Essays on Mathematics Education, August 2009 Draft, 231 page pdf document August 2009. This is the current draft of the book version, collecting the essays below. Comments welcome. Please check for updates before quoting.

The Subject

The Nature of Contemporary Core Mathematics, 61 page pdf document, January 2010. The goal of this essay is a description of modern mathematical practice, with particular emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled the deep and sweeping developments in the twentieth century. A particular concern is the significance for mathematics education: elementary education remains modeled on the mathematics of the eighteenth century and before, and use of modern methodologies might give advantages similar to those seen in mathematics. This draft is about 95\% complete, and comments are welcome.

Analysis

Math/Math-Ed Terminology Problems , 2 page pdf document, February 2009. Many common terms have very different meanings in the two communities, and sometimes neither is appropriate. The slogan ``understanding, not rote learning or mechanical calculation'', for example, has been quite influential. The math meaning for ``understand'', which is adapted to support long-term learning in math, is too strong to be a realistic goal in K-12. The weak math-ed sense is easily achieved but seems not to support long-term learning. Actual solutions will require us to transcend terminology problems.

Teaching vs. Learning in Mathematics , 5 page pdf document February 2009. Teachers seem to be far too focused on what happens on our side of the desk. It looks as though teaching and learning were never as closely linked as we wanted to think, and the gap will widen unless we focus on students and learning, particularly long--term learning, and not through the lens of teaching. Examples concern calculator arithmetic, ``clickers'', computer courseware, and diagnosis of errors.

Dysfunctional Standards Documents in Mathematics Education, 7 page pdf document December 2004, revised December 2008. Standards documents attract a great deal of attention, and reasonably so: they should provide structure and common reference points for teachers, administrators, curriculum developers, textbook writers, test developers, etc. Unfortunately current documents do a poor job with all this and it seems unlikely they will improve.

Evaluation of methods in math education , 15 page pdf document March 2006 revised October 2008. Some methodologies in education research and curriculum development seem almost designed to generate spurious findings and discourage deeper insights.

Communication Between the Mathematics and Math--Education Communities , 10 page pdf document, June 2006, edited October 2008. Communication between K-12 and college educators is needed to reverse a decline in preparation for study in technical fields. Attempts have been largely unsuccessful and sometimes so unpleasant they are described as ``wars''. We analyze obstacles and particularly try to separate linguistic differences from conflicts of underlying mindsets and priorities. Annotated lists of sample problems may offer the best approach. Another view of the issue is presented in ``Math/Math-Ed Terminology Problems'' below.

K-12 Calculator Woes , 3 page pdf document January 2009, revised February 2009. Calculators have been widely adopted in some K--12 curricula. Unfortunately the way they are used seems to cause significant learning deficits that limit further learning at higher levels. Some of the problems seem to stem from mismatch with primitive features of human learning. To appear in the Notices of the American Mathematical Society, May 2009.

The K-12 math test conundrum , 2 page pdf document January 2005. Problems with high-stakes testing; appeared in the Notices of the American Mathematical Society, April 2005. For an update see ``Beneficial high--stakes math tests: an example'', below.

Proposals

Projects with Proofs for Prospective Teachers, 24 page pdf document June 2009. This note outlines projects on fractions and area for college students who may become elementary or secondary teachers. It was written to test and illustrate ideas in [The Nature of Contemporary Core Mathematics] about proofs, definitions, abstractions and mathematical methods. Accordingly, it is a resource or starting point, and not intended to be used in this form. Comments for educators mainly concern practical aspects of learning programs that include formal definitions and proofs.

Contemporary Proofs for Mathematics Education, 21 page pdf document January 2010. In contemporary mathematical practice the primary importance of proof is the advantage it provides to users: proofs enable very high levels of reliability. This essay illustrates how a similar approach might have similar benefits in elementary education.

Beneficial high--stakes math tests: an example , 8 page pdf document, November 2008. A worked--out example is given to show how mathematical and educational insights can be incorporated in the structure of high--stakes K--12 math tests in a way that promotes better teaching practices and more effective learning. The example concerns symbolic skills deficits seen in students from calculator--oriented K--12 programs.

Tracks in a Math Course , 8 page pdf document, January 1998, revised June 2006, October 2008. Variation in student interest, preparation, and performance is usually accommodated by offering courses at several different levels and placing students in them at the beginning of the term. This practice has serious drawbacks that might be avoided by reversing the placement strategy. In a tracked course students enroll in a combined course, sort themselves into tracks according to performance, and the decision about the level they receive credit for is made at the end of the term. Resource constraints will make this approach impractical in many cases, but when it can be used it could significantly improve outcomes.

Student Computing in Mathematics: Interface Design , 13 page pdf document, January 2009, The first in a series on a computing environment designed to support learning in mathematics and other technical areas. They draw on many years experience with students working with computers and in computer environments, discovering unexpected learning problems and trying to fix them. The main point is that human learning is quite complex and as we move away from the tightly--bundled package of hand calculation in traditional classrooms the full complexity is coming into play. There are more ways for learning to fail than most of us imagined; many are different from the things educators traditionally look for and are hard to recognize; and underlying causes are obscure. This article concerns basic student--computer interactions. Among many other things we see that standard cut-and-paste can undercut some learning objectives and has to be modified. The sequel (not yet available) concerns computational functionality. Careful limitations are needed to avoid turning the subject into keystroke sequences. These articles are very speculative and intended as starting points for further investigation, not a fixed prescription for a final product.

Task--oriented Math Education , 26 page pdf document, October 2008. ``Learning tasks'' on which students work independently with support by helpers and web materials provide an approach to math education. Experience at the Math Emporium at Virginia Tech demonstrates educational effectiveness at the college level and suggests it should work in upper grades in K--12. Implementation would be tricky so the factors involved are considered carefully and in detail. Benefits could include significant improvement in the quality and effects of high--stakes tests. Many of the educational advantages come from giving students more choices and more control over their learning.

Web Sites

AMS Working Group on Preparation for Technical Careers, Annotated problem lists to help guide preparation of K--12 students for ambitious college--level education in math, science and engineering.

Mathematics Emporium at Virginia Tech. This facility supports computer--based and computer--tested math courses for more than 5,000 students each semester.

Reports and drafts of essays

Integral calculus outcomes, Spring 2006 , 12 page pdf file May 2006. Technical report on analysis of performance on common-time final and class tests, including comparisons of computer-tested and traditional sections. Five-year review with over 8,000 students coming summer 2009.

Lessons from the Emporium 1: Goals and economics , With Michael Williams, 6 page pdf file November 2003. The Math Emporium at Virginia Tech is an undergraduate math learning facility with over 500 computers and a large help staff. This is the first of three articles on "lessons": things we initially did not fully appreciate but over the last six years have learned are vital for success. This installment deals with economic constraints and how they influence educational methods and goals.

Lessons from the Emporium 2: Help for computer-based learning , With Michael Williams, 4 page pdf file November 2003. This installment describes the one-on-one help system we have evolved to support the learning process.

Software-generated test questions , PDF file April 2001. Computer-based testing has the potential to lighten faculty workloads and improve learning and the classroom atmosphere. Much work has been done on the mechanics of testing usually assuming that questions will be drawn from a database constructed by hand. The point here is that the educational wisdom used in making up questions can sometimes be encoded in software to provide large numbers of high-quality questions quickly and accurately.

Faculty interview assessment of student group work , 4 page pdf file, August 2000. Interviews by faculty are a potentially valuable way to encourage and assess work by groups of students. An analysis of problems with current practice, and how they might be avoided.

Assessing costs and benefits in NSF educational grants , 2 page pdf file, January 2000. Lack of cost assessment in NSF educational grants may be pushing development in wrong directions.

On-line help , 6 pages, March 1998. A grant proposal funded by the Center for Innovation in Learning, for development of on-line help facilities for use in the Emporium and with off-site math courses. The model is the student-initiated session at the student work area, as evolved in the Emporium, rather than a class or tutoring situation. Cost is a major constraint since equipment has to be provided at each student work area. This requires a careful analysis of what is really essential to a help session.

Emporium supervisor's manual , 9 page pdf file, February 1998. A guide to supervising the help staff on the Emporium floor. Topics are floor management (basic procedures); assignments and training (use people for what they can do well); and assessment (orient toward improvement, keep it internal).