and implementing new
programs for the Society
Candidates should have an earned Ph.D. in one of the
mathematical sciences and some administrative experience. A
strong interest in professional programs and services is essential.
The appointment will be for three to ﬁ ve years, with possible
renewal, and will commence in January 2014. The starting date
and length of term are negotiable. The Associate Executive
Director position is full time, but applications are welcome from
individuals taking leaves of absence from another position. Salary is
negotiable and will be commensurate with experience.
Teaching Practices into
Blueprint or Fantasy?
ow do you teach mathematics in your
college classes? Are undergraduate
classes taught the same as graduate
classes? Do you teach mathematics
as you were taught? Are your instruc-
tional practices aligned with what research says
about how students learn mathematics? Do you
use evidence-based instructional practices? That
is, does evidence exist that your instructional
practices help students learn the mathematics you
are teaching? How often do you exchange ideas
about teaching mathematics with colleagues? The
purpose of this paper is to encourage discussion
among college faculty about effective instructional
practices with the intent of improving the teaching
and student learning of mathematics.
The Need to Improve Mathematics
Since the days of E. H. Moore and his 1902 presi-
dential address to the AMS there have been
many calls for improved teaching of mathemat-
ics . More recently David Bressoud in the MAA
Launchings  shares some ways that physicists
have worked to utilize evidence-based teaching
methods in their classes. Bressoud also provides
resources that represent multiple perspectives
from mathematicians engaged in the scholarship
of teaching. Yet, despite the desire to promote
better mathematics learning in STEM courses and
the growing body of evidence-based research that
has implications for teaching practices, changing
collegiate instructional practices has been slow
and does not happen easily .
In September 2011 the Association of American
Universities announced a five-year initiative to im-
prove the quality of undergraduate teaching and
learning in science, technology, engineering, and
mathematics (STEM) at its member institutions.
In February 2012 the President’s Council of Advi-
sors on Science and Technology (PCAST) issued
a report  that forecast the need for producing
“1 million more college graduates in STEM fields”
during the next decade. Among other things, the
report noted the important role that college math-
ematics courses play in either opening or closing
the doors to different STEM fields. To address this
issue, the PCAST report called for more research
into the best ways to teach and learn mathematics
and urged “widespread adoption of empirically
validated teaching practices” by STEM faculty in
higher education. It also recommended the launch
of “a national experiment in postsecondary math-
ematics education to address the mathematics-
In June 2012 the National Science Founda-
tion issued a Dear Colleague Letter for Widening
Implementation and Demonstration of Evidence-
based Reforms (WIDER) calling for proposals. The
program is intended to promote improvement in
undergraduate STEM instructional practices and
bring to scale successful instructional practices
within and across departments. This is just one of
the initiatives designed to stimulate greater knowl-
edge of and widespread use of evidence-based
instructional practices. This initiative provides
mathematics departments an opportunity to focus
on teaching practices. The visual image of most
collegiate-level mathematics courses (regardless
Robert Reys is Curators’ Professor Emeritus of Mathemat-
ics Education at the University of Missouri-Columbia. His
email address is email@example.com.
of its accuracy) is a professor writing on a black/
whiteboard with his/her back to the class while stu-
dents are copying feverishly and others have their
hand raised but their questions going unanswered
. There are many factors that influence the in-
structional methods faculty members employ in
their teaching practices. Probably the single most
influential factor impacting how teachers teach is
their prior experience as a student . Stigler and
Hiebert claim that breaking away from the cultural
model of mathematics teaching that is so prevalent
in the United States, typically lecture and note
taking, is a major challenge. In the research related
to evidence-based teaching in STEM undergradu-
ate courses, it is worth noting that the focus has
been almost exclusively on science-related courses
, . Despite the work of R. L. Moore and the
legacy of the Moore method, it can be argued that
inquiry-based learning is much more prevalent in
science courses than in mathematics courses at
the undergraduate level , , . Perhaps op-
portunities to engage in laboratory environments
provides experiences that engage learners and
promotes more inquiry-based practices in science
and engineering, whereas mathematics is too often
viewed as a spectator sport; i.e., show me what to
do or tell me how to do it, and I will memorize and
practice the procedures. Even though there have
been many calls by professional organizations to
make the learning of mathematics a sense-making
activity (National Council of Teachers of Math-
ematics, Mathematical Association of America,
American Mathematical Society), for many college
students mathematics learning is viewed as assum-
ing a passive rather than an active role in learning.
Challenges Related to Changing
What are some of the hurdles to be cleared in
promoting evidence-based change in teaching
practices in collegiate mathematics classes? Here
are a few broad challenges:
We tend to teach as we have been taught. If
lecture is the primary model experienced for learn-
ing mathematics, as Stigler and Hiebert suggest,
it is difficult to break this cycle. Since professors
teaching mathematics courses were successful in
learning mathematics via a lecture method, it is
difficult for them to relate to learning difficulties
of their students. A frequently voiced sentiment
is, “if it ain’t broke, don’t fix it.” However, other
people, particularly those outside mathematics,
who cite the high rate of attrition in college math-
ematics courses argue that “actually it is broke”
and there are some fundamental problems regard-
ing the learning and teaching of mathematics that
need to be addressed.
Doctoral programs in mathematics are heavily
weighted toward research, with limited systematic
attention to teaching. In order to do original re-
search in mathematics, one needs to delve deeply
into a topic within a discipline. Completion of a
Ph.D. in mathematics requires extensive research
that focuses on mathematics, and thus few doc-
toral programs in mathematics have specific
requirements regarding competence in teaching.1
The goal of covering many chapters each semester
still exists, and fixed pacing requirements often
make it difficult for instructors to break away
from lectures and engage their students in open
discussions about the mathematical concepts
being examined. Furthermore, heavy emphasis on
mathematics research leaves little time for TAs to
focus on evidenced-based instructional practices.
In fact, Project NExT2 was established in part to
help provide some support for further develop-
ment of teaching skills by new faculty members
in mathematics departments.
The climate and environment of the department/
college/institution provides a powerful context to
either support or discourage emphasis on good
teaching. Some research suggests this is the single
most important factor inhibiting change in teach-
ing practices , . For example, does schedul-
ing of large classes with several hundred students
limit the use of evidence-based instructional op-
tions? Are specific service courses, such as busi-
ness calculus or linear algebra, viewed as service
courses? If so, it may discourage a focus on inquiry
when a course is supposed to help students learn
specific skills. Does the chair/dean/chancellor en-
courage and value good teaching? Are their values
made clear? Are there support systems for help-
ing faculty learn about evidence-based research
related to effective teaching practices? Is there
motivation/incentive to engage in learning about
different instructional models?
The reward system in comprehensive research-
oriented institutions favors faculty members who
gain national visibility via their research and schol-
arship rather than based on the quality of their
teaching. An average or below-average teacher
may be promoted because of a sterling record
of scholarship. However, it is rare (virtually
Nearly fifty years ago while I was a doctoral student,
I taught college algebra and calculus for two years as
a teaching assistant in a mathematics department at a
research university. During that time the extent of instruc-
tional support was limited to giving the TAs a book and
specific expectations about what chapters to cover. We
did not know what classes we were assigned to teach until
the night before classes started! Thank goodness progress
has been made and support for TAs has improved in
mathematics departments since I was a TA.
Project NExT (New Experiences in Teaching) is a program
founded over twenty years ago and is sponsored by the
Mathematical Association of America. For more informa-
tion see http://archives.math.utk.edu/projnext/.
impossible) for a faculty member who is a great
teacher but has done little research in mathemat-
ics to be promoted in a research-oriented institu-
tion. Thus, the argument offered by Earnest Boyer
 more than twenty years ago for rewarding
the scholarship of teaching has not made much
Faculty members face pressures throughout their
careers but may react differently depending on the
stage of their career. While publication pressure is
usually the heaviest early in their careers, once that
hurdle is cleared faculty members may focus at-
tention elsewhere, including teaching . So while
posttenured faculty members may be interested in
teaching, their instructional approaches may have
become so entrenched that significant change is
very difficult . On the other hand, some senior
faculty members choose to give more attention
to improving teaching practices. This has been
sparked in some cases by calls for collaboration
between higher-education faculty and K–12 school
systems (e.g., NSF Math Science Partnerships).
Institutions of higher education have created new
types of faculty appointments to address teaching.
This may be viewed as a blessing or a curse. New
faculty tracks have been created for specialized
roles (e.g., adjunct, clinical, visiting, fixed-term,
nonregular, and postdocs). Many of these appoint-
ments are designed to address the teaching needs
of institutions, so it seems reasonable that these
people would be highly motivated to learn about
and use evidence-based teaching practices ,
. However, hiring “others” to focus on teaching,
may in fact decrease interest by faculty in tenure-
track positions to explore ways to improve their
In an effort to gain information about what
some institutions know about the teaching prac-
tices employed in their mathematics classes, I
contacted a department chair and a provost at
two research-oriented institutions. I know both of
these people well. I felt I could be honest with them
and that they would be honest in their responses.
Questions Asked and What Was Learned from
Here is how I situated the discussion: I am trying to
learn about the instructional practices being used
in undergraduate mathematics classes at your uni-
versities. I want to identify issues and challenges
that would most likely be encountered in gather-
ing information about instructional practices used
by your mathematics faculty members teaching
undergraduate courses. I then asked:
Do you regularly collect information about
instructional practices used in teaching your
undergraduate mathematics courses at your in-
stitution? (This would include teachers who are
graduate students as well as adjunct and regular
Neither institution collected any data on teach-
ing practices. The provost said their institution
collects information to support a state man-
date to provide an “Institution Effectiveness
Plan”, but this amounted to collecting course syl-
labi that highlight content. While some of these
syllabi provide clues about instructional practices,
there is no specific requirement or expectation
that instructional practices be addressed. While
each institution gathers course evaluations from
students, they have no current structure that
systematically collects information about teach-
ing practices nor do they have any mechanism in
place that communicates information about best
or effective teaching practices to their faculty
More specifically, the mathematics department
chair said: “No, we currently do not have such
information. The department is initiating ways to
collect such information in a more formal manner
than ‘coffee room’ chatter about what somebody
did in class. We are planning to have a depart-
ment ‘retreat’ (a couple of hours) this year where
interested faculty will share their encouraging
teaching practices, and this is actually a pretty
big step for us.”
What do you think would be the most sig-
nificant barriers in getting faculty members to
participate in efforts to learn about evidence-based
Each of the administrators mentioned lack
of time and uncertainty of the perceived value/
importance of the activity. The mathematics de-
partment chair went on to say:
“Many mathematicians are so ‘involved in their
research’ that their teaching is something they
do; there is a belief that teaching is inherently
good (they teach the way they were taught and
that IS GOOD, by definition)—the students are
ill-prepared and that causes the problem. The
department culture about teaching is the most
significant barrier as I see it. Promotion requires
publications/grants, etc.; teaching is regarded
as good unless there is some major issue that
surfaces…good teaching practices rarely have an
important role in the process. So, the culture has
to change; it must respect more than traditional
thinking about research contributions.
“I have also found that many research math-
ematicians have really good ideas about teaching
and learning—and they practice these regularly,
but they really don’t share them with colleagues
in part because they don’t have an opportunity
to do so. As the culture changes and instruction
is understood to become part of the equation for
advancement, faculty will want to be better teach-
ers and share what they do.”
Are there incentives to systematically collect
information about teaching practices used by
mathematics faculty members?
The provost suggested an appeal to professional
responsibility. She also addressed the importance
of selling this need to top-level administrators
and convincing them how this information would
be useful to them in making policy decisions. She
said the data collection would need to include all
STEM disciplines and not be limited to a single
department. Furthermore, any such survey would
need to be free of implied or stated value judg-
ments regarding particular instructional practices.
The need to avoid any bias with regard to teach-
ing practices was reiterated by the mathematics
department chair. In addition, the mathematics
“In reality, there are probably not any incen-
tives. The culture has to change. It is rare that a
person will get a good raise because they were a
good teacher. We give a lot of lip service to good
teaching, but there is no concerted effort to en-
courage or reward it in the department.”
This quote prompted a reviewer of this paper
to say, “We need a culture that encourages faculty
to think about the effectiveness of their teaching,
to share their personal insights, and to provide
support that enables faculty to adopt easily
implementable ideas and helps faculty monitor
the effectiveness of what they are doing.” I say,
Amen! Although, I don’t think we should limit
ourselves to “easily implementable ideas”, as some
of the evidence-based practices may not be easy
Where to from Here?
This discussion provides a limited glimpse about
mathematics teaching, including perspectives
from administrators from a Carnegie doctoral/
research university and the other from a research
university (high research activity). Yet these
perspectives represent only two of more than
3,000 four-year institutions of higher education
and none of the nearly 2,000 two-year institu-
tions. This discussion suggests that structuring
a framework that will provide accurate profiles
of the current teaching practices in mathematics
courses in various institutions is a big challenge.
Using that information to inform faculty members
in institutions of higher education about evidence-
based teaching practices and helping to stimulate
systematic change in teaching practices are likely
to pose a far more demanding challenge.
The stakes of not doing everything possible to
improve mathematics teaching and learning are
high. Millions of students are taking courses in
mathematics at institutions of higher education
this year. Whether the courses are remedial, satis-
fying a general education requirement, calculus or
beyond, there is increasing interest and pressure
to make learning of mathematics meaningful and
to help students make sense of whatever math-
ematics is being learned. While students have a
responsibility to seek understanding of the math-
ematics, faculty members have a responsibility to
utilize evidence-based teaching practices to help
facilitate mathematics learning.
So what might a mathematics department
do? One valuable first step would be to have an
open and frank discussion about teaching among
members of the department. This might include:
addressing some of the opening questions, sharing
teaching approaches and techniques that profes-
sors use, visiting and observing other professors
as they teach, learning more about how people
learn, and examining evidence-based instructional
practices that have been shown to be successful.
Any or all of the above might be eye opening and
intellectually stimulating and perhaps most im-
portantly be recognition that the department is
serious about improving the teaching and learning
of mathematics at its institution.
As the department chair interviewed said, “The
culture has to change.” The big question is how
and whether his statement reflects a blueprint or
fantasy for his institution and thousands of other
mathematics departments throughout the U.S.
My hope is that this article will stimulate some
discussion about your departmental culture and
help bring teaching and learning of mathematics
to the forefront.
1. A. H. Schoenfeld, When good teaching leads to bad
results: The disasters of “well taught” mathematics
courses, Educational Psychologist 23 (1988), 145–166.
2. D. Bressoud, MAA Launchings (September 17, 2012).
3. A. E. Austin, M. Connolly, C. Pfund, D. L. Gillian-
Daniel, and R. Mathieu, Preparing STEM doc-
toral students for future faculty careers, in R. G.
Baldwin (ed.), Improving the Climate for Undergradu-
ate Teaching and Learning in STEM Fields, Jossey-Bass,
San Francisco, CA, 2009.
4. President’s Council of Advisors on Science and Tech-
nology-PCAST (August 29, 2012), Engage to excel:
Producing one million additional college graduates
with degrees in science, technology, engineering, and
mathematics. See http://www.whitehouse.gov/
5. M. Cirillo and B. Herbel-Eisenmann, “Mathemati-
cians would say it this way.” An investigation of
teachers’ framing of mathematicians, School Science
and Mathematics 111 (2011), 68–77.
6. J. Stigler and J. Hiebert, The Teaching Gap, Simon &
Schuster, New York, 2000.
7. A. E. Austin, Promoting evidence-based change in un-
dergraduate science education, paper commissioned
by the National Academies National Research Council
Useful Information for Deans and Department Chairs,
Jossey-Bass, San Francisco, CA, 1995, pp. 47–63.
, Institutional and departmental cultures and
the relationship between teaching and research, in
J. Braxton (ed.), Faculty Teaching and Research: Is
There a Conflict? Jossey-Bass, San Francisco, 1996,
14. E. L. Boyer, Scholarship Reconsidered: Priorities of the
Professoriate, Carnegie Foundation for the Advance-
ment of Teaching, Princeton, NJ, 1990.
15. A. Neumann, Professing to Learn: Creating Tenured
Lives and Careers in the American Research Univer-
sity, The Johns Hopkins Press, Baltimore, MD, 2009.
16. A. E. Austin and R. G. Baldwin, Faculty motivation
for teaching, in P. Seldin (ed.), Improving College
Teaching, Anker Publishing Co., Boston, MA, 1995,
17. A. E. Austin, Creating a bridge to the future: Pre-
paring new faculty to face changing expectations in
a shifting context, Review of Higher Education 26
the public sphere are perfectly represented by
these trials. Thus they serve as ideal illustrations of
these errors and of the drastic consequences that
faulty reasoning has on real lives” (p. x). The au-
thors’ strategy is to identify common mathematical
errors and then illustrate how those errors arose
in trials. They seek to accomplish two goals: first,
to impress upon the general public the importance
of being able to “distinguish whether the numbers
brandished in our faces are legitimately providing
information or being misused for dangerous ends”;
second, “to identify the most important errors that
have actually occurred” so that such mistakes can
be eliminated in the future.
These are worthy if anodyne goals, and I would
not dare argue against them. But the claims that
Schneps and Colmez make are strong ones and
prompt many questions. Do they adequately
support their contention that mathematics has a
“disastrous record of causing judicial error?” How
influential are mathematical arguments, anyway?
Are mathematical arguments more problematic
Board on Science Education, National Academy Press,
Washington, DC, 2011.
8. L. Deslauriers, E. Schelew, and C. Wieman, Improv-
ing learning in a large enrollment physics class, Sci-
ence 332 (2011), 862–864.
9. L. B. Flick, P. Sadri, P. D. Morrell, C. Wainwright,
and A. Schepige, A cross discipline study of reform
teaching by university science and mathematics
faculty, School Science and Mathematics 109 (2009),
10. C. Rasmussen, K. Marrongelle, and O. N. Kwon,
A framework for interpreting inquiry oriented teach-
ing, paper presented at the Annual Meeting of the
American Educational Research Association, San
Diego, CA, 2009.
11. C. Wainwright, P. D. Morrell, L. B. Flick, and
A. Schepige, Observation of reform teaching in un-
dergraduate level mathematics and science courses,
School Science and Mathematics 104 (2004), 322–335.
12. A. E. Austin, Understanding and assessing faculty
cultures and climates, in M. K. Kinnick (ed.), Providing
Math on Trial
Leila Schneps and Coralie Colmez
Basic Books, 2013
US$26.99, 272 pages
In Math on Trial, Leila Schneps and Coralie Col-
mez write about the abuse of mathematical argu-
ments in criminal trials and how these flawed
arguments “have sent innocent people to prison”
(p. ix). Indeed, people “saw their lives ripped apart
by simple mathematical errors.” The purpose of
focusing on these errors, despite mathematics’
“relatively rare use in trials” (p. x), is “that many of
the common mathematical fallacies that pervade
Burden of Proof: A Review of
Math on Trial
Reviewed by Paul H. Edelman
Paul H. Edelman is professor of mathematics and law
at Vanderbilt University. His email address is paul.
The author thanks Ed Cheng, Chris Slobogin, and Suzanna
Sherry for helpful comments.
Documents you may be interested
Documents you may be interested