Seb Schmoller: A report from Keith Devlin’s and Coursera’s “Introduction to Mathematical Thinking” MOOC

This report was first published on 14 April 2013 in Seb Schmoller’s Fortnightly Mailing, and is reproduced here with Seb’s permission.

I’m six or so weeks into Keith Devlin’s 10 week Introduction to Mathematical Thinking, along with some tens of thousands of others.

Here is a longish thumbnail sketch of the design of the course, followed by two appendices. Appendix 1 concerns peer review. Appendix 2 is what the course web site has to say about grading and certificates of completion.


Attempting to prove that the square root of 2 is irrational

Comments, questions and corrections would be most welcome.

1. The course is advertised as needing about 10 study hours per week. This is about right: though in my case I had to skimp a lot while I was on holiday, other than wrestling unsuccessfully with a proof that had been set as course work, the non-fruit of which is shown above.

2. There are one or two 15-30 minute video lectures per week. All have so far has consisted of a mixture of Devlin talking confidently and intently to camera in sort of “newscaster” mode, and Devlin explaining things with pen and paper, very much in the vein of the 2011 Norvig and Thrun AI course. Devlin has retained his East Yorkshire/Hull accent, which has the effect of making him sound local. To me at least.

3. The videos have some nice technical features. For example, you can play them at up to 1.75 x normal speed whilst just about retaining Devlin’s vocal register correctly; handwriting is cleverly sped up to keep pace with Devlin’s voice.

4. At several points during each video there are embedded correct/incorrect answer quizzes. These range from quite challenging to almost silly checks on whether you’ve been listening. Each quiz allows you three attempts before you are taken to an explanation of the “correct” response.

5. The embedded quizzes also exist separately from the videos, which is useful for practice or revision purposes.

6. There is an accompanying self-published 80-page book by Devlin also called Introduction to Mathematical Thinking, which I bought online for £6.29. I have made use of the book particularly whilst travelling, but I could have managed without it. My guess is that most students have not bought it.

7. Each week there are one or two assignments. These are 1-4 page PDFs – with poor typograph, especially if printed from within Chrome or Firefox – with perhaps half a dozen questions relating to the material that has already been covered. They seem to be based mainly on the questions that appear in the book. (This is no bad thing.)

8. Assignments are not submitted for marking (but a helpful feedback video is made available in the next week in which Devlin explains how to answer a selection of the questions). In the first few weeks of the course Devlin puts a very prominent amount of emphasis on the need for all students to discuss the course with others in an informally established study group. In my case I chanced on and joined a Google Group called “Mathematical Thinking UK Discussion Group”. This initially had about 40 members. About seven were helpfully active in the first three weeks, but the study group has seemingly since ceased to function. So I am on my own: it feels a bit late in the day to try to find another study group, nor to attempt to breath life into this one.

9. Each week there is a multiple-choice machine-marked problem-set, for which there is a tight deadline, and which we are required to complete alone, though this is not enforceable.

10. The final exam (yet to take) will be marked by peer-grading; and during week six (the current week) we received our first bit of training in the peer-review process, with one of today’s machine-marked problem-set being to grade a short mathematical proof, guided by a rubric provided by Devlin. This felt rewarding to do in its own right. And of course the method itself can be used at very large scale. [For more on this see Appendix 1 below; there is also a discussion about peer-based marking in the Ufi Trust’s May 2012 Scaling Up report, in which I had a hand.]

11. About every two or three days, at least during the first few weeks of the course, all students get a “do not reply” email from Devlin. This might tell us that new material has been published, or it might draw our attention to a discussion in the Coursera-provided course forum that Devlin considers relevant. For example, a discussion had cropped up about the decision to pen and paper rather than Udacity or Khan Academy style digitized writing using a tablet. Devlin had contributed to the discussion and we were encouraged to read the thread. Devlin also seems to have decided that there are benefits for learners in exposing to learners the thinking that has gone into course design, and we’ve occasionally been encouraged to review a set of short (~one to ~six minute) videos – so far there are about 30 of these. Some of these focus on design decisions about the course. Others are fragments of Devlin in action teaching at Stanford.

How am I finding it?

You can tell from my tone in the sketch above that I am enjoying the course and that I admire its careful design and overall structure. I am finding the course considerably less discussive than I did Peter Norvig and Sebastian Thrun’s AI course. This is partly because the video lectures are longer and more formal; and partly because there seems to be less active discussion in the course-provided discussion forums, possibly on account of the way in which students have been encouraged to make their own arrangements, which was far less the case with the AI course. (But their could be other reasons. The overall cohort size is smaller perhaps by a factor of ~5. The kinds of people enrolled on the course may be less prone to discuss things. Mathematical thinking may be less discussable than AI. The AI course had much more of a pioneering, “frontier” feel to it.)

But what this course shares with the AI course is the feature that struck me so forcefully in 2011: the feeling that you are getting one-to-one personal tuition from a very skilled and interesting teacher. It may not quite besitting in a bar with a really smart friend, but it is a very far cry indeed from traditional online distance learning.

Devlin and colleagues from Stanford’s School of Education are actively using this course as a research test-bed. I hope that the fruits of this work are put into the public domain in due course.

Note. This wide-ranging 28 December 2012 interview with Keith Devlin by maths blogger “Shecky Riemann” maybe of interest.

Appendix 1 – Peer review process

[Source: the Coursera web site; last accessed 14 April 2013; last modified on 27 February 2013]

NOTE: This procedure is experimental, and will be under regular review throughout the course. Based on what we observe and learn, we may make adjustments. Any change will be described on the course website.


Peer grading is the process of having students grade the assigned work of other students according to a grading rubric that has been pre-defined by the instructor.

In classes where enrollment numbers are in the thousands, and where student assignments cannot be graded by a computer, peer-grading is the most efficient way of helping students receive scores and feedback.

In this course, we use peer evaluation to grade the Final Exam.


Though some initially find it tedious, while others are intimidated by it, peer grading is known to have positive effects on learning. Students have been shown to learn better when they are asked to actively compare their answers with those of their peers. In one study on peer grading, students who self-graded outperformed students who were graded by instructors.Sadler and Good, 2006 [PDF]

Because of its proven beneficial effects on the learning of university-level mathematics, in this course, we use the grading process as a learning device long before the Final Exam. In some of the later Problem Sets you will be asked to grade sample assignment solutions (using the course rubric), and your grading will be assessed relative to that of the instructor. You will then be able to view a video in which the instructor explains his grading, so you may compare your grading with that of an expert.

Peer grading of the Final Exam works like this. As a student you submit your completed examination. Until the grading deadline, you have unlimited chances to re-submit the work, with no penalties for re-submission. When the submission deadline is up, you (and all the other students) undergo a training process in which you go through a small number of grading exercises. Once you have passed the training process, you then peer-grade several submissions from other students in the class, guided by the rubric. You then self-grade your own assignment, guided by the same rubric. Once you have completed peer- and self-grading, you can see your results for the work. You will receive your final grade on the work, as well as the breakdown of each question.

One consequence of having you undergo the training process is that you will become a proficient grader who understand the requirements of the assignment, and can grade a peer’s submission within an acceptable range of the instructor’s grade on the same assignment. But in the process, you will find that you start to understand the material much more deeply. As a result, successful completion of the training process carries credit towards your final grade.


  1. Where can I find the grading rubric for this course? A link to the rubric will be provided whenever you need it, but you can see it here [not in this Appendix – SS].
  2. Can I submit my assignment in any language? No, we cannot support multiple languages at this time; your submission must be in English or you will receive a grade of zero from your peers.
  3. Will my assignment be anonymous? Please refer to the honor code and privacy policy page.
  4. Can I opt out of the grading exercise? Course completion does not require participation in the peer review process. However, for this material, you will learn a lot by reviewing the work of others. In order to be receive a grade for the final exam (and thus be eligible for distinction), you must take part in the peer review process.
  5. If I miss the grading deadline, can I receive late credit? Sorry, at this stage, for logistical reasons, we are not able to extend the deadlines or give late credit in any part of the peer review process.
  6. Can I get a re-grade on my assignment? Sorry, at this stage, for logistical reasons, we are not able to allow re-grades.
  7. What do I do if I see a student’s assignment containing obscene, hateful, or otherwise abusive content? Contact the staff and the student will be removed from the course.


Here is a more detailed overview of the various phases of the peer grading exercise. Each phase is compulsory, and requires completion of the previous phase before moving on to the next.

Phase 1 (Complete exam): In this phase, you complete the exam. You will be given the rubric together with instructions. Pay careful attention to the rubric, as later on you will be grading your own exam submission, as well as your peers, using these metrics. (Even after you have hit submit, you are allowed to re-submit anytime before the submission deadline, with no penalty to your grade. Only your final (re-)submission will be counted when the grading deadline is up.)

Phase 2 (Training): During this phase, you will be given up to five sample exam submissions, to practice grading with. Each sample exam submission counts as one grading exercise. Passing a grading exercise: In order to pass a grading exercise, you need to grade approximately 80% of the questions within 20% of the instructor’s grade. Once you complete a grading exercise, you will be told whether you passed or failed, and will be given feedback on your scores. Passing training: Training consists of a series of up to five grading exercises. Once you pass a grading exercise, you are deemed to have passed training. If you do not pass training by the fifth attempt, your own submission can still be peer graded and you will receive a grade for your work. If you pass the training exercise but feel you need more practice, you can continue to grade any remaining sample submissions if you wish. Passing training carries credit towards your final grade.

Phase 3 (Grading): Once you have passed training, you will grade three of your classmates’ exams. Once you have finished grading these exams, you will move on to self-evaluation. Self Evaluation: You will be given your own exam to grade, using the same rubric as before.

Phase 4 (Results): In this phase, your final grade and your grade breakdown for each question will be released to you. The grade you receive for each question will be calculated from a combination of your peer-graded score and your self-graded score, according to the formula described below.

  1. How a peer grade is calculated: We will take the median of all the peer grades. The reason for taking the median instead of the average of all peer grades is to reduce unreliable grades – grades that are overly high or low will have a much smaller influence on the final median grade, than if the average were taken.
  2. If the peer grade and self-grade are within range (that range being determined by the instructor and TAs): we will take the higher grade.
  3. If the peer grade and self-grade differ significantly, we will take the peer grade to increase grading accuracy.
  4. Special case: If an instructor or TA has graded an exam, this grade may override the peer-grade or self-grade.

Appendix 2 Grading and Certificates of Completion

[Source: the Coursera web site; last accessed 14 April 2013]

Certificate of Completion. To receive a Certificate of Completion, you have to view all the lectures, complete all the in-lecture quizzes, and complete the problem sets with an adequate aggregate grade, as logged by the system. Determination of what constitutes an adequate grade for the problem sets will be made by the instructor and the TA, and will depend on the overall performance of the class as a whole. The intention is that at least 80% of the students who stay with the course to the end will receive a certificate of completion. Completion does not require taking the exam and participating in the peer review process.

Completion with Distinction. Distinction depends on achieving sufficiently high scores in the problem sets and the final exam. Determination of what constitutes sufficiently high scores will be made by the instructor and the TA’s, and will depend on the overall performance of those who complete all course requirements, including the final exam and the peer review process. The intention is that about 20% of students who receive a Certificate of Completion will receive a designation of Distinction. (Distinction is thus meant to indicate what the word suggests.)

Though you are strongly encouraged to work together on understanding the course content and attempting the regular assignments, you should work alone on the weekly Problem Sets and on the Final Exam, as they are intended to measure your individual performance.

Or simply enjoy the ride. If you find you don’t have the time to do the quizzes, assignments, or the problem sets, you’re still more than welcome to just stay and watch the videos!


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