new developments and trends in tertiary mathematics education: or, more of the same?
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New developments and trends intertiary mathematics education: or,more of the same?Annie Seldena New Mexico State University, Department of MathematicalSciences, MSC 3MB, P.O. Box 30001, Las Cruces, NM, 88003-0001,USAPublished online: 20 Feb 2007.
To cite this article: Annie Selden (2005) New developments and trends in tertiary mathematicseducation: or, more of the same?, International Journal of Mathematical Education in Science andTechnology, 36:2-3, 131-147, DOI: 10.1080/00207390412331317040
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International Journal of Mathematical Education inScience and Technology, Vol. 36, Nos. 2–3, 2005, 131–147
New developments and trends in tertiary mathematics education:
Or, more of the same?
ANNIE SELDEN
New Mexico State University, Department of Mathematical Sciences,MSC 3MB, P.O. Box 30001, Las Cruces, NM, 88003-0001, USA
(Received 15 June 2004)
This paper examines four developments, or trends, in tertiary mathematicseducation today: (1) technology as an engine driving pedagogical change; (2) theoften difficult transition for students from secondary to tertiary mathematics;(3) the responsibilities of mathematicians for the mathematical preparation offuture teachers, and (4) the potential impact on teaching of research into theteaching and learning of tertiary mathematics.
1. Introduction
There are many developments and trends, ranging from the ever-present influence,
or ‘push’, of technology to the challenges of dealing with the often difficult secondary
to tertiary transition, that one could consider. At the sessions of Topic Study Group
3 (TSG3) at the 2004 Tenth International Congress on Mathematical Education
(ICME-10), a host of issues were discussed including teaching and learning using the
World Wide Web and computer software packages, the mathematical education of
in-service and pre-service teachers, students’ cognition and difficulties related to
understanding specific topics, design of specific courses, and measuring students’
understandings. Still there are additional interesting developments one might
consider: the needs of different clientele, e.g. engineering, science, or business
students; the increasing use of cooperative groups and project work;
the advantages/disadvantages of different assessment methods; the advantages/
disadvantages of different teaching methods; non-university tertiary education,
e.g. two-year community and technical colleges; the transition from lower division,
more computational, courses to upper division, more abstract, often proof oriented,
courses; the increased demand for online delivery of courses and entire degree
programmes; service learning and other specialized courses that provide students
with ‘real world’ experiences in schools or the workplace; and accountability.
However, this discussion will be limited to observations on technology, the
secondary/tertiary transition, the mathematical education of teachers, and research
in the teaching/learning of undergraduate mathematics.1
International Journal of Mathematical Education in Science and TechnologyISSN 0020–739X print/ISSN 1464–5211 online # 2005 Taylor & Francis Ltd
http://www.tandf.co.uk/journalsDOI: 10.1080/00207390412331317040
Email: [email protected] shorter version of this paper was presented at the 2004 ICME-10 Topic Study Group 3(TSG3): New Developments and Trends in Mathematics Education at Tertiary Level.
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2. Technology – an engine driving pedagogical change
Mathematicians have been teaching tertiary mathematics with modern technologyfor a while now. Graphing calculators go back to the early 1970s. Even the WorldWide Web (WWW) was available before the year 2000. However, the availabilityand use of the World Wide Web are proliferating exceedingly rapidly, evenin developing countries such as India and China. For example, in 2001, theMassachusetts Institute of Technology (MIT) in the US announced it would makevirtually all its courses freely available on the Web for non-commercial use. Also,enrolment in undergraduate and graduate degree programmes at the US-basedfor-profit University of Phoenix Online reached 50,000 in 2002. Such courses andprogrammes are available worldwide.
Are there any really new, smarter ways to use the WWW to enhance the teaching/learning of mathematics? Or, are traditional mathematics courses simply beingtransferred online to save money for university administrations, or to reach studentswho can’t travel to university campuses? While the first generation of Web courseswas often uninspired, more recent versions take fuller advantage of the Web’sunique capabilities (e.g. Java Applets, chat rooms, bulletin boards). Also, there existsa huge number of Web resources not tied to particular courses, ranging frominformational sites on history of mathematics and mathematical olympiads toexploration and demonstration sites.
The delivery of online courses will almost certainly continue to get moresophisticated and more interactive, something that is especially good for self-directedlearners and those who need ‘just in time’ information. While it is generally concededthat development of such courses takes much instructor time, they are not asephemeral as traditional classroom-delivered university courses because theyremain on the Web for some time. There have been suggestions for makingWeb-based courses much more interactive and responsive to students. For example,when video lectures, notes, and power point presentations are coordinated anddivided into smaller searchable clips, students can skip what they already knowor review topics not well understood. This, together with automated questionanswering and follow-up suggestions, provides a high degree of interactivity.(For information on computer science courses delivered in this way, see [1].)
2.1. Evaluating online mathematical courses and resources
Reliability of information is always an issue on the Web so knowledge ofwho is offering courses and how to evaluate them is critical. Engelbrecht andHarding [2, 3] provide information (including URLs), as well as a taxonomy, thatwould be especially helpful for those wanting to know ‘what’s out there’ math-ematically. More significantly, they have devised an interesting two-dimensional,visual way of displaying the scope and content of online mathematics courses.With this2 or any other system, there remains a question: Who will serve as the‘course critics’ to implement such a system? It is not just a matter of having anevaluation system, but also a question of who will be providing those evaluations.
2Engelbrecht and Harding’s intent was for designers of online course to use their system toevaluate the content and coverage of their own courses (Engelbrecht, personalcommunication), but the question remains.
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Another method, currently being implemented by the Mathematical Association
of America (MAA) for its recently published CUPM Curriculum Guide 20043 is a
set of supplementary Illustrative Resources on the Web. While exact details are still
being discussed, there will be an editorial board and an attempt to annotate and
evaluate, as well as just link to, mathematical sites. It is presumed that the
sponsorship of the MAA, the existence of an editorial board, and signed annotations
will provide a reasonable degree of dependability and accurate information.
2.2. Other technology for mathematics
The uses of technology in mathematics teaching extend well beyond online courses,
online degree programmes, and Web resources. There are also computer algebra
systems (CAS) and other mathematical software for computers, as well as various
graphing and other calculators. Are mathematicians making better use of CAS and
other software? Hillel [4, 5] provides an overview of how technology is being used in
beginning linear algebra courses. Technology can be used to compute (e.g. matrix
inverses are extraordinarily difficult to compute by hand), to reinforce, clarify,
anticipate, or get acquainted with ideas, and to discover and investigate phenomena.
While many of the uses of the computer are essentially computational, enabling
students to investigate problems involving ‘messy’ real world data, others are meant
to facilitate both procedural and conceptual learning of mathematical topics.
For example, Dubinsky [6] and colleagues make use of the computer, with appro-
priate pedagogy, well-structured materials, and the mathematical programming
language ISETL, to enhance undergraduate students’ understanding of difficult
abstract algebra concepts like quotient group [7, 8].
It is also clear that different implementations of a single software package
can lead to very diverse results – some intended, some unintended. Design
teams in the mathematics departments of two UK universities, dubbed NU and
SU, used Mathematica in an attempt to help new science and engineering entrants,
having widely varying mathematical backgrounds, ‘learn essential mathematical
skills and concepts’. At NU, the team produced some 80 to 90 screens of hypertext
information, intended to reduce the amount of programming expected, yet
containing examples of techniques; students were also given multiple-choice
exercises, scored by the computer. At SU students were presented carefully
sequenced tasks in Mathematica notebooks, along with some partially specified
‘templates’. The NU team wanted to make the computer more ‘friendly’, and
consequently, the Mathematica syntax was less accessible. While the NU approach
realized some success, it ‘did not (and did not try to) achieve the aim of
fostering a more general mathematical empowerment’. That aim was more in
line with the objectives of the SU design team and ‘they were partially successful’.
The aims, it seems, determined, at least to some extent, the student outcomes
(see [9]).
3The full CUPM Curriculum Guide 2004 can be found at www.maa.org/cupm/. The 22-pageExecutive Summary can be found at: http://www.maa.org/cupm/summary.pdf. TheCurriculum Guide itself was a project four years in the making; it began by holding a seriesof workshops with ‘client disciplines’ and soliciting discussion papers.
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3. The secondary/tertiary transition – what are the problems?
Are incoming tertiary students really different now? Has/Will the integrationof technology (e.g. every 7th grader in the US state of Maine is provided witha laptop) change the way students think? There are those who speak of distributedknowledge, pointing out that knowledge is a joint product of the tools used anda person’s mind. Has anyone investigated this purported change in students’cognitive processes? Are mathematicians just voicing the age-old complaint thatstudents of today aren’t like students of yesterday (i.e. like the mathematicians thinkthey were)?
One general complaint about incoming tertiary students is that their secondarymathematics learning is ‘surface learning’, whereas at university they are expected toengage in deep learning of concepts. In discussing the situation at Hong Konguniversities, Luk [10] sees four factors that currently exacerbate the often difficultsecondary/tertiary transition in mathematics: (1) less well-prepared, or perhaps morejob-oriented, incoming students; (2) the fast pace of topics in first-year mathematicscourses; (3) the mathematical rigour expected at university; and (4) the localexamination system.
3.1. Ideas that need reconceptualization or reorganization
Not only are incoming students less well-prepared, the pace faster, and more rigourexpected, but also at university, students frequently encounter new waysof conceptualizing previously well-known concepts – ways that can conflict withwell-honed prior mathematical practices. Such new ways of thinking often requirestudents to make quite difficult reconstructions of their mathematical knowledge.For example, in secondary school geometry, the tangent line to a circle is oftendefined to be that unique straight line that touches the circle at just one point andis perpendicular to the radius at the point of contact. However, upon coming tocalculus or beginning real analysis, the tangent to a function at a point is defined asthe limit of approximating secant lines, and somewhat later, as the line whose slopeis given by the value of the derivative at that point. Research in France by Castela(as reported in [11], pp. 209–210) found that many of the 372 secondary studentsquestioned, after having studied analysis for a year, had great difficulty determiningfrom a graph whether a given line was tangent to a particular function at aninflection point or a cusp.
Another example is provided by the treatment of equality in analysis. Whereasin secondary school algebra and trigonometry, students have grown used toproving that two expressions are equal by transforming one into another usingknown equivalences, in analysis one often proves two numbers a and b are equalby showing that for every >0, one has |a�b|<. That this idea is not an easy oneis indicated by a French study in which more than 40% of entering universitystudents thought that if the absolute value of the difference of two numbers A andB is less than 1/N for every positive integer N, then they are not equal, onlyinfinitely close [12, p. 1379].
3.2. Calls for more adequate preparation of secondary students
Calls for more adequate preparation of secondary students have been made for sometime. The question of whether any progress is being made in preparing secondary
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students for university, or even for work, was considered by two recent US studies,
one by the American Association of Universities and the other by the American
Diploma Project.4 These projects resulted in two reports: Standards for Success and
Ready or Not: Creating a High School Diploma That Counts. Standards for Success
identifies what students at US research universities must do in order to succeed in
six entry-level academic areas, including mathematics, whereas Ready or Not,
documents ‘must have’ competencies for secondary graduates ‘to succeed in post
secondary education or in high-performance, high-growth jobs’. As well as stressing
the importance of taking at least three years of algebra and geometry in secondary
school – a standard not yet reached for approximately 50% of US 18 year olds,
Standards for Success stresses ‘habits of mind’ such as making appropriate estimates,
using calculators accurately, and understanding the role of proof. In contrast,
for employment in today’s data rich economy, Ready or Not also emphasizes topics
such as data representation, correlation, sampling bias, and conditional probability,
as well as algebra and geometry. However, according to former MAA President
Lynn Steen [13] who reviewed these reports, ‘neither report adds much new to what
the authors of A Nation at Risk [14] said so forcefully 20 years ago’.
3.3. How are secondary students actually being prepared?
While reports like Standards for Success and Ready or Not call for more well-
prepared secondary students to enable success at both university and the workplace,
the actual facts ‘on the ground’ seem to indicate that expectations, including those in
mathematics, are staying the same or declining for secondary school students around
the world. For example, Nishimari [15] alludes to a cutting of 30% of the quantity
from the course of study in Japanese elementary and middle schools, as well as to
survey results showing a deterioration in attitudes and basic mathematical skills and
thinking abilities of entering Japanese tertiary students, as perceived by their
university mathematics teachers. Hoyles, Newman and Noss [16] considered the
UK situation and conducted a case study of a single university. While noting that
the conceptual gap between secondary and tertiary education exists but is not new,
they implicate the trend to a more utilitarian higher education in the UK and the
attempt by policymakers to increase the number of secondary students prepared
to enter university by allowing a single A-level mathematics examination for
mathematical, as well as other studies. The latter, while rendering mathematical
study ostensibly more accessible, is seen as exacerbating the secondary/tertiary
transition.
Recently, the US state of Florida, under pressure to reduce class size to no more
than 25 students, passed a law allowing secondary students to earn a fast-track
high school diploma in just three years, rather than four, by taking six fewer credits5
(courses) and also reduced the requirement for entering its state universities from
19 to 18 credits. This was done despite several years of complaints from university
and community college administrators that too many high school graduates were
4For further information, see http://www.s4s.org/understanding.php and http://www.achieve.org/achieve.nsf/AmericanDiplomaProject?openform.5One credit traditionally represents attendance at one scheduled 45-minute periodof instruction every day throughout a semester.
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entering unprepared for college work.6 The reasoning seems to be that the extra costof smaller classes might actually be offset by savings from having some students leavesecondary school earlier, albeit with a diminished diploma.
Similarly, the US state of Tennessee’s Higher Education Commission hasrecently reduced the number of credit hours for a bachelors degree from a previoushigh of 132 credits hours for some majors to exactly 120 for all majors in thehope that the change will increase graduation rates and thus help ease the financialstrain on the system.7
Given such reductions in numbers of course credits for graduation from bothsecondary school and university, one is tempted to ask: in response to budgetarypressures, as a cost cutting measure, is the secondary–tertiary transition sometimesbeing dealt with by reducing the mathematical content, and perhaps the quality,of both individual courses and entire programmes of study?
Perhaps to indicate the depth of the problem, in a recent report provocativelytitled A Nation at Rest: The American Way of Doing Homework [17], one finds thatdespite some popular talk-show worries about the ‘onslaught of homework’, themajority of US secondary and middle school students spend less than one hourper day on homework, regardless of age, and this has not increased in the last25 years, except for a few brief post-Sputnik years. With fewer hours required tograduate and most students spending less than one hour per day on homework,how can one expect students to be prepared for entrance to university?
3.4. How universities are dealing with the transition
McMaster University in Canada has dealt with this transition by: (1) preparinga Mathematics Review Manual for incoming students: (2) administering a Math-ematics Background Survey to incoming students in the first week of classes and(3) redesigning the first-year Calculus and other courses [18]. Attempts to deal withthe transition in Hong Kong include instituting ‘bridge’ courses and changing first-year mathematics courses to include more topics from secondary mathematics [10].It is further suggested that using the computer to make abstract mathematics moreconcrete, starting where the students are, and instituting more teacher-student andstudent-student communication might help.
Data from the University of the Witwatersrand in South Africa show thedecreasing number of mathematics majors and the increasing number of math-ematics minors [19]. One way this has been dealt with has been to develop a moreapplied version of that university’s mathematics programme that includes fewerabstract/theory courses and more computation and applied courses. One questionseems to be: how does one provide an appropriate curriculum for students who wishto join the workforce, rather than continue on to doctoral studies, especially whenno funds or rewards are provided? This brings up a longstanding, unansweredquestion: How should the ‘scholarship of teaching’ be rewarded?8
6For details, see http://www.cnn.com/2003/EDUCATION/09/08/early.graduation.ap/ orhttp://www.sptimes.com/2003/09/16/Opinion/Fast_track_to_nowhere.shtml.7See http://dailybeacon.utk.edu/article.php/12946.8For example, the University of Saskatchewan hosts The Scholarship of Teaching and Learningwebsite that discusses the legacy of Ernest Boyer, the person who is credited with introducingthe ideas of the scholarship of discovery, the scholarship of integration, the scholarship ofapplication, and the scholarship of teaching. See http://www.usask.ca/tlc/sotl/sotl_boyer.html.
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There is currently an international project that aims to analyse and documentthe secondary–tertiary transition from various perspectives and identify anddisseminate best practices in order to develop effective strategies for dealing withthe transition; its main instigator is Miroslav Lovric of Canada.
3.5. Two current contradictory trends
It seems as though there are two contradictory trends, or approaches, to thesecondary-tertiary transition coming from two different sources. On the one hand,the larger academic community advocates, through published reports and standards,that secondary school graduates be better prepared mathematically for bothuniversity and the workplace by taking more, and better, mathematics courses.However, the reality seems to be that political and budgetary pressures, as evidencedby the actions of some legislatures and administrative bodies, have reduced coursecredit requirements for both high school diplomas and university degrees. Whilesuch actions are often couched in positive terms of students being ready for theworkplace or a desire to increase graduation rates, the real reason appears to be tosave money. ‘Money in [US] state politics plays a pivotal role in shaping publicpolicy in individual states and across the nation’ [20].
4. Mathematics courses for pre-service teachers
While preparation of pre-service teachers has been a perennial concern, what seem tobe receiving more attention lately is the mathematical portion of that preparationand the concomitant responsibilities mathematicians share for that preparation.Perhaps galvanized by the 1999 publication of, and response by mathematicians to,the book Knowing and Teaching Elementary Mathematics: Teachers’ Understandingof Fundamental Mathematics in China and the United States by Liping Ma [21], theUS mathematical community has responded with new initiatives.
4.1. What mathematics should pre-service teachers know andhow should they know it?
In 2001, the Conference Board of the Mathematical Sciences (CBMS) in the US,published The Mathematical Education of Teachers (MET) report [22] calling fora ‘rethinking of the mathematical education of prospective teachers withinmathematical sciences departments at US two- and four-year colleges and univer-sities’. It went on to say ‘too many prospective teachers enter college (i.e. theirtertiary education) with insufficient understanding of school mathematics, have littlecollege instruction focused on the mathematics they will teach, and then enter theirclassrooms inadequately prepared to teach mathematics’. For example, the reportcalls for elementary teachers to have a deep understanding of place value, for middlegrade teachers to have a deep understanding of proportion, and high school teachersto have a deep understanding of functions. General recommendations include: thatmathematics courses for prospective teachers should develop the habits of mind of amathematical thinker; that mathematics in the middle grades (grades 5–8) should betaught by mathematics specialists; that prospective elementary teachers be requiredto take at least nine semester-hours on fundamental ideas of elementary school
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mathematics; that prospective middle grade teachers of mathematics should be
required to take at least 21 semester-hours of mathematics; and that prospective
high school teachers of mathematics should be required to complete the equivalent
of an undergraduate major in mathematics, including a 6-hour capstone course.
The MET report was addressed to mathematicians in mathematics depart-
ments, regarding the mathematics content that they teach to prospective teachers.
In order to address this report and prepare mathematicians to teach pre-service
teachers better, the Mathematical Association of America (MAA) is currently
conducting a multi-dimensional NSF-funded programme, called Preparing
Mathematicians to Educate Teachers (PMET), that includes faculty development
workshops for mathematicians, mini-grants for them to develop new courses
for preservice teachers, establishment of regional networks, and provision of
information and resources.9
4.2. Suggestions for specific changes in the teachingof mathematical content for teachers
Besides the very general recommendation that prospective teachers need math-
ematics courses that develop a deep understanding of the mathematics they
will teach, the MET report makes some very specific recommendations such
as, ‘Prospective teachers at all levels need experience justifying conjectures with
informal, but valid arguments if they are to make mathematical reasoning and proof
a part of their teaching. Future high school teachers must develop a sound
understanding of what it means to write a formal proof.’
That reform of teacher education must include the mathematics content
courses is also emphasized by Wittmann [23] who notes that currently, in such
content courses, relevant subject matter for teaching may not be covered at all or
the presentations are too formalistic. He suggests that what is needed is the doing
of mathematics, not the presentation of ready-made mathematics. Wittmann
describes two ‘operative proofs’ in elementary number theory using non-formal
representations such as the number line, patterns of dots, and tables. While such
proofs may appear clumsy to a professional mathematician, they are useful for
acquainting student teachers with both non-formal representations and
non-formal means of communication that will serve them well with their future
pupils. Wittmann’s operative proofs bear a certain resemblance to ‘generic
proofs’10 in number theory for prospective secondary teachers as described by
Rowland [24], the difference perhaps being Wittmann’s suggested use of
non-formal representations.
Another suggestion for increasing student interest is to include more math-
ematical modelling in courses for teachers of all levels, including university teachers.
This has been tried with varying success using workshops for secondary and
university mathematics teachers in Uruguay [25]. Lesh and Doerr [26] in a recent
edited book, also suggest that students at all levels need to engage in model
9Information on PMET can be obtained from the project’s home page at: http://www.maa.org/pmet/.10The classic proof that
ffiffiffi
2p
is irrational is a ‘generic proof ’ since it immediately generalizes tothe proof that
ffiffiffi
pp
is irrational for any prime p.
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development, and for that purpose, teachers need model-eliciting activities,11 that is,
mathematically rich yet realistic and meaningful situations for which students
express, test, and revise their thinking.
Teachers also need to be adept at both routine and non-routine problem solving
in a variety of content areas. Gorev, Gurevich and Barabash [28] report success
in getting students to solve non-routine geometry problems using dynamic geometry
environments, provided they matched the computer tool to the students’ abilities,
both mathematically and computationally. The stronger students (i.e. higher Van
Hiele levels) were able to recall properties of geometrical figures and put them
together to solve non-routine problems, whereas the weaker students ‘profited from
the possibility to visualize parameters essential to understand the problem’.
School teachers, especially secondary teachers, need to know the role that formal
definitions occupy within mathematics. Too often pre-service teachers and other
majors recite definitions, yet fail to use them when asked to solve problems or prove
theorems. As Edwards andWard [29] point out, the distinction between mathematical
definitions (also referred to as stipulated or analytic definitions) and many dictionary
definitions (also referred to as descriptive, extracted, or synthetic definitions) need to
be made clear. In addition to merely explaining definitions, one can engage students
in the defining process. For example, when using Henderson’s investigational geo-
metry text [30], one can begin with a definition of triangle initially useful in the
Euclidean plane, on the sphere, and on the hyperbolic plane, but eventually students
will notice that the usual side-angle-side theorem (SAS) is not true for all triangles on
the sphere. At this point, they can be brought to see the need for, and participate in
developing, a definition of ‘small triangle’ for which SAS remains true on the sphere.
The mathematical education of teachers should include specific mathematical
content knowledge, learned in a way so as to be both relevant to, and go beyond, the
level of their pupils. The case study of Ms. Daniels (pseudonym), who was a
mathematics major before deciding to become a middle school teacher and found
herself unable to explain to her pupils why the ‘invert-and-multiply’ rule holds, as
well as where � comes from, is indicative of the problem [31, 32]. The question is: are
traditional mathematics courses, intended either for those going on to become
research mathematicians or for those preparing to become employed in business,
serving pre-service teachers well? It seems that, at least for middle school teachers,
some mathematics departments have decided the answer is no. Some university
mathematics departments are beginning to develop specialized courses and pro-
grammes especially designed for middle school mathematics teachers (e.g. Arizona
State University and NewMexico State University in the US).12 It remains to be seen
11In addition to espousing this view, the Lesh and Doerr book [25] takes a very broad view ofmodelling. As well as discussing modelling and model-eliciting activities for students, there arechapters on modelling the learner, modelling teacher development, teachers’ models of teach-ing/learning, etc. Similarly, Leikin [27] has models of teachers’ interactions with their mentors,allowing construction of teacher profiles, and models of teachers’ flexibility in real classroomsituations.12The University of North Carolina at Chapel Hill in the US, through its School of Education,will offer an M.Ed. in (K-8) Mathematics Education. For a description of their existing M.Ed.in (K-5) Mathematics Education, with such content courses as Revisiting Real Numbers,Topics in Geometry, and Topics in Algebra and Algebraic Reasoning, see http://www.unc.edu/depts/ed/med_exp/K-5mathematics.html.
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how widespread and effective individual university programmes, and the efforts ofmathematical organizations like the MAA through its PMET initiative, will be inpreparing ‘knowledgeable, flexible teachers who are able to effectively educatestudents in mathematics, using a variety of current and future mathematicscurricula’ [22].
5. Research into the teaching and learning of tertiary mathematics
Lastly, we address what this author foresees, and hopes, will be a trend, namely thatmore teachers at the tertiary level will become ‘informed by’ and ‘use’ results fromsuch research in their teaching. While research in undergraduate mathematicseducation (RUME) is a relatively young field, there are some ideas/results thatcan already inform mathematicians. While some research is mainly empirical [31, 32],other research is more theoretical. For example, after closely examining howmathematicians solve problems, Carlson and Bloom [33] proposed a detailedmulti-dimensional problem-solving framework, amplifying the work of Schoenfeld[34]. Other research is devoted to individual cognitive questions like: why arethe concepts of prime and irrational hard for students [35, 36]? Still other researchersinvestigate the social and cognitive aspects of learning in university classroomssuch as differential equations [37].
Some research can shed light on developments discussed in this essay. Forexample, some research concerns the uses of technology in curricula, askingquestions like: how is the use of dynamic geometry software related to the vanHiele levels of students [28]? Other research has focused on the secondary/tertiarytransition. For example, Hoyles, Newman and Noss [16], considered the UKsituation and conducted a case study at one university; they noted factors thathave exacerbated the transition, namely the trend to a more utilitarian highereducation and the attempt by UK policymakers to increase the number of secondarystudents prepared to enter university by instituting a single A-level mathematicsexperience. In the US, a large NSF-funded project [38] is studying transitions froma more student-centred point of view, investigating how students experience andcope with the transition (1) from a reform secondary curriculum called Core-PlusMathematics13 to traditional university precalculus and calculus, and (2) fromtraditional secondary programmes to the reform calculus known as HarvardConsortium Calculus. This project is considering students’ achievement, theirlearning of key concepts, their daily experience, their career and educational goals,their beliefs about mathematics and themselves as learners, and their strategiesof adjustment.
5.1. What such research can tell mathematicians
Often people regard as ‘easy’ or ‘natural’ what they have already mastered, havingforgotten their own sometimes tortuous path to expertize; this seems especiallytrue of mathematical concepts. For example, one mathematics department chair
13For details, see http://www.wmich.edu/cpmp/.
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said to a colleague who specializes in RUME, ‘I can’t understand why calculusis hard.’
Yet consider the following two parallel questions:
. What makes new mathematics research hard for mathematicians? Surely theyare not just lazy (as is sometime claimed of students).
. What makes new mathematical topics hard for students? Surely, for the mostpart, they are not just lazy.
Certain glib answers that one sometimes hears from some tertiary mathematicsteachers can’t be the entire answer: Students are lazy. They don’t do their homework.They work too much at jobs to support their cars and apartments and don’t havetime left over to study.
When students are studying calculus, and other mathematical topics, for the firsttime, it should be clear that they are not already experts. Many topics in calculus,and related areas, give students difficulty and have been investigated – variable [39],function [40, 41], limit [42, 43], derivative [44], and integral [45]. And it is not justcalculus that causes students difficulties. More advanced mathematical topics,perhaps not surprisingly, cause difficulties often related to their abstraction andhave been investigated. These include linear algebra [46], differential equations[47, 48], abstract algebra [6] and real analysis [49]. Beyond describing students’conceptual difficulties, there has been research aimed at measuring students’understandings in the context of non-routine problem solving [50].
Furthermore, in addition to understanding specific facts and concepts, there arevarious thinking processes (a.k.a., habits of mind, mathematical practices) thatmathematics students need to learn. Some of these processes have been investigatedfor both students and mathematicians: mathematical problem solving [33, 34],proof and proving [51], logical thinking [52, 53], changing representations [54],and advanced mathematical thinking [55]. As well as research on students’ cognitivedifficulties and mathematical habits of mind, there is some research on effect – oftenunderstood to include beliefs, attitudes, and emotions – as it relates to both teachingand mathematical problem solving [56].
While RUME is still a young field, there are insights and concepts from whichmathematicians who teach at tertiary level might well benefit. In addition tospecific insights about which topics students find difficult and why, a majorcontribution of mathematics education research has been to provide new con-ceptualizations and new metaphors for thinking about and observing mathematicalbehaviour [57]. It is very difficult to notice patterns of behaviour or thoughtwithout having names (and the corresponding concepts) for them. One needs a lenswith which to focus on what one is seeing. For example, there is often a mismatchbetween mathematical concepts as stated in definitions and as interpreted bystudents. Further, even university mathematics students seem unaware of therole definitions play in mathematics [29]. The terms concept definition and conceptimage were introduced into the mathematics education literature to distinguishbetween a formal mathematical definition and a person’s ideas about a particularmathematical concept, such as function. An individual’s concept image is a mentalstructure consisting of all of the examples, nonexamples, facts, and relationships,etc. that he or she associates with a concept. One’s concept image need not, butmight, include the formal mathematical definition and appears to play a major rolein cognition. (See [58].)
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5.2. What such research cannot tell mathematicians
Mathematics education research does not usually provide immediately applicableprescriptive teaching information; rather it provides general guidance and hints thatmight help with teaching and curriculum design. For example, it cannot tellmathematicians which curricula or pedagogies are ‘best’ because such decisionsnecessarily involve value judgments about what is important for students to know[59]. To take a particular instance, if engineering students use a CAS such asMathematica or Maple, research cannot tell mathematicians which calculusand matrix algebra computations students should perform flawlessly by hand.It might, however, advise mathematicians on the extent to which one’s chosencurricula, once implemented, have succeeded in attaining one’s intended goals.For example, the Noss [9] study of NU and SU did some of this.
Some questions that mathematicians and administrators would like to haveanswered are not easily amenable to research. Here are some important questionsthat this author has been asked by mathematicians that affect their teaching, yet thatare hard to frame as research questions.
. Is lecturing bad? What is the evidence?
. Are small classes better than large classes? What is the evidence?
Unfortunately, such questions are too broad and ill-specified for research to answer.Referring to the first question, of course, lecturing is not bad in every situation –it depends on one’s goals for the course, it depends on how self-motivated (to learnoutside of lecture) the students are, and it might even depend on whether anappropriate text is available. In order to get answers, it might be better to linkgoals with pedagogical strategies; that is, to ask for which kinds of goals (e.g. solvingnonroutine problems, creating original proofs, competently executing computationaltechniques, finding relevant information in books when solving problems) are whichkinds of pedagogical strategies (e.g. cooperative learning, project work, whole groupdiscussion) especially suited? For example, there are some quite general results on thepositive effects of small-group work for science, mathematics, engineering, andtechnology courses at university [60].
As for the second question, it depends on what the teacher does in the smallclasses. If he/she does exactly the same thing in a small class as in a large class(e.g. just lecture), the results are unlikely to be very different. And at least onedissertation study has found this: Warren [61] found no significant difference on testsand final examination scores, the final grade averages, or the attrition rates whenfirst-semester calculus was taught by tenured university faculty members in classsizes of 87 versus 25.
Mathematicians have interesting, important questions about their teaching, butsuch questions often need to be refined in order to make them amenable to research.Observations from teaching may sometimes be discounted as only anecdotal, butthese can be important sources of possible insights and questions for research. It isimportant to keep the lines of communication open between mathematicians andmathematics education researchers.
5.3. Is anyone paying attention to such research?
That both the mathematical research and teaching communities are beginningto pay attention to mathematics education research is evidenced by recent
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articles in, for example, the Notices of the American Mathematical Society [12, 62],
the American Mathematical Monthly [29, 63], and the Irish Mathematical Society
Bulletin [64]. Furthermore, some mathematicians have designed curriculum
segments [65], or even entire curricula [7], using information and ideas gleaned
from mathematics education research.
In addition to publications in journals of mathematical societies, there is a
Special Interest Group of the Mathematical Association of American (SIGMAA
on RUME), whose purpose is ‘to foster research on learning and teaching
undergraduate mathematics and to provide a support network for those who
participate in this area of research. . . . [It] also welcomes members who teach post
secondary mathematics or are interested in the findings of RUME; such
members often provide critical assistance in the grounding of research efforts in
the realities of the classroom’.14 Another indication of the interest of mathema-
ticians is sponsorship of a panel on mathematics education research at the recent
7th Annual Legacy of R. L. Moore Conference. These conferences and the
associated website promote and disseminate information on the Moore Method15
and other inquiry-based teaching. One of the panelists at the conference was a
mathematics education researcher who is currently conducting an in-depth study
of one Moore Method course in number theory. These conferences are an
attempt to retain ‘what works’ and the researcher is attempting to understand
why it works.
Yet despite an increasing interest in RUME, many findings are neither easily
accessible to mathematicians nor ‘user-friendly’ in the sense that one can easily
draw out teaching hints and implications. As with all research, convincing
evidence for one’s findings must be presented. While this is important for
establishing the credibility of the work, these ‘get in the way’ of mathematicians
trying to find out ‘what’s out there’. While there have been some attempts to
bring this research to mathematicians in an easy-to-read expository form,16 more
are needed.
14For further information, see the MAA-hosted SIGMAA on RUME websiteRUMEonline! at: http://www.rume.org/. It contains information about the organization, aswell as a literature database, links to publishers, and other resources.15The ‘Moore Method’ developed out of the teaching practices of a single accomplishedUS mathematician, R. L. Moore, and has been continued by his students (several ofwhom went on to become presidents of the American Mathematical Society or theMathematical Association of America) and their mathematical descendents. It has beenremarkably successful in producing research mathematicians, but has also been used inundergraduate mathematics classes. In many versions, students are given definitionsand statements of theorems or conjectures and asked to prove them or provide counterex-amples. The teacher provides the structuring of the material and critiques the students’efforts, but does not lecture. While individual teachers have described their experiences[66], up to now this method has not been subjected to in-depth analysis. For moreinformation, see the Legacy of R. L. Moore website at: http://www.discovery.utexas.edu/rlm/.16See, for example, the Research Sampler columns on MAA Online at http://www.maa.org/t_and_l/sampler/research_sampler.html. In addition, SIGMAA on RUME is currentlyputting together an MAA Notes volume, tentatively called Making the Connection:Research and Teaching in Undergraduate Mathematics, that will gather together a variety ofexpository articles on RUME.
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6. The future
Will there be more of the same? Will mathematicians teach smarter or better?In mathematics education in general, and in tertiary mathematics educationin particular, it seems that changes are more evolutionary than revolutionary,more reactive (e.g. to changes in technology) than innovative. The mathematicalcommunity should surely keep ‘what works’ (provided there is evidence that it does),while responding to new pedagogical challenges and technological tools, and changewhat doesn’t work. So, mathematics education researchers, together with mathema-ticians, need to ask themselves, what works and why? This is where mathematicseducation research can play a potentially useful role in helping answer suchquestions.
It seems certain that technology will change. Whether mathematics professorsand teachers will use changes in technology wisely remains to be seen. As for thesecondary/tertiary transition, given the two opposing forces of maintaining stan-dards and of saving money by reducing courses and programmes, almost anythingcan happen. In many USA universities, the mathematical content education ofpre-service and in-service teachers can only get better. However, at least in the USA,the turnover of in-service teachers is said to be about once every five years, so unlessmathematicians do a better job of providing appropriate content courses forpre-service teachers, providing professional development for in–service teacherswill be an ongoing task. And research in undergraduate mathematics educationwill continue,17 and hopefully, provide teaching hints and insights thatmathematicians will pay some attention to.
References
[1] Zhang, D., Zhao, J.L., Zhou, L. and Nunamaker, Jr, J.F., 2004, Communications ofthe ACM, 47(5), 75–79.
[2] Engelbrecht, J. and Harding, A., 2003, Journal of Applied Mathematics and Physics, 32,Supplemenatry Issue, 57–65.
[3] Engelbrecht, J. and Harding, A., 2004, Taxonomy of online undergraduatemathematics courses. Paper presented to the Tenth International Congress onMathematical Education, Topic Study Group 3 (Copenhagen, Denmark). http://www.icme-organisers.dk/tsg03/ or http://science.up.ac.za/sams/.
[4] Hillel, J., 2004, Trends in the teaching of linear algebra and the role of technology.Paper presented to the Tenth International Congress on Mathematical Education,Topic Study Group 3 (Copenhagen, Denmark). http://www.icme-organisers.dk/tsg03/.
[5] Hillel, J., 2001, In: D. Holton (Ed.) The Teaching and Learning of Mathematics atUniversity Level: An ICMI Study (Dordrecht: Kluwer Academic Publishers), pp. 371–380.
[6] Dubinsky, E. (Guest Ed.), 1997, Special Issue: Journal of Mathematical Behavior, 16(3),181–309.
[7] Dubinsky, E. and Leron, U., 1994, Learning Abstract Algebra with ISETL (New York:Springer-Verlag).
[8] Leron, U. and Dubinsky, E., 1995, The American Mathematical Monthly, 102, 227–242.[9] Noss, R., 1998, International Journal of Mathematical Education in Science and
Technology, 30, 373–388.
17For some thoughts on the future directions of tertiary mathematics education research, seeSelden and Selden [67].
144 A. Selden
Dow
nloa
ded
by [
Prin
ceto
n U
nive
rsity
] at
21:
04 2
4 A
ugus
t 201
3
[10] Luk, H.S., 2004, The gap between secondary school and university mathematics.International Journal of Mathematical Education in Science and Technology, 36(2–3),159–172.
[11] Artigue, M., 1992, In: W. Geeslin and K. Graham (Eds) Proceedings of the SixteenthConference of the International Group for the Psychology of Mathematics Education, Vol. 3(Durham, NH: University of New Hampshire), pp. 195–216.
[12] Artigue, M., 1999, Notices of the American Mathematical Society, 46, 1377–1385.[13] Steen, L., 2004, Education Week, 23(30), 34, 36.[14] A Nation at Risk: The Imperative for Educational Reform, 1983 (Washington, D.C.:
The Commission on Excellence in Education.) Also available online at http://www.ed.gov/pubs/NatAtRisk/.
[15] Nishimori, T., 2004, Paper presented to the Tenth International Congresson Mathematical Education, Topic Study Group 3 (Copenhagen, Denmark). http://www.icme-organisers.dk/tsg03/.
[16] Hoyles, C., Newman, K. and Noss, R., 2001, International Journal of MathematicalEducation in Science and Technology, 32, 829–845.
[17] Gill, B.P. and Schlossman, S.L., 2003, Educational Evaluation and Policy Analysis, 25,319–337.
[18] Kajander, A. and Lovric, M., 2005, Transition from secondary to tertiary mathematics,McMaster University experience. International Journal of Mathematical Education inScience and Technology, 36(2–3), 147–158.
[19] Hockman, M., 2005, Curriculum design and tertiary education. International Journal ofMathematical Education in Science and Technology, 36(2–3), 173–189.
[20] INSTITUTE ON MONEY IN STATE POLITICS, 2004. Why Look at Money in StatePolitics? http://www.followthemoney.org/.
[21] Ma, L., 1999, Knowing and Teaching Elementary Mathematics: Teachers’ Understandingof Fundamental Mathematics in China and the United States (Mahwah, NJ: LawrenceErlbaum Associates).
[22] CONFERENCE BOARD OF THE MATHEMATICAL SCIENCES, 2001,The Mathematical Education of Teachers (Providence, RI: American MathematicalSociety).
[23] Wittmann, E., 2004, Learning mathematics for teaching mathematics: The notionof operative proof. Topic Study Group 3, ICME 10, http://www.icme-10.dk.
[24] Rowland, T., 2002, In: S.R. Campbell and R. Zazkis (Eds) Learning and TeachingNumber Theory: Research in Cognition and Instruction (Westport, CT: Ablex Publishing),pp. 157–183.
[25] Martinez-Luaces, V.E., 2005, Engaging secondary school and university teachersin modelling: some experiences in South American countries. International Journal ofMathematical Education in Science and Technology, 36(2–3), 191–203.
[26] Lesh, R. and Doerr, H. (Eds), 2003, Beyond Constructivism: Models and ModelingPerspectives on Mathematics Problem Solving, Learning, and Teaching (Mahwah, NJ:Lawrence Erlbaum Associate).
[27] Leikin, R., 2005, Qualities of professional dialogue: connecting graduate research onteaching and the undergraduate teachers programme. International Journal ofMathematical Education in Science and Technology, 36(2–3), 233–252.
[28] Gorev, D., Gurevich, I. and Barabash, M., 2005, The computer as an aid in thedevelopment of geometrical proficiency: a differential approach. International Journal ofMathematical Education in Science and Technology, 36(2–3), 285–300.
[29] Edwards, B.S. and Ward, M.B., 2004, The American Mathematical Monthly, 111,411–424.
[30] Henderson, D.W., 2001, Experiencing Geometry in Euclidean, Spherical and HyperbolicSpaces, 2nd Edn (Upper Saddle River, NJ: Prentice Hall).
[31] Borko, H., Eisenhart, M., Brown, C.A., Underhill, R.G., Jones, D. and Agard, P.C.,1992, Journal for Research in Mathematics Education, 23, 194–222.
[32] Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D. and Agard, P., 1993,Journal for Research in Mathematics Education, 24(1), 8–40.
[33] Carlson, M.P. and Bloom, I., 2005, Educational Studies in Mathematics, 58(1), 45–76.[34] Schoenfeld, A.H., 1985, Mathematical Problem Solving (Orlando, FL: Academic Press).
Developments and trends in tertiary mathematics education 145
Dow
nloa
ded
by [
Prin
ceto
n U
nive
rsity
] at
21:
04 2
4 A
ugus
t 201
3
[35] Zazkis, R., 2005, Representing numbers: prime and irrational. International Journal ofMathematical Education in Science and Technology, 36(2–3), 205–215.
[36] Zazkis R. and Liljedahl, P., 2004, Journal for Research in Mathematics Education, 35,164–186.
[37] Yackel, E., Rasmussen, C. and King, K., 2000, Journal of Mathematical Behavior, 19,275–287.
[38] Star, J., Berk, D., Burdell, C., Hoffmann, A.J., Lazarovici, V., Lewis, G. and Smith, J.,2001, How can we study them? In: R. Speiser, C.A. Maher and C.N. Walter (Eds)Proceedings of the Twenty-Third Annual Meeting, North American Chapter ofthe International Group for the Psychology of Mathematics Education (Columbus,OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education),pp. 117–128.
[39] Trigueros, M. and Ursini, S., 2003, In: A. Selden, E. Dubinsky, G. Harel and F. Hitt(Eds) Research in Collegiate Mathematics Education V (Providence, RI: AmericanMathematical Society), pp. 1–29.
[40] Carlson, M.P., 1998, In: A.H. Schoenfeld, J. Kaput and E. Dubinsky (Eds) Research inCollegiate Mathematics Education III (Providence, RI: American Mathematical Society),pp. 114–162.
[41] Thompson, P.W., 1991, In: E. Dubinsky, A.H. Schoenfeld and J. Kaput (Eds) Research inCollegiate Mathematics Education I (Providence, RI: American Mathematical Society),pp. 21–44.
[42] Sierpinska, A., 1985, Recherches in Didactique des Mathematiques, 6(1), 5–68.[43] Williams, S.R., 1991, Journal for Research in Mathematics Education, 22, 219–236.[44] Zandieh, M.J., 2000, In: E. Dubinsky, A.H. Schoenfeld and J. Kaput (Eds) Research in
Collegiate Mathematics Education IV (Providence, RI: American Mathematical Society),pp. 103–127.
[45] Thompson, P.W., 1994, Educational Studies in Mathematics, 26, 229–274.[46] Dorier, J.-L. (Ed.), 2000, On the Teaching of Linear Algebra (Dordrecht: Kluwer
Academic Publishers).[47] Artigue, M., 1992, In: G. Harel and E. Dubinsky (Eds) The Concept of Function: Aspects
of Epistemology and Pedagogy (Washington, DC: Mathematical Association of America),pp. 109–132.
[48] Rasmussen, C.L., 2001, The Journal of Mathematical Behavior, 20(1), 55–87.[49] Alcock, L. and Simpson, A., 2001, In: D. Holton (Ed.) The Teaching and Learning of
Mathematics at University Level: An ICMI Study (Dordrecht: Kluwer AcademicPublishers), pp. 99–111.
[50] Kannemeyer, L., 2005, A reference framework for measuring student’s understandingin a first year calculus course. International Journal of Mathematical Education in Scienceand Technology, 36(2–3), 267–283.
[51] Weber, K., 2001, Educational Studies in Mathematics, 48(1), 101–119.[52] Selden, J. and Selden, A., 1995, Educational Studies in Mathematics, 29, 123–151.[53] Dubinsky, E. and Yiparaki, O., 2000, In: E. Dubinsky, A.H. Schoenfeld and J. Kaput
(Eds) Research in Collegiate Mathematics Education IV (Providence, RI: AmericanMathematical Society), pp. 239–289.
[54] Duval, R., 1995, Semiosis et Pensee Humaine. Registres Semiotiques et ApprentissagesIntellectuels (Berne: Peter Lang).
[55] Selden, A. and Selden, J. (Guest Eds), 2005, Mathematical Thinking and Learning,7(1), 1–73.
[56] Liljedahl, P., 2004, Mathematical discovery and affect; The effect of AHA! experienceson undergraduate mathematics students. International Journal of Mathematical Educationin Science and Technology, 36(2–3), 217–232.
[57] Davis, R.B., 1990, In: G. Booker, P. Cobb and T. N. de Mendicuti (Eds) Proceedings ofthe Fourteenth Conference of the International Group for the Psychology of MathematicsEducation (Mexico City: CINVESTAV), 1–24.
[58] Tall, D. and Vinner, S., 1981, Educational Studies in Mathematics, 12, 151–169.[59] Hiebert, J., 1999, Journal for Research in Mathematics Education, 30, 3–19.[60] Springer, L., Stanne, M.E. and Donovan, S.S., 1999, Review of Educational Research,
69(1), 21–51.
146 A. Selden
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Prin
ceto
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] at
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3
[61] Warren, E.N., 1988, The association between class size, achievement, and opinionsof university students in first-semester calculus, PhD Dissertation, The Universityof North Texas.
[62] Schoenfeld, A.H., 2000, Notices of the American Mathematical Society, 47, 641–649.[63] Epp, S.S., 2003, The American Mathematical Monthly, 110, 886–899.[64] Meehan, M., 2002, Irish Mathematical Society Bulletin, 49, 71–82.[65] Safuanov, I.S., 2004, The genetic approach to the teaching of algebra at university.
International Journal of Mathematical Education in Science and Technology, 36(2–3),253–266.
[66] Jones, F.B., 1977, The American Mathematical Monthly, 84(4), 273–278.[67] Selden, A. and Selden, J., 2001, In: D. Holton (Ed.) The Teaching and Learning of
Mathematics at University Level: An ICMI Study (Dordrecht: Kluwer AcademicPublishers), pp. 237–254.
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