Miscellanea

Some things I’ve been reading about:

• Concept inventories: These are tests designed to determine the extent to which people have an understanding of basic concepts in a particular area. The Wikipedia article on concept inventory provides a decent introduction. Concept inventories were introduced in physics, with the Force Concept Inventory (FCI) being used to determine people’s understanding of mechanics. Here’s the Google Scholar results for Force Concept Inventory, including this paper by Hestene, Wells, and Swackhamer that describes the detailed construction of the inventory. The force concept inventory led Eric Mazur, an experimental physicist at Harvard University, to change his style of teaching introductory physics. Here is a YouTube talk by Mazur where he describes how the concept inventory led him to change his style of teaching introductory physics courses.

• Open notebook science: Here is the Wikipedia entry on open notebook science, replete with links to various discussions of the subject. The UsefulChem blog has plenty of discussions and links related to open notebook science. Here is Michael Nielsen’s article/blog post on the future of science, with discussions of open notebook science and related ideas.

• Moore method: Much of inquiry-based learning (IBL) in college mathematics courses is based on the Moore method and its derivatives. The Moore method was pioneered by topologist Robert L. Moore at the University of Texas, and is often also called the Texas Method. The idea is that the instructor, instead of teaching students, gives them problems to solve on their own and listens to them as they attempt to present their solutions to their peers and the instructor. Here’s the Wikipedia article on the Moore method. Here is the Legacy of R. L. Moore project and here is the University of Texas Discovery Learning Project. There’s a three-part video series (1, 2 and 3) about the Moore method. You can also view this book about the Moore method (limited preview via Google Books). See also this Math Overflow discussion on the Moore method.

• Cognitive load theory: Here is Sweller and Chandler’s original paper (1991) on the subject and here are the Google Scholar results on the query. Cognitive load theory attempts to look at learning in terms of the cognitive load imposed on the learner. It identifies three kinds of load: intrinsic load, which is the load that naturally arises from trying to learn, extrinsic load, which is the load that arises due to distractions and does not help with learning, and germane load, which is load that the learner takes on to get a deeper understanding and form better connections within the material. The goal of good instruction should be to assess how much intrinsic load a given learning task entails, try to minimize the extrinsic load, and use whatever extra space there is to add in germane load.

Planetmath and Mathworld losing out to Wikipedia

As I’ve mentioned earlier (such as here), it has seemed to me for some time that both Planetmath and Mathworld are losing out as Internet-based mathematical references to Wikipedia. I don’t expect that there has been absolute decline in the traffic to these websites — the growth of the Internet by leaps and bounds would mean that a website has to make a fairly active effort to lose users. But I do expect that more and more of the new users who are coming in are treating Wikipedia more as a first source of information.

From the preliminary results of a recent SurveyMonkey survey: out of 29 respondents (most of them people pursuing or qualified in mathematics), 24 claimed to have used Wikipedia often, and the remaining 5 to have used it occasionally for mathematical reference information. For Planetmath, 22 people claimed to have used it occasionally, and 7 to have heard of it but not used it. For Mathworld, 5 people claimed to have used it often, 16 to have used it occasionally, 7 to have heard of it but not used it, and 1 to have never heard of it.

In response to a question on how Planetmath and Mathworld compare with Wikipedia, the nine free responses were:

• Planetmath now loads too slowly, so I seldom use it. Even when I did use it often, I found that the articles tended either to be too incomplete, or too tailored to people who are already experts. Many Mathworld pages are still much better than many Wikipedia pages, but Wikipedia is more comprehensive.

• planet math is slow!!! mathworld has more graphics and less maths.

• Material is generally better presented on Mathworld, but the topics are more limited.

• Not as much information as Wikipedia and not as well arranged

• I usually find Wikipedia to be my first source and only go to Mathworld or Planetmath if wikipedia fails me. I guess that means Wikipedia is better.

• Planetmath is dying, Mathworld is static.

• I find wikipedia much more useful than Mathworld. Mathworld’s pages are very technical, which is not what I am looking for on the internet usually. Usually I am looking for someone’s nice conceptual understanding of a topic or definition (through nice examples), and Wikipedia usually has lots of these.

• Planetmath is quite useful to find proofs. Mathworld is very specialised, but it has a few nice bits of information sometimes. They seem to both be quite stagnant compared to Wikipedia.

• They can’t keep up. There was probably a time when PlanetMath was a better reference than Wikipedia, but it’s fading fast. I think their ownership model isn’t conducive to long term quality.

(See the responses to these and other questions here and take the survey here).

The general consensus does seem to confirm my suspicions. Why is Wikipedia gaining? Here are the broad classes of explanations:

1. It’s just a self-reinforcing process. The more people hear about and link to Wikipedia, the more people are likely to read it, the more people are likely to edit it and improve it. But if that’s the case, why did Wikipedia ever get ahead of Mathworld and Planetmath? Two reasons: (i) its more radically open editing rules (ii) Wikipedia covers many areas other than mathematics, so people come to the site more in general. Also, since it covers many areas other than mathematics, it can better cover content straddling mathematics and other areas, such as biographies of mathematicians, and historical information that is relevant to mathematics. This creates a larger, strongly internally linked, repository of information.

2. Planetmath’s owner-centric model (as mentioned in one of the responses) where each entry is owned by one person, does not create a conducive environment for the gradual growth and improvement of entries.

3. The appearance of content is better on Wikipedia. Prettier symbols, faster loading, better internal links, better search. This is definitely an advantage over Planetmath, which has slow load times in the experience of many users (as indicated by the comments above) though perhaps not so much over Mathworld.

4. Google weights Wikipedia higher (because of the larger size of the website and the fact that a lot of people link to Wikipedia). This is related to (1).

5. The people in charge of Mathworld and Planetmath simply lost interest. Mathworld is largely run by Eric Weisstein, an employee at Wolfram, who seems to have recently been trying to integrate metadata about mathematical theorems and conjectures into Wolfram Alpha. Developing Mathworld continually to a point of excellence does not seem to have been a top priorty for Weisstein or his employer Wolfram Research (that hosts Mathworld) over the last few years. The people running Planetmath also may have become less interested in continually innovating.

Given all this, is Wikipedia the best in terms of: (i) the current product or (ii) the process of arriving at the product? While I’m far from a Wikipedia evangelist, I think that the answer to (ii) is roughly yes if you’re thinking of broad appeal. Anything which beats Wikipedia will probably do so by being more narrowly focused, but it may then not be of much interest to people outside that domain. A host of many such different niche references may together beat out Wikipedia for people who care enough to learn about a multiplicity of references. For those who just want one reference website, Wikipedia will continue to be the place of choice in the near future (i.e., the next 3-4 years at least, in my opinion).

Currently, Wikipedia is an uneasy mix of precise technical information and motivational paragraphs. It makes little use of metadata to organize its information; on the other hand, it is easy to edit and join in. The mathematics entries cannot be radically changed in a way that would make them radically different in appearance from the articles on the rest of the site. This opens up many niche possibilities, some of which are being explored:

1. Lab notebooks, where people store a bunch of thoughts about a topic, without attempts to organize them into something very coherent. Here, good metadata and tagging conventions could allow these random lab notebook-type jottings to cohere into an easily accessible reference. This would be the mathematical version of open notebook science, a practice that is slowly spreading in some of the experimental sciences. nLab (the n-category lab) is one example of a “lab notebook” in the mathematical context. This is great for motivation, and also for understanding the minds of mathematicians and the process of mathematical reasoning.

2. Something that focuses on a particular aspect of mathematical activity. For instance, Tricki, called the Tricks Wiki, focuses on tricks. Other references may focus on formulas, others may focus on counterexamples, yet others (such as the AIMath wiki on localization techniques and equivariant cohomology) may focus simply on providing extensive bibliographies. Somewhat more developed examples include the Dispersive Wiki and complexity zoo (actually, a computer science topic, but similar in nature to a lot of mathematics). Some may focus on exotic tricks of relevance to a particular mathematical discipline. There is some cross-over with lab notebooks, as the tricks become more and more exotic and the writing becomes more and more spontaneous and less subject to organization into an article.

3. Highly structured content rich in metadata that is intended to provide definitions, proofs and clarify analogies/relations. Examples include the Group Properties Wiki [DISCLOSURE: I started it and am the primary contributor] which concentrates on group theory. The flip side is that the high degree of organization uses subject-specific structures and hence must be concentrated on a particular narrow subject.

There are probably many other niches waiting to be filled. And there may also be close susbstitutes for reference sites that weren’t created as references. For instance, Math Overflow, though not a reference site, may play the role of a reference site once it accumulates a huge number of questions and answers and adopts better search and specific tagging capabilities. Similarly, thirty years from now, the contents of Terry Tao’s weblog may contain a bit on virtually every mathematical topic, in the same way as Marginal Revolution have a bit on almost all basic economic topics (I say “thirty years” because economics is in many ways a smaller subject than mathematics).

Survey on math resources on the Internet

I have put up a medium-length survey on the use of math resources on the Internet. The survey is a preliminary one — the results of this will be used in designing more sophisticated surveys. Take the survey here:

http://www.surveymonkey.com/s/T2CBTZJ.

You can view the survey results here (but don’t do this before taking the survey!):

http://www.surveymonkey.com/sr.aspx?sm=aaYWJUl4DF9hjvM4AqrQxhgmev4PNRiYa8MJbwhLs9w_3d.