Using Failure As A Friendly Tool For Learning

Current grading policies at most schools are intended to measure learning, but they are counterproductive in that they actually inhibit learning.

Taking a typical university mathematics course as an example, students might have a handful of assignments, a mid-term test, and a final exam. The number of graded items is small because universities are strapped for resources, and they can’t afford to grade more student work. There is also an issue of professor workload, as creating assignments takes work. (I realize that online quizzes are gaining in popularity, but they are not yet typical. And online quizzes share a problem with many assignment problems: The latter are often of the technical variety, because graders are often assistants with little capacity to judge higher levels of thinking.)

Because the number of graded items is so small, because grades are all-important, and therefore students are under a lot of pressure to achieve high grades, students typically spend an enormous amount of time on these small number of items. However, this is a poor way to learn mathematics; one rather ought to spend a lot of time on a large number of different exercises and problems, together with reading the textbook and reading around the subject.

It does no good to complain that students ought to have better time-management skills and should be doing all that is needed for effective learning as a matter of course. Most of them have not been prepared to be good learners by their high-school experience, and there is little in the university experience that trains them to be effective learners, particularly in first-year, when they need the most attention. Placing students in classes of 500, talking at them for a few hours per week, and then placing them under severe time constraints and pressuring them to perform on just a few tests is not conducive to good learning. The current system of undergraduate education does very little to train students, only samples their panicky attempts to keep their heads above water.

Rather than blame students, we need to look at the goals of our education system. If we are to train a generation of highly creative problem solvers that will help to move humanity off the path to destruction, then the current system is entirely inadequate. The current system, which assumes that students are doing the right things and then merely samples the results, has nothing to do with training and everything to do with throwing students into the deep end and seeing who sinks and who swims.

The ills of our education system are multifarious, but they are well-signified by our typical grading philosophy. Students are heavily penalized for making mistakes, which inhibits effective learning and promotes fear and anxiety. Students should be guided towards creating something of value that is of interest to them; in going through the creative process a number of times, students will master their own good creative habits, and will learn to value mistakes as an essential part of the process, rather than be paralyzed in fear.

The natural learning process involves playful exploration, which is not supported by our current typical learning environments; they rather encourage obedience. In a system that supports playful exploration, mistakes and failures would be treated honestly for what they are: essential elements of the learning process. Instead, students view them as permanent black marks on their records, which might as well be labels that are cruelly tattooed onto their skins.

It’s high time that we trained students to use mistakes and outright failures as friends … as friendly tools that aid the ultimate success of our plans. A number of recent articles highlight this theme; consider, for example, Please Don’t Focus on Failure, by Matthew E. May (hat-tip to Steve Miranda, who directs a school (PSCS) that focuses on creating a safe environment for students to “identify, cultivate, and express their passions”).


About Santo D'Agostino

I have taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects while teaching physics at my local university. I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. Further resources, and online tutoring, can be found at my other site
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7 Responses to Using Failure As A Friendly Tool For Learning

  1. bradleyjolly says:

    I must say, Santo, that reading this as a university student myself, I have to agree. Although I don’t study Mathematics, it seems it is this way across all boards, and it does, indeed, cause a panic on keeping our heads above water. Nice piece.

    • Hi Bradley,

      Thanks for the feedback, and sorry to hear that this is your experience, too. I know students in the humanities who are evaluated with a few small quizzes and then a major essay per semester. In large first-year classes, it can happen that the essay is turned in mid-semseter and then is handed back at the end of the semester, because they have a single TA mark the enormous number of papers for consistency. This has no formative function … you can’t learn from the essay and then develop your skills to improve your final grade, because the result of your essay IS your grade, end of story. So I agree with you, this is a problem across all subject areas.

      All I can say is hang in there! Try to keep your enthusiasm up by engaging in some activities just for the joy of them, and try not to let your formal studies dampen that enthusiasm. Your personal enthusiasms are precious and will sustain you for a lifetime with good nurturing and development.

      All the best!

  2. Hi Santo,

    As someone who has been to quite a few maths courses – specifically the advanced level ones and a few of the “standard” ones at the University of Waterloo (the advanced courses teach the same content as the standard ones except with a more rigourous introduction and more upper year content) – I would like to add some of my experiences into the mix.

    (It’s a fairly long post, so skip to the end if you just want to know my main points)

    In the standard maths courses, usually those filled to the brim with around 300-400 students, competition was fierce with my peers and I really felt no one around me was really willing to just play around with the problems and offer up new perspectives. Once a problem was done, we just left it, went with the solution from the first person who solved it, and moved on. This is quite understandable as we were constantly bombarded with course content, expected to pick up material with little motivation behind the proofs and propositions, and expected to creatively come up with solutions.

    Now, like you said, I can’t really blame the professors or the students. The professors are under constant pressure to come up with research results, create tests and exams, and not to mention deal with the dozen or so students that visit their office hours regularly. The students are also under constant pressure to preform well in their course and from the many people I have talked to, a lot of the motivation comes from keeping their average above a certain percentage to maintain a scholarship or to make sure they look desireable to potential employers. Coupled with the fact that midterms are usually worth more than all the assignments combined, as well as a final that is worth more than 60% one’s mark, it’s very easy to see how stressed a student can be under this situation.

    Contrasting the standard maths courses is the advanced level ones which are usually between 15-25 students after second year (usually starting with 80 in the first year offerings). Since the class was really small and the questions extremely difficult, the class as a group had to get together weekly to complete assignments. During these 4-6 person sessions, all of which I attended, every one of us tended to offer up a new perspective on solving a question and very often, we would come back to the questions to see if they offered any hints on solving future questions. This was one of things I enjoyed more about the advanced classes when compared to the standard ones.

    In regards to the content of the course, the proofs and theorems were explained in great detailed but at a very quick pace, which really forced us to spend time after class to really understand them. I even remembered how a few of the senior undergrads told us the professor would always add in an extra upper year topic just so he could throw in a few Putnam questions on the exam (haha). To offset this difficulty, all advanced courses implement a “bell curving” system and alternate grade weightings to make sure our marks were at par with the marks that we were expected to get under the standard courses. Although, I put bell curving in quotes because usually the professors will just randomly adjust our marks based on how well he thinks we understand the material. Even the TA’s in the course give out free marks because of the difficulty of the content.

    (If TL;DR see below)

    All in all, I would say I really would prefer an education that forces its students to learn outside of the classroom, challenges them, and allows them to genuinely appreciate the subject even at the cost of lower marks (I tended to get about 5-10% lower in advanced courses than standard ones) than one that is just there to propel students on to higher mathematics without the proper preparation.

    • Hi Stochastic Seeker,

      Thanks for your thoughtful and very detailed comments! It’s always good to hear the perspective of current students.

      I was particularly struck by your description of the group problem-solving sessions that were so useful, with everyone sharing their viewpoints.

      My hypothesis is this: All of the useful and fun things that you can accomplish in undergraduate mathematics education can be accomplished more effectively and more humanely if we eliminate grading.

      I don’t say to eliminate feedback or record-keeping, but to eliminate our current cruel system of grading, along with high-pressure testing in severely time-limited (12 or 13 week semesters) courses.

      Have students work on projects of their own choosing, guide them and support them in completing them effectively, and keep records on which of the technical components (skills/knowledge) that they master. A future professional school/employer will have all the information they need by looking at the record and the work done by the student, and this will be a much more useful description than a grade.

      What do you think about this? Consider the professor you describe, who put in Putnam-level problems, but then had to monkey with the grades … why? Why not just eliminate the grades entirely, and then the professor would be free to put in whatever he or she would like; students would recognize it as a challenge, and could tackle it as a challenge, without fear that their grades would suffer.

      Down with grades!

      • Haha, well with the advanced classes, one of the benefits of the class was that we could never fail (unless you were trying) if any effort was put into our work. According to our professor, he would always arrange it so that our class average would be 81-85% depending on how well the class did as a whole (i.e. if everyone did well, it was an 85% and if they did poorly, 81%); we were rewarded for just being brave enough to take the course.

        As an example, the average for the midterm was around 61-63% but when our professor saw how no one actually failed, he was delighted enough to boost our marks so the average was 83%. We even had a few students scoring over 100% because of this. I also heard from people who have also taken the course before where one time the average for the final was around 40% (apparently 2 Putnam questions were on there) but the professor was still generous enough make sure that the average would be above 80%.

        Because of this, the professor regularly went off on wild tangents in class, posted many weakly-related (but very interesting) problems, and often put in topics not on the syllabus (like having spherical geometry in a linear algebra course). We as a class really enjoyed that part of the course.

        For now, I believe this is as close as we are going to get to “grade-less” courses at the moment, because at least with this method we are drastically reducing the risk of low grades but decreasing the chances of high grades (i.e. lowering the variance of grades).

        As far as I know, a lot of the co-op employers at Waterloo are seldomly concerned about grades (unless they’re in the 95% ranges) and tend to focus on what differentiates you from the crowd. So I completely agree with you on how if we reduce or remove the stress that comes with the current system, then students will become more adventurous with their studies and naturally look into more projects that will emphasize their skills to potential employers.

  3. Hichem BEM says:

    Right on, Santo! excellent observations.

    I am experimenting this term with an online assignment system for precisely the very reasons you give: allow the student to try a larger amount of practice problems; get instant feedback; get directions for extra reading and new perspectives on failed questions; allow an additional submission to assess learning after failure. I’ll keep you posted on the results.

    We miss you here!!


    • Thanks for the kind words, Hichem! And I miss you all as well!

      I would be delighted to hear how students benefit from your new assignment system. It’s heartening to hear from a leader within a university who is striving to improve mathematics education.

      It would be interesting to run a mathematics program (i.e., just for mathematics majors) without grades, using a mastery system, with students working at their own pace. This would allow students to catch up without penalty if their high-school background is deficient. The intended result would be graduates who are superbly well-prepared for further studies or for a mathematics-related career.

      All the best wishes, Hichem!


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