Contemporary Proofs for Mathematics Education, 21 page pdf document. 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. Proceedings of the ICMI Study 19 on Proof in Elementary Education, Taipei 2009; proceedings published March 2012.

A Science-of-Learning approach to Mathematics Education, 21 page pdf document (May 2011, published Notices of the AMS October 2011 ) A scientifically disciplined, learning-oriented approach to mathematics education is described and illustrated through examples. Mathematicians tend to focus on teacher education, with an implicit presumption that the current teacher corps is not sufficiently competent. The analysis here suggests that the teachers are competent enough, but the methodology they have been taught to use is incompetent. In other words, significantly better outcomes are unlikely without profound changes in educational philosophy and teacher-education programs.

Supplement to: A Science-of-Learning approach to Mathematics Education, 14 page pdf document (September 2011), Further examples and discussion, including comparisons with articles in the March 2011 Special Issue on Education of the Notices of the American Mathematical Society.

A Revolution in Mathematics? What really happened a century ago, and why it matters today 11 page pdf, October 2011. (published in Notices of the AMS January 2012) I argue that a revolution in core mathematics occurred between 1890 and 1930. The nature of the change is sketched and the reasons for its apparent obscurity are discussed in some detail. A better understanding is urgently needed to avoid losing some of the gains of the last century, and as a resource for education. This is an extract of a much more detailed account in ``Contributions to the Science of Mathematics'', on the history/structure page.

What should students get from Calculus? (And how can we provide it?) 3 page pdf, October 2011, to appear in the AMS Notices. Keith Stroyan, in the September 2011 Notices, suggests extended projects on applications. I suggest a rather different approach for students in science and engineering. There is a lot of commonality beneath the differences, however. Both focus on student needs more than content, and differences in detail reflect needs of different student populations. Resource constraints are likely to prevent widespread use of either, except perhaps as enrichments for largely computer-based courses.

Reform mathematics education is counterproductive for high-tech careers (and its not the teachers) (version 5) 16 page pdf, May 2012. K-12 preparation for technical careers has been declining for decades and has taken a nosedive in the last five years or so. This essay describes the problems, explains why they are problems, and locates sources in reform practice.

Very low functionality goals are an overarching problem. Innovations undertaken without discipline imposed by functionality goals have led to many new problems: weak symbolic, abstraction, number, and geometric senses from the way graphing calculators are used, and weak logical and reasoning skills from the `discovery' approach to learning.

A corollary is that we should stop blaming teachers: competent teaching with reform methodology should give bad outcomes. Another conclusion is that we need tech-oriented tracks, not an entire tech-oriented curriculum.

Book Draft

Contributions to a Science of Mathematical Learning, 294 page pdf draft, March 2011. This is the mostly-current draft, but due for an overhaul. Please check (or ask) for updates before quoting.

Related

Neural Mapping of Mathematical Activity, 6 page pdf document, April 2011. This is a proposal for detailed mapping of neural implementation of elementary mathematical functionality. Mathematics is a good candidate because it is a learned activity that is highly structured and can provide detailed external work records. Carrying out the proposal will still require refined neuroscience techniques, particularly in magnetoencephalography.

Old versions of book chapters

Proof Projects for 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 [Contributions to a Science of Contemporary 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.

Neuroscience Experiments for Mathematics Education, 23 page pdf document February 2011.

These proposed experiments contrast standard educational approaches that seem to cause problems, with alternatives adapted to mathematical and cognitive structure. Many of the alternatives have been used successfully in individual cases, so the job of neuroscience is not so much to detect the effects as to quantify them and clarify the mechanisms. The tools of neuroscience are difficult and indirect so specific tasks are designed to maximize expectation of clear outcomes.

These studies should have significant implications for educational practice.

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.

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, Published Notices of the AMS May 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.

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''.

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).