Research in Scientific Computing in Undergraduate Education

06. Teaching mathematics interactively

Preliminary Research Proposal accepted for RUME conference

Gary Davis and Sigal Gottlieb had their preliminary research proposal  accepted for the Mathematical Association of America’s Research in Undergraduate Mathematics Education (RUME) conference, Raleigh, NC, February 28-March 1, 2008.

Here is a copy (with identfiying details edited out, for purposes of review):

Preliminary Research ReportInteractive teaching and computational mathematics: Promoting mathematical conceptualization and competence


We report on an innovative methodology and analytical approach to study the effects of interactive teaching of computationally oriented undergraduate mathematics. Previous studies indicate that a more interactive approach to teaching has a number of positive effects on student motivation, engagement and learning. Emphasizing computational aspects of mathematics in undergraduate mathematics courses provides students with the opportunity to experiment, the need to analyze and compare algorithms, and data to reflect upon. Web-based blogging software allows students to develop their mathematical writing to be more in line with professional scientific standards. The significance of focusing on the conjunction of interactive teaching and computational mathematics is that it is vital that mathematics majors are motivated to learn and understand at a deep level the new mathematics required of them as successful twenty-first century professional mathematicians, scientists, and engineers. We describe the data we are currently collecting and a number of research issues to be addressed by this data.


We discuss research questions related to two significant issues in contemporary undergraduate mathematics education: the need for, and usefulness of, a more interactive approach to teaching and learning mathematics, and the increasing importance of a computational approach to mathematics. The relevance of both these issues has been discussed in recent literature. Interactive mathematics, science and engineering teaching is teaching that deemphasizes lecturing to, and talking at, students, and engages them maximally in active participation in problem solving or project work, usually carried out in groups or teams. Several studies have noted the beneficial aspects of interactive teaching, including longer and better student discussion, student willingness to volunteer answers and ask questions, increased student confidence in their own views and answers, more thorough and longer-lasting understanding, increased potential to respond to individual student needs, increased conceptual understanding, and increased ability to apply knowledge in novel settings (Bonwell & Eison, 1991; Rodger, 1995; Hake, 1998; Croft & Ward, 2002; Dijk & Jochems, 2002; Cahyadi, 2004, 2007).

A computational approach to mathematics has increased dramatically in breadth and depth in recent years (Corless & Jeffery, 1997; Meza, 2007; Yang, 2008). This is due both to the availability of more powerful hardware and dedicated software, and to the development of new and powerful algorithms. The computational revolution is impacting many aspects of mathematics: discrete mathematics, particularly graph theory; algebra; number theory; differential equations; partial differential equations; and statistics, to name some of the more prominent.


Computational mathematics involves students working on modern computing machinery, and using up to date mathematical software, typically MATLAB, Maple, Mathematica, or open source software such as Octave or Sage. Additionally, we have begun to experiment with Web-based blogging software. Weblogs, or blogs, for short, were established to facilitate the exchange of news via the Web, often on a daily basis. There are many blog hosts, and we chose to use Word Press ( principally because this open-source site offers full support for AMS-LaTeX. Additionally, Word Press allows extensive use of graphics and embedding of videos (see, as an example, our website: web address).

Weblog software provides students with a working space, available to them, individually, to us as instructors, and to other students, where they can compose mathematics that, thanks to LaTeX, actually looks like mathematics. Part of our theoretical perspective on engaging students is that the act of writing mathematics, and refining that writing for assessment and for sharing with other students, will promote deeper understanding of the mathematics, as well as a greater involvement in it, a higher degree of ownership of the work, and greater sharing and discussion of work by students (Bonwell & Eison, 1991; Sterrett, 1992). Computation gives students something to do (compute) and then something to write about (their computations).


Can computational experience, in which students carry out computations to selected problems, coupled with writing about those computations in a professional mathematical format, using LaTeX, promote deeper mathematical conceptualization and competence? Given a broadly, rather than narrowly, defined task can students utilize computation to appropriately address the task? Does the act of writing mathematics on a public Weblog, using LaTeX, promote a higher standard of mathematical writing, and is this writing indicative of a higher standard of mathematical engagement and competence than is usual? For which students is this combination of computation and interactive teaching most productive, and under what circumstances?


To this point we have data from several classes of students. This data is in the form of student writing and edits in several classes: numerical analysis (web address), differential equations (web address), and mathematical inquiry (web address). Preliminary analysis of this data indicates a higher degree of engagement and motivation to finish assignments completely for most, but not all students, and to do this more accurately, than is typical of student work we have experienced in recent years, for which we also have written data. This is in line with several of the studies we cited in the Introduction. Additionally we are noticing the following features:

Students are not leaving class at the appointed time; typically they leave only when the next instructor enters the classroom. This is in marked contrast to previous semesters in which students seem to have an instinctive appreciation of the end of class, and act accordingly. Students are sharing work – walking around in class to look at other student writing, or going on-line to view other student work. Students are learning from, but not directly copying, each other. The need to produce a written product, using LaTeX for the mathematics, and incorporating graphics, in a coherent story appears to focus students on an understanding of the depth or lack of it, of their own knowledge. Most often they take it upon themselves to find out what they do not know, or ask a question of the instructor: “What’s an eigenvalue, again?” “How do you do a matrix in LaTeX?” for example. Students are paying attention when the instructor speaks. This is in marked contrast to other semesters in which many students will typically be inattentive, passive, or off-task. This observation may have to do with the lower frequency with which an instructor talks in this more interactive approach to teaching. That is something we will measure in future semesters.By the end of Fall semester we will have collected data relating to student perception of this more interactive approach to teaching computationally based mathematics, as well as begun the analysis of the depth of student work. To this point we are noticing a marked qualitative difference in student engagement and understanding. The slower pace of a class, induced by students actively working and writing each class period, seems to promote a deeper, more reflective and careful approach to both doing and understanding mathematics. Documenting and providing evidence for that is our next task.


To this point we are the only two faculty in the Department of Mathematics utilizing the interactive approach we have outlined here. Other faculty have shown an interest in our approach. Our aim is to continue to share student work with our colleagues – always available at the click of a mouse – and to get them to come to our classes and participate informally when they show interest. Currently we are confident that the results of our approach are strong enough to spread to other faculty, Whether and how this approach infuses through the Department by example is another thing we will obtain data for, and analyze at a later date.REFERENCES

Bonwell, C. C. & Eison, J. A. (1991) Active Learning: Creating Excitement in the Classroom. The George Washington University, School of Education and Human Development, Washington, DC, ASHE-ERIC Higher Education Report No.1.Cahyadi, V. (2004) The effect of interactive engagement teaching on student understanding of introductory physics at the Faculty of Engineering, University of Surabaya, Indonesia. Higher Education Research and Development, 23(4), 455-464.Cahyadi, V. (2007) Improving teaching and learning in introductory physics. Ph.D. thesis. University of Canterbury, Christchurch, New Zealand.Corless, R.M. & Jeffery, D.J. (1997) Scientific computing: One part of the revolution. Journal of Symbolic Computation, 23 (5-6), May, Pages 485-495Croft, A. & Ward, J. (2002) A modern and interactive approach to learning engineering mathematics. British Journal of Educational Technology, 32(2), 195 – 207.

Hake, R. (1998) Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses. American Journal of Physics, 66(1), 64-74.

Meza, J. (2007) Computational mathematics: Role, impact, challenges. Lawrence Berkeley National Laboratory. Berkeley, CA.

Rodger, S. H. (1995) An interactive lecture approach to teaching computer science. ACM SIGCSE Bulletin, 27(1), 278 – 282.

Sterrett, A. (Ed.) (1992) Using Writing to Teach Mathematics. MAA Notes, Volume 16. Washington, DC: The Mathematical Association of America.

Van Dijk, L. A. & Jochems, W. M. G. (2002) Changing a traditional lecturing approach into an interactive approach: Effects of interrupting the monologue in lectures. International Journal of Engineering Education, 18(3), 275-284.

Yang, X. S. (2008) Introduction to Computational Mathematics. World Scientific Publishing.

John Seely Brown: Teaching 2.0 – Doing More With Less

Teaching Mathematics Interactively: Gary E. Davis

Teachers of college level mathematics, even at elite institutions, typically stand in front of a board (black or white) or project an electronic image, and talk about mathematics in what they think, with some justification, is a clear and well-organized way. For an example of an excellent mathematician doing just that, take a look at Gilbert Strang’s MIT lectures on linear algebra (see Lecture 1 on YouTube here). There have been some excellent college level work on collaborative learning in mathematics (refs to come) but overall this work does not seem to have seriously impacted the stand-and-deliver mode of inducting undergraduate students into a mathematical topic. Commonly, the term “interactive teaching” in a mathematical context, refers to the use of interactive technology, as described in Gadanidis 2002)or McPherson & Tyson (2006). We regard this affordance for students of being able to operate interactively with on-ine software as a small part of a much more general notion of interactive teaching and learning in mathematics. Our view is that modern technological advances – of which the vehicle for this written page is an excellent example! – provide an opportunity for teachers and students to interact in the learning of mathematics in ways that were not possible previously. If mathematicians are all children of Pythagoras – the founder of a secret sect – the internet and modern communication technology has made possible the expression of such modern masters as Terry Tao and Timothy Gowers, of whom we are ardent supporters. These outstanding mathematicians are pioneering a new form of mathematical learning, discovery and communication.  How often, in the past, has it been possible to communicate almost instantly with a great living mathematical master, and receive a considered respectful response in short order?  It is this spirit of mathematical interactivity and communication that we aim to promote in our classrooms. Currently two of us – Davis & Gottlieb – are experimenting with increased interactivity in our teaching by using Word Press blog posts. We are carrying this out in a variety of courses in the Fall 2008 semester: Ordinary Differential Equations, Mathematical Inquiry, and Technology in Mathematics Education (Davis), and Women in Mathematics and Numerical Analysis (Gottlieb).


Graham, T., Rowlands, S., Jennings, S. & English, J. (1999) Towards whole-class interactive teaching. Teaching Mathematics and its Applications, 18(2):50-60.


This paper considers an approach to addressing the decline in the level of achievement of British pupils in mathematics. It looks in detail at the differences between the teaching methods of Britain and Hungary, as research studies have indicated the high level of achievement of Hungarian pupils in mathematics. The paper outlines three theoretical perspectives (radical constructivism, social constructivism and Vygotsky’s zone of proximal development) that are helpful in considering the important differences. The major differences are considered under four categories: expectation and consistency, assessment, continuity and differentiated teaching. The paper proposes the method of whole-class interactive teaching as a way forward that would improve pupils’ achievement, and gives practical suggestions for developing such a teaching strategy.

Gadanidis, G. (2002). The Effect of Interactive Applets in Mathematics Teaching. In G. Richards (Ed.), Proceedings of World Conference on E-Learning in Corporate, Government, Healthcare, and Higher Education 2002 (pp. 1481-1484). Chesapeake, VA: AACE. [effect_of_interactive_applets_in_mathematics_teaching]


Interactive applets, which are typically web-based, are the most recent manifestation of technological tools used in mathematics education. This paper briefly reports on recent case studies of three grades 5-6 and one grade 10 teacher where the use of web-based applets may have acted as pedagogical models for teachers’ classroom practice, shifting the teaching focus from the learning of isolated concepts to algebraic relationships among concepts. The paper also describes an upcoming study of the affect of the availability of interactive applets on the pedagogical thinking of teachers.

McPherson, R. F., & Tyson, V. (2006). Creating your own interactive computer-based algebra teaching tools: A no programming zone. Contemporary Issues in Technology and Teacher Education, 6(2), 293-301.


In this article the reader will be able to download four spreadsheet tools that interactively relate symbolic and graphical representations of four different functions and learn how to create tools for other functions. These tools uniquely display the symbolic functional representation exactly as found in textbooks. Five complete lesson activities based on the tools are included. A design tutorial is also presented. The design tutorial shows readers how to create their own interactive mathematics learning tools conforming to National Council of Teachers of Mathematics philosophies. The techniques require only built-in point-and-click commands found in most spreadsheet programs. No programming is required. Step-by-step instruction and animations lead the reader through creating a tool. The intended audience of this article is mathematics education professors, preservice teachers, and in-service teachers. These techniques are currently taught in the mathematics education methods classes at Longwood University.

Narum, Jeanne L. (2006) Transforming America’s scientific and technological infrastructure. Recommendations for urgent action. Project Kaleidoscope.


RECOMMENDATIONS FOR URGENT ACTION Focus on students now in the pipeline * support those students demonstrating promise for success in the study of science and mathematics as they enter into and pursue undergraduate studies * give each undergraduate the opportunity for personal experience with inquiry-based learning that brings him or her to a deep understanding of the nature of science, the language of mathematics, the tools of technology * extend research opportunities beyond the classroom and campus * capitalize on and celebrate the growing diversity of students in American classrooms. Focus on the future workforce * connect student learning in STEM fields to the world beyond the campus, so students appreciate the relevance of their studies and consider careers that use the skills and understandings gained from study in these fields * build regional collaborations of academe, business, and civic groups working to ensure a steady stream of graduates well-prepared for the 21st century workplace, as well as to be responsible citizens in our “flat world” * respond to contemporary calls for interdisciplinarity by nurturing and rewarding faculty who make the kind of cross-discipline connections they hope their students will make. Focus on innovation for the future * be adventurous in exploring opportunities to strengthen student learning in the STEM fields and in piloting new ideas, tools, and approaches to keep the work of transforming student learning at the cutting edge * set benchmarks (2010, 2015, 2020) against which action plans can be shaped and progress measured, at the local, regional, and national levels.


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