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Teaching and Learning Mathematics with Technology

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Teaching and Learning Mathematics with Technology Teaching and learning mathematics with technology Peter Jipsen School of Computational Sciences Chapman University May 18, 2012 Outline I Matching technology with undergraduate mathematics I Mathematics on the web I Mathematics on computers I Wikis and collaborative editing I Audience response systems I Online homework systems I What next? Teaching and learning What is eective teaching? When every student learns as much as s/he is capable of, within the given constraints When students are engaged; have frequent Aha! moments When they are able to reconstruct ideas and techniques months or years after they understood them When students feel competent enough to tutor other students, and do it well ... Technology for undergraduate mathematics Can technology help? It depends on how it is used Certainly the invention of the blackboard was a step forward But some instructors were better at using it than others Probably the invention of the Blackboard course management system was a step backwards But some students and instructors nd it useful Taking notes, doing calculations Recording what one is learning, working with ideas, formulating questions Requires writing technology Paper, pencil and eraser are good old technology Wordprocessors are bad new technology for mathematics LATEX is even worse Why? Typing math notes during a lecture in LATEX or MS Word is challenging Doing calculations or drawings in a word processor in real time is more dicult Typing notes in a non-math class is easy for anyone who can type well Notes available in digital form with little post-processing, can be shared easily Can use handwritten notes for math classes, scan them later Or use a tablet PC, or new solution: use a tablet (well, iPad for now) Can use Lyx, a free open-source what-you-see-is-what-you-get LATEX-based wordprocessor Easy to install on MS Windows, Apple OSX, Linux Look and feel similar to MS Word 2010, but free and available on all computers Why not use a Computer Algebra System? Mathematica / Maple / Sage / Scientic Notebook / Mathcad Expensive and/or hard-to-use not suitable as learning tools Wikis Collaborative learning tools, getting students involved in online content creation Used a wiki in a discrete math class several years ago All students had access to all parts of the wiki All pages are versioned; all users had to log in; edits are logged Used for course notes, with annotations and links to other online content Used for homework, assigned ALL exercises, one for each student to post Had to also solve 4 other exercises from each section, comment on the posted solutions The problem What should one use for math notation on a wiki? In 2004 no suitable solution was available How to make computers display and understand e.g.: sin−1 plog e π e = 2 Even Wikipedia had no solution (with 200000 articles in 2004) [still does not have a good solution; it advocated using italics and <math>LATEX</math>; now stuck with a huge legacy problem] Possible solutions −1 p π sin loge e = 2 I ArcSin[Sqrt[Log[E]]]==Pi/2 Mathematica I Math.asin(Math.sqrt(Math.log(Math.E)))==Math.Pi/2 JavaScript (the native browser language) I \sin^{-1}\sqrt{\log_e e}=\frac{\pi}{2} MathJax or LATEX I <msup><mo>sin</mo><mrow><mo>-</mo><mn>1</mn></mrow> </msup><msqrt><msub><mo>log</mo><mi>e</mi></msub><mi>e</mi> </msqrt><mo>=</mo><mfrac><mi>&pi;</mi><mn>2</mn></mfrac> Presentation MathML I sin^-1sqrt(log_e e)=pi/2 ASCIIMath Aims of a convenient linear math notation I Close to standard mathematics I Motto: if it looks like math, it should work I Easy to read I Easy to type I Formulas should be short I No obscure syntax errors I Syntax easy to dene and remember I Mostly language independent I Simple to extend or modify (localization) ASCIIMath: A linear math notation with 8 syntax rules; designed in 2004 Examples of ASCIIMath π lim tan−1x = lim_(x->oo)tan^-1x = pi/2 x!1 2 P1 1 π2 sum_(n=1)^oo 1/n^2 = pi^2/6 n=1 n2 = 6 p 1 1 x2dx π int_-1^1sqrt(1-x^2)dx = pi/2 −1 − = 2 ´ [0; 1) = fx 2 R : 0 ≤ x < 1g [0,1) = {x in RR : 0<=x<1} These examples carry students a long way ASCII: American Standard Code for Information Interchange ASCIIMathML I Implemented in a single JavaScript le (~1000 lines of code) I Conversion to MathML is done as the web page loads I Makes MathML work in HTML in Firefox and IE (was a big deal in 2004) A I Includes a subset of LTEX (with some text formatting) I Search online for ASCIIMath, download for free How about visual content A graph can be included by typing e.g. agraph plot(sin(x)) endagraph Each graph can be modied by the reader directly on the webpage, to allow for a hands-on learning experience Interactive dynamic pictures are possible with minimal programming Ease of use: An auto-math-recognize mode allows formulas and text to be freely mixed E.g. Let m,b in RR and consider the line y = m x + b. Produces: Let m; b 2 R and consider the line y = mx + b. ASCIIMath is widely used I Downloaded by thousands of users around the globe (17900 hits on Google) I Integrated into many wikis (MediaWiki, PmWiki,...), blogs (Blogger), course management systems (Moodle), Homework systems (WebWorK, IMathAS), Editors (TinyMCE, Xinha) I Includes ASCIIsvg and a scientic calculator I ASCIIMath serverside in PHP [Chan 2004], Perl [Nodine 2006] A I Better LTEX support [Woodall 2006, J. 2007] I Interactive graphs [J. 2007] I Included in MathJax 2.0 [Lippman, Cervone 2012] ASCIIMathML addresses the following problems I Few (school/undergrad) students know how to type mathematics I Most math homework is handwritten I Math tests are often multiple choice (presentation is not tested) I Mathematics seems oldfashioned to computer savvy youth I Mathematics education is aected negatively I Dicult to help students by email or chat I Online interactive math content is low I Ironically, it's hard to do math on a computer! Issues MathML is still not available natively in Chrome or Internet Explorer MathML is nally part of HTML5 (previously only valid in XHTML) Does not display as nicely as LATEX MathJax (formerly JSMath) does a great job, now handles ASCIIMath in ALL browsers ASCIIMath is very similar to MS Word 2010 math language but not identical ASCIIMath is not (yet) an ocial standard Does not handle full LATEX, so can't easily copy-paste from/to wikis Back to learning math with technology Audience Response Systems (Clickers) in Undergraduate Mathematics Started using them in 2009 in Calculus I Once a week (at most) Works well as attendance check Still some technological issues Students should have their own clicker (mine didn't) Clickers TurningPoint works best from within PowerPoint (essentially a deal-breaker) Fortunately TurningPoint Anywhere works with any displayed content The most important variable is the quality of the questions that one poses Writing good clicker questions is dicult Hard to tell if students learn more, but they denitely pay more attention I recommend clickers if you can get a set of Good Questions matched with your course and teaching approach Alternatives to clickers There are many choices for clickers, inexpensive open-source would be best Cell-phones with SMS texting are a recent alternative (polleverywhere.com, smspoll.net) Smartphones or laptops with web access (WiFi) can also be used Good solution also for online classes Can't ignore the rst choice for student communication (texting) Twitter, Gmail chat, Google Docs has also been used in this setting More eective technology Online homework systems I have used WebWorK since 2002, MyMathLab since 2012 Webassign, WileyPlus, IMathAS are other choices WebWorK is free, open-source, large database of problems targeting main undergraduate math courses (Calculus I-III, Linear Algebra, Discrete Math, ...) Based on Perl + TEX + PDF + Linux MyMathLab is from Pearson, $82 for 3 semesters with ebook from Pearson Based on Blackboard + MathXL + Flash + Mathematica CDF Flash and CDF do not work on the iPad; ebook is hard to print; help button makes questions too easy Personal observations It doesn't matter which online homework system is used It does matter HOW it is used Need to convince students to treat these systems like ordinary handwritten graded homework Require a homework notebook in which all work is recorded, to be inspected e.g. at the end of each month Or require written homework handed in with printout of questions Using Wolfram Alpha is not a problem, solution still needs to be written Give timed quizzes (outside of class, no online help), allow students to redo next day,... Improved my students' skills and saved me some time Possible Improvements These systems need to become math-aware, not just answer oriented They need to provide editor windows to write out a solution, with math syntax/grammar checker for correct notation and logic Should allow justications for steps Should prevent copy-paste (from outside) Should work with stylus input for convenience and graphs/diagrams iPad apps like Notability and GoodReader are way better than Tablet PC programs Some Considerations Prefer to use technology that students can aord and use on their own (e.g. not Tablet PCs or active whiteboards) Use tablets for taking notes, annotating slides, giving presentations Prefer to use technology for engaging projects rather than just drills Use ebooks that provide LATEX source, encourage students to edit it Future developments Smart phones Tablets ! real time bidirectional access to projector screen ebooks that work better than printed books, open content licenses, LATEX source, ..
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