## Future modules

- 2018-2019 Semester 1: Proof (YSC2209)
- 2018-2019 Semester 1: Scientific Inquiry 2 (YCC2137)

## Past modules

### Yale-NUS College

- 2017-2018 Semester 2: Proof (YSC2209)
- 2017-2018 Semester 2: MCS capstone seminar (YSC4103)
- 2017-2018 Semester 1: Ordinary & partial differential equations (YSC3230)
- 2017-2018 Semester 1: Proof (YSC2209)
- 2017-2018 Semester 1: Applied calculus (YSC1211)
- 2016-2017 Semester 2: Ordinary & partial differential equations (YSC3230)
- 2016-2017 Semester 2: Proof (YSC2209)
- 2016-2017 Semester 1: Proof (YSC2209)

### University of Michigan

- 2016 Winter: Boundary value problems (Math 454)
- 2015 Fall: Calculus 1 (Math 115)

### University of Cincinnati

- 2014 Fall: Coordinated Calculus 1 with precalculus review (Math 1060)
- 2014 Fall: Calculus 1 (Math 1061)
- 2014 Summer: Introduction to discrete mathematics (Math 1071)
- 2014 Summer: Calculus 1 (Math 1061)
- 2013 Fall: Calculus 1 with precalculus review (Math 1060)

### University of Crete

- 2012 Autumn: Fokas' method for linear initial-boundary value problems

### University of Reading

- 2012 Spring: Real Analysis 2 (MA2AN2)
- 2011 Autumn: Real Analysis 1 (MA1AN1)
- 2007–2010: support teaching in Real Analysis 1 (MA1AN1) and Algebra 1 (MA1AL1)

## Teaching tools

I am always looking for technological solutions to pedagogical problems. These are some of the tools I use to simplify my job as an instructor, with descriptions of the problems they were developed to solve.

Each of these tools was hacked together very quickly, and is not feature-complete. If you spot any bugs, or would like to extend the functionality of one of the tools, please send me an email.

### Group randomizer

Download: Group Randomizer

Last updated 2015-Dec-27

Group work is an essential component of modern classroom learning, and can also be very powerful in homework assignments. I found that students engage with group work assignments much more reliably if the groups are both assigned by the instructor and changed regularly.

But whenever groups are assigned, it is important to avoid isolating students belonging to certain categories within any group. For example, suppose that Orange group contains 3 women and one man, and Green group contains 1 woman and three men. Then the isolated woman in Green group has not only to contend with the same challenges as every other student, but also has the additional pressure of being forced to represent women within her group. It would be better to rebalance the groups so that either there are two women in each group, or none in one group and four in the other. The important thing is to ensure that no group contains an isolated woman.

There are other categories of students that should not be isolated in their groups. These include: students belonging to racial minorities, first generation students, international students.

This is all very well in theory, but to carefully assign groups in this way is very time consuming. The tool Group Randomizer can be used to automatically assign students to groups while avoiding isolating any student in any group, across any one, two, or three categories of your choice. The tool is an MS Excel macro-enabled workbook, written in VBA. In order to use it, you must enable macros. The first worksheet has some instructions. The tool should work with MS Excel 2010 (Windows) or 2011 (Mac) or later, but is not compatible with Google Docs, Libre Office, etc.

### Worksheet randomizer

Download: Worksheet Randomizer

Last updated 2015-Dec-27

It is often valuable to be able to generate multiple random permutations of a particular worksheet or examination. If students will sit an examination, but there are insufficient proctors/invigilators, or insufficient space in the room to space students out, it can be very difficult to discourage plagiarism. This problem is exacerbated by multiple choice tests.

One popular solution is to write 2 or 3 versions of tests, with slightly different problems on each version, by switching constants. However it is extremely difficult to ensure that the problems are of the same difficulty; experience shows that not only is solving *x*^{2}-2=0 more difficult than solving *x*^{2}-4=0, but, surprisingly, solving *x*^{2}-9=0 is significantly more difficult than solving *x*^{2}-4=0, for Calculus 1 students. Therefore writing multiple versions of tests with slightly different problems is a very bad idea.

An alternative approach is to have multiple versions of the exam, with all problems identical, but presented in a different order, and with multiple choice solutions presented in different orders. This is clearly not a panacea to plagiarism, but it makes plagiarism more difficult, thereby lowering the incentive. There are legitimate questions to be asked about whether the order in which problems appear on an exam affects students' performance, and I would like to be pointed towards studies on this, but I expect this will turn out to be the least-bad option.

The job of permuting exam papers in this way is a significant burden on faculty, but exam security concerns mean that very few people can be trusted to do it. The tool Worksheet Randomizer can be used to automatically generate many permutations of exams or worksheets, written in LaTeX. The tool is an MS Excel Macro-enabled Workbook, written in VBA. In order to use it, you must enable macros. The program is compatible with Windows versions of Excel only (2010 or later). It relies on your native pdfLaTeX compiler (eg MikTeX). Also in the zip archive are a Windows batch file (essential to the operation of the program) and extensive documentation. You should extract all 3 files from the zip archive before trying to use the program.

## Classroom materials

These are some of the pedagogical materials I have developed to help enable my students' learning.

### Calculus 1: Definite integral

Download Definite Integral

Last updated 2015-Dec-27

When students first encounter the definite integral, it is usually by far the most complicated limit they have ever seen. But the sum of the areas of a collection of rectanges is relatively easy to understand, even in the limit, so a graphical approach is particularly valuable. Moreover, if numerical integration is being taught in parallel to analytic integration, then this interpretation of the definite integral is an absolutely essential learning outcome. However it is tempting to deemphasize the graphical model for the Riemann integral, because drawing lots of rectangles under curves takes a very long time!

The purpose of this classroom tool is to automate the arduous process of drawing rectangles and calculating the sum of their (signed) areas. In addition to the MS Excel macro-enabled workbook, I included some problem sheets designed for groups to complete using the program, which focus mainly on interpreting over- and under-estimates. I have found that one computer between two students is optimal, with students working in groups of 4 (2 computers). Too many computers clutter up the workspace, but only 1 computer results in competition to use the computer. The workbook is compatible with MS Excel 2010 (Windows) or 2011 (Mac) or later, but is not compatible with Google Docs, Libre Office, etc.

Students can play with the formula in cell C1 and press PLOT to see the new curve. The formula is just a normal excel formula, except that it does not have a preceeding =, and the cell reference you would usually include is replaced by a lower-case x. Note that if you want to use an excel function with an x in its name (eg MAX) you should be sure to capitalize that function; enter MAX, not max. Cell C7 contains a lower-case l, r, or c, to specify left, right, or central Riemann sums. The program is currently incapable of displaying upper or lower Riemann sums.

### Upper division modules: How to read a mathematics textbook

Download How to read an upper division mathematics textbook

Last updated 2015-Dec-31

We tell our students "to prepare for each class, you should read the corresponding (sub)section(s) of the textbook, according to the course calendar, because in class we will be challenging you to solve problems." But I've never before tried to explain what the phrase "read the textbook" actually means. I think it is reasonable to assume our Seniors (year 4) know that skim-reading on the bus is insufficient. But do the Juniors (year 3) know that we expect them to do the exercises? I doubt it. So I tried to articulate clearly what I mean by "read the textbook", to someone who has little/no experience reading mathematics, but perhaps has plenty of experience reading an Engineering/Science textbook. I was struck by what a complex and alien process this is, and how long it takes to explain it properly.